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--- abstract: | We have performed cosmological simulations in a $\Lambda$CDM cosmology with and without radiative cooling, in order to study the effect of cooling on the cluster scaling laws. Our simulations consist of 4.1 million particles each of gas and dark matter within a box-size of 100 $h^{-1}$Mpc and the run with cooling is the largest of its kind to have been evolved to $z=0$. Our cluster catalogues both consist of over 400 objects and are complete in mass down to $\sim 10^{13} h^{-1} {\rm M_{\odot}}$. We contrast the emission-weighted temperature-mass ($T_{\rm ew}-M$) and bolometric luminosity-temperature ($L_{\rm bol}-T_{\rm ew}$) relations for the simulations at $z=0$. We find that radiative cooling increases the temperature of intracluster gas and decreases its total luminosity, in agreement with the results of Pearce et al. Furthermore, the temperature dependence of these effects flattens the slope of the $T_{\rm ew}-M$ relation and steepens the slope of the $L_{\rm bol}-T_{\rm ew}$ relation. Inclusion of radiative cooling in the simulations is sufficient to reproduce the observed X-ray scaling relations without requiring excessive non-gravitational energy injection. author: - 'O. Muanwong, P. A. Thomas, S. T. Kay, F. R. Pearce and H. M. P. Couchman' nocite: - '[@TMP01]' - '[@RiT01]' title: 'The effect of radiative cooling on scaling laws of X-ray groups and clusters' --- Introduction {#sec:introduction} ============ The mass of clusters of galaxies[^1] is dominated by dark matter. The evolution of the dark matter halo population is now well-understood both theoretically [@LaC94] and numerically [@JFW01]. The halos themselves are approximately self-similar and may be described in their inner regions by a one-parameter model [@NFW97] with a concentration parameter that is a slow function of mass. (Note, however, that there are significant deviations from this simple profile in the outer parts of clusters; see Thomas et al.2001.) The intracluster medium (ICM) does not share the approximate self-similarity of the dark matter. This is expressed most clearly in the X-ray luminosity-temperature relation. For pure bremsstrahlung emission, the bolometric X-ray luminosity should scale with temperature as $L_{\rm x}\propto T_{\rm x}^2$ (the inclusion of line emission flattens this relation), whereas observations indicate a much steeper temperature dependence, especially for low-mass systems [@EdS91; @DSJ93; @PBE96; @WJF97; @XuW00]. At high temperatures part of this steepening is due to a central cooling flow, but removing the cooling flow component still does not reconcile the observations with the self-similar prediction [@AlF98; @Mar98]. The reason for the departure from self-similarity is that the gas is not as centrally-concentrated in clusters as the dark matter, which is best physically expressed as an increase in entropy of the innermost gas [@EvH91; @Kai91; @Bow97]. The most obvious explanation for this is that there has been some form of energy injection. The amount of energy required depends upon the density of the gas at the time when the heating occured. @LPC00 argue for heating prior to cluster collapse and estimate a value of 0.3keV per particle. @ToN01 show that this can be lowered to 0.1keV per particle by heating at the optimal time (when the gas is at its minimum density) but most other studies that consider heating within collapsed halos require much higher values of 1–3keV per particle [@WFN99; @BBB00; @Low00]. There are two likely sources for any excess energy: stellar winds/supernovae and active galactic nuclei (AGN). It is known for CDM models that some feedback of energy into the intergalactic medium must occur in order to prevent the “cooling catastrophe”, in which the majority of the baryons in the Universe cool and form stars at high redshift [@WhF91; @Col91; @BVM92]. If the heating efficiency is high, supernovae can inject an energy of order 0.3keV per particle into the intergalactic medium, but they do not do so in an optimal way. Various authors, all of whom consider realistic, but different, models for the build-up of structure, conclude that supernovae are unable to provide the required excess entropy [@VaS99; @WFN99; @BBB00]. Heating of the intergalactic medium by quasars is also not without its problems as the heating must occur at just the right time in order not to overly suppress galaxy formation. Alternatively, the heating may arise from AGN buried within individual galaxies [@BBB00]. Radiative cooling results in the removal of low-entropy gas in the cluster core and thus leads to an overall increase in temperature of the ICM [@ThC92]. The amount of cooling that takes place in a cooling flow after the final assembly of a cluster is insufficient to explain the observations [@BBB00], but a more realistic model in which cooling occurs at every stage of the collapse hierarchy can give entropy increases equivalent to an excess energy of 1–2keV per particle [@WFN99]—in essence most of the cooling occurs in galaxy-sized halos before the formation of the cluster. @Bry00 has developed a simple model in which low-entropy gas is removed from the cluster core and the surrounding gas is assumed not to have cooled at all. Although this model has obvious deficiencies, it predicts the correct luminosity-temperature relation from 0.5–10keV. Simulations of cluster formation including radiative energy loss have been carried out by @PTC00. They showed that the gas is slightly heated and that the luminosity is greatly reduced (except in cooling flow clusters), in agreement with expectations. However, the simulations were of limited resolution and covered only a small range in cluster mass. We are now undertaking a programme of simulations to extend these results over a wider mass-range and to contrast the properties of clusters in different cosmologies. In this letter we report results at $z=0$ from two simulations of a 100$h^{-1}$Mpc box in the  cosmology, one with and one without radiative cooling. The simulations and cluster extraction method are described in Section \[sec:method\] and the results are presented in Section \[sec:results\]. We summarize our conclusions in Section \[sec:conclusions\]. method {#sec:method} ====== The simulations {#sec:simulations} --------------- We have carried out two simulations with 160$^3$ particles each of gas and dark matter within cubical volumes of side 100$h^{-1}$Mpc. The cosmological parameters were as follows: density parameter, $\Omega_0=0.35$; cosmological constant, $\Lambda_0=\Lambda/3H_0^2=0.65$; Hubble parameter, $h=H_0/100$kms$^{-1}$Mpc$^{-1}=0.71$; baryon density parameter, $\Omega_{\rm b}h^2=0.019$; power spectrum shape parameter, $\Gamma=0.21$; and a linearly-extrapolated root-mean-square dispersion of the density fluctuations on a scale 8${h^{-1}}$Mpc, ${\sigma_8}= 0.90$. With these parameters, the gas and dark matter particle masses are approximately $2.6\times10^9$ and $2.1\times10^{10}h^{-1}{\rm M_{\odot}}$, respectively. This mass is below the @StW97 limit above which numerical heating dominates cooling. The runs were started at a redshift, $z=50$ and evolved to the present day, $z=0$. The gravitational softening was fixed at $50\,h^{-1}$kpc in comoving co-ordinates until $z=1$ and thereafter held constant at 25$\,h^{-1}$kpc in physical co-ordinates. This softening is sufficient to prevent two-body relaxation [@TMP01] and we have checked that the gas and dark-matter have similar specific energy profiles in clusters drawn from the non-radiative simulation. The only difference in the two runs was that one of them included radiative cooling. For this run, we used the cooling tables of @SuD93 and assumed a uniform but time-varying metallicity of $Z=0.3\,(t/t_0)\,Z_\odot$, where $t/t_0$ is the age of the Universe in units of the current time. This time-varying metallicity is meant to crudely mimic the gradual enrichment of the ICM by stars, but more importantly it results in a global cooled gas fraction at the end of the simulation of approximately 20 per cent. This is the maximum value inferred from the observations of clusters [@BPB01] which means that our simulation will represent the largest effect that cooling alone is likely to have on the intracluster medium. We use a parallel version of the [hydra]{} $N$-body/SPH code as described by @CTP95 and @PeC97 except that the SPH equations have been modified to use the pairwise artificial viscosity of @MoG83—for a test of different SPH formalisms see @TTP00. We decouple the hot and cold gas in the manner described by @PJF99 [see also Ritchie & Thomas 2001 ] to prevent artificial overcooling of hot gas onto the central cluster galaxies. Groups of 13 or more cold, dense gas particles (with $\delta>500$ and $T<$12000K) are merged together to form collisionless [*galaxy*]{} particles, which can only grow via the accretion of more gas. Not only does this save considerable computational effort, it also prevents small objects from being artifically disrupted within cluster potentials. The cluster catalogue {#sec:catalogue} --------------------- Initially we identify clusters in our simulation by searching for groups of dark matter particles within an isodensity contour of 200, as described in @TCC98. We work with a preliminary catalogue of all objects with more than 250 particles, then retain only those which have a total mass within the virial radius exceeding $M_{\rm lim}\approx1.18\times10^{13}h^{-1}{{\mbox{$M_\odot$}}}$, corresponding to 500 particles of each species. The use of a small mass for the preliminary cluster selection ensures that our catalogue is complete. We have checked that using a different isodensity threshold, a different selection algorithm, or using gas particles instead of dark-matter particles to define the cluster, leads to an almost identical cluster catalogue—the only difference being the merger or otherwise of a small number of binary clusters. We define the centre of the cluster to be the position of the densest dark matter particle. Because the density parameter of the real universe is not known, we choose to average properties of the clusters within spheres that enclose an overdensity of 200 relative to the critical density (whereas for this cosmology the virial radius corresponds to an overdensity relative to critical of about 110). Our final catalogues consist of 427 and 428 clusters in the radiative and non-radiative simulations, respectively. Cluster X-ray properties {#subsec:xrayprop} ------------------------ We calculate the bolometric luminosity of each cluster using $$L_{\rm bol} = \sum_{i} \, {m_i \, \rho_i \over (\mu m_{\rm H})^2} \Lambda(T_i,Z), \label{eqn:lx}$$ where the subscript $i$ denotes the sum over all gas particles within radius $r_{200}$ that have temperatures, $T_i>10^{5}{\rm K}$, masses $m_i$ and densities, $\rho_i$; we assume a mean molecular mass $\mu m_{\rm H}=10^{-24}$g and an emissivity, $\Lambda(T_i,Z(t))$, that is the same function used by [hydra]{} to calculate cooling rates, as discussed in Section \[sec:simulations\]. Note that, for temperatures below about $10^7$K, line emission is important and the cooling rates are substantially higher than the contribution from bremsstrahlung alone. The emission-weighted temperature of each cluster is calculated as $$T_{\rm ew} = {\sum_{i} \, m_i \, \rho_i \Lambda(T_i,Z) \, T_i \over \sum_{i} \, m_i \, \rho_i \Lambda(T_i,Z)}.$$ Many of the smaller clusters in our catalogues have emission-weighted temperatures that are below 0.5keV which means that the majority of their emission will emerge at energies that are below the X-ray bands. We do not attempt to calculate the emission in any particular X-ray passband in this paper but instead quote bolometric luminosities and emission-weighted temperatures. Results {#sec:results} ======= Temperature-mass relations {#sec:mt} -------------------------- =6.5cm We plot the emission-weighted temperature-mass relation for both cluster samples in Fig. \[fig:tm200\]. Results from the simulations with and without cooling are illustrated using crosses and filled circles, respectively. The broken lines are power-law fits to the observational relation as determined by @HMS99, using mass estimates from galaxy velocity dispersions (dashed), X-ray temperature profiles (dash-dotted), the isothermal $\beta$-model (dotted) and the surface brightness deprojection method (dash-triple dotted). For the simulation with radiative cooling the relationship between ${T_{ew}}$ and $M$ is an approximate power law $$\label{eq:mtxcool} kT_{\rm ew} = 1.91\left(\frac{M_{200}}{10^{14} h^{-1}{\rm M_{\odot}}}\right)^{0.58} {\rm keV},$$ and for the simulation without radiative cooling $$\label{eq:mtxnocool} kT_{\rm ew} = 0.98\left(\frac{M_{200}}{10^{14} h^{-1}{\rm M_{\odot}}}\right)^{0.67} {\rm keV}.$$ These should be compared with the virial relation (shown as the solid line on the figure) $$\label{eq:mtvir} kT_{\rm vir} = 1.61\left(\frac{M_{200}}{10^{14} h^{-1}{\rm M_{\odot}}}\right)^{0.67} {\rm keV}.$$ (Note that no attempt has been made when making these fits to reproduce the observational selection effects. They should therefore be regarded as rough guides rather than precise predictions.) The clusters from the non-cooling run mostly have emission-weighted temperatures that are much lower than the virial values. The reason for this is that their emission is dominated by high-density, low-temperature gas in the core of the cluster (this temperature drop in the core is simply due to the fact that the specific energy profile of the gas mimics that of the dynamically-dominant dark matter). Small fluctuations in the core properties of the clusters (which are not well-resolved by our simulations) lead to a large scatter in predicted temperatures. It is important to note that these non-radiative simulations do not provide sensible predictions for the observed properties of real clusters—the core gas has a short cooling time and would not in reality persist in the intracluster medium for the lifetime of the cluster. We note that our non-radiative $T_{\rm ew}-M$ relation has a lower normalization than found by @TMP01 for simulations in the $\tau$CDM cosmology because there it was assumed that the emission is purely bremsstrahlung, which underestimates cooling rates below $10^7$K and places less weight on the innermost region of clusters. In contrast to the non-radiative run, the clusters from the radiative simulation have emission-weighted temperatures that show less scatter and that exceed the virial values. The reason for this is that the core gas has cooled to low temperatures and been removed from the intracluster medium, leaving behind higher-entropy, higher-temperature gas—an effect that is more pronounced in lower-mass clusters. The emission is no longer dominated by the core and is well-resolved by our simulations. The clusters provide an adequate fit to the observational data given the large uncertainty in the latter as evidenced by the various broken lines in Fig. \[fig:tm200\]. Luminosity-temperature relations {#sec:lt} -------------------------------- =6.5cm Fig. \[fig:lt200\] illustrates X-ray luminosity-temperature relations for both simulations, again using crosses and filled circles for radiative and non-radiative clusters respectively. We have trimmed the original catalogues by selecting clusters only with temperatures above 0.35 and 0.75keV for the non-radiative and radiative cooling catalogues respectively, to assure completeness in temperature. The broken lines illustrate best-fit power-law relations as determined by @XuW00 for their collected group sample (dashed line), cluster sample (dot-dashed line) and combined sample (dotted line). (Note that the units for the normalisation of the relations in Table 2 of Xue & Wu 2000 are misquoted and should be a factor of 10 larger, in agreement with their Table 1 and Figure 1.) A power-law fit to results from the radiative simulation gives $$L_{\rm bol} = 9.0\times10^{41} \left(\frac{T_{\rm ew}}{1\,{\rm keV}}\right)^{3.3} \, h^{-2} \, {\rm erg \, s^{-1}}, \label{eq:lxtxcool}$$ and for the non-radiative simulation $$L_{\rm bol} = 1.2\times10^{44} \left(\frac{T_{\rm x}}{1\,{\rm keV}}\right)^{1.9} \, h^{-2} \, {\rm erg \, s^{-1}}. \label{eq:lxtxnocool}$$ (As stated in section \[sec:mt\], these fits should not be taken as precise calibrations.) Again, both the slope and normalization differ between the two simulations. The non-radiative simulation gives a slope close to 2, as predicted by self-similar scaling relations (the relation is slightly flatter due to the inclusion of line emission). However, the radiative simulation gives a significantly steeper slope of $3.3$. The normalization of the relation is around a factor of 10–100 lower in the radiative simulation than in the non-radiative simulation. Qualitatively, these differences are the same as were found by @PTC00, for a smaller cluster sample. The high entropy gas that replaces cooled material in the radiative simulation is hotter and less dense than the gas in the non-radiative simulation. The change in density has the greater effect, since the X-ray emissivity is a slow function of temperature but is proportional to the square of the gas density. The combination of the increase in temperature and the decrease in luminosity of the clusters causes the substantial shift in the $L_{\rm bol}-T_{\rm ew}$ relation. The results from the radiative simulation are in good agreement with the best-fit relations of @XuW00. If anything, we predict luminosities that are are a factor of 2–3 too low, although there is still significant uncertainty in the observational determinations. Conclusions {#sec:conclusions} =========== In this letter, we have presented results from an ongoing programme to measure the evolution of X-ray cluster properties for a range of physical and cosmological models. Specifically, we have presented the current-day ($z=0$) emission-weighted temperature-mass ($T_{\rm ew}-M$) and bolometric luminosity-temperature ($L_{\rm bol}-T_{\rm ew}$) relations from two simulations of a $\Lambda$CDM cosmology, one with and one without radiative cooling. The $T_{\rm ew}-M$ relation is significantly different in non-radiative and radiative simulations, with the latter in reasonable agreement with observational determinations. In the non-cooling simulation, the emission is dominated by cold, dense gas in the cores of the clusters; radiative cooling removes this gas from the intracluster medium (converting it into stars) and replaces it with higher-entropy, hotter material. This effect is more prevalent in lower-mass systems and so flattens the temperature-mass relation. The $L_{\rm bol}-T_{\rm ew}$ relation is also significantly different between the two simulations. The high entropy material in the radiative simulation is less dense than the material it replaces and causes the X-ray luminosity of clusters to decrease by around a factor of 100 at $T_{\rm ew} = 1$keV. The slope of the relation in the non-radiative simulation is 1.9, similar to self-similar predictions. However, the slope of the radiative relation is significantly steeper, 3.3, again due to the differential effect of cooling with halo temperature. The radiative simulation is in much closer agreement with the observations, both in the slope and normalization of the relation. In this paper, we have used bolometric luminosities and emission-weighted temperatures. For the low-temperature clusters found in the non-radiative simulations, these will differ significantly from properties measured in any particular X-ray band. However, we wish to emphasize that to attempt to correct for this is misleading as the use of such simulations is wrong in principle—the low-entropy gas in the cores of these clusters has a short cooling time and will not be present in real systems. The global fraction of cooled gas (and stars) in the radiative simulation, 20 per cent, is higher than suggested by observations of the $K$-band galaxy luminosity function [@BPB01]. More importantly, the fraction of cooled gas within the clusters varies between about one-third and two-thirds with decreasing mass. While these values are not convincingly ruled out by observations, most people would also regard these as high values—in which case our simulation can be treated as an upper bound on the effect of radiative cooling. In this paper, we have deliberately ignored the effect of non-gravitational heating upon the gas. In reality, we know that there must be heating associated with star-formation and metal-enrichment of the intracluster medium. This will raise the entropy of the gas and reduce the amount of cooling that is required to match the observations. However, we do not regard the case for significant heating by AGN or very efficient supernovae feedback as proven. The simulations described in this paper were carried out on the Cray-T3E at the Edinburgh Parallel Computing Centre as part of the Virgo Consortium investigations of cosmological structure formation. OM is supported by a DPST Scholarship from the Thai government; PAT is a PPARC Lecturer Fellow. Allen S. W., Fabian A. C., 1998, , 297, L57 Balogh M. L., Pearce F. R., Bower R. G., Kay S. T., 2001, , submitted Blanchard A., Valls–Gabaud D., Mamon G. A., 1992, , 264, 365 Bower R. G., 1997, , 288, 355 Bower R. G., Benson A. J., Baugh C. M., Cole S., Frenk C. S., Lacey C. G., 2000, , in press (astro-ph/0006109) Bryan G. L., 2000, , 544, L1 Cole S., 1991, , 367, 45 Couchman H. M. P., Thomas P. A., Pearce F. R., 1995, , 452, 797 David L. P., Slyz A., Jones C., Forman W., Vrtilek S. D., 1993, , 412, 479 Edge A. C., Stewart G. C., 1991, , 252, 414 Evrard A. E., Henry J. P., 1991, , 383, 95 Horner D. J., Mushotzky R. F., Scharf C. A., 1999, , 520, 78 Jenkins A., Frenk C. S., White S. D. M., Colberg J. M., Cole S., Evrard A. E., Couchman H. M. P., Yoshida N., 2001, , 321, 372 Kaiser N., 1991, , 383, 104 Lacey C., Cole S., 1994, , 271, 676 Lloyd-Davies E. J., Ponman T. J., Cannon D. B., 2000, , 315, 689 Lowenstein M., 2000, , 532, 17 Markevitch M., 1998, , 504, 27 Monaghan J. J., Gingold R. A., 1983, J. Comp. Phys., 52, 375 Navarro J. F., Frenk C. S., White S. D. M., 1997, , 490, 493 Pearce F. R., Couchman H. M. P., 1997, New Astron., 2, 411 Pearce F. R. [et al.]{} (the Virgo Consortium), 1999, , 521, L99 Pearce F. R., Thomas P. A., Couchman H. M. P., Edge A. C., 2000, , 317, 1029 Ponman T. L., Bourner P. D. J., Ebeling H., Böhringer H., 1996, , 283, 690 Ritchie B. W., Thomas P. A., 2001, , in press (astro-ph/0005357) Steinmetz M., White S. D. M., 1997, , 288, 545 Sutherland R. S., Dopita M. A., 1993, , 88, 253 Thacker R. J., Tittley E. R., Pearce F. R., Couchman H. M. P., Thomas P. A., 2000, , 319, 619 Thomas P. A. [et al.]{} (the Virgo Consortium), 1998, , 296, 1061 Thomas P. A., Couchman H. M. P., 1992, , 257, 11 Thomas P. A., Muanwong O., Pearce F. R., Couchman H. M. P., Edge A. C., Jenkins A., Onuora L., 2001 , in press (astro-ph/0007348) Tozzi P., Norman C., 2001, , 546, 63 Valageas P., Silk J., 1999, , 350, 725 White S. D. M., Frenk C. S., 1991, , 379, 52 White D. A., Jones C., Forman W., 1997, , 292, 419 Wu K. K. S., Fabian A. C., Nulsen P. E. J., 2000, , 318, 889 Xue Y.-J., Wu X.-P., 2000, , 538, 65 [^1]: Throughout this paper we will not distinguish between groups and clusters of galaxies but will use the term clusters to stand for both

--- abstract: 'Let $\phi$ be a 3CNF formula with $n$ variables and $m$ clauses. A simple nonconstructive argument shows that when $m$ is sufficiently large compared to $n$, most 3CNF formulas are not satisfiable. It is an open question whether there is an efficient refutation algorithm that for most such formulas proves that they are not satisfiable. A possible approach to refute a formula $\phi$ is: first, translate it into a graph $G_{\phi}$ using a generic reduction from 3-SAT to max-IS, then bound the maximum independent set of $G_{\phi}$ using the Lovász $\vartheta$ function. If the $\vartheta$ function returns a value $<m$, this is a certificate for the unsatisfiability of $\phi$. We show that for random formulas with $m<n^{3/2 -o(1)}$ clauses, the above approach fails, i.e. the $\vartheta$ function is likely to return a value of $m$.' author: - Uriel Feige - Eran Ofek title: Random 3CNF formulas elude the Lovász theta function --- \[section\] \[thm\][Corollary]{} \[thm\][Lemma]{} \[thm\][Theorem]{} \[thm\][Proposition]{} \[thm\][Conjecture]{} \[thm\][Definition]{} =1 =1 Introduction {#sec: introduction} ============ A 3CNF formula $\phi$ over $n$ variables is a set of $m$ clauses, where each clause contains exactly $3$ literals. A formula $\phi$ is satisfiable if there is an assignment to its $n$ variables that sets at least one literal in every clause to “true”. The 3-SAT problem of deciding whether an input 3CNF formula $\phi$ is satisfiable is NP-hard. In this paper we consider a certain heuristic for 3-SAT. A heuristic for satisfiability may try to find a satisfying assignment for an input formula $\phi$ if one exists. A refutation heuristic may try to prove that no satisfying assignment exists. How does one measure the quality of a refutation heuristic? A possible test may be to check how good the heuristic is on a random input. But then, how do we generate a random unsatisfiable formula? To answer this question we review some known properties of random 3CNF formulas. The satisfiability property has the following interesting threshold behavior. Let $\phi$ be a random 3CNF formula with $n$ variables and $cn$ clauses (each new clause is chosen independently and uniformly from the set of all possible clauses). As the parameter $c$ governing the density of the formula is increased, it becomes less likely that $\phi$ is satisfiable, as there are more constraints to satisfy. In [@FriedgutBo99] it is shown that there exists $c_n$ such that for $c < c_n(1-{\epsilon})$ almost surely $\phi$ is satisfiable, and for $c > c_n(1+{\epsilon})$, $\phi$ is almost surely unsatisfiable (for some ${\epsilon}$ which tends to zero as $n$ increases). It is also known that $3.52 < c_n < 4.596$ [@KaporisKiLa03; @HajiaghayiSo03; @JansonStVa00] and it is widely believed that $c_n$ converge to some constant $c$. We will use random formulas with $cn$ clauses (for $c > c_n(1 + {\epsilon})$) to measure the performance of a refutation heuristic. Notice that for any $n$, as $c$ is increased (for $c>c_n(1+{\epsilon})$), the algorithmic problem of refutation becomes less difficult since we can always ignore a fixed fraction of the clauses. In this paper we analyse a semidefinite programming based refutation algorithm which was introduced at [@Feige02], and show that for random formulas of certain densities (well above the satisfiability threshold) this algorithm fails. The algorithm itself is simple to describe (to readers familiar with some of the previous work). 1. Given an input 3CNF formula $\phi$, apply to it a standard reduction from max 3-SAT to maximum independent set, resulting in a graph $G_{\phi}$. The size of the maximum independent set in $G_{\phi}$ is equal to the maximum number of clauses that can be simultaneously satisfied in $\phi$. 2. Compute the Lovász $\vartheta$ function of the graph $G_{\phi}$. This provides an upper bound on the size of the maximum independent set of $G_{\phi}$. 3. If $\vartheta(G_{\phi}) < m$, then output “unsatisfiable”. Otherwise return “do not know”. We now describe the graph $G_{\phi}$ in more detail. Recall that for a 3CNF clause, there are seven different assignments to its three literals that satisfy the clause. For each clause of $\phi$ the graph contains a clique of $7$ vertices, which we call a *cloud*. Hence $G_{\phi}$ contains $7m$ vertices. Each vertex of the clause cloud is associated with a different assignment to the three literals of the clause that satisfies the clause. Vertices of different clouds are connected by an edge if they are associated with contradicting assignments. (Namely, if there is a variable that is assigned to true by one of the assignments and to false by the other. For the same reason, the vertices within a cloud form a clique.) The $\vartheta$ function of any graph $G$ upper bounds the maximum independent set in it, and can be computed in polynomial time up to arbitrary precision, using semidefinite programming. The fact that the vertices of $G_{\phi}$ can be covered by $m$ cliques implies that $\vartheta(G_{\phi}) \leq m$. Thus, if $\phi$ is satisfiable then the value of the theta function will be exactly $m$. If the value of the theta function is $< m$ then $\phi$ is unsatisfiable. The above algorithm has one sided error, in the sense that it will never say “unsatisfiable” on a satisfiable formula, but for some unsatisfiable formulas it will fail to output “unsatisfiable”. If for some formula $\phi$ the algorithm outputs ‘unsatisfiable’, then the algorithm execution on $\phi$ is a witness for the unsatisfiability of $\phi$. Our main result is that for random 3CNF formula $\phi$ with $m < n^{3/2 -o(1)}$ clauses it is very likely that $\vartheta(G_{\phi})=m$. Related work ------------ A possible approach for refuting a formula $\phi$ is to find a resolution proof for the unsatisfiability of $\phi$. However, Chvatal and Szemeredi [@ChvatalSz88] proved that a resolution proof of a random 3CNF formula with linear number of clauses is almost surely of exponential size. A result of a similar flavor for denser formulas was given by Ben-Sasson and Wigderson [@Ben-SassonWi01] who showed that a random formula with $n^{3/2 -{\epsilon}}$ clauses almost surely requires a resolution proof of size $2^{\Omega(n^{{\epsilon}/(1-{\epsilon})})}$. These lower bounds imply that finding a resolution proof for a random formula is computationally inefficient. A simple refutation algorithm can be used to refute random instances with $cn^2$ clauses, when $c > 2/3$. This is done by selecting all the clauses that contain a variable $x$. Fixing $x$ to be true leaves about half of the selected clauses as a random 2-cnf formula with roughly $3cn/2 > n$ clauses. This formula is unlikely to be satisfiable, and its nonsatisfiability can be verified by a polynomial time algorithm for 2SAT. The same can be done when fixing $x$ to be false. A spectral approach introduced by Goerdt and Krivelevich [@GoerdtKr01] gave a significant improvement and reduced the bound to $(\log n)^7 \cdot n^{k}$ clauses for efficient refutation of $2k$-cnf formulas. This was later improved by [@CojaGoLaSc03], [@FeigeOf04] that showed how to efficiently refute a random $2k$-cnf instances with at least $cn^{k}$ clauses. The basic approach for refutation of $2k$-cnf formulas was later extended in [@FriedmanGoKr01],[@GoerdtLa03],[@FeigeOf04] to handle also random 3CNF formulas with $n^{3/2+{\epsilon}}, \text{poly}(\log n) \cdot n^{3/2}, cn^{3/2}$ clauses respectively. Our current result gives a somewhat weak indication that spectral methods can not break the $n^{3/2-o(1)}$ barrier. Further motivation for studying efficient refutability of random 3CNF formulas is given in [@Feige02]. There it is shown that if there is no polynomial time refutation heuristic that works for most 3CNF formulas with $cn$ clauses (where $c$ is an arbitrarily large constant) then certain combinatorial optimization problems (like minimum graph bisection, the dense $k$-subgraph, and others) have no polynomial time approximation schemes. It is an open question whether it is NP-hard to approximate these problems arbitrarily well, though further evidence that these problems are indeed hard to approximate is given in [@khot]. The algorithm considered in the current paper for refuting $\phi$ by computing $\vartheta(G_{\phi})$ was presented in [@Feige02]. There is was shown that when $m < n^{2-o(1)}$, almost surely $\vartheta(G_{\phi}) \geq (1 -o(1))m$. Our current work overcomes a difficulty that prevented the approach of [@Feige02] to show that $\vartheta(G_{\phi}) = m$, not even for formulas $\phi$ with a linear number of clauses. The difficulty was the existence of pairs of clauses that share two variables. Related algorithms for refuting CNF formulas were analysed in [@BGHMP; @AAT]. There the authors considered a certain linear programming relaxation of the satisfiability problem, and successive tightenings of this relaxation via the operators of Lovasz and Schrijver. The authors of [@AAT] show that in order to refute a random 3CNF formula with $cn$ clauses (where $c$ is a sufficiently large constant) one has to apply $\Omega(n)$ rounds of the Lovasz-Schrijver operator to the initial relaxation. Our results deal only with the Lovasz $\vartheta$ function which lies at the lowest level of the Lovasz-Schrijver hierarchy (for maximum independent set relaxation). In this respect, the results in [@AAT] are stronger than ours. However, we believe that our results are of independent interest. (In fact, they were obtained independently of and roughly concurrently with the results of [@AAT].) One superficial difference is that we consider denser 3CNF formulas. This difference is only superficial, because also the results of [@AAT] extend to denser formulas, by limiting them to the lower levels of the Lovasz-Schrijver hierarchy. A more substantial difference is that the staring point of [@AAT], which is a linear program relaxation of 3CNF, is different from ours. We first apply a reduction to the 3CNF formula, inducing a graph, and only then apply the Lovasz $\vartheta$ function to the induced graph. It is not obvious (at least for us) what is the minimal $i$ for which the $i$-th relaxation used in [@AAT] is stronger than the relaxation we use (such $i$ exists since the $n$-th relaxation always returns the correct answer). And finally, there are differences between our proof techniques and those of [@AAT]. We present a solution to the vector formulation of the $\vartheta$ function, whereas [@AAT] present a solution to the matrix formulation of their relaxation. Results ======= Instead of working with $G_{\phi}$ we work with an induced subgraph of $G_{\phi}$ that is derived from $G_{\phi}$ by retaining in each clause cloud only the vertices corresponding to satisfying 3XOR assignments of the clause. Namely, for each clause we keep those four vertices that are associated with assignments that satisfy an odd number of literals in the clause. We call this subgraph $G_{\phi}^{3xor}$. Since $G_{\phi}^{xor}$ is an induced subgraph of $G_{\phi}$ it follows (by known monotonicity properties of the theta function) that $\vartheta(G_{\phi}^{xor}) \leq \vartheta(G_{\phi})$. We show that when $m \leq n^{3/2 -o(1)}$ w.h.p. $\vartheta(G_{\phi}^{xor}) =m$, which by the above discussion implies that also $\vartheta(G_{\phi}) = m$. \[thm: main\] Let $\phi$ be a random 3CNF formula with $m =o (n^{\frac{3}{2} - \frac{22 \log \log n}{\log n}})$ clauses and $n$ variables. With high probability $\vartheta(G_{\phi}^{xor}) = m$. Let $\phi$ be a random 3CNF formula with $m =o (n^{\frac{3}{2} - \frac{22 \log \log n}{\log n}})$ clauses and $n$ variables. With high probability $\vartheta(G_{\phi}) = m$. For $G_{\phi}^{xor}$ our results are nearly optimal in terms of the density of the underlying 3CNF formula $\phi$. \[thm: refutation for dense\] Let $\phi$ be a random 3CNF formula with $m \geq cn^{3/2}$ clauses and $n$ variables, where $c$ is a sufficiently large constant. With high probability $\vartheta(G_{\phi}^{xor}) < m$. We suspect that when $m \geq cn^{3/2}$ then also $\vartheta(G_{\phi}) < m$, although we did not prove it (when $m \geq cn^{3/2}$ there are other refutation methods that succeed, see [@FeigeOf04] for details). For convenience, from now on we will refer to the $\vartheta(G_{\phi}^{xor})$ also as $SDP(\phi)$. We prove Theorem \[thm: main\] in two steps. First we introduce a simple refutation proof system that we call narrow Gauss Elimination $3$ (in short GE3) and prove that it is stronger then $SDP(\phi)$, i.e. if $\phi$ cannot be refuted by GE3 then $SDP(\phi) =m$. We then show that a random 3XOR formula with $m =o (n^{\frac{3}{2} - \frac{22 \log \log n}{\log n}})$ clauses almost surely cannot be refuted by GE3. The GE3 proof system works as follows. It receives as input a system of linear equations modulo 2, where every equation has at most three literals. It succeeds in refuting the system of linear equations if it manages to derive the equation $0 = 1$. A new equation can be derived only if it contains at most three variables, and it is the result of adding exactly two existing equations and simplifying the result mod $2$. By simplifying modulo 2 we mean that $1 \pm 1 = 0$, $x_i \pm x_i = 0$ and $x_i \pm \bar{x}_i = 1$, for every variable $i$. To clarify the derivation rule of GE3, consider the following three linear equations: $x_1 + x_2 + x_3 = 1$, $x_1 + x_4 + x_5 = 1$ and $x_2 + x_4 + x_6 = 1$. No new equation can be derived by the GE3 proof system, because adding any two equations produces an equation with four variables. In particular, also the equation $x_3 + x_5 + x_6 = 1$ cannot be derived, even though it contains only three variables and is implied by the original equations (by adding the three of them). Observe that if an equation $e_1$ containing only two variables is derived in GE3 (say, $x_1 + x_2 = 0$), then in every other equation $e_2$ we can use GE3 to replace the occurrence of one of the variables by the other, by adding $e_1$ and $e_2$. The proof of Theorem \[thm: main\] is an immediate consequence of the following two lemmas. \[lemma: GE3 reduction\] Let $\phi$ be any formula with $m$ clauses. If $\phi$ cannot be refuted by $GE3$ then $SDP(\phi) = m$. \[lemma: GE3 is weak\] Let $\phi$ be a random 3XOR formula with $n$ variables and $m =o (n^{\frac{3}{2} - \frac{22 \log \log n}{\log n}})$ clauses. With high probability GE3 cannot refute $\phi$. SDP formulation of the $\vartheta$ function {#sec: SDP theta} =========================================== For each vertex $i$ we assign a vector $v_i$. There is also a special vector $v_0$. The semidefinite program is: $$\begin{aligned} &\max \sum_{i=1}^{n} {\langle v_0,v_i \rangle} &\text{subject to:} \nonumber \\ & &{\langle v_0,v_0 \rangle} = 1\\ &\text{for every $i\geq 1$:} &{\langle v_i,v_i \rangle} = {\langle v_i,v_0 \rangle}\label{eqn: 2}\\ &\text{for every pair $i,j$:} &{\langle v_i,v_j \rangle} \geq 0 \label{eqn: 3}\\ &\text{for any edge $(i,j)$:} &{\langle v_i,v_j \rangle} = 0 \label{eqn: 4}\end{aligned}$$ Instantiating the above semi-definite program for the graph $G_{\phi}^{xor}$ we derive the following semi-definite program, in which for clause $i$ there are $4$ assignment vectors $v_{i}^j$, one for every assignment of its three variables that satisfies an odd number of literals in the clause. $$\begin{aligned} &\max \sum_{\substack{i=1..m,\\j=1..4}} {\langle v_0,v_i^j \rangle} &\text{subject to:} \nonumber \\ & &{\langle v_0,v_0 \rangle} = 1\\ &\text{for every vector:} &{\langle v_i^j,v_i^j \rangle} = {\langle v_i^j,v_0 \rangle}\label{eqn: 6}\\ &\text{for every pair of vectors:} &{\langle v_i^j,v_k^l \rangle} \geq 0 \label{eqn: 7}\\ &\text{for every pair of contradicting vectors:} &{\langle v_i^j,v_k^l \rangle} = 0 \label{eqn: 8}\end{aligned}$$ (A pair of vectors is contradicting if there is some variable that the assignment associated with one of the vectors sets to true, and the assignment associated with the other vector assigns to false.) The value of the second semi-definite program is at most $m$ because every clause cloud forms a clique. As the following known Lemma shows, the contribution of a clique to the objective function is at most $1$. \[lemma: theta clique cover\] Let $v_0$ be a unit vector and let $v_1,v_2,v_3,v_4$ be orthogonal vectors, such that ${\langle v_i,v_i \rangle} = {\langle v_i,v_0 \rangle}$ for all $i$. Then $\sum_{i=1}^4 {\langle v_0,v_i \rangle} \leq 1$. Since $v_0$ is a unit vector and $v_1,v_2,v_3,v_4$ are orthogonal, it holds that $\sum_{i=1}^4 {\langle v_0,\frac{v_i}{{\| v_i \|_{_{}}}} \rangle}^2 \leq 1$. It thus follows that $$\begin{aligned} & \sum_{i=1}^4 {\langle v_0,v_i \rangle} =\sum_{i=1}^4 {\| v_i \|_{_{}}} {\langle v_0,\frac{v_i}{{\| v_i \|_{_{}}}} \rangle} = \sum_{i=1}^4 {\langle v_0,\frac{v_i}{{\| v_i \|_{_{}}}} \rangle}^2 \leq 1,\end{aligned}$$ where the last equality follows from ${\| v_i \|_{_{}}}^2 = {\langle v_0,v_i \rangle}$. Note that Lemma \[lemma: theta clique cover\] implies that for any graph $G$, if the vertices of $G$ can be covered by $p$ cliques, then $\vartheta(G) \leq p$. Proofs ====== We will use the SDP formulation of the $\vartheta$ function as appears in Section \[sec: SDP theta\]. Apply the derivation rule of the GE3 system as long as new equations are generated by it. Since the number of possible equations with at most three variables is $O(n^3)$, then this procedure must end. Assume that the equation $0 = 1$ could not be derived. Hence we are left with equations containing one variable (meaning that the value of this variable must be fixed to a constant), two variables (meaning that their values must be identical, or sum up to 1, depending on the free constant in the equation), or three variables. The information that GE3 derives about $\phi$ allows us to partition all literals into equivalence classes of the form: $$\begin{aligned} S_1:~~ &x_1 = x_{18} = \ldots = \bar{x}_9 &(\bar{S}_1:~~ \bar{x}_1 = \bar{x}_{18} = \ldots = x_9) \nonumber\\ S_2:~~ &x_4 = x_{20} = \ldots = x_5 &(\bar{S}_2:~~\bar{x}_4 = \bar{x}_{20} = \ldots = \bar{x}_5) \nonumber\\ &~~~~. &.~~~~~~~~~ \\ &~~~~. &.~~~~~~~~~ \nonumber\\ S_9:~~ &\bar{x}_2 = x_{21}= \dots = x_{30} &(\bar{S}_9:~~ x_2= \bar{x}_{21}= \dots = \bar{x}_{30} )\nonumber \\ &~~~~. &.~~~~~~~~~ \nonumber\\ S_l:~~ &1=x_6 = \bar{x}_{11}= \ldots x_8 &(\bar{S}_l:~~0=\bar{x}_6 = x_{11}= \ldots \bar{x}_8) \nonumber\\ \nonumber\end{aligned}$$ Notice that each equivalence class $S_i$ has a “mirror” part $\bar{S}_i$; we think of these two parts as one class. A class might contain only one variable. We call a variable *fixed* if it belongs either to $S_l$ or to the mirror of $S_l$. Other variables are called *free*. Similarly, except $S_l$ which is fixed, all other classes are free. A variable is fixed if and only if the GE3 refutation system can derive a clause containing only this variable (equal to a constant). Two free variables belong to the same class if and only if the GE3 system can derive a clause containing only these two variables. Each original clause of $\phi$ is of one of the following types: 1. It contains three free variables, each of them has distinct equivalence class. 2. It contains one fixed variable and two free variables from the same equivalence class. 3. It contains three fixed variables. We now explain why the above three types cover all clauses. If a clause has no fixed variable then its variables must be from distinct classes (type 1), as otherwise two of them will cancel out and cause the other variable to be fixed. If a clause has exactly one fixed variable then the other two belong to the same class and they are free (type 2). A clause cannot have exactly two fixed variables as the remaining variable will be also fixed (thus the remaining case is type 3). We will now give values to the vectors corresponding to all clauses. These vectors will satisfy the SDP constraints and will also give a value of $m$. An assignment for a clause that contradicts the information gathered by GE3 is called *illegal*; otherwise it is *legal*. For example, for the equivalence classes given above, an assignment such as $x_{11} = 1$ is illegal because it contradicts $S_l$, also an assignment such as $(x_1,x_9,x_{11}) = (1,1,0)$ is illegal because it contradicts $S_1$. We will use the following guidelines: - Each vector has $l$ coordinates, numbered from 0 to $l-1$. For $1 \le i \le l-1$, coordinate $i$ will correspond to free class $i$ - A clause vector that corresponds to an illegal assignment will be set to the zero vector $\vec{0}$. For a clause of type (1) the clause cloud will have four assignments with non zero vectors, for a clause of type (2) there will be two assignments, and for a clause of type (3) there will be one assignment. - Let $c$ be a clause that has $i$ different free classes ($i \in \{0,1,3\}$). The vectors corresponding to legal assignments of $c$ will have exactly $1 + i$ non zero entries. The only non-zero coordinates are $0$ and the coordinates corresponding to the indices of the free classes. Notice that the second bullet can be interpreted as removing from $G_{\phi}^{xor}$ all the vertices corresponding to illegal assignments. Thus from now we will assume that such vertices are indeed removed from $G_{\phi}^{xor}$. To simplify the notation in the remainder of the proof, we do the following. With each subclass $S_i$ we associate a literal $s_i$ (and with $\bar{S}_i$ we associate $\bar{s}_i$). We translate each clause $c=(\bar{x}_9,x_2,x_5)$ into a new clause $\tilde{c}=(s_1,\bar{s}_9, s_2)$ by replacing each literal $x_i$ of $c$ with the literal corresponding the unique subclass which contains $x_i$. Note that the subclass literal replacing the literal $x_i$ may have polarity opposite to $x_i$ (if for example $x_i \in \bar{S}_i$). The new induced formula $\tilde{\phi}$ may contain some clauses with multiplicity $>1$ as well as clauses in which some variable appears more than once (e.g. $(s_1,s_1,s_8)$). We will now define a homomorphism $f$ from $G_\phi^{xor}$ to $G_{\tilde{\phi}}^{xor}$, which implies that $\vartheta(G_\phi^{xor}) \geq \vartheta(G_{\tilde{\phi}}^{xor})$ (a homomorphism $f:G \rightarrow H$ maps the vertices of $G$ into the vertices of $H$ while preserving the edge relation, i.e. if $(u,v) \in E(G)$ then $(f(u),f(v)) \in E(H)$). Recall that each clause $c = (\bar{x}_9,x_2,x_5)$ of $\phi$ has a unique corresponding clause $\tilde{c} = (s_1,\bar{s}_9, s_2)$ of $\tilde{\phi}$ (although other copies of $(s_1,\bar{s}_9, s_2)$ may exist in $\tilde{\phi}$). The map $f$ is defined only for legal satisfying assignments of $\phi$ (we already removed from $G_{\phi}^{xor}$ all the non legal assignments). $f$ maps the vertices (assignments) in the clause cloud of $c$ to vertices (assignments) in the clause cloud of $\tilde{c}$ as follows:\ for a **legal** satisfying assignment of $c$, say $(\bar{x}_9 ,x_2,x_5)= (1,1,1)$, we replace each literal $x_i$ with its corresponding class literal and leave the values as is. For example if $\bar{x}_9 \in S_1, x_2 \in \bar{S}_9, x_5 \in S_2$ then $f$ maps the assignment $(\bar{x}_9 ,x_2,x_5) = (1,1,1)$ into $(s_1,\bar{s}_9, s_2) = (1,1,1)$. It is not hard to see that $f$ maps a legal satisfying assignment for $c$ into an assignment for $\tilde{c}$ that is both satisfying and noncontradictory (meaning for example that it will not result in one occurrence of $s_1$ being set to 0 and the other being set to 1). The assignment $f$ returns must be non contradictory as otherwise $\phi$ can be refuted by GE3. Note that GE3 can not refute $\tilde{\phi}$ nor can it derive an equation like $s_i = s_j$, for $i \neq j$. From here on we show a SDP solution to $G_{\tilde{\phi}}^{xor}$. The vector $v_0$ is set to be $(1,0,\ldots,0)$. The remaining vector assignments are as follows, divided by the clause types: 1. Type (1), three free distinct classes. Assume the clause is $\tilde{c} = (\bar{s}_{1},s_2,s_4)$. The vector assignments will be: $$\begin{aligned} &v^{\tilde{c}}_{(\bar{s}_{1},s_2,s_4) =(1,1,1)} = (\nicefrac{1}{4},~ - \nicefrac{1}{4},~~~~~\nicefrac{1}{4},~~~0,~~~~\nicefrac{1}{4},~~~0,~\ldots,0)\\ &v^{\tilde{c}}_{(\bar{s}_{1},s_2,s_4)=(1,0,0)} = (\nicefrac{1}{4},~ - \nicefrac{1}{4},~-\nicefrac{1}{4},~~~0,~-\nicefrac{1}{4},~~~0,~\ldots,0)\\ &v^{\tilde{c}}_{(\bar{s}_{1},s_2,s_4)=(0,1,0)} = (\nicefrac{1}{4}, ~~~~~\nicefrac{1}{4},~~~~~\nicefrac{1}{4},~~~0,~-\nicefrac{1}{4},~~~0,~\ldots,0)\\ &v^{\tilde{c}}_{(\bar{s}_{1},s_2,s_4)=(0,0,1)} = (\nicefrac{1}{4}, ~~~~~\nicefrac{1}{4},~-\nicefrac{1}{4},~~~0,~~~~~\nicefrac{1}{4},~~~0,~\ldots,0)\end{aligned}$$ 2. Type (2), one fixed class and two occurrences of some free class. Hence the equation has exactly two satisfying assignments. One assignment would get a vector that has $1/2$ in its 0 coordinate and $1/2$ on the coordinate corresponding to the free class, and the other would get a vector that has $1/2$ in its 0 coordinate and $-1/2$ on the coordinate corresponding to the free class. For example, for the clause $\tilde{c} = (s_l,s_2,s_2)$ the vectors would be: $$\begin{aligned} &v^{\tilde{c}}_{(s_l,s_2,s_2) = (1,1,1)} = (\nicefrac{1}{2} ,~ 0 ,~~~~ \nicefrac{1}{2},~~~ 0,~\ldots,0)\\ &v^{\tilde{c}}_{(s_l,s_2,s_2) = (1,0,0)} = (\nicefrac{1}{2},~ 0,~ -\nicefrac{1}{2} ,~~~ 0,~\ldots,0)\end{aligned}$$ 3. Type (3), three fixed classes. Assume $\tilde{c} = (s_l,\bar{s}_l,\bar{s}_l)$. In this case the only non-zero vector is: $$\begin{aligned} &v^{\tilde{c}}_{(s_l,\bar{s}_l,\bar{s}_l) = (1,0,0)} = (1,0 ,0,0,\ldots,0)\end{aligned}$$ We next show that the above vector configuration is a valid solution of the $\vartheta$ function of $G_{\tilde{\phi}}^{xor}$ (it is easy to see that the above solution has value of $m$). Constraints of type hold because of the special form of non-zero vectors. The fact that constraints of type hold will be implicit in our proof that constraints of type hold, and is omitted. Hence we will only consider now constraints of type . Observe first that within every clause cloud constraints of type hold. Hence it remains to check for pairs of different clauses that have an $s$ variable in common. Let $\tilde{c}_1,\tilde{c}_2$ be two clauses that intersect. We continue by case analysis according to the number of distinct $s$ variables shared by $\tilde{c}_1,\tilde{c}_2$. 1. Three distinct variables are shared: since GE3 did not deduce $0=1$ the clauses are identical and (and ) hold from the fact that it holds for each cloud separately. 2. Two distinct variables are shared: using GE3 we deduce that also the third variable is shared and this case was already handled. 3. Exactly one variables is shared: for simplicity, assume that each of the clauses contain $3$ different variables and say $s_i$ is the shared variable. The only two indices that contribute to the inner product sum are $0$ and (possibly) $i$. If $s_i$ is fixed the assignments cannot be contradictory and the sum is strictly positive (only coordinate $0$ contribute to the sum). Assume that $s_i$ is free. Consider the case in which in each clause the other two literals are also free. If the vectors are of contradicting assignments the sum will be $(1/4)(1/4) + (-1/4)(1/4)$ (or $(1/4)(1/4) + (1/4)(-1/4)$). If the vectors are not of contradicting assignments, the sum is strictly positive. Note that also in the other cases where one of the clauses contains only one or two different $s$ variables, a similar argument works. We follow the line of proof given at [@Ben-SassonWi01] with some simplifications that can be applied in our case. We use the following definitions from [@Ben-SassonWi01]. Let $A,B$ be any two formulas. $A \models B$ if every satisfying assignment for $A$ is a satisfying assignment for $B$, or equivalently, every non-satisfying assignment of $B$ is also a non-satisfying assignment of $A$. Let $\phi$ be a formula (collection of clauses) and let $C$ be any clause. We use $\mu_{\phi}(C)$ to denote the minimum size subformula of $\phi$ that implies $C$, i.e. $\mu_{\phi}(C) {\stackrel{\vartriangle}{=}}\min_{ \phi' \subseteq \phi} |\{\phi' \models C\}|$. As $\phi$ is known from the context (and fixed) we use $\mu(C)$ instead of $\mu_{\phi}(C)$. The function $\mu$ is sub-additive, meaning that if $A,B \models C$ then $\mu(C) \leq \mu(A) + \mu(B)$. We use $0$ to denote a contradiction (the empty clause). A simple counting argument shows that any subformula of $\phi$ of size smaller than $k {\stackrel{\vartriangle}{=}}\frac{\log n}{4\log \log n}$ is satisfiable; see Lemma \[lemma: random formula properties\]. Thus, $\mu(0) \geq k$. From the sub-additivity of $\mu$, it follows that any GE3 proof of $0$ contains some clause $C$ for which $\frac{k}{3} \leq \mu(C) \leq \frac{2k}{3}$ (the explanation is as follows. The derivation of $0$ can be described by a tree in which every leaf has a label that equals to some clause of $\phi$ and the root has a label that equals $0$. For each leaf label, say $A$, it holds that $\mu(A)=1$ and for the root label $0$ it holds $\mu(0) \geq k$). In other words, the minimal subformula $E'$ that implies $C$ is of size in $[\frac{k}{3},\frac{2k}{3}]$. The subformula $E'$ (as any other subformula of $\phi$, whose size in $[\frac{k}{3},\frac{2k}{3}]$, see Lemma \[lemma: random formula properties\]) has at least $4$ *special variables*, each of them appears in exactly one clause of $E'$. We show in the next paragraph that each of these $4$ special variables must be in $C$. This implies that $C$ cannot be derived in GE3, contradicting the assumption that GE3 refutes $\phi$. Let $x$ be a special variable that belongs to some clause $f$ of $E'$ (and not to any other clause in $E'$). From the minimality of $E'$, there exists an assignment $\alpha$ such that $f(\alpha) = C(\alpha) = 0$ but for any other clause $g \in E'$ it holds that $g(\alpha)=1$ (as otherwise $E' \setminus \{f\} \models C$). By contradiction, assume that $x \not \in C$. Changing the value of $\alpha$ only on $x$ leaves $C$ unsatisfied. Yet, $f$ becomes satisfied and any other clause of $E'$ remains satisfied because $x$ appears only on $f$. We deduce that after changing $\alpha$ only on $x$ the subformula $E'$ becomes satisfied while $C$ is not, this is a contradiction to $E' \models C$. \[lemma: random formula properties\] Let $\phi$ be a random formula with $m = o\left( n^{\frac{3}{2} - \frac{22 \log \log n}{\log n}} \right)$ clauses. Set $k = \frac{\log n}{4\log \log n}$. With high probability the following properties hold. 1. Any subformula of $\phi$ of size $k$ is satisfiable. 2. Any subformula $E' \subset \phi$, whose size is in $[\frac{k}{3},\frac{2k}{3}]$, has at least $4$ variables, each of them belongs to exactly one clause of $E'$. We show that any small subformula $\phi'$ is satisfiable by showing that in any such small subformula, the number of variables is at least the number of clauses. By Hall’s marriage theorem, in any such subformula $\phi'$ there is a matching from the variables to the clauses that covers all the clauses, which implies that $\phi'$ is satisfiable. We now analyse the first event (proving part 1 of the lemma). Consider $k$ clauses chosen at random. The probability that they contain less than $k$ different variables is bounded by the probability of the following event: when throwing $3k$ balls into $n$ bins, the set of non empty bins is $<k$. Thus the probability for the first event is at most $$\begin{aligned} & {m \choose k} \sum_{i=1}^{k-1} {n \choose i} \left( \frac{i}{n}\right)^{3k} \leq 2 \left( \frac{me}{k} \right)^k \left( \frac{ne}{k-1} \right)^{k-1} \left( \frac{k-1}{n}\right)^{3k} \\ & \leq 2\frac{e^{2k-1}(k-1)^{k+1}}{n} \left( \frac{m}{n^2}\right)^{k} \leq o(1) \left( \frac{m}{n^2}\right)^{k}.\end{aligned}$$ (the first inequality is because the sum is geometric with ratio $\geq \frac{en}{k}$, the last inequality holds for $k = \frac{\log n}{4\log \log n}$). We now bound the probability of the second event (part $2$ of the lemma). Fix $l$ to be in the interval $[\frac{k}{3},\frac{2k}{3}]$. Consider $l$ clauses chosen at random. The probability that they contain less than $4$ special variables equals the probability of the following event. When throwing $l$ triplets of balls into $n$ bins (where each triplet of balls choose three different bins) there are less than $4$ bins that contain exactly one ball. Notice that if the balls fall into more than $3(l+1)/2$ bins, there must be at least $4$ bins that contain exactly one ball. The probability is thus bounded by $$\begin{aligned} {m \choose l} \sum_{i=1}^{3(l+1)/2} {n \choose i} \left( \frac{{i \choose 3}}{{n \choose 3}}\right)^{l} &\leq 2 \left( \frac{me}{l} \right)^l \left( \frac{ne}{3(l+1)/2} \right)^{3(l+1)/2} \left( 1.01 \frac{3(l+1)/2}{n}\right)^{3l} \\ & \leq l^{4l} \left( \frac{m}{n^{\frac{3}{2}(1 - \frac{1}{l})}}\right)^{l} \leq \left( \frac{ml^4}{n^{\frac{3}{2}(1 - \frac{1}{l})}}\right)^{l} .\end{aligned}$$ To cover all possible values of $l \in [\frac{k}{3},\frac{2k}{3}]$ we multiply the last term by $k$. The induced bound is $o(1)$ for $m = o\left( n^{\frac{3}{2} - \frac{22 \log \log n}{\log n}} \right)$. A simple probabilistic argument shows that if $c$ is large enough, $\phi$ is likely to contains four clauses of the following form (see Lemma \[lemma: four clauses\]): $$\begin{aligned} &c_1 = (x_1,x_2,x_3) & c_3 = (x_5,x_6,x_3)\\ &c_2 = (x_1,x_2,x_4) & c_4 = (x_5,x_6,\bar{x}_4)\end{aligned}$$ The above four clauses are contradictory (summing all of them give $1=0$ modulus $2$). The $\vartheta$ function of the graph induced only by these $4$ clauses has a value of $\approx 3.4142 < 4$. This bound was experimentally derived by running a semi-definite programming package on Matlab. The adjacency matrix we used is: $$\begin{MAT}(e,3pt,3pt)[3pt]{cccccccccccccccc} 0 &1 &1 &1 &0 &1 &1 &1 &0 &1 &1 &0 &0 &0 &0 &0\\ 1 &0 &1 &1 &1 &0 &1 &1 &1 &0 &0 &1 &0 &0 &0 &0\\ 1 &1 &0 &1 &1 &1 &0 &1 &1 &0 &0 &1 &0 &0 &0 &0\\ 1 &1 &1 &0 &1 &1 &1 &0 &0 &1 &1 &0 &0 &0 &0 &0\\ 0 &1 &1 &1 &0 &1 &1 &1 &0 &0 &0 &0 &1 &0 &0 &1\\ 1 &0 &1 &1 &1 &0 &1 &1 &0 &0 &0 &0 &0 &1 &1 &0\\ 1 &1 &0 &1 &1 &1 &0 &1 &0 &0 &0 &0 &0 &1 &1 &0\\ 1 &1 &1 &0 &1 &1 &1 &0 &0 &0 &0 &0 &1 &0 &0 &1\\ 0 &1 &1 &0 &0 &0 &0 &0 &0 &1 &1 &1 &0 &1 &1 &1\\ 1 &0 &0 &1 &0 &0 &0 &0 &1 &0 &1 &1 &1 &0 &1 &1\\ 1 &0 &0 &1 &0 &0 &0 &0 &1 &1 &0 &1 &1 &1 &0 &1\\ 0 &1 &1 &0 &0 &0 &0 &0 &1 &1 &1 &0 &1 &1 &1 &0\\ 0 &0 &0 &0 &1 &0 &0 &1 &0 &1 &1 &1 &0 &1 &1 &1\\ 0 &0 &0 &0 &0 &1 &1 &0 &1 &0 &1 &1 &1 &0 &1 &1\\ 0 &0 &0 &0 &0 &1 &1 &0 &1 &1 &0 &1 &1 &1 &0 &1\\ 0 &0 &0 &0 &1 &0 &0 &1 &1 &1 &1 &0 &1 &1 &1 &0\\ \end{MAT}$$ vertices $1,2,3,4$ correspond to $c_1$ ,vertices $5,6,7,8$ correspond to clause $c_2$, vertices $9,10,11,12$ correspond to clause $c_3$ and vertices $13,14,15,16$ correspond to clause $c_4$: aaa = aaa = aaa = aaa = aaaaa = aaa = aaa = aaa = aaa = aaa $x_1$ $x_2$ $x_3$ $x_1$ $x_2$ $x_4$\ $v_1$ 1 1 1 $v_5$ 1 1 1\ $v_2$ 0 1 0 $v_6$ 0 1 0\ $v_3$ 1 0 0 $v_7$ 1 0 0\ $v_4$ 0 0 1 $v_8$ 0 0 1\ aaa = aaa = aaa = aaa = aaaaa = aaa = aaa = aaa = aaa = aaa $x_5$ $x_6$ $x_3$ $x_5$ $x_6$ $\bar{x}_4$\ $v_9$ 1 1 1 $v_{13}$ 1 1 1\ $v_{10}$ 0 1 0 $v_{14}$ 0 1 0\ $v_{11}$ 1 0 0 $v_{15}$ 1 0 0\ $v_{12}$ 0 0 1 $v_{16}$ 0 0 1\ The $\vartheta$ function of $G_{\phi}$ must be smaller than $<m$ as the remaining graph (without the clouds of $c_1,c_2,c_3,c_4$) can be covered by $m-4$ cliques. \[lemma: four clauses\] Let $\phi$ be a random formula with $n$ variables and $m=cn^{3/2}$ random clauses. Almost surely $\phi$ contains four clauses of the form: $$\begin{aligned} &c_1 = (x_1,x_2,x_3) & c_3 = (x_5,x_6,x_3)\\ &c_2 = (x_1,x_2,x_4) & c_4 = (x_5,x_6,\bar{x}_4)\end{aligned}$$ We say that $a(n) \sim b(n)$ if $\lim_{n \rightarrow \infty} \frac{a(n)}{b(n)} =1$. A pair of clauses is said to *match* if the two clauses share the same first and second literal. The expected number of matched pairs in $\phi$ is $$\begin{aligned} {m \choose 2} \frac{1}{2n}~\frac{1}{2n-2} \sim \frac{c^2n^3}{2} \frac{1}{4n^2} = \frac{c^2n}{8}.\end{aligned}$$ Furthermore, it can be shown that w.h.p. $\phi$ contains $\sim \frac{c^2n}{8}$ matched pairs such that each clause of $\phi$ participates in at most one pair of matching clauses (a standard use of the second moment, see for example [@FeigeOf04] for a proof). Assume we have $\sim \frac{c^2n}{8}$ matched pairs. For any such pair the third literal in each of them is still random. Fix two matched pairs $c_1,c_2$ and $c_3,c_4$. With probability $\sim \frac{1}{4n^2}$ the third literal of $c_1$ and $c_3$ is the same and the third literal of $c_2$ is opposite from the third literal of $c_4$. It thus follows that the expected number of two pairs of the form $$\begin{aligned} &c_1 = (x_1,x_2,x_3) & c_3 = (x_5,x_6,x_3)\\ &c_2 = (x_1,x_2,x_4) & c_4 = (x_5,x_6,\bar{x}_4),\end{aligned}$$ is $$\begin{aligned} \sim \frac{1}{2} \left(\frac{c^2n}{8} \right)^2 \frac{1}{4n^2} \sim \frac{c^4}{8^3}.\end{aligned}$$ Using standard techniques (such as the second moment), it can be shown that almost surely $\phi$ contains four clauses of this form. Details are omitted. Acknowledgements {#acknowledgements .unnumbered} ================ This work was supported in part by a grant from the German-Israeli Foundation for Scientific Research and Development (G.I.F.). [99]{} M. Alekhnovich, S. Arora and I. Tourlakis. Towards strong nonapproximability results in the Lovasz-Schrijver hierarchy. STOC 2005, 294–303. J. Buresh-Oppenheim, N. Galesi, S. Hoory, A. Magen and T. Pitassi. Rank bounds and integrality gaps for cutting plane procedures. FOCS 2003. E. Ben-Sasson and A. Wigderson. Short proofs are narrow -—resolution made simple. J. ACM, 48(2):149–169, 2001. V. Chvatal and E. Szemeredi. Many hard examples for resolution. J. ACM, 35(4):759–768, Oct 1988. A. Coja-Oghlan, A. Goerdt, A. Lanka, and F. Schadlich. Certifying unsatisfiability of random 2k-sat formulas using approximation techniques. FCT 2003, 15–26. U. Feige. Relations between average case complexity and approximation complexity. STOC 2002, 534–543. U. Feige and E. Ofek. Easily refutable subformulas of large random 3cnf formulas. ICALP 2004, 519–530. E. Friedgut and J. Bourgain. Sharp thresholds of graph properties, and the k-sat problem. J. of the American Mathematical Society, 12(4):1017–1054, 1999. J. Friedman, A. Goerdt, and M. Krivelevich. Recognizing more unsatisfiable random 3-sat instances efficiently. Technical report, 2003. A. Goerdt and M. Krivelevich. Efficient recognition of random unsatisfiable k-[SAT]{} instances by spectral methods. STACS 2001, 294–304. A. Goerdt and A. Lanka. Recognizing more random 3-sat instances efficiently. Manuscript, 2003. M. Hajiaghayi and G.B. Sorkin. The satisfiability threshold for random 3-SAT is at least 3.52. http://arxiv.org/abs/math.CO/0310193, 2003. S. Janson, Y. C. Stamatiou, and M. Vamvakari. Bounding the unsatisfiability threshold of random 3-sat. Random Structures and Algorithms, 17(2):103–116, 2000. A.C. Kaporis, L.M. Kirousis, and E.G. Lalas. Selecting complementary pairs of literals. LICS 2003. S. Khot. Ruling Out PTAS for Graph Min-Bisection, Densest Subgraph and Bipartite Clique. FOCS 2004, 136–145.

--- abstract: 'We consider passive scalar convected by multi-scale random velocity field with short yet finite temporal correlations. Taking Kraichnan’s limit of a white Gaussian velocity as a zero approximation we develop perturbation theory with respect to a small correlation time and small non-Gaussianity of the velocity. We derive the renormalization (due to temporal correlations and non-Gaussianity) of the operator of turbulent diffusion. That allows us to calculate the respective corrections to the anomalous scaling exponents of the scalar field and show that they continuously depend on velocity correlation time and the degree of non-Gaussianity. The scalar exponents are thus non universal as was predicted by Shraiman and Siggia on a phenomenological ground (CRAS [**321**]{}, 279, 1995).' address: | $^a$ Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel\ $^{b}$ Landau Institute for Theoretical Physics, Moscow, Kosygina 2, 117940, Russia author: - 'M. Chertkov$^a$, G. Falkovich$^a$ and V. Lebedev$^{a,b}$' title: | Non-universality of the scaling exponents\ of a passive scalar convected by a random flow --- [2]{} The most striking feature of turbulence is its intermittent spatial and temporal behavior. Statistically, intermittency means substantial non-Gaussianity. For developed turbulence, where the correlation functions are scale invariant at the inertial interval of scales, the intermittency is manifested as an anomalous scaling of correlation functions. That means that some random field $\theta({\bf r},t)$ has the structure functions $S_{2n}\!=\!\langle [\theta(t,{\bf r}_1)\!-\!\theta(t,{\bf r}_2)]^{2n}\rangle \propto r_{12}^{\zeta_{2n}}$ with the exponents $\zeta_{2n}$ that are not equal to $n\zeta_2$. As a result, the degree of non-Gaussianity, that may be characterized by the ratio $S_{2n}/S_2^n$, depends on scale. Experiments and simulations show that the intermittency and anomalous scaling of the scalar field passively convected by a fluid are much stronger pronounced than the intermittency of the velocity field itself [@84AHGA; @91Sre; @94HS; @95KYC]. It is in the problem of passive scalar where consistent analytic theory of an anomalous scaling starts to appear [@95KYC; @68Kra-a; @95GK; @95CFKLb; @95SS]. It is intuitively clear that the physical reason for scalar intermittency is a spatial inhomogeneity of the advecting velocity. The analysis of the velocity field with smooth inhomogeneity shows, however, that there is no anomalous scaling of the scalar whatever be the (finite) temporal correlations of the velocity [@68Kra-a; @94SS; @95CFKLa]. Analytic treatment of a non-smooth multi-scale velocity was possible hitherto only in the so-called Kraichnan’s problem of white advected scalar [@68Kra-a] where the correlation functions satisfy closed linear equations of the second order [@94SS]. It has been shown [@95GK; @95CFKLb] that, even without any temporal correlations, spatial non-smoothness of the velocity provides for an anomalous scaling of the scalar. The anomalous parts appeared as zero modes of the operator of turbulent diffusion and entered the correlation functions due to matching conditions at the pumping scale [@95GK; @95CFKLb; @95SS]. The coefficients at the modes were thus pumping-dependent while the form of any zero mode was universal i.e. determined only by the exponent of the velocity spectrum and space dimensionality. In particular, the exponents $\zeta_n$ of the scalar were universal for the delta-correlated velocity. Now, what is the role of velocity temporal behavior in building up intermittency of the scalar field? It was argued phenomenologically by Shraiman and Siggia [@95SS] that the exponents of the scalar field depend on more details of the velocity statistics “than just exponents”. Here, we consider the simplest possible generalization of Kraichnan’s problem and consistently derive the equations for scalar correlation functions in the case of short yet finite velocity correlation time $\tau_r$ which is supposed to be a power function of the scale $r$. The behavior of the ratio $\tau_r/t_r$ is important, where $t_r$ is the turnover time at the scale $r$. If the ratio tends to zero at decreasing $r$ then we return to the $\delta$-correlated case. If the ratio increases at decreasing $r$ we encounter the problem of the quenched disorder type which should be considered separately. We consider the marginal case of a complete self-similarity where $\epsilon=\tau_r/t_r$ does not depend of $r$ and formulate the perturbation theory regarding the ratio as the small parameter of our theory. We show that $\zeta_2$ does not depend on $\epsilon$ while $\zeta_n$ for $n>2$ are $\epsilon$-dependent that is the set of the exponents is non universal along with the prediction of [@95SS]. The principal difference between the second and higher correlation functions is naturally explained on the language of zero modes: there is no zero mode (except constant) for the pair correlator while the zero modes of the high correlators depend on the precise form of the operator of turbulent diffusion which is $\epsilon$-dependent. This is formally similar to what has been discovered by Kadanoff, Wegner and Polyakov in the theory of phase transitions: the critical exponents continuously depend on the amplitude of the operator term with dimension $d$ added to the Hamiltonian [@71KW; @73Pol]. Note that the below results cannot be directly applied to the description of scalar advection by a Kolmogorov turbulence: because of sweeping effect, the different-time velocity statistics is not scale-invariant in the Eulerian frame [@64Kr]. Our use of scale-invariant velocity is intended to establish the general fact of the sensitive dependence of scalar exponents on the velocity statistics. The advection of passive scalar $\theta(t,{\bf r})$ by an incompressible flow is governed by the equations $$(\partial_t\!-\!\hat{P})\theta\!=\!\phi,\quad \hat{P}(t)\!=\!-\!v^\alpha\nabla^\alpha \!+\!\kappa\triangle,\quad \nabla^\alpha v^\alpha\!=\!0, \label{em}$$ where $\kappa$ is the coefficient of molecular diffusion. The advecting velocity ${\bf v}$ and the source $\phi$ are independent random functions. A formal solution of (\[em\]) is $$\begin{aligned} && \theta(t,{\bf r})=\int_{-\infty}^t dt_1 {\sl T}\exp\biggl(\int_{t_1}^{t} dt'\, \hat{P}(t')\biggr)\phi(t_1,{\bf r}) \,, \label{a2a} \end{aligned}$$ where ${\sl T}\exp$ designates the chronologically ordered exponent. From (\[a2a\]) it follows $$\begin{aligned} && F_n(t,{\bf r}_1,\dots,{\bf r}_{2n})\equiv \langle\theta(t,{\bf r}_1)\dots\theta(t,{\bf r}_{2n})\rangle \nonumber \\ && =\int\limits_{-\infty}^t\!\!\! dt_1 \dots \int\limits_{-\infty}^t\!\!\! dt_{2n} \hat{\cal A}\, \Biggl\langle \prod_{i=1}^{2n} \phi(t_i,{\bf r}_i)\Biggr\rangle \,, \label{co} \\ && \hat{\cal A}=\langle \hat Q \rangle \,, \quad \hat Q(t)=\prod_{i=1}^{2n}{\sl T}\exp \left(\int_{t_i}^t dt'_i\hat P(t'_i,{\bf r}_i)\right) \,. \label{ti1} \end{aligned}$$ Differentiating $\hat{\cal A}$ over the current time $t$, one gets $$\partial_t{\hat{\cal A}}= \left\langle\hat{\cal P}(t)\hat Q(t)\right\rangle \,, \quad \hat{\cal P}=\kappa\nabla_i^2-v_i^\alpha\nabla_i^\alpha \,, \label{ti2}$$ where ${\bf v}_i={\bf v}(t,{\bf r}_i)$ and $\nabla_i=\partial/\partial{\bf r}_i$. Here and below summation over both repeated vector superscripts and subscripts enumerating points ${\bf r}_i$ is implied. The identity (\[ti2\]) can be brought to the form $$\partial_t\hat{\cal A}(t) =\kappa\nabla^2_i\hat{\cal A}(t) +\int\limits_0^\infty dt' \hat{\cal N}(t')\hat{\cal A}(t-t') \,, \label{co1}$$ where $\hat{\cal N}(t)$ is to be found. The decay of $\hat{\cal N}(t)$ is determined by the velocity correlation time $\tau_r$ which is supposed to be much smaller than the spectral transfer time characteristic of (\[co\]). It is the reason why the upper limit in (\[co1\]) can be substituted by infinity. We shall find the first terms of the expansion of $\hat{\cal N}$ in $\tau_r$. Let us first examine the Gaussian contribution to $\partial_t\hat{\cal A}$ related to reducible correlation functions of ${\bf v}$ $$\int^t d\tilde t\, \underline{v}_i^\alpha(t)\nabla_i^\alpha \bigg\langle{\sl T}\exp \Big[\int\limits_{\tilde t}^t dt' \hat{\cal P}(t')\Big] \underline{v}_j^\beta(\tilde t)\nabla^\beta_j \hat Q(\tilde t)\bigg\rangle_{\rm G} , \label{ti3}$$ where the product $\underline{v}_i^\alpha(t)\underline{v}_j^\beta(\tilde t)$ should be substituted by the pair correlation function $\langle v^\alpha(t,{\bf r}_i)v^\beta(\tilde t,{\bf r}_j)\rangle$. The integrand is nonzero for $t-\tilde t\leq\tau_r$ and consequently ${\sl T}\exp \Big[\int_{\tilde t}^t dt'\hat{\cal P}(t')\Big]$ can be expanded over $t-\tilde t$. The zero term gives $$\hat{\cal N}_0(t)=\langle v^\alpha_i(t)\nabla^\alpha_i v^\beta_j(0)\nabla^\beta_j \rangle \,. \label{co2}$$ The first and the second terms of the expansion produce the linear in $\tau_r$ contribution $$\begin{aligned} && \hat{\cal N}_1(t)\!=\! \int\limits_0^t\! dt_1\biggl[\int\limits_0^{t_1}\! dt_2 [\underline{v}_i^\alpha(t)\nabla_i^\alpha \overline v_k^\gamma(t_1)\nabla_k^\gamma \underline{v}_j^\beta(t_2)\nabla^\beta_j \overline v_m^\mu(0)\nabla_m^\mu \nonumber \\ && +\underline{v}_i^\alpha(t)\nabla_i^\alpha \overline v_k^\gamma(t_1)\nabla_k^\gamma \overline v_m^\mu(t_2)\nabla_m^\mu \underline{v}_j^\beta(0)\nabla^\beta_j] \label{nz} \\ && +\kappa t_1\underline{v}_i^\alpha(t_1)\nabla_i^\alpha\nabla_k^2 \underline{v}_j^\beta(0)\nabla_j^\beta\biggr], \nonumber \end{aligned}$$ where the products $\overline v\,\overline v$ and $\underline v\,\underline v$ should be substituted by the corresponding pair correlation functions. For a short-correlated velocity field, the leading non-Gaussian contribution to the correlation functions of ${\bf v}$ is determined by the irreducible part of the fourth-order correlation function of ${\bf v}$. Generalizing the trick leading from (\[ti2\]) to (\[ti3\]) we obtain the non-Gaussian term $\hat{\cal N}_{\rm nG}(t)$: $$\!\!\int_0^t\!\!dt_1\!\!\int_0^{t_1}\!\!d t_2 \langle\!\langle\! v_i^\alpha(t)\nabla_i^\alpha v_k^\mu(t_1)\nabla_k^\mu v_j^\beta(t_2)\nabla_j^\beta v_n^\gamma(0)\nabla_n^\gamma \!\rangle\!\rangle\,, \label{nog}$$ where double angular brackets stand for the cumulant. The operator $\hat{\cal A}(t)$ is exponential in time $$\hat{\cal A}(t)=\exp\bigl[(t-t_0)(\kappa\nabla^2_i +\hat{\cal L})\bigr]\hat{\cal A}(t_0) \label{th}$$ asymptotically at $t-t_0\gg\tau_r$. Substituting (\[th\]) into (\[co1\]), expanding $\exp(t'\hat{\cal L})$ and keeping only the principal terms we find the operator of turbulent diffusion $$\begin{aligned} && \hat{\cal L}=\hat{\cal L}_0+\hat{\cal L}_1 +\hat{\cal L}'_1+\hat{\cal L}_{nG}\, \label{tj2} \\ && \hat{\cal L}_{\{0,1,nG\}}\! =\!\!\int_0^\infty\!dt\,\hat {\cal N}_{\{0,1,nG\}}(t),\, \ \hat{\cal L}'_1\!=\!\!-\!\! \int_0^\infty\!dt\, t {\cal N}_0(t)\hat{\cal L}_0 \,. \nonumber \end{aligned}$$ Using (\[co\],\[th\]) we can obtain the expression for $\partial_t F_n$. For the pumping $\delta$-correlated in time, one gets $$\begin{aligned} && \partial_t F_n(t,{\bf r}_1,\dots,{\bf r}_{2n}) -\hat{\cal L}F_n(t,{\bf r}_1,\dots,{\bf r}_{2n}) \nonumber \\ && =\hat{\cal M}[\chi_{12}F_{n-1}(t,{\bf r}_3,\dots, {\bf r}_{2n})+{\rm permutations}] \,. \label{eq} \end{aligned}$$ Here, the function $\chi(r_{12})=\int dt\,\langle\phi(t,{\bf r}_1)\phi(0,{\bf r}_2)\rangle$ decays on the pumping scale $L$ and $\chi(0)$ is the production rate of $\theta^2$. The operator $\hat{\cal M}$ in (\[eq\]) can be estimated as $\hat{\cal A}(\tau_L)$. The account of temporal correlations of the pumping (which can be done perturbatively as long as the pumping correlation time is much less than the time of scalar transfer) results in an extra renormalization of $\hat{\cal M}$ operator. Its explicit form is unimportant for what follows. Indeed, the balance equation (\[eq\]) contains the renormalization (due to velocity temporal correlations and non-Gaussianity) of all three relevant quantities: pumping, turbulent diffusion and molecular diffusion (the last term in $\hat {\cal N}_1$). We discuss here only the scaling exponents in the convective interval of scales (see below) that are determined solely by the form of the operator of turbulent diffusion $\hat{\cal L}$. Let us consider the pair correlation function of the velocity to be scale-invariant: $$\begin{aligned} && \langle[v^\alpha(t,{\bf r})\!-\!v^\alpha(0,{\bf 0})] [v^\beta(t,{\bf r})\!-\!v^\beta(0,{\bf 0})]\rangle\!=\!2 K^{\alpha\beta}(t,r), \nonumber \\ && K^{\alpha\beta}\!=\!\frac{D r^{2-\gamma}}{\tau_r}\biggl[ \biggl(\!\delta^{\alpha\beta}\! -\!\frac{r^\alpha r^\beta}{r^2}\!\biggr)g_\bot \biggl(\!{|t|\over\tau_r}\!\biggr)\!+\! \delta^{\alpha\beta}g_\|\biggl(\!{|t|\over\tau_r}\!\biggr)\biggr] \label{vel} \end{aligned}$$ with the correlation time $\tau_r\!=\!\tau_L (r/L)^z$. Dimensionless functions $g_\bot$ and $g_\|$ satisfy the incompressibility condition $(d\!-\!1)g_\bot(x)\!=\!zx^{a}d[x^{1-a}g_\|(x)]/dx$ where $a\!=\!({2\!-\!\gamma})/{z}-1$. Their normalization is fixed by below expressions (\[ko\],\[kw\]). The main term in (\[tj2\]) is [@94SS] $$\begin{aligned} && \hat{\cal L}_0=-\sum_{ij} {\cal K}_0^{\alpha\beta}(r_{ij})\nabla_i^\alpha\nabla_j^\beta \, \quad {\cal K}_0^{\alpha\beta} =2\int\limits_0^\infty dt\,{\cal K}^{\alpha\beta}(t)\, \label{lo} \\ && {\cal K}_0^{\alpha\beta}=Dr^{2-\gamma} \left(\frac{d+1-\gamma}{2-\gamma}\delta^{\alpha\beta} -\frac{r^\alpha r^\beta}{r^2}\right) \,. \label{ko} \end{aligned}$$ The expressions (\[lo\],\[ko\]) lead to the following turnover time $t_r\!=\!(2\!-\!\gamma)r^\gamma/D\gamma d(d\!-\!1)$ obtained for the delta-correlated case [@68Kra-a; @95CFKLb]. Our marginal case corresponds to $z=\gamma$ and the small parameter of the perturbation theory is thus $$\epsilon={D\tau_L}L^{-\gamma}{\gamma d(d-1)}/{(2-\gamma)}\ll 1. \label{c4}$$ Note that $\epsilon$ contains $d^2$ which tells us that the space dimensionality should not be very large for the approximation of a short correlation to be valid: the characteristic time of the scalar transfer (proportional to $d^{-2}$) should be larger than the correlation time. Starting from the expression for the pair velocity correlator (\[vel\]) we can obtain the first Gaussian $\epsilon$-correction to (\[lo\]). Calculating (\[nz\]) and then integrals in (\[tj2\]) we find $$\begin{aligned} && \hat{\cal L}_1\!+\!\hat{\cal L}_1^\prime\!=\! \frac{1}{2}\sum_{i,j,k}K_{0;ij}^{\alpha\beta} K_{1;ik}^{\mu\nu;\alpha} \nabla_i^{\mu}\nabla_j^{\beta}\nabla_k^{\nu}\!-\! \frac{1}{2}\sum_{i,j}{\cal B}_{ij}^{\mu\nu} \nabla_i^{\mu}\nabla_j^{\nu}\!-\! \frac{\kappa}{2}\sum_{i,j,k}\nabla^2_k {\cal K}_{1;ij}^{\alpha\beta}\nabla_i^\alpha \nabla_j^\beta, \quad {\cal B}^{\mu\nu}({\bf r})\!= \!K_{1;ij}^{\alpha\mu;\beta} K_{0;ij}^{\beta\nu;\alpha}\!\! -\!K_{1;ij}^{\alpha\beta} K_{0;ij}^{\mu\nu;\alpha\beta}\!\! \nonumber \\ && + 2\!\int_0^\infty\!\! dt_1\nabla_{\bf r}^{\alpha} \nabla_{\bf r}^{\beta}\biggr[ \int_{t_1}^\infty\!\!\! dt_2 K^{\alpha\beta}(t_2;{\bf r}) \int_{t_1}^\infty\!\!\! dt_3 K^{\mu\nu}(t_3;{\bf r}) -\int_{t_1}^\infty\!\!\! dt_2 K^{\alpha\mu}(t_2;{\bf r}) \int_{t_1}^\infty\!\!\! dt_3 K^{\beta\nu}(t_3;{\bf r})\biggr], \label{c9}\end{aligned}$$ [2]{} where $K^{\alpha\beta;\mu}_n\equiv\nabla_{\bf r}^{\mu} K^{\alpha\beta}_n$, $K^{\alpha\beta;\mu\nu}_n\equiv \nabla_{\bf r}^{\mu} \nabla_{\bf r}^{\nu} K^{\alpha\beta}_n$ and $$\begin{aligned} && K_1^{\alpha\beta}\!({\bf r})\!=\! 2\!\int\limits_0^\infty\! dt K^{\alpha\beta}(t,{\bf r})\!=\!Dr^{2-\gamma}\tau_r \left(\!\frac{d\!+\!1}{2}\delta^{\alpha\beta}\!- \!\frac{r^\alpha r^\beta}{r^2}\right)\,, \nonumber \\&& {\cal B}^{\alpha\beta}({\bf r})=\epsilon D r^{2-\gamma}\left[ b_\|\delta^{\alpha\beta}+b_\bot\left(\delta^{\alpha\beta}- \frac{r^\alpha r^\beta}{r^2}\right)\right]\ . \label{kw} \end{aligned}$$ Now we can analyze the equation (\[eq\]) for $F_n$. At the convective interval of scales $L\!\gg\! r\!\gg\! [\kappa(2\!-\!\gamma)/D(d\!-\!1)]^{1/(2\!-\!\gamma)}$, the molecular diffusion term can be dropped: it is enough to require $F_n=0$ at $r=0$ [@95GK; @95CFKLb]. Here, the zero modes of $\hat{\cal L}$ are responsible for the anomalous scaling of $F_{n}$. The scaling exponents of the bare operator $\hat{\cal L}_0$ and the perturbation operator $\hat{\cal L}_1+ \hat{\cal L}_1^\prime$ coincide. For a self-similar velocity statistics, the non-Gaussian contribution $\hat{\cal L}_{nG}$ has the same scaling too. The first consequence is that the exponent of the pair correlation function is $\gamma$ at arbitrary finite order in $\epsilon$ for any $\gamma$ and $d$. Indeed, there is no zero mode of the two-point $\hat{\cal L}$ with a nonzero positive exponent that could provide an anomaly. Contrary, for $n>2$, the account of the $\epsilon$-contributions to the bare operator $\hat{\cal L}_0$ should produce obviously $\epsilon$-dependent corrections to the exponents of zero modes and consequently $\epsilon$-dependent anomalous scaling. To illustrate the above conclusion about $\tau$-dependence of the scalar exponents, let us give an example where the calculation can be done explicitly. We consider a large dimensionality (the limit $\gamma\gg (2-\gamma)/d$ solved in [@95CFKLb; @95CF] for $\tau=0$) while assuming that, in addition to (\[c4\]), $1/d\gg\epsilon$ (it will be seen below how the parameter $\gamma$ enters the condition). The leading (in $d$) terms of the bare and the Gaussian perturbative operators in terms of relative distances $r_{ij}$ are as follows \[multiplied by $(2\!-\!\gamma)/dD$\] $$\begin{aligned} &&\hat{\cal L}_{0,0}\!\!=\!\! d\!\sum\limits_{i>j}\!r_{ij}^{1-\gamma}\partial_{r_{ij}},\ \hat{\cal L}_{0,1}\!=\!\! \sum_{i>j}r_{ij}^{1-\gamma}\!\bigl(r_{ij}\partial^2_{ij}\!-\! \gamma\partial_{ij}\bigr)\!\!-\!\! \frac{1}{2}\!\!\sum_{i,j,p,q}\!r_{ij}^{2-\gamma}\frac{{\bf r}_{ip}{\bf r}_{jq}} {r_{ip}r_{jq}}\partial_{ip}\partial_{jq},\nonumber\\&& \hat{\cal L}_{1,0}= {\epsilon d(2\!-\!\gamma)\over\gamma}\! \left[\sum_{k>l}\!r_{kl}\partial_{kl}\! +\!\gamma\!-\!1\!+\!\!2b_\|^{(0)}\!\right]\! \sum_{i>j}\!\!r_{ij}^{1-\gamma}\partial_{ij}, \nonumber\\&& \hat{\cal L}_{1,1}\! =\! {\epsilon (\gamma-2)\over8\gamma}\biggl[\!\sum\! \biggl(\frac{{\bf r}_{ip}{\bf r}_{jq}}{r_{ip}r_{jq}} r_{ij}^2r_{kl}^{1-\gamma}\partial_{ip}\partial_{jq}\partial_{kl}\!+\! \frac{{\bf r}_{kp}{\bf r}_{lq}}{r_{kp}r_{lq}} r_{ij}r_{kl}^{2-\gamma}\partial_{kp}\partial_{lq}\partial_{ij}\biggr)\! +\!4\!\sum\!\frac{{\bf r}_{jp}{\bf r}_{ik}}{r_{ik}r_{jp}} \biggl(r_{ij}^{2-\gamma}\!+\!\frac{2-\gamma}{2}r_{ij}^2r_{ik}^{-\gamma}\biggr) \partial_{jp}\partial_{ik} \nonumber \\ && +4{b_\|^{(0)}}\!\sum\! r_{ij}^{2-\gamma}\frac{{\bf r}_{ip}{\bf r}_{jq}}{r_{ip}r_{jq}} \partial_{ip}\partial_{jq}\!+\! 16\!\sum_{i>j,k>l}\!r_{ij}^{1-\gamma}r_{kl}\partial_{ij} \partial_{kl}\!+\! \left(16\!-\!4\gamma\!-\!8\bigl(b_\|^{(1)}\!+\!b_\bot^{(0)}\!-\! b_\|^{(0)}\bigr)\right)\!\sum_{i>j}\! r_{ij}^{1-\gamma}\partial_{ij}\biggr], \label{Lpp} \end{aligned}$$ [2]{} The summation is performed over $n(n-1)/2$ distances which are independent variables if $d\!>\!n\!-\!2$. For the chosen form of $K^{\alpha\beta}(t,{\bf r})$, $b_\|\to (2-\gamma)( b_\|^{(0)}d^3+b_\|^{(1)}d^2)$, $b_\bot\to (2-\gamma) b_\bot^{(0)}d^2$ at $d\to\infty$, with $d$-independent constants $b_{\|,\bot}^{(i)}$. First, we calculate the corrections to the exponents related to the Gaussian correction (\[Lpp\]) and then discuss the corrections due to non-Gaussianity. Solving the equation for the pair correlation function one can check that $\zeta_2\!=\!\gamma$ is independent of $\epsilon$ and $d$. Then, we consider the four-point correlation function ($i,j,k,l=1\ldots4$). To get the main contribution at $r\ll L$ one has to perturb the bare zero mode of $\hat{\cal L}_{0,0}+\hat{\cal L}_{0,1}$. In the limit under consideration, it is enough to consider only the mode $$Z_0=\!\sum_{\{i,j,k,l\}}\! (r^\gamma_{ij}\!-\!r^\gamma_{kl})^2 -1/2 \!\sum_{\{i,j,k\}}\! (r^\gamma_{ij}\!-\!r^\gamma_{ik})^2, \label{zm1}$$ with the leading exponent $\Delta_4(0)=4(2-\gamma)/d$ found in [@95CFKLb] by $1/d$-expansion. The first $\epsilon$-correction to (\[zm1\]) can be obtained by applying the operator $-\hat{\cal L}_{0,0}^{-1} (\hat{\cal L}_{1,0}\hat{\cal L}_{0,0}^{-1}\hat{\cal L}_{0,1} +\hat{\cal L}_{0,1}\hat{\cal L}_{0,0}^{-1}\hat{\cal L}_{1,0})$ to the mode $Z_0$. The correction to the anomalous exponent is determined by the coefficient at $\ln(L/r)$ in the first $\epsilon$-contribution to $Z_0$, it is $$\Delta_4(\epsilon)=\Delta_4(0)+{\epsilon(2-\gamma)\over d\gamma} \bigl(4+6\gamma-2\gamma^2\bigr)\ . \label{corr}$$ The sign of the correction is positive for $0\!<\!\gamma\!<\!2$. We can also calculate $\tau$-related corrections for the high-order functions by using the technique developed in [@95CF] for finding the largest anomalous exponents. For $n\!\ll\!\gamma d$, $$\Delta_{2n}(\epsilon)={n(n-1)}\Delta_4(\epsilon)/2. \label{ne}$$ The scaling exponents thus depend not only on purely dimensionless quantities $\gamma$ and $d$ yet also on a dimensionless ratio of dimensional quantities. In other words, the exponents depend on the form of the structural functions $g_\bot$ and $g_\|$. Considering opposite hierarchy of small parameters $1/d\!\ll\!\epsilon $ and neglecting $1/d$ corrections, one finds $\zeta_{2n}\!=\!n\zeta_2$ at any order in $\epsilon$: temporal correlations by themselves do not produce an anomalous scaling if it is absent in the uncorrelated case. In this limit, the anomalous exponents appear only in the next $1/d$ order and are proportional to $\epsilon d$ – see [@chert] for the details. Now let us discuss the non-Gaussian contribution to the anomalous exponents. We denote by $\epsilon_4$ the ratio of the cumulant to the fourth-order correlator and consider the limit $1\gg 1/d\gg\epsilon_4 d^3$. Using the expressions (\[nog\],\[tj2\],\[eq\]) we conclude that the contribution to $\Delta_n$ is proportional to $n(n-1)\epsilon_4 d^2$ for $n\ll\gamma d$. To conclude, we learned that the scalar exponents are sensitive to the details of the velocity statistics. The existence of two different contributions (due to temporal correlations and non-Gaussianity of the velocity) makes it possible that there exist some classes of the statistics with special relations between the contributions (some remarkable cancelations, for instance), their analysis is left for future studies. Hopefully, real turbulent flows belong to those classes and analytic expressions for the passive scalar exponents can be found some day. We are grateful to E. Balkovsky, G. Eyink, I. Kolokolov and B. Shraiman for useful discussions. This work was partly supported by the Clore Foundation (M.C.), by the Rashi Foundation (G.F.) and by the Minerva Center for Nonlinear Physics (V.L.). R. Antonia, E. Hopfinger, Y. 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--- author: - 'V. F. Cardone' - 'E. Piedipalumbo' - 'C. Tortora' date: 'Receveid / Accepted ' title: Some astrophysical implications of dark matter and gas profiles in a new galaxy clusters model --- Introduction ============ Being the largest bound structures in the universe, galaxy clusters occupy a special position in the hierarchy of cosmic structures in many respects. They can be detected at high redshift because of the presence of hundreds of galaxies and hot X-ray emitting gas and, therefore, they appear to be ideal tools for studying large scale structure, testing the theories of structure formation and extracting invaluable cosmological information (see, e.g., [@BG01; @RBN02]). To first order, galaxy clusters may be described as large dark matter haloes since this component represents up to $80\%$ of the cluster mass. Cluster properties may thus be investigated by means of numerical simulations performed in the standard framework of hierarchical CDM structure formation. The impressive growth in processing speeds of computers in recent years has allowed us to deeply investigate this issue leading to a strong debate about this fundamental topic. While there is a general consensus that relaxed galaxy clusters exhibit a density profile that is well described by a double power law with outer asymptotic slope $-3$, there is still an open controversy about the value of the inner asymptotic slope $\beta$ with proposed values mainly in the range $\sim 1.0 - 1.5$ ([@NFW97; @TBW97; @M98; @JS02; @P03; @N03]). On the other hand, a similar controversy has arisen over the question whether such cusps are indeed observed in galaxies (see, e.g., [@S03] and references therein). However, on galaxy scale, the effect of baryonic collapse and astrophysical feedback processes (such as supernova explosions) may alter significantly the dark halo structure thus complicating the interpretation of the observations. Strongly lensed arcs in galaxy clusters probe the gravitational potential on scales ($r \sim 50 - 100 \ {\rm kpc}$) large enough to avoid baryonic contamination and are thus an ideal tool to investigate this puzzling question. In particular, radial arcs probe the slope of the mass profile at their positions, while tangential arcs constrain the total mass within their radial distance from the cluster centre. Moreover, a measurement of the velocity dispersion essentially fixes the mass divided by the radius even if it is worth stressing that this estimate is reliable only for relaxed clusters. These observables can then be combined to obtain a powerful method to extract information on the cluster structure. It is worth noting that the second most important component of a galaxy cluster, namely the gaseous intracluster medium (ICM), is often neglected in numerical simulations. In the usual approach, the ICM distribution is determined [*a posteriori*]{} from the dark matter density profile imposing the hydrodynamical equilibrium and assuming an isothermal or polytropic equation of state for the gas ([@KS01; @Asca03]). However, such an approach is somewhat biased since it relies on hypotheses (isothermality and hydrodynamical equilibrium) that do not hold in real galaxy clusters. On the other hand, it is also possible to determine the radial mass profile of both dark matter and ICM directly from simulations explicitly taking into account the gas component. This is the approach followed in a recent paper by Rasia et al. (2004). Using an extended set of high resolution non radiative hydrodynamic simulations and assuming spherical symmetry, these authors have first derived the phase space density of the dark matter particles and then given fitting formulae for the density profile and the velocity dispersion thus allowing them to verify the dynamical equilibrium of the system. Turning then to the hot gas component, they have derived analytic expressions for the density structure, the temperature profile and the velocity dispersion of the ICM without imposing any [*a priori*]{} hypotheses on the gas dynamical status or its equation of state. In particular, Rasia et al. have shown that the isothermality hypothesis breaks down at distances from the centre larger than $\sim 0.2 R_v$, with $R_v$ the virial radius of the cluster. The Rasia et al. model (hereafter RTM model) presents some peculiarities that make it different from the other models available in literature. Moreover, all the relevant quantities of both the dark matter and gas have been derived in a self-consistent way free of any bias induced by aprioristic hypotheses on the dynamical state of the system. In particular, the knowledge of the gas profile allows one to resort to a completely different (and complementary) observable. With temperature of the order of few keV, the ICM gas is dense and hot enough that clusters are luminous X-ray sources with the bulk of the X-rays being produced as bremsstrahlung radiation ([@Sar98]). Electrons in the ICM are not only scattered by ions, but may themselves Compton scatter photons of the cosmic microwave background radiation (CMBR) giving rise to the Sunyaev-Zel’dovich (SZ) effect (see [@Bir99] for a comprehensive review). The temperature decrement due to the SZ effect is able to provide information on the cluster structure, on the motions of galaxy clusters relative to the Hubble flow and on the Hubble flow itself and the cosmological constants that characterize it ([@Reese; @Mauro03]). In order to investigate if the RTM model is a viable one, a direct comparison with the main cluster observables (both from lensing and SZ effect) is needed. As a first step, one has to study the lensing properties of the RTM model and to calculate the SZ effect taking care of the peculiar temperature profile. This is the aim of the present paper. In Sect. 2 we evaluate the deflection angle and the lensing potential of the spherically symmetric RTM model. Radial and tangential arcs form near the position of the critical curves. Therefore, Sect. 3 is devoted to a detailed investigation of the critical curves structure of the model with a particular emphasis on how these properties depend on the model parameters. The effect of taking into account the contribution of the brightest cluster galaxy to the lensing potential is investigated in Sect. 4, while the impact of deviations from spherical symmetry or tidal perturbations from nearby clusters are mimicked by an external shear and discussed in Sect. 5. Having been proposed recently, the RTM model has to be compared with the previous proposals. In particular, in Sect. 6, we compare some of its lensing properties with those of the NFW model investigating possible systematic errors in the virial mass estimate. The computation of the SZ effect due to the distribution of the ICM in the RTM model is presented in Sect. 7, while the results are compared to the prediction of both the $\beta$ model and the NFW model in Sect. 8 where we also discuss the detectability of RTM clusters in SZ survey. The details of the numerical simulations on which the RTM model is based may induce systematic errors on the main results. Some qualitative comments on this topic are presented in Sect. 9. We summarize and conclude in Sect. 10. Deflection angle and lensing potential ====================================== Let us adopt a rectangular coordinate system $(x, y, z)$ with origin in the cluster centre and let $(r, \theta, \phi)$ be the usual spherical coordinates. The mass density profile of the RTM model is ([@RTM03]): $$\rho(r) = \rho_0 \rho_b \left [ \frac{r}{R_v} \left ( x_p + \frac{r}{R_v} \right )^{1.5} \right ]^{-1} \label{eq: rhortm}$$ with $\rho_b = \Omega_M \rho_{crit}$ the present day mean matter density of the universe and: $$\rho_0 = \frac{(1 - f_b) \Delta_v}{6 [ (1 + 2 x_p)/(1 + x_p)^{1/2} - 2 x_p^{1/2} ]}$$ where $\Delta_v$ is the virial overdensity specified by the cosmological model and the term $f_b$ is the average baryonic fraction used to properly weight the dark matter component in the cluster. Following Rasia et al. (2004), we set $f_b = 0.097$ as obtained by averaging over their sample of simulated clusters. The RTM model is fully characterized by two parameters, namely the dimensionless scale radius $x_p$ (or the concentration $c_{RTM} = 1/x_p$) and the virial radius $R_v$. However, it is more convenient to express $R_v$ in terms of the virial mass (i.e. the total cluster mass) $M_v$ using the following relation: $$R_v = \left ( \frac{3 M_v}{4 \pi \Delta_v \rho_b} \right )^{1/3} \ . \label{eq: rvmv}$$ We will assume that the model is spherically symmetric so that all the lensing quantities will depend only on the projected radius $R = (x^2 + y^2)^{1/2}$. This is the same approximation used in Rasia et al. (2004) to obtain the density profile in Eq.(\[eq: rhortm\]). While useful in the computations, this approximation is not a serious limitation to our analysis since the results for the circular case may be immediately generalized to flattened models by means of numerical integration ([@Schramm; @K01]). Moreover, we will investigate later the impact of deviations from circular symmetry by adding a shear term to the lensing potential. However, we stress that the results for the circularly symmetric models allow us to obtain a picture of the main properties of the RTM model as a lens. As a first step to investigate the lensing properties of the RTM model, we have to evaluate the corresponding surface mass density. Starting from the definition: $$\Sigma(x, y) = \int_{-\infty}^{\infty}{\rho(x, y, z) dz}$$ and using a convenient transformation to spherical coordinates, we get: $$\begin{aligned} \Sigma(\xi) & = & 2 \ \rho_0 \ \rho_b \ R_v \int_{0}^{\pi/2}{\frac{1}{\sin{\theta}} \left ( x_p + \frac{\xi}{\sin{\theta}} \right )^{-1.5} d\theta} \nonumber \\ ~ & = & \Sigma_v \ \times \ \frac{{\cal{S}}(\xi, x_p)}{{\cal{S}}(1, x_p)} \label{eq: sigmartm}\end{aligned}$$ with $\xi = R/R_v$ and $\Sigma_v$ the surface density at the virial radius given by: $$\Sigma_v \equiv \Sigma(\xi = 1) = \sqrt{\frac{8}{\pi}} \ \rho_0 \ \rho_b R_v {\cal{S}}(1, x_p) \label{eq: defsigmav}$$ and we have defined the function: $$\begin{aligned} {\cal{S}} & = & \xi^{-5/2} \left \{ \left [ \Gamma\left( \frac{3}{4} \right ) \right ]^2 {_2F_1 \left [ \left \{ \frac{3}{4}, \frac{3}{4} \right \}; \left \{ \frac{1}{2} \right \}; \frac{x_p^2}{\xi^2}, \right ]} \xi - \right . \nonumber \\ ~ & ~ & \left . - 2 x_p \left [ \Gamma\left( \frac{5}{4} \right ) \right ]^2 {_2F_1 \left [ \left \{ \frac{5}{4}, \frac{5}{4} \right \}; \left \{ \frac{3}{2} \right \}; \frac{x_p^2}{\xi^2}, \right ]} \right \} \ . \label{eq: defesse}\end{aligned}$$ In the previous equation, $\Gamma(\zeta)$ is the actual $\Gamma$ function and ${_pF_q[\{a_1, \ldots, a_p\}, \{b_1, \ldots, b_q\}, \zeta)}$ is the generalized hypergeometric function[^1] ([@GR80]). Having obtained the surface mass density, it is now straightforward to compute the deflection angle. Because of the circular symmetry in the lens plane, the deflection angle is purely radial and its amplitude is ([@SEF]): $$\alpha = \frac{2}{R} \int_{0}^{R}{\frac{\Sigma(R')}{\Sigma_{crit}} R' dR'} \label{eq: defalpha}$$ with $\Sigma_{{\rm crit}} = c^2 D_s/4 \pi G D_l D_{ls}$ the critical density for lensing and $D_s$, $D_l$, $D_{ls}$ are the angular diameter distances between observer and source, observer and lens and lens and source, respectively. Inserting Eq.(\[eq: sigmartm\]) into Eq.(\[eq: defalpha\]), we get: $$\alpha(\xi) = \alpha_v \times \frac{{\cal{F}}(\xi, x_p)}{{\cal{F}}(1, x_p)} \label{eq: alphartm}$$ with $\alpha_v$ the deflection angle at the virial radius given by: $$\alpha_v \equiv \alpha(\xi = 1) = \frac{4 \Sigma_v R_v}{\Sigma_{crit}} \frac{{\cal{F}}(1, x_p)}{{\cal{S}}(1, x_p)} \label{eq: alphav}$$ and we have introduced the function: $$\begin{aligned} {\cal{F}} & = & \frac{1}{\xi} \left \{ \left [ \Gamma\left(\frac{3}{4} \right ) \right ]^2 {_2F_1 \left [ \left \{ - \frac{1}{4}, \frac{3}{4} \right \}; \left \{ \frac{1}{2} \right \}; \frac{x_p^2}{\xi^2}, \right ]} \xi^{1/2} + \right . \nonumber \\ ~ & ~ & + 2 x_p \left [ \Gamma\left(\frac{5}{4} \right ) \right ]^2 {_2F_1 \left [ \left \{ \frac{1}{4}, \frac{5}{4} \right \}; \left \{ \frac{3}{2} \right \}; \frac{x_p^2}{\xi^2}, \right ]} \xi^{-1/2} - \nonumber \\ ~ & ~ & \left . - \frac{}{} \sqrt{2 \pi x_p} \ \right \} \ . \label{eq: defeffe}\end{aligned}$$ It is interesting to observe that only the parameter $x_p$ of the RTM model determines the behaviour with the dimensionless radius $\xi$ of the scaled deflection angle $\alpha/\alpha_v$, while the virial radius $R_v$ (and thus the total mass $M_v$) enters only as a scaling factor. As an example, Fig.\[fig: alphavsx\] shows $\alpha/\alpha_v$ for three values of $x_p$. This plot may be qualitatively explained as follows. Lower values of $x_p$ correspond to higher values of the concentration $c_{RTM}$ and hence to more mass within a fixed radius. Since, for a given position in the lens plane, the deflection angle scales with the projected mass within $\xi$, it turns out that $\alpha/\alpha_v$ is higher for lower values of $x_p$ as Fig.\[fig: alphavsx\] shows. This qualitative discussion also explains the behaviour of $\alpha_v$ with $x_p$ that is shown in Fig.\[fig: alphavvsxp\]. A comment is in order here to understand how this plot has been obtained. Eq.(\[eq: alphav\]) shows that $\alpha_v$ depends on the cluster parameters $(M_v, x_p)$, the lens and source redshift $(z_l, z_s)$ and the background cosmological model. We adopt a flat $\Lambda$CDM model with $(\Omega_M, \Omega_{\Lambda}, h) = (0.3, 0.7, 0.72)$ giving $\Delta_v = 324$ ([@ECF96]). We set $(z_l, z_s) = (0.313, 1.502)$ as for the real cluster lens MS2137-23 (Sand et al. 2002, 2004). Unless otherwise stated, the same values for the cosmological parameters and the lens and source redshift will be used throughout the paper. To obtain the plot in Fig.\[fig: alphavvsxp\], we have fixed $M_v = 7.5 \times 10^{14} \ {\rm M_{\odot}}$, but the results for other values of $M_v$ may be easily scaled observing that $\alpha_v \propto \Sigma_v R_v \propto R_v^2 \propto M_{v}^{2/3}$. Let us now derive the lensing potential $\psi(R)$ for the RTM model. To this aim, one should solve the two dimensional Poisson equation ([@SEF]): $$\nabla^2 \psi = 2 \kappa \label{eq: poisson}$$ with $\kappa = \Sigma/\Sigma_{crit}$ the convergence. However, because of the circular symmetry in the lens plane, it is also: $$\alpha(R) = \frac{d\psi}{dR} \rightarrow \psi(R) = \int{\alpha(R) dR} \ . \label{eq: alphapsi}$$ Inserting Eq.(\[eq: alphartm\]) into Eq.(\[eq: alphapsi\]) and integrating, we find: $$\psi(\xi) = \psi_v \times \frac{{\cal{P}}(\xi, x_p)}{{\cal{P}}(1, x_p)} \label{eq: psirtm}$$ with: $$\psi_v \equiv \psi(\xi = 1) = \alpha_v R_v {\cal{P}}(1, x_p) \ , \label{eq: psiv}$$ $$\begin{aligned} {\cal{P}} & = & - \frac{\Gamma\left ( - \frac{1}{4} \right ) \Gamma\left ( \frac{3}{4} \right )}{2} {_2F_1 \left [ \left \{ - \frac{1}{4}, - \frac{1}{4} \right \}; \left \{ \frac{1}{2} \right \}; \frac{x_p^2}{\xi^2}, \right ]} \xi^{1/2} - \nonumber \\ ~ & ~ & - x_p \Gamma\left ( \frac{1}{4} \right ) \Gamma\left ( \frac{5}{4} \right ) {_2F_1 \left [ \left \{ \frac{1}{4}, \frac{1}{4} \right \}; \left \{ \frac{3}{2} \right \}; \frac{x_p^2}{\xi^2}, \right ]} \xi^{-1/2} - \nonumber \\ ~ & ~ & - 2 \sqrt{2 \pi x_p} \ln{\xi} \ . \label{eq: defpi}\end{aligned}$$ Finally, let us consider the lens equations. Adopting polar coordinates $(R, \vartheta)$ in the lens plane with $\vartheta$ measured counterclockwise from North, the time delay of a light ray deflected by the gravitational field of the cluster lens is: $$\begin{aligned} \Delta t & = & h^{-1} \tau_{100} \ {\times} \nonumber \\ ~ & ~ & \left [ \frac{1}{2} R^2 - R R_s \cos{(\vartheta - \vartheta_s)} + \frac{1}{2} R_s^2 - \psi(R, \vartheta) \right ] \label{eq: timedelaygen}\end{aligned}$$ where $(R, \vartheta)$ is the image position, $(R_s, \vartheta_s)$ the unknown source position and we have defined: $$\tau_{100} = \left(\frac{D_{l} D_{s}}{D_{ls}}\right) \frac{(1 + z_l)}{c} \ . \label{eq: taucento}$$ According to the Fermat principle, the images lie at the minima of $\Delta t$, so that the lens equations may be simply obtained by minimizing $\Delta t$. Inserting Eq.(\[eq: psirtm\]) into Eq.(\[eq: timedelaygen\]) and differentiating, we get: $$\xi - \xi_s \cos{(\vartheta - \vartheta_s)} = \frac{\alpha_v}{R_v} \times \frac{{\cal{F}}(\xi, x_p)}{{\cal{F}}(1, x_p)} \label{eq: lenseqa}$$ $$\xi_s \sin{(\vartheta - \vartheta_s)} = 0 \ . \label{eq: lenseqb}$$ with $\xi_s = R_s/R_v$. Eq.(\[eq: lenseqb\]) has two solutions. The first is $\xi_s = 0$, i.e. lens, source and observer are perfectly aligned. The only image is the Einstein ring that we will discuss in much detail in the next section. The second solution is obtained for $\sin{\vartheta - \vartheta_s} = 0 \iff \vartheta = \vartheta_s + m \pi$ with $m= 0, 1$. In this case, we get two images[^2] symmetrically placed with respect to the lens centre. The radial coordinate $\xi_1$ of the first image (i.e., the one with $\vartheta = \vartheta_s$) is obtained by solving Eq.(\[eq: lenseqa\]) with $\cos{(\vartheta - \vartheta_s)} = 1$, while the second has a distance $\xi_2$ from the lens centre obtained by solving Eq.(\[eq: lenseqa\]) with $\cos{(\vartheta - \vartheta_s)} = -1$. These equations may be solved numerically provided that the cluster parameters have been fixed. Up to now, there is only one multiple image system in which the lens is a cluster galaxy, namely the recently discovered SDSSJ1004+4112 ([@SDSSlens1; @SDSSlens2]), while for all the other multiply imaged quasar the lens is a galaxy (see, e.g., the CASTLES web page, [@CASTLES]). Since the RTM model has not been tested on galactic scale (given the mass range probed by the simulations employed by Rasia et al. 2003), we prefer to not discuss further the formation of multiple images. Critical curves =============== The most spectacular effect of lensing by a galaxy cluster is the formation of giant arcs (see, e.g., [@Kneib96] for the textbook example of A2218). These are very luminous and highly distorted images of a source galaxy whose position is near one of the critical curves of the lensing potential. The position of the arcs in a given lensing system allows to strongly constrain the cluster mass distribution and can also be used to determine the cosmological parameters ([@Mauro02]). Moreover, arc statistics is a promising and efficient tool to discriminate among different cosmological models and theories of structure formation. It is thus quite interesting to investigate the number (and the type) of arcs the RTM model may form. To this aim, let us first remember the expression for the magnification $\mu$ of a source due to the lensing effect. It is ([@SEF]): $$\mu = \frac{1}{\det{A}} = \frac{1}{(1 - \psi_{xx}) (1 - \psi_{yy}) - \psi_{xy}^2} = \frac{1}{\lambda_r \lambda_t} \label{eq: defmu}$$ where $A$ is the amplification matrix (that is the jacobian matrix of the lens mapping) and $(\lambda_r, \lambda_t)$ for a circularly symmetric model are given as: $$\lambda_r = 1 - \frac{d\alpha}{dR} \ , \label{eq: deflambdar}$$ $$\lambda_t = 1 - \frac{\alpha}{R} \ . \label{eq: deflambdat}$$ Inserting Eq.(\[eq: alphartm\]) into Eqs.(\[eq: deflambdar\]), (\[eq: deflambdat\]), we get the corresponding quantities for the RTM model that we do not explicitly report here for sake of shortness. The critical curves are the loci where $\det{A} = 0$. This condition is satisfied by imposing $\lambda_r = 0$ or $\lambda_t = 0$. The second equation implicitly defines the tangential critical curves. It is easy to show that: $$\lambda_t = 0 \rightarrow \xi_E = \frac{\alpha_v}{R_v} \times \frac{{\cal{F}}(\xi_E, x_p)}{{\cal{F}}(1, x_p)} \label{eq: findxie}$$ with $\xi_E \equiv R_E/R_v$ and $R_E$ the Einstein radius. Eq.(\[eq: findxie\]) may be solved numerically for fixed values of the cluster model parameters. Fig.\[fig: reconts\] shows the contours of equal $R_E$ in the $(\log{M_v}, x_p)$ plane[^3] with higher values of $R_E$ corresponding to lower curves in the plot. We have only considered clusters with $\log{M_v} \in (14.5, 15.5)$ because this is (approximately) the mass range probed in Rasia et al. (2004), while we will (usually but not always) consider $x_p \in (0.01/7.13, 2/7.13)$ since Rasia et al. (2004) states that the average value of $\langle c_{RTM} \rangle = \langle 1/x_p \rangle \simeq 7.13$ over their cluster sample. Fig.\[fig: reconts\] shows that highly concentrated clusters (i.e., with small values of $x_p$) give rise to tangential critical curves that are more distant from the cluster centre. Thus, one could qualitatively conclude that only RTM models with lower values of $x_p$ are able to produce tangential arcs. However, one should also take into account that, as expected, for a fixed $x_p$ the Einstein radius increases with $M_v$. Therefore, RTM models with high values of $x_p$ could still produce tangential arcs with large radii provided that the mass is large enough and that it is possible to extrapolate the RTM model outside the mass range probed by the simulations. As said above, radial critical curves are implicitly defined by the condition $\lambda_r = 0$. Inserting Eq.(\[eq: sigmartm\]) and (\[eq: alphartm\]) into Eq.(\[eq: deflambdar\]), we get $R_{rad}$, the radial critical curve distance from the cluster centre. To investigate how $R_{rad}$ depends on the model parameters, we plot in Fig.\[fig: lr\] the contours of equal $R_{rad}$ in the $(\log{M_v}, x_p)$ plane. The behaviour of $R_{rad}$ with the $\log{M_v}$ is qualitatively similar to that of the Einstein radius $R_E$, but the opposite holds for the dependence on $x_p$. For a given value of $\log{M_v}$, $R_{rad}$ increases with $x_p$, while $R_{E}$ decreases. As a result, highly concentrated RTM models (i.e. models with small $x_p$ and henceforth high $c_{RTM}$) give rise to tangential arcs situated at large distances from the cluster centre, but the radial arc lies in the very inner regions of the cluster. As a simple application of these results, we consider the case of the real cluster lens MS2137-23 in which both a tangential and a radial arc have been observed with $R_E = 15.35$arcsec and $R_{rad} = 4.5$arcsec (Sand et al. 2002, 2004). An easy way to find out the values of the RTM model parameters able to fit the arc positions in MS2137-23 is to draw the contour levels in the plane $(\log{M_v}, x_p)$ for $R_E$ and $R_{rad}$ equal to the values quoted above and look for the intersection point between these two curves. It turns out that the best fit parameters are $(\log{M_v}, x_p) = (14.7, 0.019)$, i.e. $M_v = 5.0 \times 10^{14} \ M_{\odot}$ and $c_{RTM} \simeq 53$. As a further test, we also consider the case of the real cluster lens RXJ1133 at redshift $z_L = 0.394$ ([@STSE03]). Both a radial and a tangential arc are observed with $z_S = 1.544$ and $(R_{rad}, R_E) = (3.2, 10.9) \ arcsec$ that may be obtained by describing the cluster with a RTM model with best fit parameters $(\log{M_v}, x_p) = (14.5, 0.018)$, i.e. $M_v = 3.2 \times 10^{14} \ {\rm M_{\odot}}$ and $c_{RTM} \simeq 56$. The values of the concentration are quite high if compared to $\langle c_{RTM} \rangle \simeq 7.13$ found by Rasia et al. (2004) over their sample of simulated clusters. However, this could not be considered an evidence against the RTM model. Actually, we have only considered the spherical case, while it is well known that also a small cluster ellipticity changes significantly the position of the critical curves (see, e.g., [@BartMen03]). Moreover, one should also take into account the impact on the critical curves of the galaxy lying at the centre of the cluster gravitational potential and of other eventual substructures. That is why we do not speculate further on the high $c_{RTM}$ values needed to reproduce the arcs positions in MS2137-23 and RXJ1133, while a detailed comparison with observations will be presented elsewhere. Adding a bright cluster galaxy ============================== Some recent studies have highlighted the importance of considering the brightest cluster galaxy (hereafter BCG) when investigating the lensing properties of a cluster ([@MBM03; @STSE03]). It is thus interesting to study how the critical curves of the RTM model are affected by the addition of a BCG. To this aim, we place the galaxy exactly at the centre of the cluster and model it using the Hernquist profile whose mass density is ([@H90]): $$\rho(r) = \frac{\rho_s}{r/r_s \ (1 + r/r_s)^3} \label{eq: rhohern}$$ with $\rho_s$ a characteristic density and $r_s$ a scale radius. The Hernquist profile has the notable property that its projected density well approximates the $R^{1/4}$ law ([@deV48]) provided that the effective radius is related to the scale radius of the Hernquist model by the relation: $R_e \simeq 1.81 r_s$. The model is fully characterized by two parameters that we choose to be $R_e$ and the total mass $M_h$ given by: $$M_h = 2 \pi r_s^3 \ \rho_s \ . \label{eq: masshern}$$ Assuming spherical symmetry, the deflection angle of the galaxy is ([@K01]): $$\alpha_H(\sigma) = 2 \kappa_s r_s \ \frac{\sigma [1 - {\cal{H}}(\sigma)]}{\sigma^2 - 1} \label{eq: alphahern}$$ with $\sigma = R/r_s$ and $\kappa_s = \rho_s r_s/\Sigma_{crit}$ and we have defined: $${\cal{H}}(\sigma) = \cases{ {1 \over \sqrt{\sigma^2-1}}\,\mbox{tan}^{-1} \sqrt{ \sigma^2-1 } & $(\sigma>1)$ \cr {1 \over \sqrt{1-\sigma^2}}\,\mbox{tanh}^{-1}\sqrt{ 1-\sigma^2 } & $(\sigma<1)$ \cr 1 & $(\sigma=1)$ \ . \cr } \label{eq: defacca}$$ Since the standard theory of lensing is developed in the weak field limit, the total deflection angle is simply the sum of the contributions from the cluster and the BCG. Hence, the total magnification may be written as: $$\mu = \frac{1}{(\lambda_{r}^{H} + \lambda_{r}^{RTM} - 1) (\lambda_{t}^{H} + \lambda_{t}^{RTM} - 1)} \label{eq: mutot}$$ with $\lambda_r$ and $\lambda_t$ given by Eqs.(\[eq: deflambdar\]) and (\[eq: deflambdat\]) respectively and quantities with the superscript [*“H”*]{} ([*“RTM”*]{}) refers to the Hernquist (RTM) model. The radial and tangential arcs radii $R_{rad}$ and $R_E$ are defined as those radii vanishing the first and the second term respectively of the denominator in Eq.(\[eq: mutot\]). To study where the critical curves form, we plot in Fig.\[fig: detA-RTM+Hern\] the loci in the $(R, \log{M_v})$ plane where the total magnification formally diverges having arbitrarily fixed $x_p = 0.5/7.13$ and $R_e = 24.80$kpc as for the BCG in MS2137-23 ([@STSE03]). The solid line refers to the case with no BCG, while the dashed one shows how the curves are modified by the addition of a BCG with total mass[^4] $M_h = 5 \times 10^{12} \ {\rm M_{\odot}}$. The number of critical curves is still two, but their position is affected by the presence of BCG with the distance between them increased with respect to the case with no BCG. In particular, while $R_{rad}$ is slightly smaller or higher depending on the ratio between the mass of galaxy and that of the cluster, the Einstein radius $R_E$ significantly increases. This is expected since $R_E$ is proportional to the total mass within the tangential critical curve so that, adding the BCG mass, $R_E$, gets obviously higher. Fig.\[fig: detA-RTM+Hern-bis\] is similar to Fig.\[fig: detA-RTM+Hern\], but now the cluster mass is set as $M_v = 1.125 \times 10^{15} \ {\rm M_{\odot}}$ and we let $x_p$ changing, while the BCG parameters are fixed as before. The increase of $R_E$ is still visible, but what is most important to note is the possibility to have radial and critical curves with higher values of $x_p$. From Fig.\[fig: detA-RTM+Hern-bis\], one sees that, when the BCG is absent, RTM models with $x_p > 0.18$ are unable to produce radial arcs (i.e. it is $R_{rad} = 0$), while the radial critical curve appears when the BCG is taken into account in the total lensing potential even for less concentrated (i.e. with larger $x_p$) clusters. Finally, we investigate qualitatively how the constraints on the RTM model parameters are changed by the presence of a BCG. To this aim, we plot in Fig.\[fig: arcsconstr-Hern\] the constraints imposed by the presence of a radial arc at $R_{rad} = 4.5 \ {\rm arcsec}$ or a tangential arc at $R_E = 15.35 \ {\rm arcsec}$ as observed for the real cluster lens MS2137-23 (Sand et al. 2002, 2004). The results in the right panel are easy to explain qualitatively. Adding a BCG pushes up the curve in the $(\log{M_v}, x_p)$ plane so that a tangential arc at a given distance may be produced by less concentrated and less massive clusters with respect to the case with no BCG. As yet noted, $R_E$ is proportional to the total projected mass within $R_E$ itself. Since now the BCG provides part of this mass, less mass has to be contributed by the cluster and thus less massive and concentrated models are needed to obtain a given value of $R_E$. Note, however, that the deviations from the case with no BCG are quite small as expected given the high mass ratio between the cluster and the galaxy. On the other hand, the left panel shows that adding a BCG requires more concentrated and massive clusters to produce a radial arc at a given $R_{rad}$ with respect to the case with no BCG. It is also worth noting that the constraints from the position of the radial arc are more sensitive to the presence (or absence) of the BCG (see, e.g., the distance between the solid and dashed line in the left panel compared to the same in the right one). This is expected since the radial critical curve is innermost and thus probes a range that is more sensitive to the inner structure of the cluster where the BCG plays a more significant role. The impact of the shear ======================= Up to now, we have investigated the lensing properties of the RTM model (with and without a central BCG) assuming spherical symmetry of the mass distribution. However, real clusters are moderately elliptical and it is well known that even small ellipticities may alter significantly the lensing properties of a given model. In particular, taking into account deviations from spherical symmetry is very important when trying to extract constraints on the model parameters from the position of the lensed arcs in real systems (as clearly demonstrated, for instance, in [@BartMen03; @DalKee03]). On the other hand, even when assuming spherical symmetry, it is important to take into account also the effect of substructures in the cluster mass distribution and possible tidal deformations due to nearby clusters. Finally, the large scale structure as a whole could also have a not negligible effect ([@KKS97]). To the lowest order, all these effects may be mimicked by adding an external shear to the lensing potential which is now written as: $$\psi(R, \vartheta) = \psi_{RTM}(R) - \frac{1}{2} \gamma R^2 \cos{2(\vartheta - \vartheta_{\gamma})} \label{eq: psitot}$$ with $\psi_{RTM}$ given by Eq.(\[eq: psirtm\]) and $(\gamma, \vartheta_{\gamma})$ the shear strength and position angle. Without loss of generality we assume that the shear is oriented along the major axis so that it is $\vartheta_{\gamma} = 0$. Fig.\[fig: shearplot\] shows the critical curves and the caustics for the lensing potential in Eq.(\[eq: psitot\]). To be quantitative, we also report in Table1 the quantity: $$\Delta \xi_i = 100 \times \frac{\xi_i(\gamma) - \xi_i(\gamma = 0)}{\xi_i(\gamma = 0)}$$ with $\xi_i$ the length of the radial or tangential critical curve along the $i$-axis (with $i = x, y$). In the case with no shear (upper left panel), the two critical curves are spherical, while the tangential caustic is the origin and the radial one is a circle. The effect of the shear is to deform the critical curves into ellipses, while the inner caustic (corresponding to the tangential critical curve) takes a diamond shape and the external one becomes elliptical. In particular, the radial critical curve is more and more elongated along the major axis[^5] as the shear strength increases. Quantitatively, this could be seen in Table1 where $\Delta \xi_x$ is positive and increases with $\gamma$, while $\Delta \xi_y$ is negative and different from $\Delta \xi_x$. This is what corresponds graphically to an ellipse more and more elongated along the major axis as $\gamma$ increases. On the other hand, the tangential critical curve may also deviate from the elliptical symmetry taking a dumbell shape as $\gamma$ gets higher. As a consequence, the higher is $\gamma$, the higher is the the radial arc distance (measured along the major axis), $R_{rad}$, from the cluster centre for fixed values of the RTM model parameters. A similar result holds also for $R_E$. It is worth noting that these effects are qualitatively the same whatever is the value of $x_p$, but are more pronounced (i.e. $\Delta \xi_i$ is larger) for lower values of $x_p$ (i.e. higher concentrations) as can be seen from Table1. $\gamma$ ---------- ---------------- ---------------- ---------------- ----------------   $\Delta \xi_x$ $\Delta \xi_y$ $\Delta \xi_x$ $\Delta \xi_y$ 0.05 -8 8 8 -5 0.10 -14 18 16 -13 0.15 -21 28 25 -19 0.05 -12 13 13 -12 0.10 -22 29 28 -22 0.15 -31 46 45 -31 : The impact of the shear on the length along the major an minor axes of the tangential and radial critical curves. The upper half of the table refers to RTM model with $x_p = 1/\langle c_{RTM} \rangle$, while for the lower half it is $x_p = 0.5 \langle c_{RTM} \rangle$. In both cases, the virial mass is set to $M_v = 1.125 \times 10^{15} \ {\rm M_{\odot}}$. Note that these results are in qualitative agreement with the approximate analytical treatment presented in Bartelmann & Meneghetti (2004). Actually, these authors used a different approach deforming the lens model so that the isocountour lines of the lensing potential are ellipses with ellipticity $\varepsilon$. Elliptical deformation of the lensing potential leads to dumbell shaped surface mass distribution for values of $\varepsilon > 0.2$. Even if clusters are highly structured, similar mass models are quite unrealistic so that we have preferred not to follow this approach. On the other hand, it is possible to show that an elliptical potential $\psi(x^2 + y^2/q^2)$ with an on axis shear and axial ratio $q$ produces the same image configuration as a pure elliptical potential with axis ratio $q' = q \sqrt{(1-\gamma)/(1+\gamma)}$ without shear ([@W96]). In our case, this means that using the lensing potential given by Eq.(\[eq: psitot\]) is equivalent to deform the RTM model such that the lensing isopotential contours are ellipses with axis ratio $q' = \sqrt{(1-\gamma)/(1+\gamma)}$. This shows the complete equivalence among our approach and that of Bartelmann & Meneghetti (2004). Comparison with the NFW model ============================= It is interesting to compare the lensing properties of the RTM model with those of the model proposed by Navarro, Frenck & White (1997, hereafter NFW) and mostly used in literature. Using the same normalization as in Rasia et al. (2004), the density profile of the NFW model is: $$\rho = \frac{\rho_{0,NFW} \rho_b}{(x/x_s) (1 + x/x_s)^2} \label{eq: rhonfw}$$ with $x = r/R_v$, $x_s \equiv 1/c_{NFW}$, $c_{NFW}$ the concentration of the NFW model and: $$\rho_{0,NFW} = \frac{(1 - f_b) \Delta_v}{3 \left [ \ln{(1 + c_{NFW})} - c_{NFW}/(1 + c_{NFW}) \right ]} \ . \label{eq: rhoznfw}$$ The deflection angle for the NFW model may be conveniently written as ([@B96; @K01]): $$\alpha_{NFW}(\xi) = \frac{\alpha_{v}^{NFW}}{\xi} \times \frac{\ln{(c_{NFW} \xi/2)} + {\cal{H}}(c_{NFW} \xi)}{\ln{(c_{NFW}/2)} + {\cal{H}}(c_{NFW})} \label{eq: alphanfw}$$ having defined: $$\begin{aligned} \alpha_{v}^{NFW} & = & \frac{4 (1 - f_b) \Delta_v \rho_b}{3 \Sigma_{crit}} \times R_v \times \nonumber \\ ~ & ~ & \frac{\ln{(c_{NFW}/2)} + {\cal{H}}(c_{NFW})}{\ln{(1 + c_{NFW})} - c_{NFW}/(1 + c_{NFW})} \label{eq: defavnfw}\end{aligned}$$ with $R_v$ expressed in [*arcsec*]{}. Using Eqs.(\[eq: alphartm\]), (\[eq: alphav\]), (\[eq: alphanfw\]) and (\[eq: defavnfw\]) and the definition of virial radius, we get the following expression for the ratio between the deflection angle of the NFW and RTM model: $$\begin{aligned} {\cal{R}}_{\alpha} \equiv \frac{\alpha_{NFW}}{\alpha_{RTM}} & = & \frac{2 \sqrt{\pi/8}}{\xi} \left ( \frac{M_{v}^{NFW}}{M_v} \right )^{1/3} \times \nonumber \\ ~ & ~ & \frac{(1 + 2 x_p)/\sqrt{1 + x_p} - 2 \sqrt{x_p}}{\ln{(1 + c_{NFW})} - c_{NFW}/(1 + c_{NFW})} \times \nonumber \\ ~ & ~ & \frac{\ln{(c_{NFW} \xi/2)} + {\cal{H}}(c_{NFW} \xi)}{{\cal{F}}(\xi, x_p)} \ . \label{eq: ratio}\end{aligned}$$ It is worth noting that the ratio does not depend on the redshift of lens and source as it is expected since it is related to the different density profiles of the two models which is, of course, the same at all redshifts. Moreover, to better compare the lensing properties of the two models, it is meaningful to assume that the virial mass is the same so that, at a given radius $\xi$, ${\cal{R}}_{\alpha}$ only depends on the concentrations $c_{RTM}$ and $c_{NFW}$ of the two models. Fig.\[fig: ratiovsxi\] shows ${\cal{R}}_{\alpha}(\xi)$ setting $c_{RTM} = 7.13$ and $c_{NFW} = 6.8$ as determined by Rasia et al. (2004) averaging over their simulated clusters sample. The deflection angle of the NFW model turns out to be higher than that of the RTM model until $\xi < 0.6$, while ${\cal{R}}_{\alpha} < 1$ for light rays impacting in the outer region of the halo. This result is simply related to the different mass profile of the two models with $M_{NFW}(\xi)/M_{RTM}(\xi)$ being larger than 1 in the inner regions for this choice of $(c_{NFW}, c_{RTM})$. According to some authors, the NFW model is a one parameter model since it is possible to relate the concentration $c_{NFW}$ to the virial mass even if this relation has a quite large scatter. Following Bullock et al. (2001), we adopt: $$c_{NFW} = 15 - 3.3 \log{\frac{M_v}{10^{12} h^{-1} \ {\rm M_{\odot}}}} \label{eq: cvsmv}$$ and plot, in Fig.\[fig: ratiomass\], ${\cal{R}}_{\alpha}$ for three different values of the virial mass. The qualitative behaviour is the same, but it is more pronounced for lower mass models corresponding, according to Eq.(\[eq: cvsmv\]), to lower concentrations. The NFW model is used in most of the studies of the structure of dark matter haloes in galaxy clusters. It is therefore interesting to investigate what is the error induced by using the NFW model to describe a cluster that is actually better described by the RTM model. As a straightforward example, we consider the estimate of the virial mass from the size of the Einstein radius. To this aim, let us look at Fig.\[fig: ratiore\] where we plot the contour level curves in the $(\log{M_v}, c_{NFW})$ plane for the Einstein radius of the NFW model corresponding to values of $R_E$ evaluated for RTM models. Let us consider, for instance, the RTM model with $(M_v, x_p) = (1.5 \times 10^{15} \ {\rm M_{\odot}}, 1/7.13)$ for which one obtains the fourth line (from left) in the left panel of Fig.\[fig: ratiore\]. If one assumes that Eq.(\[eq: cvsmv\]) holds, then one should estimate the NFW model parameters from the intersection point of the dashed curve with the fourth line thus grossly underestimating the mass. Actually, even if one does not use Eq.(\[eq: cvsmv\]), $M_v$ turns out to be underestimated for the values of $c_{NFW}$ in the plot. To get the correct value of $M_v$, one should select a value of $c_{RTM}$ that is unrealistically low for a galaxy cluster. We thus conclude that using the NFW model to study a cluster that is intrinsically described by the RTM model leads to underestimate the virial mass by an amount that depends on the concentration of both the NFW and the RTM model. In principle, one should compare the lensing properties of the NFW and of the RTM models by also including the effects of the brightest cluster galaxy and of the shear. However, this should increase the number of parameters to eight: two for the NFW model, two for the RTM model, the BCG mass and scale radius and the shear strength and orientation. It is likely that some degeneracies could occur among parameters rendering a comparison between the NFW and the RTM model meaningless in this case so that we prefer to not perform this test. The thermal Sunyaev-Zel’dovich effect ===================================== At the beginning of Seventies, Sunyaev and Zel’dovich (1970, 1972) suggested that the cosmic microwave background radiation (CMBR) can be scattered by the trapped hot intracluster electrons giving rise to a measurable distortion of its spectrum. This (inverse) Compton scattering (now referred to as [*Sunyaev-Zel’dovich effect*]{}, hereafter SZE) has been recognized in the last two decades as an important tool for cosmological and astrophysical studies ([@Bir99]). More recently, the SZE has been used to investigate several physical properties of the gas with much attention devoted to the geometry of its density profile as well as its thermodynamical status. In particular, it has been shown that the [*canonical*]{} hypotheses of spherical symmetry and isothermal temperature profile may induce errors (of order up to $30\%$) on the estimated values of different parameters such as the Hubble constant ([@RSM97; @jetzer1]). The physics of the SZE is quite simple to understand. A gas of electrons in hydrostatic equilibrium within the gravitational potential of a cluster will have a temperature $T_e \simeq G M m_p/(2 k_B R_{eff}) \sim {\rm few \ keV}$, with $M$ and $R_{eff}$ typical values of the total mass and size of a cluster. At this temperature, the thermal emission in X-ray is composed of thermal bremsstrahlung and line radiation processes. Electrons in the intracluster gas are not only scattered by ions, but can themselves scatter photon of the CMBR giving rise, in average, to a slight change in the photon energy. Because of this inverse Thomson scattering, an overall change in brightness of the CMBR is observed. As a consequence, the SZE is localized and visible in and towards clusters of galaxies having an X-ray emission strong enough to be detectable. In the non relativistic limit, the scattering process can be described by the Kompaneets equation $$\frac{\partial n}{\partial y} = \frac{1}{p_e^2} \frac{\partial}{\partial p_e} \left [ p_e^4 \left ( \frac{\partial n}{\partial p_e} + n + n^2 \right ) \right ] \label{eq:sze1}$$ which describes the change in the occupation number of photons $n(\nu)$. In Eq.(\[eq:sze1\]), we have introduced the dimensionless variable[^6] $p_e = h \nu_e/k_B T_e$, while $y$ is the so called Comptonization parameter defined as: $$y \equiv \int{\frac{k_B T_e}{m_e c^2} n_e \sigma_T dl} \label{eq: defycomp}$$ being $\sigma_T$ the Thompson scattering cross section and $n_e$ the electrons number density and the integral is performed along the line of sight. Assuming that the photons distribution after the scattering is close to the equilibrium one, we have that: $$\frac{\partial n}{\partial p_e}\gg n , n^2 \ ,$$ so that Eq.(\[eq:sze1\]) simplifies to $$\frac{\partial n}{\partial y} = \frac{1}{p^2} \left( p^4 \frac{\partial n}{\partial p}\right) \ . \label{eq:sze3}$$ having also replaced $p_e$ with $p$ because of the homogeneity. By solving this equation in the quasi equilibrium hypothesis, it is possible to obtain both the variation $\Delta n$ in the occupation number with respect to the equilibrium value $n_0$ and the corresponding shift in temperature $\Delta T$. It is: $$\frac{\Delta n}{n_0} = \frac{y \ p \ e^p}{e^p -1} \left[ p \coth \left( \frac{p}{2}\right) - 4 \right] \ , \label{eq:sze4}$$ $$\frac{\Delta T}{T_0} = y\left[ p \, \mbox {coth}\, \left(\frac{p}{2} \right) -4 \right] \equiv y \ g(p)\ , \label{eq:sze5}$$ where $g(p)$ is the SZE frequency spectrum and we have considered that $T_e$ ($\sim 10^7\, K$) is much higher than the CMBR temperature $T_0 \simeq 2.7K$. In the limit of low frequencies, we get the useful approximated expression: $$\frac{\Delta T}{T} = -2 y \ . \label{eq:sze5bis}$$ Eq.(\[eq:sze5bis\]) allows to evaluate the shift in temperature due to the SZE provided that the gas number density $n_e(r)$ and the temperature profiles $T_e(r)$ are given. For the RTM model, it is ([@RTM03]): $$n_e(s) = \frac{\rho_{g,0} \ \rho_b}{\mu \ m_p} (s + x_p)^{-2.5} \ , \label{eq: nertm}$$ $$T_e(s) = \frac{T_0 T_v s^{0.016}}{\left ( s^4 + x_p^4 \right )^{0.13}} \label{eq: tertm}$$ where $s = r/R_v$, $\mu$ is the mean molecular weight, $m_p$ the proton mass, $T_v$ the virial temperature and $\rho_{g,0}$ a normalization density given by: $$\rho_{g,0} = \displaystyle{ \frac{f_b \Delta_v} {3 \left [ \displaystyle{\frac{2 + 10 x_p + (40/3) x_p^2 + (16/3) x_p^3}{(1 + x_p)^{2.5}}} - \displaystyle{\frac{16}{3} x_p^{1/2}} \right ]}} \label{eq: rhozgas}$$ Finally, in Eqs.(\[eq: nertm\]) and (\[eq: tertm\]), $x_p$ and $T_0$ are fitting parameters to be determined on a cluster by cluster basis. In particular, we set $(T_0, x_p) = (0.255, 10^{-0.51})$ as found by Rasia et al. (2004) for their set of simulated clusters. In the following, we analyze the implications of the gas density and temperature profile of the RTM model on the SZE. Contrary to the lensing applications, we limit our analysis to the spherically symmetric case, without considering any ellipticity in the profiles. The reason for this choice is the fact that the gas profiles have been deduced by the mean of hydrodynamical (and not simply N-body) simulations, so that the hypothesis of axial symmetry for the density profile is not enough to assure a [*similar*]{} elliptical temperature profile. The gas profile and the structure integral ------------------------------------------ The radial dependence of cluster profiles is becoming a testing ground for models of structure formation and for our understanding of gas dynamics in galaxy clusters. Actually, the formation of structures is believed to be driven by some hierarchical development, which leads to the prediction of self similar scalings between systems of different masses and at different epochs. Moreover, the intracluster gas is generally assumed to be isothermal and in hydrostatical equilibrium. From the observational point of view, however, the situation is rather controversial and yet undeterminated: X-rays observations of poor clusters fall belove the self similar expectations, and even if the isothermal distribution is often a reasonable approximation to the actual observed clusters, some clusters show not isothermal distribution ([@jetzer]). It turns out that the emerging temperature profile is one where the temperature increases from the center to some characteristic radius, and then decreases again. The central temperature decrements has been much discussed in terms of cooling flows, while the outer temperature decrement seems to be confirmed both observationally and numerically. On the other hand, the temperature profile described in Eq.(\[eq: tertm\]) reproduces quite well some of these observational features: it shows an isothermal core up to $0.2 \,R_v$, followed by a steep decrease that reaches a factor two lower around the virial radius; the density profiles are self-similar roughly $s > 0.06$, while the gas becomes flatter in the inner region. However, these non-trivial clusters profiles need to be observationally tested. The observations of the SZE, which are becoming increasingly accurate, can be used to probe these properties. Here we analyze the radial dependence of the SZE observables, deserving the comparison with the observational data in the X-ray and SZE domain to a forthcoming paper. The temperature shift may be evaluated inserting Eqs.(\[eq: nertm\]) and (\[eq: tertm\]) into Eqs.(\[eq:sze5bis\]) and (\[eq: defycomp\]) thus obtaining: $$\frac{\Delta T}{T_0} = - \frac{2 k_B \sigma_T T_{e_0} \, n_{e_0}}{m_e c^2} \ {\times} \ \eta \ , \label{eq:sze7}$$ with $\eta$ the so called [*structure integral*]{} defined as: $$\eta = 2 \int_0^l {n_e(s)\over n_{e_0}}{T_e(s)\over T_{e_0}} dl' \label{eq:sze8}$$ which, for a given density and temperature profile, depends only on the geometry and the extension of the cluster along the line of sight. In Eq.(\[eq:sze8\]), $l$ is the maximum extension of the gas along the line of sight. Measuring the lengths in units of the virial radius, it is $l = 1$ since the RTM model is truncated at $R_v$. Note that, usually, one takes $l \rightarrow \infty$ thus introducing a systematic bias whose effect we will examine later. A simple geometrical argument converts the integral in Eq.(\[eq:sze8\]) in angular form introducing the angular diameter distance, $D_A$, to the cluster: $$\begin{aligned} \eta & = & 2 \theta_v D_A \int_0^1{n_e(\chi) T_e(\chi) dl'} \nonumber \\ ~ & = & 2 \theta_v D_A \int_{0}^{\sqrt{1 + \xi^2}}{n_e(\chi) T_e(\chi) \frac{\chi d\chi}{\sqrt{\chi^2 - \xi^2}}} \label{eq: etaint}\end{aligned}$$ with $\chi= (l^2 + x^2 + y^2)/R_v^2$, and $\xi^2 = (x^2 + y^2)/R_v^2$. The integral in Eq.(\[eq: etaint\]) can be further simplified introducing the auxiliary variable $$\tilde{\alpha} = \frac{1 + s^2}{1 + \sqrt{\xi}} \ .$$ The structure integral $\eta$, once evaluated, allows to calculate the comptonization parameter $y$, and then the temperature shift $\Delta T/T$, according to the formula (\[eq:sze5bis\]). Adopting a flat $\Lambda$CDM model with $(h, \Omega_m) = (0.7, 0.3)$ and typical values for the virial radius and the virial temperature ($R_v \simeq 2 h^{-1} \ {\rm Mpc}$, $T_v \simeq 8 - 9 \ {\rm keV}$), we get a central number density $n_{e,0} = 5.07 \ {\rm cm^{-3}}$ and a shift in temperature at the cluster centre $\Delta T \simeq 9.3 \ {\rm mK}$. Let us now investigate in more detail how the peculiarities of the model affect the structure integral (and hence the comptonization parameter and the temperature shift). First, we consider the effect of the finite extension of the RTM model. To this aim, in Fig.\[geo\], we plot the structure integral $\eta$ as function of $\xi$ with the upper (lower) curve obtained assuming $l = 1$ ($l \rightarrow \infty$) in Eq.(\[eq:sze8\]). It is worth noting that the usual hypothesis of infinite extension may lead to significantly underestimate the SZE effect of the model by an amount that depends on the value of $\xi$. However, since what is usually measured is the temperature shift at the cluster centre, the relative error is not dramatic being less than $\sim 10\%$ for $\xi < 0.2$ as it is shown in Fig.\[error\]. Note, however, that the error due to the finite cluster extension for the RTM model is lower than the corresponding one for the standard $\beta$ model ([@jetzer]). The temperature profile of the RTM model is approximately isothermal up to $\sim 0.2 R_v$. Moreover, the isothermality hypothesis is often used in SZE computations. It is thus interesting to investigate what is the systematic error induced by the simplifying assumption $T(r) = T_{e,0}$ for the RTM model. In this case, the structure integral may be analytically expressed in terms of hypergeometric functions: $$\eta_{iso}(s) = 2 \theta_v D_A {\cal{Z}}(s) \label{eq: isoeta}$$ with $$\begin{aligned} {\cal{Z}}(s) & = & \frac{1}{s^{1.5}} \left ( 1.19814 _2F_1 \left [ \{1.75, 1.75\}; \{0.5\}; \frac{0.09}{s^2} \right ] \right ) \nonumber \\ ~ & ~ & - \frac{1}{s^{2.5}} \left ( 0.65514 _2F_1\left [ \{2.25, 1.25\}; \{1.5\}; \frac{0.09}{s^2} \right ] \right ) \nonumber \ . \label{eq: defh}\end{aligned}$$ Fig.\[iso\] shows that there is a dramatic change in the structure integral (and thus in the SZE temperature shift) for the RTM model if we use an isothermal temperature profile instead of that given by Eq.(\[eq: tertm\]). Actually, even if the RTM model has an almost isothermal core up to $r = 0.2 \,R_v$, assuming an isothermal profile leads to a large error in the the structure integral. In particular, in the inner region of the cluster, where the SZE effect is measured, $\eta_{iso}$ is more than $30\%$ lower than the true $\eta$ thus leading to a similar error on the temperature shift. Moreover, this error is larger than the one due to the finite extension of the cluster that dominates in the outer regions. Comparison with [*standard*]{} gas profiles =========================================== The peculiar non-isothermal temperature profile of the RTM model makes it different from most of the parametrizations used to describe the gas properties in galaxy clusters. It is thus particularly interesting to compare the SZ signal for the RTM model with that from more [*standard*]{} gas profiles. As interesting examples, we will consider the $\beta$ and the NFW models. The $\beta$-model ----------------- The X-ray surface brightness in galaxy clusters is commonly fitted using the so called $\beta$-model ([@cff76]) whose density profile is: $$\frac{\rho}{\rho_b} = \rho_{0 \beta} x_c^{-3\beta} \left ( 1 + \frac{x^2}{x_c^2} \right )^{-\frac{3 \beta}{2}} \label{beta}$$ where $x_c = r_c/R_v$ is the core radius in units of the virial radius, $\rho_{0 \beta}$ is related to the central electron number density $n_{e 0} = \rho_b \rho_{0 \beta}/(\mu m_p)$ and $\beta$ is a fitting parameter lying in the range $0.5 \le \beta \le 1$. The comptonization parameter for such a model is: $$y = \frac{2 k_B \sigma_T}{m_e c^2} \int_{0}^{L}{n_e T_e dl} = \frac{k_B \sigma_T n_{e 0} T_{e 0}}{m_e c^2} \eta_{\beta} \label{ybeta}$$ with $\eta_{\beta}$ the structure integral given as: $$\begin{aligned} \eta_{\beta} & = & R_v x_c^{1 - \beta} \int_{\tilde{\xi}}^{\tilde{\xi} + \tilde{\xi}_L} {\frac{(1 + \tilde{\xi}')^{-3 \beta/2}}{\sqrt{\tilde{\xi}' - \tilde{\xi}}} d\tilde{\xi}'} \nonumber \\ ~ & = & R_v x_c^{1 - 3 \beta} \left ( 1 + \tilde{\xi}_L \right )^{-3 \beta/2} \ \times \nonumber \\ ~ & \times & {_2F_1}\left [ \left \{ \frac{1}{2}, \frac{3}{2} \beta \right \}; \left \{ \frac{3}{2} \right \}; - \frac{1}{x_c^2(1 + \tilde{\xi}_L)} \right ] \ , \label{etabeta}\end{aligned}$$ where we have defined $\tilde{\xi} \equiv x^2/x_c^2$. In order to compare the RTM and $\beta$ SZE signal we evaluate the quantity $\epsilon_y= 1 - y_{\beta}/y_{RTM}$. To this aim, it is convenient to first reparametrize the RTM structure integral as follows: $$\begin{aligned} \eta_{RTM} = R_v x_{p_1}^{-1.5} \int_{\chi}^{\chi + \chi_L} {\frac{\chi'^{0.008} \left ( 1 + \sqrt{\chi'} \right )^{-2.5} d\chi'}{\sqrt{\chi' - \chi} \left ( x_{p_2}^4 + \chi'^2 x_{p_1}^4 \right )^{0.13}}} \label{etartm}\end{aligned}$$ with $x_{p_1} = 0.04$, $x_{p_2} = 10^{-0.51}$ and $\chi = x^2/x_{p_1}^2$. With these settings, we get: $$\begin{aligned} \epsilon_y & = 1 - & \frac{\left ( 1 + \tilde{\xi} \right )^{- 3 \beta/2}}{x_{p_1}^{-1.5} x_c^{3 \beta - 1}} \ {_2F_1}\left [ \left \{ \frac{1}{2}, \frac{3}{2} \beta \right \}; \left \{ \frac{3}{2} \right \}; - \frac{1}{x_c^2(1 + \tilde{\xi}_L)} \right ] \nonumber \\ ~ & ~ & \times \ \left ( \int_{\chi}^{\chi + \chi_L} {\frac{\chi'^{0.008} \left ( 1 + \sqrt{\chi'} \right )^{-2.5} d\chi'}{\sqrt{\chi' - \chi} \left ( x_{p_2}^4 + \chi'^2 x_{p_1}^4 \right )^{0.13}}} \right )^{-1} \ .\end{aligned}$$ To estimate $\epsilon_y$, we have first to choose reasonable values for the $\beta$-model parameters. To this aim, we first set $\rho_{0 \beta} = \rho_{0}$, with $\rho_0$ the characteristic density of the RTM model, and then choose $(\beta, x_c)$ by fitting the $\beta$-model to the RTM density profile. We obtain as best fit values $(\beta, x_c) = (0.74, 0.04)$ in qualitative agreement with $(\beta, x_c) = (0.7, 10^{-1.53})$ given in Rasia et al. (2004). With this choice of parameters, the two profiles agree quite well, within few $\%$, in the inner regions, while the relative error increases up to $\sim 10\%$ in the outer zone. Nonetheless, due to the radically different temperature profiles, the SZE signal is quite different as can be seen in Fig.\[deltaeta\_rtmbeta\]. Actually, $\epsilon_y$ turns out to be significantly different from unity even in the region $\xi \le 0.8$ where the two models fit each other quite accurately. In particular, we find that the SZE signal is larger for the RTM than for the $\beta$-model everywhere but in the extreme outer regions of the cluster where the situation is reversed. Actually, the most interesting implications of the non isothermal temperature profile for the RTM model with respect the standard $\beta$-model concerns the [*detectability*]{} of a cluster SZE signal, and therefore the statistics of the Sunyaev-Żeldovich clusters in different cosmological models. Even if all of the physics of the effect is coded in the Compton $y$ parameter, it is the total flux density from the cluster that is requested from the observational point of view. This is found by integrating the comptonization parameter over all the cluster face obtaining: $$Y = \int{y(\vec{\theta}) d^2\vec{\theta}} \label{detection}$$ being $\vec{\theta}$ the angular position on the sky. Since $y$ is dimensionless, $Y$ is effectively a solid angle. A caveat is in order here. Any SZE clusters survey (as for instance Plank) has some fixed angular resolution, which will not allow to spatially resolve low mass clusters, and even high mass cluster can be barely resolved ([@Aghanim97]). Therefore, a background $y_b$ Compton parameter will be present and will be dominated by low mass clusters since their higher number density overcompensates their lower individual contributions. It is also worth noting that an isotropic background would not matter since it could be completely removed. Because of this noise, the detectability of a SZ cluster will depend on the average background fluctuations that can be estimated as ([@bartelmann]): $$\bar{y} = \int{y(\vec{\theta'}) b(\vec{\theta} - \vec{\theta'}) d^2\vec{\theta}'} \label{detection2}$$ being $b(\vec{\theta})$ the cluster beam profile. A cluster is assumed to be detectable if its integrated, beam-convolved Compton $y$ parameter is sufficiently large, i.e.: $$\bar{Y} = \int{\bar{y}(\vec{\theta}) d^2\vec{\theta}} \ge \bar{Y}_{min}$$ where the integral has to be evaluated over the area where the integrand sufficiently exceeds the background fluctuations. However, in the following, we will neglect for simplicity the background noise, i.e. we will simply consider $Y$ rather than $\bar{Y}$. Even within such a simplified situation, we will discuss some aspects of the SZE detectability of RTM clusters, which can be easily generalized in presence of the background noise. If the gas temperature profile is isothermal, the integrated SZE flux calculated according to Eq.(\[detection\]) may be simply related to the cluster temperature weighted mass divided by $D_A^2$, being $D_A$ the angular diameter distance. Actually, in an isothermal regime, being $d\Omega = dA/D_A^2$, Eq.(\[detection\]) becomes: $$Y = \int{y(\vec{\theta}) d^2\vec{\theta}} \propto \frac{N_e \langle T_e \rangle}{D_A^2} \propto \frac{M \langle T_e \rangle}{D_A^2} \label{detectioniso}$$ being $N_e$ the total number of electrons in the cluster, $\langle T_e \rangle$ the mean electron temperature (which appears in an isothermal profile), and $M$ the total mass of the cluster (or the gas mass $M_{g}= f_{g} M$). From Eq.(\[detectioniso\]), it turns out that an SZE survey detects all clusters above some mass threshold which thus has a crucial role for the estimate of the SZE cluster counts and its cosmological and astrophysical applications ([@carl2002]). However, for the RTM model, the temperature profile is approximately isothermal only in the inner regions ($r \le 0.2 R_v$) so that the question of the detectability has to be significantly revisited. As a first step, let us compare $Y_{RTM}$, the exact (i.e. computed using the correct temperature profile) integrated SZE flux for the RTM model, with $Y_{RTM}^{iso}$ that is evaluated under the isothermal approximation. These quantities are given as: $$Y_{RTM} \propto 4 \pi \int_{0}^{X}{\frac{x^{2.016}}{(x_{p_1} + x)^{2.5} (x_{p_2}^4 + x^4)^{0.13}} dx} \ , \label{detrtm}$$ $$Y_{RTM}^{iso} \propto 5.33 ({\cal{Y}} - x_{p_1}^{1/2}) \label{detrtmiso}$$ with: $${\cal{Y}} = \frac{x_{p_1}^3 + X \left [ x_{p_1} \left ( 2.5 x_{p_1} + 1.875 X \right ) + 0.375 X^2 \right ]} {(x_{p_1} + x)^{2.5}} \ . \label{eq: detrtmiso}$$ with $X = r/R_v$. Fig.\[delta\_YRTMiso\] shows how $\epsilon_Y = 1 - Y_{RTM}^{iso}/Y_{RTM}$ increases with the distance from the cluster centre being of order $25\%$ at the virial radius. As a result, Eq.(\[detectioniso\]) for a RTM cluster is replaced by: $$Y \propto \frac{M_{eff} \langle T_e \rangle}{D_A^2} = \frac{\nu M \langle T_e \rangle}{D_A^2} \label{detectionrtmexact}$$ being $\nu = Y_{RTM}/Y_{RTM}^{iso}$. Because of the coefficient $\nu > 1$, the SZE detectability of a RTM cluster is naturally improved: for a fixed threshold value $Y_{lim}$, it is possible to detect less massive clusters than in the isothermal case. As a consequence, a RTM cluster is also detectable at lower mass regimes than a $\beta$-model cluster, as shown from a direct comparison. To this aim, we first remind tha the integrated SZE flux for a $\beta$-model is: $$Y_{\beta} \propto \frac{\xi^2}{3} \times {_2F_1}\left [ \left \{ \frac{3}{2}, \frac{3}{2} \beta \right \}; \left \{ \frac{5}{2} \right \}, - \frac{\xi^2}{x_c^2} \right ] \ .$$ In Fig.\[det\_rtm\_beta\], we plot the relative deviation of the best fit $\beta$-model and the corresponding RTM model. It turns out, indeed, that the integrated SZE flux is rather higher for a RTM cluster. The NFW model ------------- Even if it is mostly used to describe the dark matter rather than the gas distribution in galaxy clusters, it is nonetheless interesting to compare the SZE predictions for the RTM model with the same quantities evaluated for the NFW model. As a first step, we fit the NFW model to the RTM one obtaining $c_{NFW} = 5.99$ as best fit value for the concentration parameter. In agreement with Rasia et al. (2004), we find that the NFW model slightly overestimates the gas density in the range $0.04 \le x \le 0.4$, while it works sufficiently well in the outer regions. Assuming an isothermal profile for the gas temperature, the comptonization parameter for the NFW model turns out to be: $$\begin{aligned} y_{NFW} & \propto & \left [ \left ( X^2 - x_s^2 \right ) \left ( x_s + \sqrt{L^2 + X^2} \right ) \sqrt{x_s^4 - x_s^2 X^2} \right ]^{-1} \nonumber \\ ~ & \times & x_s^3 \left \{ L \sqrt{x_s^2 - X^2} + x_s^2 \left ( 1 + \frac{\sqrt{L^2 + X^2}}{x_s} \right ) \ \times \right . \nonumber \\ ~ & ~ & \left . \ \ \ \ \ \ln{\left [ \frac{\left ( X^2 - x_s^2 \right ) {\cal{Y}}_1}{{\cal{Y}}_2 {\cal{Y}}_3} \right ]} \right \} \ , \label{ynfw}\end{aligned}$$ with: $${\cal{Y}}_1 = x_s \left ( X^2 + x_s \sqrt{L^2 + X^2} \right ) - L \sqrt{x_s^4 - x_s^2 X^2} \ ,$$ $${\cal{Y}}_2 = \frac{X \left (X^2 - x_s^2 \right )}{x_s \sqrt{x_s^4 - x_s^2 X^2}} \ ,$$ $${\cal{Y}}_3 = x_s^3 \left ( x_s + \sqrt{L^2 + X^2} \right ) \sqrt{x_s^2 - X^2} \ ,$$ having denoted with $x_s = 1/c_{NFW}$, $L$ the cluster length along the line of sight and $X = r/R_v$. In Fig.\[rely\_rtmnfw\], we compare the SZE signal predicted from both models, plotting the relative discrepancy $\epsilon_y = 1 - y_{NFW}/y_{RTM}$ between the comptonization parameters. We see that the SZE signal from a RTM cluster exceeds that from a NFW model as yet observed when comparing to the $\beta$-model. The RTM model thus emerges as the most effectively detectable also at larger distances from the cluster centre. This is yet more clear from Fig.\[deltaY\_rtmnfw\] where we compare the integrated SZE flux for the RTM and NFW models. A short comment on possible systematic errors ============================================= There is a potential caveat about the RTM model that could affect the main results we have discussed insofar. The simulations that have been considered by Rasia et al. (2004) to develop the model are based non radiative hydrodynamics so that cooling flows and cold blobs that may eventually be along the line of sight are not reproduced. Taking into account these effects is a difficult task, but we expect, qualitatively, that cooling flows and conduction should lower the central temperature and thus increase the SZE signal at the centre. However, we stress that detailed simulations also taking into account star formation and feedback processes are needed to investigate how the gas temperature profile (and thus the SZE signal) are affected. Some preliminary results have been presented ([@DJSBR04]), but the effect on the SZE flux has still to be investigated. As a final remark, it is worth noting that merging of clusters has not been considered, but it is likely that this does not affect the main results. Actually, the sample of simulated clusters analyzed by Rasia et al. (2004) comprises both relaxed, unrelaxed and post-merging systems and the RTM model turns out to be a good fit to the full sample which is an evidence strongly suggesting that merging effects do not alter significantly the cluster structure. As a result, the lensing properties of the model are likely to be affected only when the merging is in progress in which case an external shear could mimic to first order the deviations from spherical symmetry of the outer regions of the dark matter halo. A stronger effect is expected for the impact of merging on the SZE signal and the X-ray emission, since they depend on $n_{e_0}$ and $n_{e_0}^2$ respectively. Although further detailed simulations are needed to quantitatively address this question, some partial analytical results have been already obtained for some special mergers regimes, when the presence of cold fronts marks the late merging stages: namely the transonic and the subsonic mergers. It turns out that in the transonic regime the frequency spectrum of the SZE signal $g(p)$ changes, due to a shock particle re-acceleration mechanism, depending on the concentration, which induces a new electron population. As net effect, the crossover changes still up to $\sim 10\%$. In the subsonic case, instead, $g(p)$ remains unchanged, but the amplitude of the SZE signal is enhanced in a not negligible way, mainly in the interior regions of the cluster where it reaches also $\sim 30 - 40 \%$ ([@koch04]). Moreover, it is well known ([@torri04]) that the X-ray luminosity overall increases during merging so that it is likely that, as net effect, the SZE signal is enhanced during cluster merging events. Conclusions =========== Being detectable at high redshift, galaxy clusters are promising tools for determining cosmological parameters and testing theories of structures formation. Hydrodynamical simulations are able to predict not only the dark matter mass distribution, but also the density law and the temperature profile of the gas component thus allowing a study of both the lensing properties and the Sunyaev-Zel’dovich effect due to the cluster. This has been the aim of the present paper where we have applied this study to the RTM model, recently proposed by Rasia et al. (2004) on the basis of the results of a large set of high resolution hydrodynamical simulations. Assuming spherical symmetry in the density profile, we have derived the main lensing properties of the RTM model evaluating the deflection angle and the lensing potential that allows us to write down the lens equations. Most of the work has been devoted to a detailed investigation of the critical curves structure of the RTM model since it is the position of radial and tangential arcs (that forms just near the critical curves) that gives the most useful constraints on the cluster parameters. The main results are summarized below. 1. [The RTM model always gives rise to both radial and tangential arcs, but their distance from the cluster centre is comparable to that observed in real systems only for values of the concentration much higher than typically predicted by the numerical simulations of Rasia et al. (2004).]{} 2. [Adding a giant elliptical galaxy (described with the Hernquist profile) to the cluster potential increases the distance between the radial and the tangential critical curves and allows to form observable radial arcs for lower values of the cluster concentration.]{} 3. [The shape of the critical curves may be significantly altered by a perturbing shear (mimicking an internal ellipticity or the effect of tidal perturbations) thus changing the positions of both radial and tangential arcs by an amount that can be up to $\sim 30\%$ for a shear strength $\gamma = 0.15$ (see Table 1).]{} 4. [Fitting an NFW model to a cluster that is intrinsically described by the RTM model leads to overestimate the cluster mass by an amount that depends on the concentrations $c_{RTM}$ and $c_{NFW}$ used in the modeling.]{} The work presented should be complemented by a detailed study also taking into account the presence of substructures (predicted by the CDM paradigm of structures formation) in the dark halo cluster. The total lensing potential thus should be made of the sum of the contributions from the elliptical RTM model, the external shear (due, e.g., to tidal perturbations or large scale structure), the bright cluster galaxy and the satellite haloes. This approach is quite complicated given the high number of unknown parameters entering the modeling, but one could resort to X-ray data to constrain (at least) the RTM parameters $(M_v, x_p)$. However, the most compelling test is a direct comparison with real systems showing tangential and radial arcs. To this aim, adding the BCG contribution (and eventually the shear term) to the RTM lensing potential should give a sufficiently accurate cluster model to be compared with the data on the arcs position. Having determined the galaxy scalelength and surface brightness from photometric observations, this method should allow us to test whether the RTM model may reproduce the observed arcs positions and to constrain its parameters. This will be performed in a future paper. One of the most interesting feature of the RTM model is that the temperature profile is approximately isothermal only up to $0.2 \ R_v$. We have investigated the implications of the RTM model for the intracluster gas on the SZE evaluating the structure integral that determines the temperature decrement. We have thus taken into account both the finite extension of the model and its peculiar temperature profile estimating the errors induced by the usually adopted simplifications of infinite extension and isothermal temperature. The main results are as follows. 1. [Neglecting the finite extension of the cluster systematically underestimates $\Delta T/T$ by an amount that is less than $10\%$ in the inner regions of the cluster so that it may be neglected in a first order analysis.]{} 2. [Using $T = T_ {e,0}$ as temperature profile instead of that found by RTM leads to underestimate $\Delta T/T$ up to $30\%$ (in absolute value) in the centre.]{} 3. [The comptonization parameter $y$ for the RTM model is higher than that of both the $\beta$ and the NFW models even if the parameters are chosen in such a way that the gas density is well fitted by the three models. Using the best fitting $\beta$ (NFW) model instead of the correct RTM one underestimates $y$ up to $\sim 40\%$ ($\sim 52\%$, respectively) in the inner cluster regions.]{} 4. [The non isothermal temperature profile leads to an integrated SZ flux which is higher for the RTM model than for both the $\beta$ and NFW models by an amount that depends on the distance from the cluster centre, but can be as high as $\sim 45\%$ and $\sim 90\%$ for the $\beta$ and NFW models respectively. As a result, less massive clusters should be detected in SZE survey if the RTM model is indeed the correct one.]{} As a final remark, we would like to stress that, in our opinion, a combined analysis (from the theoretical and observational point of view) of the both the lensing properties and the SZ temperature decrement could be the best method to validate a given cluster model independently on the peculiarities of the numerical simulations inspiring it. 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--- abstract: 'Audio tagging is the task of predicting the presence or absence of sound classes within an audio clip. Previous work in audio tagging focused on relatively small datasets limited to recognising a small number of sound classes. We investigate audio tagging on AudioSet, which is a dataset consisting of over 2 million audio clips and 527 classes. AudioSet is weakly labelled, in that only the presence or absence of sound classes is known for each clip, while the onset and offset times are unknown. To address the weakly-labelled audio tagging problem, we propose attention neural networks as a way to attend the most salient parts of an audio clip. We bridge the connection between attention neural networks and multiple instance learning (MIL) methods, and propose decision-level and feature-level attention neural networks for audio tagging. We investigate attention neural networks modelled by different functions, depths and widths. Experiments on AudioSet show that the feature-level attention neural network achieves a state-of-the-art mean average precision (mAP) of 0.369, outperforming the best multiple instance learning (MIL) method of 0.317 and Google’s deep neural network baseline of 0.314. In addition, we discover that the audio tagging performance on AudioSet embedding features has a weak correlation with the number of training samples and the quality of labels of each sound class.' author: - | Qiuqiang Kong, , Changsong Yu, Yong Xu, , Turab Iqbal,\ Wenwu Wang,  and Mark D. Plumbley, [^1] [^2] [^3] [^4] [^5] bibliography: - 'refs.bib' title: | Weakly Labelled AudioSet Tagging\ with Attention Neural Networks --- Audio tagging, AudioSet, attention neural network, weakly labelled data, multiple instance learning. Introduction ============ Audio tagging is the task of predicting the tags of an audio clip. Audio tagging is a multi-class tagging problem to predict zero, one or multiple tags for an audio clip. As a specific task of audio tagging, audio scene classification often involves the prediction of only one label in an audio clip, i.e. the type of environment in which the sound is present. In this paper, we focus on audio tagging. Audio tagging has many applications such as music tagging [@fu2011survey] and information retrieval [@typke2005survey]. An example of audio tagging that has attracted significant attention in recent years is the classification of environmental sounds, that is, predicting the scenes where they are recorded. For instance, the Detection and Classification of Acoustic Scenes and Events (DCASE) challenges [@dcase2013; @dcase2015; @dcase2016; @mesaros2017dcase] consist of tasks from a variety of domains, such as DCASE 2018 Task 1 classification of outdoor sounds, DCASE 2017 Task4 tagging of street sounds and DCASE 2016 Task4 tagging of domestic sounds. These challenges provide labelled datasets, so it is possible to use supervised learning algorithms for audio tagging. However, many audio tagging datasets are relatively small [@dcase2013; @dcase2015; @dcase2016; @mesaros2017dcase], ranging from hundreds to thousands of training samples, while modern machine learning methods such as deep learning [@deep-learning-lecun; @deep-learning-schmidhuber] often benefit greatly from larger dataset for training. In 2017, a large-scale dataset called *AudioSet* [@audioset] was released by Google. AudioSet consists of audio clips extracted from YouTube videos, and is the first dataset that achieves a similar scale to the well-known ImageNet [@deng2009imagenet] dataset in computer vision. The current version (v1) of AudioSet consists of 2,084,320 audio clips organised into a hierarchical ontology with 527 predefined sound classes in total. Each audio clip in AudioSet is approximately 10 seconds in length, leading to 5800 hours of audio in total. AudioSet provides an opportunity for researchers to investigate a large and broad variety of sounds instead of being limited to small datasets with limited sound classes. One challenge of AudioSet tagging is that AudioSet is a weakly-labelled dataset (WLD) [@kumar2016audio; @kong2017joint]. That is, for each audio clip in the dataset, only the presence or the absence of sound classes is indicated, while the onset and offset times are unknown. In previous work in audio tagging, an audio clip is usually split into segments and each segment is assigned with the label of the audio clip [@kong2016deep]. However, as the onset and offset of sound events are unknown so such label assignment can be incorrect. For example, a transient sound event may only appear a short time in a long audio recording. The duration of sound events can be very different and there is no prior knowledge of their duration. Different from ImageNet [@deng2009imagenet] for image classification where objects are usually centered and have similar scale, in AudioSet the duration of sound events may vary a lot. To illustrate, Fig. \[fig:wav\] from top to bottom shows: the log mel spectrogram of a 10-second audio clip[^6]; AudioSet bottleneck features [@audioset] extracted by a pre-trained VGGish convolutional network followed by a principal component analysis (PCA); weak labels of the audio including “music”, “chuckle”, “snicker” and “speech’. In contrast to WLD, strongly labelled data (SLD) refers to the data labelled with both the presence of sound classes as well as their onset and offset times. For example, the sound event detection tasks in DCASE challenge 2013, 2016, 2017 [@dcase2013; @dcase2016; @mesaros2017dcase] provide SLD. However, labelling onset and offset times of sound events is time-consuming, so these strongly labelled datasets are usually limited to a relatively small size [@dcase2013; @dcase2016; @mesaros2017dcase], which may limit the performance of deep neural networks that require large data to train a good model. ![From top to bottom: Log mel spectrogram of a 10-second audio clip; AudioSet bottleneck features extracted by a pre-trained VGGish convolutional neural network followed by a principle component analysis (PCA) [@hershey2017cnn]; Weak labels of the audio clip. There are no onset and offset times of the sound classes. []{data-label="fig:wav"}](figs/waveform.pdf){width="\columnwidth"} In this paper, we train an audio tagging system on the large-scale weakly labelled AudioSet. We bridge our previously proposed attention neural networks [@kong2017audio; @yu2018multi] with multiple instance learning (MIL) [@maron1998framework] and propose decision-level and feature-level attention neural networks for audio tagging. The contributions of this paper include the following: - Decision-level and feature-level attention neural networks are proposed for audio tagging; - Attention neural networks modelled by different functions, widths and depth are investigated; - The impact of the number of training samples per class on the audio tagging performance is studied; - The impact of the quality of labels on the audio tagging performance is studied. This paper is organised as follows. Section II introduces audio tagging with weakly labelled data. Section III introduces our previously proposed attention neural networks [@kong2017audio; @yu2018multi]. Section IV introduces multiple instance learning. Section V reviews attention neural networks under the MIL framework and proposes decision-level and feature-level attention models. Section VI shows the experimental results. Section VII concludes and forecasts future work. Audio Tagging with weakly labelled data ======================================= Audio tagging has attracted much research interests in recent years. For example, the tagging of the CHiME Home dataset [@foster2015chime], the UrbanSound dataset [@salamon2014dataset] and datasets from the Detection and Classification of Acoustic Scenes and Events (DCASE) challenges in 2013 [@stowell2015detection], 2016 [@mesaros2016tut], 2017 [@mesaros2017dcase] and 2018 [@mesaros2018multi]. The DCASE 2018 Challenge includes acoustic scene classification [@mesaros2018multi], general purpose audio tagging [@fonseca2018general] and bird audio detection [@stowell2018automatic] tasks. Mel frequency cepstral coefficients (MFCC) [@li2001classification; @uzkent2012non; @eghbal2016cp] have been widely used as features to build audio tagging systems. Other features used for audio tagging include pitch features [@uzkent2012non] and I-vectors [@eghbal2016cp]. Classifiers include such as Gaussian mixture models (GMMs) [@aucouturier2007bag] and support vector machines [@sigtia2016automatic]. Recently, neural networks have been used for audio tagging with mel spectrograms as input features. A variety of neural network methods including fully-connected neural networks [@kong2016deep], convolutional neural networks (CNNs) [@cakir2016domestic; @choi2016automatic; @hershey2017cnn] and convolutional recurrent neural networks (CRNNs) [@choi2017convolutional; @xu2017large] have been explored for audio tagging. For sound localization, an identify, locate and separate model [@parekh2018identify] was proposed for audio-visual object extraction in large video collections using weak supervision. A WLD consists of a set of *bags*, where each bag is a collection of instances. For a particular sound class, a positive bag contains at least one positive instance, while a negative bag contains no positive instances. We denote the $ n $-th bag in the dataset as $ B_{n}=\left \{ \mathbf{x}_{n1}, ..., \mathbf{x}_{nT_{n}} \right \} $, where $ T_{n} $ is the number of instances in the bag. An instance $ \mathbf{x}_{nt} \in \mathbb{R}^{M} $ in the bag has a dimension of $ M $. A WLD can be denoted as $ D = \left \{ B_{n}, \mathbf{y}_{n} \right \}_{n=1}^{N} $, where $ \mathbf{y}_{n} \in \{0, 1\}^{K} $ denotes the tags of bag $ B_{n} $, and $ K $ and $ N $ are the number of classes and training samples, respectively. In WLD, each bag $ B_{n} $ has associated tags but we do not know the tags of individual instances $ \mathbf{x}_{nt} $ within the bag [@amores2013multiple]. For example, in the AudioSet dataset, a bag consists of instances that are bottleneck features obtained by inputting a logmel to a pre-trained VGGish convolutional neural network. In the following sections, we omit the training example index $ n $ and the time index $ t $ to simplify notation. Previous audio tagging systems using WLD have been based on segment based methods. Each segment is called an instance and are assigned the tags inherited from the audio clip. During training, instance-level classifiers are trained on individual instances. During inference, bag-level predictions are obtained by aggregating the instance-level predictions [@kong2016deep]. Recently, convolutional neural networks have been applied to audio tagging [@choi2017convolutional], where the log spectrogram of an audio clip is used as input to a CNN classifier without predicting the individual instances explicitly. Attention neural networks have been proposed for AudioSet tagging in [@kong2017audio; @yu2018multi]. Later, a clip-level and segment-level model with attention supervision was proposed in [@chou2018learning]. Audio tagging with attention neural networks ============================================ Segment based methods {#section:bow} --------------------- R3: In segment based methods, an audio clip is split into segments and each segment is assigned the tags inherited from the audio clip. In MIL, each segment is called an instance. An instance-level classifier $ f $ is trained on the individual instances: $ f: \mathbf{x} \mapsto f(\mathbf{x}) $, where $ f(\mathbf{x}) \in [0, 1]^{K} $ predicts the presence probabilities of sound classes. The function $ f $ depends on a set of learnable parameters that can be optimised using gradient descent methods with the loss function $$\label{eq:bag_of_words_loss} l(f(\mathbf{x}), \mathbf{y}) = d(f(\mathbf{x}), \mathbf{y}),$$ where $ \mathbf{y} \in \{0, 1\}^{K} $ are the tags of the instance $ \mathbf{x} $ and $ d(\cdot, \cdot) $ is a loss function. For instance, it could be binary cross-entropy for multi-class tagging, given by $$\label{eq:binary_crossentropy} d(f(\mathbf{x}), \mathbf{y}) = - \sum_{k=1}^{K} [ y_{k} \text{log}f(\mathbf{x})_{k} + (1 - y_{k}) \text{log} (1 - f(\mathbf{x})_{k} ].$$ In inference, the prediction of a bag is obtained by aggregating the predictions of individual instances in the bag such as by majority voting [@kong2016deep]. The segment based model has been applied to many tasks such as information retrieval [@pancoast2012bag] due to its simplicity and efficiency. However, the assumption that all instances inherit the tags of a bag is incorrect. For example, some sound events may only occur for a short time in an audio clip. Attention neural networks ------------------------- Attention neural networks were first proposed for natural language processing [@sankaran2016temporal; @luong2015effective], where the words in a sentence are attended differently for machine translation. Attention neural networks are designed to attend to important words and ignore irrelevant words. Attention models have also been applied to computer vision, such as image captioning [@attention-xu] and information retrieval [@liu2015content]. We proposed attention neural networks for audio tagging and sound event detection with WLD in [@kong2017audio; @xu2017large]: these were ranked first in the DCASE 2017 Task 4 challenge [@xu2017large]. In a similar way to the segment based model, attention neural networks build an instance-level classifier $ f(\mathbf{x}) $ for individual instances $ \mathbf{x} $. In contrast to the segment based model, attention neural networks do not assume that instances in a bag have the same tags as the bag. As a result, there is no instance-level ground truth for supervised learning using (\[eq:bag\_of\_words\_loss\]). To solve this problem, we aggregate the instance-level predictions $ f(\mathbf{x}) $ to a bag-level prediction $ F(B) $ given by $$\label{eq:att_func} F(B)_{k} = \sum_{\mathbf{x} \in B} p(\mathbf{x})_{k} f(\mathbf{x})_{k},$$ where $ p(\mathbf{x})_{k} $ is a weight of $ f(\mathbf{x})_{k} $ that we refer to as an *attention function*. The attention function $ p(\mathbf{x})_{k} $ should satisfy $$\label{eq:probability_constraint} \sum_{\mathbf{x} \in B} p(\mathbf{x})_{k} = 1,$$ so that the bag-level prediction can be seen as a weighted sum of the instance-level predictions. Both the attention function $ p(\mathbf{x}) $ and the instance-level classifier $ f(\mathbf{x}) $ depend on a set of learnable parameters. The attention function $ p(\mathbf{x})_{k} $ controls how much a prediction $ f(\mathbf{x})_{k} $ should be attended. Large $ p(\mathbf{x})_{k} $ indicates that $ f(\mathbf{x})_{k} $ should be attended, while small $ p(\mathbf{x})_{k} $ indicates that $ f(\mathbf{x})_{k} $ should be ignored. To satisfy (\[eq:probability\_constraint\]), the attention function $ p(\mathbf{x})_{k} $ can be modelled as $$\label{eq:decision_level_normalize} p(\mathbf{x})_{k} = v(\mathbf{x})_{k} / \sum_{\mathbf{x} \in B} v(\mathbf{x})_{k},$$ where $ v(\cdot) $ can be any non-negative function to ensure that $ p(\cdot) $ is a probability. An extension of the attention neural network in (\[eq:att\_func\]) is the multi-level attention model [@yu2018multi], where multiple attention modules are applied to utilise the hierarchical information of neural networks: $$\label{eq:multi_att} F(B) = g(F_{1}(B), ..., F_{L}(B)),$$ where $ F_{l}(B) $ is the output of the $ l $-th attention module and $ L $ is the number of attention modules. Each $ F_{l}(B) $ can be modeled by (\[eq:att\_func\]). Then a mapping $ g $ is used to map from the predictions of $ L $ attention modules to the final prediction of a bag. The multi-level attention neural network has achieved state-of-the-art performance in AudioSet tagging. In the next section, we show that the attention neural networks explored above can be categorised into an MIL framework. Multiple instance learning ========================== Multiple instance learning (MIL) [@dietterich1997solving; @maron1998framework] is a type of supervised learning method. Instead of receiving a set of labelled instances, the learner receives a set of labelled bags. MIL methods have many applications. For example, in [@dietterich1997solving], MIL is used to predict whether new molecules are qualified to make some new drug, where molecules may have many alternative low-energy states, but only one, or some of them, are qualified to make a drug. Inspired by the MIL methods, a sound event detection system trained on WLD [@kumar2016audio] was proposed. General MIL methods include the expectation-maximization diversity density (EM-DD) method [@zhang2002dd], support vector machine (SVM) methods [@andrews2003support] and neural network MIL methods [@zhou2002neural; @wang2018revisiting]. In [@wang2018comparison], several MIL pooling methods were investigated in audio tagging. Attention-based deep multiple instance learning is proposed in [@ilse2018attention]. In [@amores2013multiple], MIL methods are grouped into three categories: the instance space (IS) methods, where the discriminative information is considered to lie at the instance-level; the bag space (BS) methods, where the discriminative information is considered to lie at the bag-level; and the embedded space (ES) methods, where each bag is mapped to a single feature vector that summarises the relevant information about a bag. We introduce the IS, BS and ES methods in more detail below. Instance space methods {#section:IS_paradigm} ---------------------- In IS methods, an instance-level classifier $ f: \mathbf{x} \mapsto f(\mathbf{x}) $ is used to predict the tags of an instance $ \mathbf{x} $, where $ f(\mathbf{x}) \in [0, 1]^{K} $ predicts the presence probabilities of sound classes. The IS methods introduce aggregation functions [@amores2013multiple] to convert an instance-level classifier $ f $ to a bag-level classifier $ F: B \mapsto [0, 1]^{K} $, given by $$\label{eq:IS_inference_aggregation} F(B) = \text{agg} \left ( \{f(\mathbf{x})\}_{\mathbf{x} \in B} \right ),$$ where $ \text{agg}(\cdot) $ is an aggregation function. The classifier $ f $ depends on a set of learnable parameters. When the IS method is trained with (\[eq:bag\_of\_words\_loss\]) in which each instance inherits the tags of the bag, the IS method is equivalent to the segment based model. On the other hand, the IS method can also be trained using the bag-level loss function: $$\label{eq:IS_loss} l(F(B), \mathbf{y}) = d(F(B), \mathbf{y}),$$ where $ \mathbf{y} \in \{ 0, 1 \}^{K} $ is the tag of the bag and $ d(\cdot, \cdot) $ is a loss function such as the binary cross-entropy in (\[eq:binary\_crossentropy\]). To model the aggregation function, the standard multiple instance (SMI) assumption and collective assumption (CA) are proposed in [@amores2013multiple]. Under the SMI assumption, a bag-level classifier can be obtained by $$\label{eq:IS_max} F(B)_{k} = \underset{\mathbf{x} \in B}{\mathrm{max}} f(\mathbf{x})_{k},$$ where the subscript $ k $ denotes the $ k $-th sound class of the instance-level prediction $ f(\mathbf{x}) $ and the bag-level prediction $ F(B) $. Under the SMI assumption, for the $ k $-th sound class, only one instance with the maximum prediction probability is chosen as a positive instance. One problem of the SMI assumption is that a positive bag may contain more than one positive instance. In SED, some sound classes such as “ambulance siren” may last for several seconds and may occur in many instances. In contrast to the SMI assumption, with the CA assumption, all the instances in a bag contribute equally to the tags of the bag. The bag-level prediction can be obtained by averaging the instance-level predictions: $$\label{eq:IS_average} F(B)=\frac{1}{\left | B \right |}\sum_{\mathbf{x} \in B}f(\mathbf{x}).$$ The symbol $ |B| $ denotes the number of instances in bag $ B $. Equation (\[eq:IS\_average\]) shows that CA is based on the assumption that all the instances in a positive bag are positive. Bag space methods {#BS_paradigm} ----------------- Instead of building an instance-level classifier, the BS methods regard a bag $ B $ as an entirety. Building a tagging model on the bags rely on a distance function $ D(\cdot, \cdot): B \times B \mapsto \mathbb{R} $. The distance function can be, for example, the Hausdorff distance [@wang2000solving]: $$\label{eq:BS} D(B_{1}, B_{2}) = \underset{\mathbf{x}_{1} \in B_{1}, \mathbf{x}_{2} \in B_{2}}{\textrm{min}}\left \| \mathbf{x}_{1} - \mathbf{x}_{2} \right \|.$$ In (\[eq:BS\]), the distance between two bags is the minimum distance between the instances in bag $ B_{1} $ and $ B_{2} $. Then this distance function can be plugged into a standard distance-based classifier such as a k-nearest neighbour (KNN) or a support vector machine (SVM) algorithm. The computational complexity of (\[eq:BS\]) is $ |B_{1}||B_{2}| $, which is larger than the IS and the ES methods described below. Embedded space methods {#section:ES_paradigm} ---------------------- Different from the IS methods, ES methods do not classify individual instances. Instead, the ES methods define an embedding mapping from a bag to an embedding vector: $$\label{eq:ES_mapping} f_{\text{emb}}: B \mapsto \mathbf{h}.$$ Then the tags of a bag is obtained by applying a function $ g $ on the embedding vector: $$\label{eq:ES_prediction} F(B) = g(\mathbf{h}).$$ The embedding mapping $ f_{\text{emb}} $ can be modelled in many ways. For example, by averaging the instances in a bag, as in the simple MI method in [@dong2006comparison]: $$\label{eq:ES_average} \mathbf{h}=\frac{1}{\left | B \right |}\sum_{\mathbf{x} \in B}\mathbf{x}.$$ Alternatively, the mapping can be obtained in terms of the max-min operations on the instances [@gartner2002multi]: $$\label{eq:ES_max} \begin{split} \begin{cases} \mathbf{h}=(a_{1}, ..., a_{M}, b_{1}, ..., b_{M}), \\ a_{m}=\underset{\mathbf{x} \in B}{\mathrm{max}} (x_{m}), \\ b_{m}=\underset{\mathbf{x} \in B}{\mathrm{min}} (x_{m}), \\ \end{cases} \end{split}$$ where $ x_{m} $ is the $ m $-th dimension of $ \mathbf{x} $. Equation (\[eq:ES\_max\]) shows that only one instance with the maximum or the minimum value is chosen for each dimension, while other instances have no contribution to the embedding vector $ \mathbf{h} $. The ES methods summarise a bag containing an arbitrary number of instances with a vector of fixed size. Similar methods have been proposed in natural language processing to summarise sentences with a variable number of words [@bahdanau2014neural]. Attention neural networks under MIL {#section:revision_attention_neural_network} =================================== In this section, we show that the previously proposed attention neural networks [@kong2017audio; @yu2018multi] belong to MIL frameworks, especially the IS methods. We refer to these attention neural networks as decision-level attention neural networks, because the prediction of a bag is obtained by aggregating the predictions of instances (see (\[eq:IS\_inference\_aggregation\])). We then propose feature-level attention neural networks inspired by the ES methods with attention in the hidden layers. Decision-level attention neural networks {#section:decision_level_att} ---------------------------------------- The IS methods predict the tags of a bag by aggregating the predictions of individual instances in the bag described in (\[eq:IS\_inference\_aggregation\]). Section \[section:IS\_paradigm\] shows that conventional IS methods are based on either the SMI assumption (see (\[eq:IS\_max\])) or the CA (see (\[eq:IS\_average\])). The problem of the SMI assumption is that only one instance in a bag is considered to be positive for a sound class while other instances are not considered. The SMI assumption is not appropriate for bags with more than one positive instance for a sound class. On the other hand, CA assumes that all instances in a positive bag are positive. CA is not appropriate for sound events that only last for a short time. To address the problems of the SMI and CA methods, a decision-level attention neural network based on the IS methods in (\[eq:IS\_inference\_aggregation\]) is proposed to learn an attention function to weight the predictions of instances in a bag, so that $$\label{eq:decision_level_agg} \begin{split} F(B)_{k} & = \text{agg}(\{f(\mathbf{x})_{k}\}_{\mathbf{x} \in B}) \\ & = \sum_{\mathbf{x} \in B} p(\mathbf{x})_{k} f(\mathbf{x})_{k}, \end{split}$$ where $ p(\mathbf{x}) $ is an attention function modelled by (\[eq:decision\_level\_normalize\]). We refer to (\[eq:decision\_level\_agg\]) as a decision-level attention neural network because the attention function $ p(\mathbf{x}) $ is multiplied with the predictions of the instances $ f(\mathbf{x}) $ to obtain the bag-level prediction. The attention function $ p(\mathbf{x}) $ controls how much the prediction of an instance $ f(\mathbf{x}) $ should be attended or ignored. Equation (\[eq:decision\_level\_agg\]) can be seen as a general case of the SMI and CA assumptions. When one instance $ \mathbf{x} $ in a bag has a value of $ p(\mathbf{x})=1 $ the other instances have values of $ p(\mathbf{x})=0 $, then (\[eq:decision\_level\_agg\]) is equivalent to the SMI assumption in (\[eq:IS\_max\]). When $ p(\mathbf{x})=\frac{1}{|B|} $ for all instances in a bag, (\[eq:decision\_level\_agg\]) is equivalent to CA. ![(a) Joint detection and classification (JDC) model; (b) Self attention neural network in [@lin2017structured]; (c) Proposed attention neural network [@kong2017audio]. The blue outlined block in (c) is called a forward (FWD) block.[]{data-label="fig:jdc_att"}](figs/jdc_att.pdf){width="\columnwidth"} Fig. \[fig:jdc\_att\] shows different ways to model the attention neural network in (\[eq:decision\_level\_agg\]). For example, Fig. \[fig:jdc\_att\](a) shows the joint detection and classification (JDC) model [@kong2017joint] with attention function $ p $ and the classifier $ f $ modelled by separate branches. Fig. \[fig:jdc\_att\](b) shows the self attention neural network [@lin2017structured] proposed in natural language processing. Fig. \[fig:jdc\_att\](c) shows the JDC improved by using shared layers for the attention function $ p $ and the classifier $ f $ before they separate in the penultimate layer [@kong2017audio]. In the attention neural networks, both $ p $ and $ f $ depend on a set of learnable parameters which can be optimised with gradient descent methods using the loss function in (\[eq:IS\_loss\]). For the proposed model in Fig. \[fig:jdc\_att\](c), the attention function $ p $ and the classifier $ f $ share the low-level layers. We denote the output of the layer before they separate as $ \mathbf{x}' $. The mapping from $ \mathbf{x} $ to $ \mathbf{x}' $ can be modelled by fully-connected layers, for example. $$\label{eq:att_fc_func} \mathbf{x}' = f_{\text{FC}}(\mathbf{x}).$$ The classifier $ f $ can be modelled by $$\label{eq:decision_level_att_f} f(\mathbf{x}) = \sigma(\mathbf{W}_{1} \mathbf{x}' + \mathbf{b}_{1}),$$ where $ \sigma(x) = 1 / (1 + e^{-x}) $ is the sigmoid function. The attention function $ p $ can be modelled by $$\label{eq:decision_level_att_p} \begin{cases} v(\mathbf{x}')_{k} = \phi_{1}(\mathbf{U}_{1}\mathbf{x}' + \mathbf{c}_{1}), \\ p(\mathbf{x})_{k} = v(\mathbf{x}')_{k} / \sum_{\mathbf{x} \in B} v(\mathbf{x}')_{k}, \\ \end{cases}$$ where $ \phi_{1} $ can be any non-negative function to ensure $ p(\mathbf{x})_{k} $ is a probability. ![(a) Decision-level single attention neural network [@kong2017audio]; (b) Decision-level multiple attention neural network [@yu2018multi]; (c) Feature-level attention neural network (proposed). []{data-label="fig:decision_feature_att"}](figs/decision_feature_att.pdf){width="\columnwidth"} Feature-level attention neural network {#section:feature_level_att} -------------------------------------- The limitation of the decision-level attention neural networks is that the attention function $ p(\mathbf{x}) $ is only applied to the prediction of the instances $ f(\mathbf{x}) $, as shown in (\[eq:decision\_level\_agg\]). In this section, we propose to apply attention to the hidden layers of a neural network. This is inspired by the ES methods in (\[eq:ES\_mapping\]), where a bag $ B $ is mapped to a fixed-size vector $ \mathbf{h} $ before being classified. We model (\[eq:ES\_mapping\]) with attention aggregation: $$\label{eq:feature_level_att_1} h_{j} = \sum_{\mathbf{x} \in B} q(\mathbf{x})_{j} u(\mathbf{x})_{j},$$ where both $ q(\mathbf{x}) \in [0, 1]^{J} $ and $ u(\mathbf{x}) \in \mathbb{R}^{J} $ have a dimension of $ J $. The embedded vector $ \mathbf{h} \in \mathbb{R}^{J} $ summarises the information of a bag. Then the tags of a bag $ B $ can be obtained by classifying the embedding vector: $$\label{eq:feature_level_att_2} F(B) = f(\mathbf{h}).$$ The probability $ q(\mathbf{x})_{j} $ in (\[eq:feature\_level\_att\_1\]) is the attention function of $ u(\mathbf{x})_{j} $ and should satisfy $$\label{eq:feature_level_constraint} \sum_{\mathbf{x} \in B} q(\mathbf{x})_{j} = 1.$$ We model $ u(\mathbf{x}) $ with $$\label{eq:feature_level_att_f} u(\mathbf{x}) = \psi(\mathbf{W}_{2} \mathbf{x}' + \mathbf{b}_{1}),$$ where $ \psi $ can be any linear or non-linear function to increase the representation ability of the model. The attention function $ q $ can be modelled by $$\label{eq:feature_level_att_q} \begin{cases} w(\mathbf{x}')_{j} = \phi_{2}(\mathbf{U}_{2}\mathbf{x}' + \mathbf{c}_{2}), \\ q(\mathbf{x})_{j} = w(\mathbf{x}')_{j} / \sum_{\mathbf{x} \in B} w(\mathbf{x}')_{j}, \\ \end{cases}$$ where $ w(\mathbf{x})_{j} $ can be any non-negative function to ensure $ q(\mathbf{x})_{j} $ is a probability. Fig. \[fig:decision\_feature\_att\] shows the decision-level single attention [@kong2017audio], decision-level multiple attention [@yu2018multi] and the proposed feature-level attention neural network. The forward (Fwd) block in Fig. \[fig:decision\_feature\_att\] is the same as the block in Fig. \[fig:jdc\_att\](c). The difference between the feature-level attention function $ q(\mathbf{x}) $ and the decision-level attention function $ p(\mathbf{x}) $ is that the dimension of $ q(\mathbf{x}) $ can be any value, while the dimension of $ p(\mathbf{x}) $ is fixed to be the number of sound classes $ K $. Therefore, the capacity of the decision-level attention neural networks is limited. With an increase in the dimension of $ q(\mathbf{x}) $, the capacity of feature-level attention neural networks is increased. The decision-level attention function attends to the predictions of instances, while the feature-level attention function attends to the features, so it is equivalent to feature selection. The multi-level attention model [@yu2018multi] in (\[eq:multi\_att\]) can be seen as a special case of the feature-level attention model, with embedding vector $ \mathbf{h} = (F_{1}(B), ..., F_{L}(B)) $. The superior performance of the multi-level attention model shows that the feature-level attention neural networks have the potential to perform better than the decision-level attention neural networks. Modeling the attention function with different non-linearity {#section:exp_different_att_func} ------------------------------------------------------------ We adopt Fig. \[fig:jdc\_att\](c) as the backbone of our attention neural networks. The attention function $ p $ and $ q $ for the decision-level and feature-level attention neural networks are obtained via non-negative functions $ \phi_{1} $ and $ \phi_{2} $, respectively. The $ \phi_{1} $ and $ \phi_{2} $ appearing in the summation term of the denominator of (\[eq:decision\_level\_att\_p\]) and (\[eq:feature\_level\_att\_q\]) may affect the optimisation of the attention neural networks. We investigate modelling $ \phi_{2} $ in the feature-level attention neural networks with different non-negative functions, including ReLU [@nair2010rectified], exponential, sigmoid, softmax and network-in-network (NIN) [@lin2013network]. We omit the evaluation of $ \phi_{1} $, as the feature-level attention neural networks outperform the decision-level attention neural networks. The ReLU function is defined as [@nair2010rectified] $$\label{eq:relu} \phi(z) = \text{max}(z, 0).$$ The exponential function is defined as $$\label{eq:exponential} \phi(z) = e^{z}.$$ The sigmoid function is defined as $$\label{eq:sigmoid} \phi(z) = \frac{1}{1 + e^{-z}}.$$ For a vector $\mathbf{z}$, the softmax function is defined as $$\label{eq:softmax} \phi(z_{j}) = \frac{e^{z_{j}}}{\sum_{k}e^{z_{k}}}.$$ The network-in-network function [@lin2013network] is defined as $$\label{eq:nin} \phi(\mathbf{z}) = \sigma(\mathbf{H}_{2}\psi(\mathbf{H}_{1}\mathbf{z}+\mathbf{d}_{1})+\mathbf{d}_{2}),$$ where $ \mathbf{H}_{1} $, $ \mathbf{H}_{2} $ are transformation matrices, $ \mathbf{d}_{1} $ and $ \mathbf{d}_{2} $ are biases, $\psi$ is ReLU nonlinearity and $ \sigma $ is the sigmoid function. ![Distribution of the number of sound classes in an audio clip. []{data-label="fig:classes_per_clip"}](figs/classes_per_clip.pdf){width="\columnwidth"} ![A VGGish CNN model is trained on the YouTube 100M dataset. Audio clips from AudioSet are given as input to the trained VGGish CNN to extract the bottleneck features, which are released by AudioSet. []{data-label="fig:feature_extraction"}](figs/feature_extraction2.pdf){width="\columnwidth"} {width="\textwidth"} Experiments =========== Dataset ------- We evaluate the proposed attention neural networks on AudioSet [@audioset], which consists of 2,084,320 10-second audio clips extracted from YouTube video with a hierarchical ontology of 527 classes in the released version (v1). We released both Keras and PyTorch implementations of our code online[^7]. AudioSet consists of a variety of sounds. AudioSet is multi-labelled, such that each audio clip may contain more than one sound class. Fig. \[fig:classes\_per\_clip\] shows the statistics of the number of sound classes in the audio clips. All audio clips contain at least one label. Out of over 2,084,320 audio clips, there are 896,045 audio clips containing one sound class, followed by around 684,166 audio clips containing two sound classes. Only 4,661 audio clips have more than 7 labels. Instead of providing raw audio waveforms, AudioSet provides bottleneck features of audio clips. The bottleneck features are extracted from the bottleneck layer of a VGGish CNN, pre-trained on 70 million audio clips from the YouTube100M dataset [@hershey2017cnn]. The VGGish CNN consists of 6 convolutional layers with kernel size of $ 3 \times 3 $ and 2 fully layers. To begin with, the 70 million training audio clips are segmented to non-overlapping 960 ms segments. Each segment inherits all tags of its parent video. Then short-time Fourier transform (STFT) is applied on each 960 ms segment with a window size of 25 ms and a hop size of 10 ms to obtain a spectrogram. Then a mel filter bank with 64 frequency bins is applied on the spectrograms followed by a logarithmic operation to obtain log mel spectrograms. Each log mel spectrogram of a segment has a shape of $ 96 \times 64 $, representing the time steps and the number of mel frequency bins. A VGGish CNN is trained on these log mel spectrograms with the 3087 most frequent labels. After training, the VGGish CNN is used as a feature extractor. By inputting an audio clip to the VGGish CNN, the outputs of the bottleneck layer are used as bottleneck features of the audio clip. The framework of AudioSet feature extraction is shown in Fig. \[fig:feature\_extraction\]. [\*[4]{}[c]{}]{} & mAP & AUC & d-prime\ Random guess & 0.005 & 0.500 & 0.000\ Google baseline [@audioset] & 0.314 & 0.959 & 2.452\ Segment based [@kong2016deep] & 0.293 & 0.960 & 2.483\ (IS) SMI assumption [@kumar2016audio] & 0.292 & 0.960 & 2.471\ (IS) Collective assumption & 0.300 & 0.964 & 2.536\ (ES) Average instances [@dong2006comparison] & **0.317** & **0.963** & **2.529**\ (ES) Max instance & 0.284 & 0.958 & 2.443\ (ES) Min instance & 0.281 & 0.956 & 2.413\ (ES) Max-min instance [@gartner2002multi] & 0.306 & 0.962 & 2.505\ Evaluation criterion -------------------- We first introduce basic statistics [@mesaros2016metrics]: true positive (TP), where both the reference and the system prediction indicate an event to be active; false negative (FN), where the reference indicates an event is active but the system prediction indicates an event is inactive; false positive (FP), where the system prediction indicates an event is active but the reference indicates it is inactive; true negative (TN), where both the reference and the system prediction indicate an event is inactive. Precision (P) and recall (R) are defined as in [@mesaros2016metrics]: $$\label{eq:prec_recall} \text{P}=\frac{\text{TP}}{\text{TP}+\text{FP}}, \qquad \text{R}=\frac{\text{TP}}{\text{TP}+\text{FN}}.$$ In addition, the false positive rate is defined as [@mesaros2016metrics] $$\label{eq:fpr} \text{FPR} = \frac{\text{FP}}{\text{FP} + \text{TN}}.$$ Following [@audioset], we adopt mean average precision (mAP), area under the curve (AUC) and d-prime as evaluation metrics. Average precision (AP) [@audioset] is defined as the area under the recall-precision curve of a specific class. The mean average precision (mAP) is the average value of AP over all classes. As AP is regardless of TN, AUC is used as a complementary metric. AUC is the area under the receiver operating characteristic (ROC) created by plotting the recall against the false positive rate (FPR) at various threshold settings for a specific class. We use mAUC to denote the average value of AUC over all classes. D-prime is a statistic used in signal detection theory that provides separation between signal and noise distributions. D-prime is obtained via a transformation of AUC and has a better dynamic range than AUC when AUC is larger than 0.9. A higher mAP, AUC and d-prime indicates a better performance. D-prime can be calculated by [@audioset]: $$\label{eq:auc} \text{d-prime}=\sqrt{2}F_{x}^{-1}(\text{AUC}),$$ where $F_{x}^{-1}$ is an inverse of the cumulative distribution function defined by $$F_{x}(x) = \int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}e^{\frac{-({x-\mu})^2}{2}}dx \tag{8} \label{eq:8}.$$ [\*[4]{}[c]{}]{} & mAP & AUC & d-prime\ Balanced data & 0.274 & 0.949 & 2.316\ Full data (no bal. training) & 0.268 & 0.950 & 2.331\ Full data (bal. training) & **0.317** & **0.963** & **2.529**\ Baseline system {#section:experiment_baseline} --------------- We build baseline systems with segment based method, IS and ES models without the attention mechanism described in Section \[section:bow\], \[section:IS\_paradigm\] and \[section:ES\_paradigm\], respectively. In the segment based model, a classifier is trained on individual instances, where each instance inherits the tags of a bag. A three-layer fully-connected neural network with 1024 hidden units and ReLU [@nair2010rectified] non-linearity is applied. Dropout [@srivastava2014dropout] with a rate of 0.5 is used to prevent overfitting. The loss function for training is given in (\[eq:bag\_of\_words\_loss\]). In inference, the prediction is obtained by averaging the prediction of individual instances. The IS models have the same structure as the segment based model. Different from the segment based model, the instance-level predictions by the IS models are aggregated to a bag-level prediction by either the SMI assumption in (\[eq:IS\_max\]) or CA in (\[eq:IS\_average\]). The loss function is calculated from (\[eq:IS\_loss\]). The ES method aggregates the instances of a bag to an embedded vector before tagging. The embedding function can be the averaging mapping in (\[eq:ES\_average\]) or max-min vector mapping in (\[eq:ES\_max\]). Then the embedded vector is input to a neural network in the same way as the segment based model. The loss function is calculated from (\[eq:IS\_loss\]). We adopt the Adam optimiser [@kingma2014adam] with a learning rate of 0.001 in training. The mini-batch size is set to 500. The networks are trained for a total number of 50,000 iterations. We average the predictions of 9 models from 10,000 to 50,000 iterations as the final prediction to ensemble and stabilise the result, which can reduce the prediction randomness caused by the model. Table \[table:baseline\] shows the tagging result of segment based method, IS and ES baseline methods. The first row shows that the random guess achieves an mAP of 0.005, an AUC of 0.500 and a d-prime of 0. The segment based model achieves an mAP of 0.293, slightly better than the IS methods with the CA and SMI assumption, with mAP of 0.300 and 0.292, respectively. The sixth to the ninth rows show that both the ES methods with averaging and the max-min instances perform better than the segment based model and IS methods. Averaging the instances performs the best in the ES methods with an mAP of 0.317, an AUC of 0.963 and a d-prime of 2.529. Data balancing -------------- AudioSet is highly imbalanced, as some sound classes such as speech and music are more frequent than others. The upper bars in Fig. \[fig:data\_distribution\] show the number of audio clips per class sorted in descending order (in log scale). The data has a long tail distribution. Music and speech appear in almost 1 million audio clips while some sounds such as gargling and toothbrush only appear in hundreds of audio clips. AudioSet provides a balanced subset consisting of 22,160 audio clips. The lower bars in Fig. \[fig:data\_distribution\] show the number of audio clips per class of the balanced subset. When training a neural network, data is loaded in mini batches. We found that without a balancing strategy, the classes with fewer samples are less likely to be selected in training. Several balancing strategies have been investigated in image classification such as balancing the frequent and infrequent classes [@shen2016relay]. In this paper, we follow the mini-batch balancing strategy [@kong2017audio] for AudioSet tagging, where each mini-batch is balanced to have approximately the same number of samples in training the neural network. {width="\textwidth"} {width="\textwidth"} We first investigate the performance of training on the balanced subset only and training on the full data. We adopt the best baseline model; that is, the ES average instances model in Section \[section:ES\_paradigm\]. Table \[table:balancing\] shows that the model trained with only the balanced subset achieves an mAP of 0.274. The model trained with the full dataset without balancing achieves an mAP of 0.268. The model trained with the balancing strategy achieves an mAP of 0.317. Fig. \[fig:balance\] shows the class-wise AP. The dashed and solid curves show the training and testing AP, respectively. In addition, Fig. \[fig:balance\] shows that the AP is not always positive related to the number of training samples. For example, when using full data for training, “bagpipes” has 1,715 audio clips but achieves an mAP of 0.884, while “outside” has 34,117 audio clips but only achieves an AP of 0.093. We discover that for a majority of sound classes, the improvement of AP is small compared when using the full dataset rather than the balanced subset. For example, there are 60 and 1,715 “bagpipes” audio clips in the balanced subset and the full dataset, respectively. Their APs are 0.873 and 0.884, respectively, indicating that collecting more data for “bagpipes” does not substantially improve its tagging result. To investigate how AP is related to the number of training samples, we calculate their Pearson correlation efficient (PCC)[^8]. PCC is a number between -1 and +1. The PCC of -1, 0, +1 indicate negative correlation, no correlation and positive correlation, respectively. The null hypothesis is that the correlation of the pair of random variables is 0. The p-value indicates the confidence when the null hypothesis is satisfied. If the p-value is lower than the conventional 0.05 the PCC is called statistically significant. Table \[table:PCC\] shows that AP and the number of training samples have a correlation with a PCC of 0.169 and the p-value is $ 9.35 \times 10 ^{-5} $, indicating that AP is only weakly positively related with the number of training samples. [\*[3]{}[c]{}]{} & PCC & p-value\ Training examples & 0.169 & 9.35 $ \times 10^{-5} $\ Labels quality & 0.230 & 7 $\times 10^{-7} $\ Noisy labels ------------ AudioSet contains noisy tags [@audioset]. That is, some tags for training may be incorrect. There are three major reasons leading to the noisy tags in AudioSet shown in [@audioset]: 1) confusing labels, where some sound classes are easily confused with others; 2) human error, where the labeling procedure may be flawed; 3) faint/non-salient sounds, where some sound are faint to recognise in an audio clip. Sound classes with a high label confidence include “christmas music” and “accordion”. Sound classes with a low label confidence include “boiling” and “bicycle”. To investigate how accurate are the ground truth tags, The authors of AudioSet conducted an internal quality assessment task where experts checked 10 random segments for most of the classes. The quality is a value between 0 and 1 measured by the percentage of correctly labelled audio clips verified by human. The quality of labels is shown in Fig. \[fig:balance\] with red plus symbols. Hyphen symbols are plotted for the classes that have not been evaluated. We discover that AP is not always correlated positively with the quality of labels. For example, our model achieves an AP of 0.754 in recognizing “harpsichord”, while the human label quality is 0.4. On the other hand, humans achieve a label quality of 1.0 in “hiccup”, but the AP of our model is 0.076. Table \[table:PCC\] shows that AP and the quality of labels have a weak PCC of 0.230, indicating AP is only weakly correlated with the quality of labels. Attention neural networks ------------------------- We evaluate the decision-level and the feature-level attention neural networks in this subsection. We adopt the architecture in Fig. \[fig:jdc\_att\](c) as our model. The output $ \mathbf{x}' $ of the layer before the attention function is obtained by (\[eq:att\_fc\_func\]). Then the decision-level and feature-level attention neural networks are modelled by (\[eq:decision\_level\_agg\]) and (\[eq:feature\_level\_att\_1\]), respectively. The first row of Table \[table:attention\] shows that the ES method with averaged instances achieves an mAP of 0.317. The second and third rows show that the JDC model in Fig. \[fig:jdc\_att\](a) and the self-attention model in Fig. \[fig:jdc\_att\](b) achieve an mAP of 0.337 and 0.324, respectively. The fourth and fifth row show that the decision-level attention neural network achieves an mAP of 0.337. The decision-level multiple attention neural network further improves this result to an mAP of 0.357. The results of the feature-level attention neural networks are shown in the bottom block of Table \[table:attention\]. The ES methods with average and maximum aggregation achieve an mAP of 0.298 and 0.343, respectively. The feature-level attention neural network achieves an mAP of 0.361, an mAUC of 0.969 and a d-prime of 2.641, outperforming the other models. One explanation is that the feature-level attention neural network can attend to or ignore the features in the feature space which further improves the capacity of the decision-level attention neural network. Fig. \[fig:aggregation\] shows the class-wise performance of the attention neural networks. The feature-level attention neural network outperforms the decision-level attention neural network and the ES method with averaged instances in a majority of sound classes. The results of all 527 sound classes are shown in Fig. \[fig:full\]. [\*[4]{}[c]{}]{} & mAP & AUC & d-prime\ Average instances [@dong2006comparison] & 0.317 & 0.963 & 2.529\ JDC [@kong2017joint] & 0.337 & 0.963 & 2.526\ Self attention [@ilse2018attention] & 0.324 & 0.962 & 2.506\ Decision-level single-attention [@kong2017audio] & 0.337 & 0.968 & 2.612\ Decision-level multi-attention [@yu2018multi] & 0.357 & 0.968 & 2.621\ Feature-level avg. pooling & 0.298 & 0.960 & 2.475\ Feature-level max pooling & 0.343 & 0.966 & 2.589\ Feature-level attention & **0.361** & **0.969** & **2.641**\ Modeling attention function with different functions ---------------------------------------------------- As described in Section \[section:exp\_different\_att\_func\], we model the attention function $ q $ of the feature-level attention neural network via a non-negative function $ \phi_{2} $. The choice of the non-negative function may affect the optimisation and result of the attention neural network. Table \[table:att\_func\] shows that the exponential, sigmoid, softmax and NIN functions achieve a similar mAP of approximately 0.360. Modeling $ \phi(\cdot) $ with ReLU is worse than with other non-linear functions.. [\*[4]{}[c]{}]{} & mAP & AUC & d-prime\ ReLU att & 0.308 & 0.963 & 2.520\ Exp. att & 0.358 & 0.969 & 2.631\ Sigmoid att & **0.361** & **0.969** & **2.641**\ Softmax att & 0.360 & 0.969 & 2.636\ NIN & 0.359 & 0.969 & 2.637\ Attention neural networks with different embedding depth and width {#section:exp_depth} ------------------------------------------------------------------ As shown in (\[eq:att\_fc\_func\]), our attention neural networks map the instances $ \mathbf{x} $ to $ \mathbf{x}' $ through several non-linear embedding layers to increase the representation ability of the instances. We model $ f_{\text{FC}} $ using the feature-level attention neural network with fully-connected layers with different depths. Table \[table:layers\] shows that the mAP increases from 0 layers and reaches a peak of 0.361 at 3 layers. More hidden layers do not increase the mAP. The reason might be that the AudioSet bottleneck features obtained by a VGGish CNN trained on YouTube100M have good separability. Therefore, there is no need to apply very deep neural networks on the AudioSet bottleneck features. On the other hand, the YouTube100M data may have a different distribution from AudioSet. As a result, the embedding mapping $ f_{\text{FC}} $ can be used as domain adaption. [\*[4]{}[c]{}]{} Depth & mAP & AUC & d-prime\ $0$ & 0.328 & 0.963 & 2.522\ $1$ & 0.356 & 0.967 & 2.605\ $2$ & 0.358 & 0.968 & 2.620\ $3$ & **0.361** & **0.969** & **2.641**\ $4$ & 0.356 & 0.969 & 2.637\ $6$ & 0.348 & 0.968 & 2.619\ $8$ & 0.339 & 0.967 & 2.595\ $10$ & 0.331 & 0.966 & 2.579\ Based on the network $ f_{\text{FC}} $ modelled with three layers in the feature-level attention neural network, we investigate the width of $ f_{\text{FC}} $. Table \[table:width\] shows that feature-level attention model with 2048 hidden units in each hidden layer achieves an mAP of 0.369, an mAUC of 0.969 and a d-prime of 2.641 is achieved, outperforming the models with 256, 512, 1024 and 4096 hidden units in each layer. On the other hand, with 4096 hidden units, the model tends to overfit, and does not outperform the model with 2048 hidden units. [\*[4]{}[c]{}]{} Hidden units & mAP & AUC & d-prime\ $256$ & 0.305 & 0.962 & 2.512\ $512$ & 0.339 & 0.967 & 2.599\ $1024$ & 0.361 & 0.969 & **2.641**\ $2048$ & **0.369** & **0.969** & 2.640\ $4096$ & 0.369 & 0.968 & 2.619\ Conclusion ========== We have presented a decision-level and a feature-level attention neural network for AudioSet tagging. We developed the connection between multiple instance learning and attention neural networks. We investigated the class-wise performance of all the 527 sound classes in AudioSet and discovered that the AudioSet tagging performance on AudioSet embedding features is only weakly correlated with the number of training examples and quality of labels, with Pearson correlation coefficients of 0.169 and 0.230, respectively. In addition, we investigated modelling the attention neural networks with different attention functions, depths and widths. Our proposed feature-level attention neural network achieves a state-of-the-art mean average precision (mAP) of 0.369 compared to the best MIL method of 0.317 and the decision-level attention neural network of 0.337. In the future, we will explore weakly labelled sound event detection on AudioSet with attention neural networks. Acknowledgment {#acknowledgment .unnumbered} ============== The authors would like to thank all anonymous reviewers for their suggestions to improve this paper. {width="\textwidth"} [Qiuqiang Kong]{} (S’17) received the B.Sc. and M.E. degrees from South China University of Technology, Guangzhou, China, in 2012 and 2015, respectively. He is currently working toward the Ph.D. degree from the University of Surrey, Guildford, U.K on sound event detection. His research topic includes sound understanding, audio signal processing and machine learning. He was nominated as the postgraduate research student of the year in University of Surrey, 2019. plus -1fil [Changsong Yu]{} received the B.E. degree from Anhalt University of Applied Sciences and M.S. degree University of Stuttgart, Germany, in 2015 and 2018, respectively. He is currently working as simultaneous localization and mapping (SLAM) algorithm engineer in HoloMatic, Beijing, China. His research interest includes deep learning and SLAM. plus -1fil [Yong Xu]{} (M’17) received the Ph.D. degree from the University of Science and Technology of China (USTC), Hefei, China, in 2015, on the topic of DNN-based speech enhancement and recognition. Currently, he is a senior research scientist in Tencent AI lab, Bellevue, USA. He once worked at the University of Surrey, U.K. as a Research Fellow from 2016 to 2018 working on sound event detection. He visited Prof. Chin-Hui Lee’s lab in Georgia Institute of Technology, USA from Sept. 2014 to May 2015. He once also worked in IFLYTEK company from 2015 to 2016 to develop far-field ASR technologies. His research interests include deep learning, speech enhancement and recognition, sound event detection, etc. He received 2018 IEEE SPS best paper award. plus -1fil [Turab Iqbal]{} received the B.Eng. degree in Electronic Engineering from the University of Surrey, U.K., in 2017. Currently, he is working towards a Ph.D. degree from the Centre for Vision, Speech and Signal Processing (CVSSP) in the University of Surrey. His research interests are mainly in machine learning using weakly labeled data for audio classification and localization. plus -1fil [Wenwu Wang]{} (M’02-SM’11) was born in Anhui, China. He received the B.Sc. degree in 1997, the M.E. degree in 2000, and the Ph.D. degree in 2002, all from Harbin Engineering University, China. He then worked in King’s College London, Cardiff University, Tao Group Ltd. (now Antix Labs Ltd.), and Creative Labs, before joining University of Surrey, UK, in May 2007, where he is currently a professor in signal processing and machine learning, and a Co-Director of the Machine Audition Lab within the Centre for Vision Speech and Signal Processing. He has been a Guest Professor at Qingdao University of Science and Technology, China, since 2018. His current research interests include blind signal processing, sparse signal processing, audio-visual signal processing, machine learning and perception, machine audition (listening), and statistical anomaly detection. He has (co)-authored over 200 publications in these areas. He served as an Associate Editor for IEEE TRANSACTIONS ON SIGNAL PROCESSING from 2014 to 2018. He is also Publication Co-Chair for ICASSP 2019, Brighton, UK. plus -1fil [Mark D. Plumbley]{} (S’88-M’90-SM’12-F’15) received the B.A.(Hons.) degree in electrical sciences and the Ph.D. degree in neural networks from University of Cambridge, Cambridge, U.K., in 1984 and 1991, respectively. Following his PhD, he became a Lecturer at King’s College London, before moving to Queen Mary University of London in 2002. He subsequently became Professor and Director of the Centre for Digital Music, before joining the University of Surrey in 2015 as Professor of Signal Processing. He is known for his work on analysis and processing of audio and music, using a wide range of signal processing techniques, including matrix factorization, sparse representations, and deep learning. He is a co-editor of the recent book on Computational Analysis of Sound Scenes and Events, and Co-Chair of the recent DCASE 2018 Workshop on Detection and Classifications of Acoustic Scenes and Events. He is a Member of the IEEE Signal Processing Society Technical Committee on Signal Processing Theory and Methods, and a Fellow of the IET and IEEE. [^1]: Manuscript received March 13, 2019; revised June 18, 2019; accepted July 11, 2019. Date of publication July 26, 2019; date of current version August 21, 2019. This work was supported in part by the EPSRC Grant EP/N014111/1 “Making Sense of Sounds”, in part by the Research Scholarship from the China Scholarship Council 201406150082, and in part by a studentship (Reference: 1976218) from the EPSRC Doctoral Training Partnership under Grant EP/N509772/1. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Alexey Ozerov. (*Qiuqiang Kong is first author.) (Corresponding author: Yong Xu.)* [^2]: Q. Kong, T. Iqbal, and M. D. Plumbley are with the Centre for Vision, Speech and Signal Processing, University of Surrey, Guildford GU2 7XH, U.K. (e-mail: q.kong@surrey.ac.uk; t.iqbal@surrey.ac.uk; m.plumbley@surrey.ac.uk). [^3]: Y. Xu is with the Tencent AI Lab, Bellevue, WA 98004 USA (e-mail: lucayongxu@tencent.com). [^4]: W. Wang is with the Centre for Vision, Speech and Signal Processing, University of Surrey, Guildford GU2 7XH, U.K., and also with Qingdao University of Science and Technology, Qingdao 266071, China (e-mail: w.wang@surrey.ac.uk). [^5]: Digital Object Identifier 10.1109/TASLP.2019.2930913 [^6]: https://www.youtube.com/embed/Wxa36SSZx8o?start=70&end=80 [^7]: https://github.com/qiuqiangkong/audiosetclassification [^8]: Given a pair of random variables $ X $ and $ Y $, the PCC is calculated as $ \frac{\text{cov(X, Y)}}{\sigma_{X}\sigma_{Y}} $, where $ \text{cov}(\cdot, \cdot) $ is the covariance of two variables and $ \sigma $ is the standard deviation of the random variables.

--- abstract: | We demonstrate how a chiral soft pion theorem (SPT) shields the scalar meson ground state isoscalar $\sigma(600-700)$ and isospinor $\kappa(800-900)$ from detection in $a_{1}\rightarrow\pi(\pi\pi)_{\mathrm{swave}}$, $\gamma\gamma\rightarrow2\pi^{0}$, $\pi^{-}p\rightarrow\pi^{-}\pi^{+}n$ and $K^{-}p\rightarrow K^{-}\pi^{+}n$ processes. While pseudoscalar meson PVV transitions are known to be determined by (only) quark loop diagrams, the above SPT also constrains scalar meson SVV transitions to be governed (only) by meson loop diagrams. We apply this latter SVV theorem to $a_{0}\rightarrow\gamma\gamma$ and $f_{0}\rightarrow\gamma\gamma$ decays. pacs: 11.30.Rd, 12.39.-x, 12.39.Ki author: - | L. Babukhadia$^{a}$, Ya. A. Berdnikov$^{b}$, A. N. Ivanov$^{b,c}$\ and M. D. Scadron$^{a}$\ $^{a}$ Physics Dept., Univ. of Arizona,\ Tucson, AZ 85721 USA\ $^{b}$ Nuclear Physics Dept., State Technical Univ.,\ 195251 St. Petersburg, Russian Federation\ $^{c}$ present address, Inst. f[ü]{}r Kernphysik, Technische Univ.,\ A–$1040$, Wien, Austria title: Chiral Shielding --- Introduction ============ The recent plethora of scalar meson papers appearing in the Los Alamos archives [@plethora] stresses once again the importance but difficulty in observing the ground state $I=0$ and $I=1/2$ scalar mesons $\sigma(600-700)$ and $\kappa(800-900)$. Although these resonances were first listed in many of the $1960$–$70$ particle data group (PDG) tables, they were later removed in the mid $1970$’s in favor of the higher mass $\epsilon(1300)$ and $\kappa(1400)$. Chiral symmetry shields the $\sigma(600-700)$ and $\kappa(800-900)$ for many different reasons which we shall discuss shortly. Given the new CLEO measurement [@cleo] of the $a_{1}(1230)\rightarrow\sigma\pi$ branching ratio based on $\tau\rightarrow\nu3\pi$ decay of $\mathrm{BR}(a_{1}\rightarrow\sigma\pi)=(16\pm4)\%$, the average PDG value of [@pdg] $\Gamma(a_{1})\sim425$ MeV then suggests a substantial partial width of size $$\label{eq:cleo} \Gamma_{\mathrm{CLEO}} (a_{1}\rightarrow\sigma\pi) \sim (0.16)(425 \ \mathrm{MeV}) = 68 \pm 33 \ \mathrm{MeV}\,.$$ This was anticipated a decade ago by Weinberg [@weinb], using mended chiral symmetry (MCS) to predict $$\label{eq:weinb} \Gamma_{\mathrm{MCS}} (a_{1}\rightarrow\sigma\pi) = 2^{-3/2}\Gamma_{\rho} \approx 53 \ \mathrm{MeV} \, .$$ Moreover, assuming chiral symmetry, the needed coupling is related to $g_{a_{1}\sigma\pi}=g_{\rho\pi\pi}\approx6$, the latter found from $\Gamma_{\rho}\approx151$ MeV. Invoking the PDG $\sigma$ mass of $\sim550$ MeV [@pdg; @tornquist] (giving $q_{\mathrm{CM}}\approx480$ MeV), one anticipates the width $$\label{eq:ours} \Gamma (a_{1}\rightarrow\sigma\pi) = \frac{1}{3} \left( g^{2}_{a_{1}\sigma\pi}/4\pi \right) \frac{q^{3}_{\mathrm{CM}}}{m^{2}_{a_{1}}} \approx 70 \ \mathrm{MeV} \, .$$ Considering the compatible (nonvanishing) $\Gamma_{a_{1}\rightarrow\sigma\pi}$ widths in Eqs. (\[eq:cleo\]–\[eq:ours\]) above, one might question (as Weinberg did in reference [@weinb]) why the PDG listed the much smaller value BR$(a_{1}\rightarrow\pi(\pi\pi)_{\mathrm{swave}})<0.7\%$ in the $1980$s or the essentially vanishing width $$\label{eq:pmone} \Gamma(a_{1}\rightarrow\pi(\pi\pi)_{\mathrm{swave}}) = 1 \pm 1 \ \mathrm{MeV}$$ in the $1990$s. Vanishing Soft Pion Theorem (SPT) ================================= To resolve this apparent contradiction, we note that there are in fact *two* Feynman graphs to consider for $a_{1}\rightarrow\pi(\pi\pi)_{\mathrm{swave}}$ decay, the “box” quark graph of Fig. $1$a and the quark “triangle” graph of Fig. $1$b (for nonstrange $u$ and $d$ quarks). In the soft pion limit for one soft pion in the $(\pi\pi)_{\mathrm{swave}}$ doublet (but not the pion outside the $(\pi\pi)_{\mathrm{swave}}$ doublet), there is a vanishing SPT [@ivanov; @ivanov1], cancelling the box graph in Fig. 1a against the triangle graph Fig. 1b in the chiral soft pion limit. Such a cancellation stems from the Dirac matrix *identity*[^1] $$\label{eq:identity} \frac{1}{\gamma\cdot p - m} 2m\gamma_{5} \frac{1}{\gamma\cdot p - m} \equiv -\gamma_{5} \frac{1}{\gamma\cdot p - m} - \frac{1}{\gamma\cdot p - m} \gamma_{5} \, .$$ We apply (\[eq:identity\]) together with the pseudoscalar pion quark (chiral) Goldberger–Treiman coupling $g_{\pi qq} = m / f_{\pi}$ for $f_{\pi} \approx 93$ MeV. This SPT for $p_{\pi}\rightarrow 0$ applied to graphs of Figs. 1–4 results in a\) $a_{1}\rightarrow\pi(\pi\pi)_{\mathrm{swave}}$: The box graph of Fig. 1a and Eq. (\[eq:identity\]) gives the amplitude as $p_{\pi}\rightarrow 0$, $$\label{eq:two} M^{box}_{a_{1}\rightarrow 3\pi} \rightarrow - \frac{1}{f_{\pi}} M(a_{1}\rightarrow \sigma\pi) \, .$$ But the additional $\sigma$ pole quark triangle graph of Fig. 1b is $$\label{eq:three} M^{tri}_{a_{1}\rightarrow 3\pi} = \frac{1}{f_{\pi}} M(a_{1}\rightarrow \sigma\pi) \, ,$$ because $2g_{\sigma\pi\pi}=(m^{2}_{\sigma}-m^{2}_{\pi})/f_{\pi}$ in the linear $\sigma$ model (L$\sigma$M). Thus the sum of (\[eq:two\]) and (\[eq:three\]) vanishes in the soft pion limit [@ivanov; @ivanov1] $$\label{eq:four} M_{a_{1}\rightarrow 3\pi}|_{total} = M^{box}_{a_{1}\rightarrow 3\pi} + M^{tri}_{a_{1}\rightarrow 3\pi} \rightarrow 0 \, ,$$ compatible with data [@pdg]: $\Gamma(a_{1}\rightarrow\pi(\pi\pi)_{\mathrm{swave}})=1\pm1$ MeV. b\) $\gamma\gamma\rightarrow2\pi^{0}|_{s=m^{2}_{\sigma}}$: Again using pseudoscalar pion-quark couplings, it was predicted [@kaloshin] five years before data appeared that this $\gamma\gamma\rightarrow2\pi^{0}$ cross section should fall to about $10$ nbarns in the $700$ MeV region. Equivalently, using the SPT theorem stemming from Eq. (\[eq:identity\]), we predict the amplitude due to the quark box plus quark triangle graphs of Fig. 2 $$\label{eq:five} \langle \pi^{0}\pi^{0}|\gamma\gamma \rangle \rightarrow \left[ - \frac{i}{f_{\pi}}\langle\sigma|\gamma\gamma\rangle + \frac{i}{f_{\pi}}\langle\sigma|\gamma\gamma\rangle \right] \rightarrow 0 \, ,$$ as $s\rightarrow m^2_{\sigma}(700)$ [@ivanov1]. This picture was supported by recent Crystal Ball data [@crystalball]. c\) $\pi^{-}p\rightarrow\pi^{-}\pi^{+}n$: The SPT stemming from Eq. (\[eq:identity\]) also suggests that the sum of the two $\pi^{+}$ peripheral–dominated $\pi^{-}p\rightarrow\pi^{-}\pi^{+}n$ amplitudes of Figs. 3 vanishes: $$\label{eq:six} M_{\pi^{-}p\rightarrow\pi^{-}\pi^{+}n}|_{per} \propto \left[ M^{box}_{\pi\pi} + M^{tri}_{\pi\pi} \right] \rightarrow 0 \, .$$ This “chirally–eaten” $\sigma(600-700)$ in Figs. 1b, 2b, 3b indeed did not appear in PDG tables prior to 1996, just as the SPT mandates. In fact the $\sigma(600-700)$ does not appear in recent Crystal Ball $\pi^{-}p\rightarrow\pi^{0}\pi^{0}n$ studies either [@nefkens]. d\) $K^{-}p\rightarrow K^{-}\pi^{+}n$: Finally the SPT due to Eq. (\[eq:identity\]) requires the sum of the two $\pi^{+}$ peripheral–dominated $K^{-}p\rightarrow K^{-}\pi^{+}n$ amplitudes of Figs. 4 to vanish, $$\label{eq:seven} M_{K^{-}p\rightarrow K^{-}\pi^{+}n}|_{per} \propto \left[ M^{box}_{K\pi} + M^{tri}_{K\pi} \right] \rightarrow 0 \, ,$$ shielding this ground state $\kappa^{0}(800-900)$ scalar in Fig. 4b. Instead the $K^{\ast}(1430)$ (excited state) scalar resonance clearly appears in LASS data [@lass]; this $K^{\ast}(1430)$ not being eaten means it also is not a true ground state obeying the SPT. An analogous disappearance of the ground state $\kappa(800-900)$ scalar occurs for the peripheral-dominated processes $K^{-}p\rightarrow \pi^{-}\pi^{+}\Lambda, \bar{K}K\Lambda$. None of the above four SPT processes depicted in Figs. 1–4 have been used by the experimentalists to observe such scalar mesons. Instead they study processes avoiding these four SPTs, e.g. J/$\psi\rightarrow \omega\pi\pi$ to isolate the $\sigma(500)$ resonance ‘bump’. In effect, the above s–wave SPTs (with quark boxes cancelling quark triangle graphs in the soft pion limit) chirally ‘eat’ the ground state $\sigma(600-700)$ and $\kappa(800-900)$ scalar mesons, justifying in part[^2] why these scalar mesons have been so difficult to isolate and identify in the past. With hindsight, the L$\sigma$M dynamically generates ground state $\sigma(650)$ and $\kappa(850)$ scalars via (one-loop-order) tadpole graphs [@nontriv]. Even though these tadpoles can be suppressed by working in the infinite momentum frame  [@imf], SU(6) mass formulae (requiring squared masses) then kinematically favor [@scadron] the (ground state) $\sigma(650)$ and $\kappa(820)$. This is another way (besides e.g. J/$\psi\rightarrow \omega\pi\pi$) to circumvent the four SPTs discussed in this section. Quark Loops versus Meson Loops ============================== In most effective chiral field theories (such as the L$\sigma$M), one usually computes consistently either quark loops alone or meson loops alone for a given process. Sometimes one must add together quark and meson loops [@nontriv]. Chiral symmetry and the SPT discussed in Sec. II actually help to put order in this morass of quarks and meson loops. Specifically for PVV transitions, the anomaly [@abj] or simply the vanishing of e.g. a meson $\pi\pi\pi$ vertex, etc. leads directly to a ‘quark loops alone’ theory [@pvv], such as for $\pi^{0}\rightarrow 2\gamma$. However for SVV transitions, it turns out that *only* meson loop graphs contribute. This SVV ‘meson loops alone’ theorem also is a direct consequence of the soft pion theorem (SPT) proved in refs. [@ivanov; @ivanov1] and reviewed in Sec. II above. Specifically we study $\gamma\gamma\rightarrow\pi^{0}\pi^{0}$ with one of the pions soft. Again the quark box plus quark triangle graphs of Figs. 2 add up to zero in the soft pion limit. Turning Fig. 2b around, if $\sigma$ (as a $2\pi$ resonance) decays to $2\gamma$, this SPT eats up the needed quark triangle due to the quark box. This leaves only the meson triangle $\sigma\rightarrow K^{+}K^{-} \rightarrow 2 \gamma$ dominating SVV decay $\sigma\rightarrow\gamma\gamma$. A more practical example of this theorem is for $a_{0}(983)\rightarrow 2\gamma$ decay. First we consider the inverse process $\gamma\gamma\rightarrow\eta\pi$, with the $\eta\pi$ final state forming an $a_{0}(983)$ resonance $\gamma\gamma\rightarrow a_{0} \rightarrow \eta\pi$. So we should begin by first considering the quark box graph for $\gamma\gamma\rightarrow a_{0}$ followed by $a_{0}\rightarrow\eta\pi$. Again these quark box plus triangle graphs vanish in the soft pion limit by the SPT of Sec. II. All that remains are the meson loop graphs for $a_{0}\rightarrow\gamma\gamma$ decay. Here $a_{0}\rightarrow K^{+}K^{-} \rightarrow 2 \gamma$ and the charged kaon loop contributes to the $a_{0}\gamma\gamma$ covariant amplitude $$\label{eq:covampl} \langle 2\gamma | a_{0} \rangle = \mathrm{M} \varepsilon_{\mu}(k') \varepsilon_{\nu}(k) (g^{\mu\nu}k'\cdot k - k'^{\mu}k^{\nu}) \,$$ where, according to ref. [@deakin], the effective amplitude $\mathrm{M}$ is given by $$\label{eq:M} | \mathrm{M}_{\mathrm{K-loop}} | = \frac{2g'\alpha}{\pi m^{2}_{a_{0}}} \left[ - \frac{1}{2} + \xi I(\xi) \right] \, ,$$ with $\xi=m^{2}_{K^{+}}/m^{2}_{a_{0}}=0.2520>1/4$. Then the loop integral becomes $$\label{eq:int_xi} I(\xi) = \int_{0}^{1} \! dy \, y \int_{0}^{1} \! dx \left[ \xi - xy(1-y) \right]^{-1} = 2 \left[ \arcsin\sqrt{1/4\xi} \right]^{2} \approx 4.39 \, .$$ Also the L$\sigma$M $a_{0}KK$ coupling ($g'$) is [@deakin; @su3] $$\label{eq:gprime} g' = (m^{2}_{a_{0}} - m^{2}_{K})/2f_{K} \approx 3.18 \ \mathrm{GeV} \, ,$$ so that the $a_{0}\gamma\gamma$ amplitude in Eq. (\[eq:M\]) is approximately $$\label{eq:Mvalue} | \mathrm{M}_{\mathrm{K-loop}} | \approx 9.27 \times 10^{-3} \ \mathrm{GeV}^{-1} \, .$$ This results in the decay width $$\label{eq:a0width} \Gamma ( a_{0} \rightarrow 2 \gamma ) = m^{3}_{a_{0}} | \mathrm{M}_{K} |^{2} / 64 \pi \approx 0.406 \ \mathrm{keV} \, .$$ The resonance $\kappa(900)$ contributes [@deakin] $10\%$ of Eq. (\[eq:Mvalue\]), reducing (\[eq:a0width\]) to $$\label{eq:a0width1} \Gamma ( a_{0} \rightarrow 2 \gamma ) \approx 0.406 \ \mathrm{keV} (0.90)^{2} \approx 0.33 \ \mathrm{keV} \, .$$ Assuming the $a_{0}$ width is ($100\%$) dominated by $a_{0}\rightarrow\eta\pi$, the PDG tables suggest $$\label{eq:a0widthPDG} \Gamma ( a_{0} \rightarrow 2 \gamma ) = \left( 0.24^{+0.08}_{-0.07} \right) \ \mathrm{keV} \, .$$ Another measured SVV decay is $f_{0}(980)\rightarrow\gamma\gamma$ with [@pdg] $$\label{eq:f0widthPDG} \Gamma ( f_{0} \rightarrow 2 \gamma ) = 0.56 \pm 0.11 \ \mathrm{keV} \, .$$ Here $\sigma-f_{0}$ mixing enters the amplitude analysis with [@su3; @scadron84] $$\label{eq:mixing} |f_{0}\rangle = \sin \phi_{s} |\mathrm{NS}\rangle + \cos \phi_{s} |\mathrm{S}\rangle \, ,$$ for $f_{0}(980)$ being mostly strange, with $\phi_{s}\approx 20^{\circ}$. The nonstrange (NS) and strange (S) quark basis states are respectively $|\mathrm{NS}\rangle = |\bar{u}u+\bar{d}d\rangle/\sqrt{2}$ and $|S\rangle=|\bar{s}s\rangle$ with singlet-octet angle $\theta_{s} = \phi_{s} - \arctan \sqrt{2}$. The angle $\phi_{s}$ can be obtained from Eq. (\[eq:mixing\]) using $\langle\sigma | f_{0} \rangle = 0 $ or $m^{2}_{\sigma_{s}} = m^{2}_{\sigma} \sin^{2} \phi_{s} + m^{2}_{f_{0}} \cos^{2} \phi_{s} $, leading to [@su3; @scadron84] $$\label{eq:mixing_angle} \phi_{s} = \arcsin \left[ \frac{m^{2}_{f_{0}} - m^{2}_{\sigma_{s}}} {m^{2}_{f_{0}} - m^{2}_{\sigma}} \right]^{1/2}\approx20^{\circ}$$ for $m_{\sigma}\approx610$ MeV and $m_{\sigma_{s}}\approx 2m_{s} \approx 940$ MeV, with constituent quark masses $m_{s} = ( m_{s} / \hat{m} ) \hat{m} \approx 470$ MeV, and $\hat{m}\approx 325$ MeV, $m_{s}/\hat{m}\approx 1.45$. Since $f_{0}(980)$ is mostly $\bar{s}s$ with $m_{f_{0}}\approx m_{a_{0}}$ [@su3], we simply scale up the width $\Gamma_{a_{0} \rightarrow \gamma\gamma} \approx 0.33$ keV in Eq. (\[eq:a0width1\]) by $2(\cos20^{\circ})^{2}$ from Eq. (\[eq:mixing\]) (the $2$ due to [@su3; @scadron84] $g_{SKK}=1/\sqrt{2}$ whereas $g_{NSKK}=1/2$): $$\label{eq:f0width} \Gamma (f_{0}\rightarrow\gamma\gamma) \approx 2 (\cos 20^{\circ})^{2} (0.33\ \mathrm{keV})\approx 0.58 \ \mathrm{keV} \, ,$$ again for a $f_{0} \rightarrow K^{+}K^{-} \rightarrow 2\gamma$ meson loop. We observe that the predictions (\[eq:a0width1\]) and (\[eq:f0width\]) are in close agreement with the $a_{0},f_{0}\rightarrow 2\gamma$ measured decay rates in (\[eq:a0widthPDG\]) and (\[eq:f0widthPDG\]), respectively. Summary ======= In Sec. I we gave one experimental and two theoretical reasons supporting the somewhat broad width $\Gamma (a_{1}\rightarrow\sigma\pi) \sim 65$ MeV. The latter appears to contradict the complementary PDG result $\Gamma (a_{1}\rightarrow\pi(\pi\pi)_{\mathrm{swave}}) = 1 \pm 1$ MeV. But in Sec. II we resolve this apparent contradiction, finding that *both* quark box and quark triangle graphs contribute to the rate $\Gamma (a_{1}\rightarrow\pi(\pi\pi)_{\mathrm{swave}})$, but the quark box–triangle sum of these amplitudes *vanishes* in the soft–pion limit. This SPT is also valid for $\sigma(\gamma\gamma\rightarrow\pi^{0}\pi^{0})$, and peripheral decay rates $\Gamma_{\mathrm{per}} (\pi^{-} p \rightarrow \pi^{-}\pi^{+} n)$, $\Gamma_{\mathrm{per}} (K^{-} p \rightarrow K^{-}\pi^{+} n)$. With hindsight, our quark loop chiral shielding SPTs in Sec. II parallel the L$\sigma$M “miraculous cancellation” eating up the $\sigma$ pole in $\pi-\pi$ scattering ref. [@fubini], reducing the low energy amplitude to Weinberg’s well-known CA–PCAC result [@weinb66]. Finally in Sec. III we turn this SPT around. 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[**A14**]{}, 4331 (1999), hep-ph/9905501. A. S. Deakin, V. Elias, D. G. C. McKeon, M. D. Scadron and A. Bramon, Mod. Phys. Lett. [**A9**]{}, 2381 (1994). R. Delbourgo and M. D. Scadron, Int. Journ. Mod. Phys. [**A13**]{}, 657 (1998), hep-ph/9807504; R. Delbourgo, D. Liu and M. D. Scadron, Phys. Lett. [**B446**]{}, 332 (1999). M. D. Scadron, Phys. Rev. [**D29**]{}, 1375 (1984). V. deAlfaro, S. Fubini, G. Furlan and C. Rossetti, “Currents in Hadron Physics”, (North Holland $1973$). See pp $324$–$327$. Also see M. D. Scadron, Eur. Phys. J. [**C6**]{}, $141$ ($1999$), hep-ph/$9710317$. S. Weinberg, Phys. Rev. Lett. [**17**]{}, $616$ ($1966$). See e.g. J. Lucio and M. Napsuciale, hep-ph/9903234. **Figure Captions.** Fig. 1: Quark $u$, $d$ box (a) and triangle (b) graphs contributing to $a_{1}\rightarrow\pi(\pi\pi)_{\mathrm{swave}}$. Fig. 2: Quark $u$, $d$ box (a) and triangle (b) graphs contributing to $\gamma\gamma\rightarrow\pi^{0}\pi^{0}$. Fig. 3: Peripheral–dominated quark $u$, $d$ box (a) and triangle (b) graphs contributing to $\pi^{-} p \rightarrow \pi^{-}\pi^{+} n$. Fig. 4: Peripheral–dominated quark $u$, $d$ box (a) and triangle (b) graphs contributing to $K^{-} p \rightarrow K^{-} \pi^{+} n$. [^1]: Equation (\[eq:identity\]) reduces to $2m\gamma_{5} = 2m\gamma_{5}$ when multiplying both sides of (\[eq:identity\]) on the lhs and rhs by $(\gamma \cdot p - m)$. [^2]: Two other reasons for suppressing these scalars are: (1) they are low mass and broad, sometimes at the edge of the phase space and (2) they are usually swamped by the nearby vectors $\rho(770)$ or $\omega(783)$ and $K^{\ast}(895)$, respectively.

--- abstract: 'Combining cognitive radio technology with user cooperation could be advantageous to both primary and secondary transmissions. In this paper, we propose a first relaying scheme for cognitive radio networks (called “Adaptive relaying scheme 1"), where one relay node can assist the primary or the secondary transmission with the objective of improving the outage probability of the secondary transmission with respect to a primary outage probability threshold. Upper bound expressions of the secondary outage probability using the proposed scheme are derived over Rayleigh fading channels. Numerical and simulation results show that the secondary outage probability using the proposed scheme is lower than that of other relaying schemes. Then, we extend the proposed scheme to the case where the relay node has the ability to decode both the primary and secondary signals and also can assist simultaneously both transmissions. Simulations show the performance improvement that can be obtained due to this extension in terms of secondary outage probability.' author: - - - bibliography: - 'IEEEabrv.bib' - 'tau.bib' title: Opportunistic Adaptive Relaying in Cognitive Radio Networks --- Introduction ============ In order to overcome the problems related to the rigid allocation of spectrum bands to few licensed operators and the under-utilization of these bands, Cognitive Radio (CR) technology has evolved in wireless communications for allowing unlicensed secondary users (SUs) to access licensed parts of the spectrum without harmfully interfering with the transmissions of the licensed primary users (PUs) taking place in the same spectrum band [@Mitola]-[@Haykin]. In underlay spectrum sharing mode, SUs are allowed to transmit simultaneously with PUs if they tune their transmission parameters (such as transmit power) to be harmless to primary transmissions. Meanwhile, user cooperation has been recognized as an interesting technique that allows to achieve increased diversity order when one or several relay nodes assist the transmission [@Laneman1]-[@Jaafar_WCM]. Consequently, combining user cooperation and cognitive radio has recently attracted attention to improve both the spectrum utilization and the transmission performances. User cooperation has been applied for the primary transmission when a cognitive secondary transmitter acts as a relay [@Yang_2009]. By doing so, the primary outage probability is improved, while SUs have more opportunities to access the licensed spectrum bands and hence secondary transmission performances can be also improved. In [@Zou], an adaptive user cooperation scheme with best-relay selection is proposed in multiple-relay Cognitive Radio Networks (CRNs) to improve the secondary outage performance while satisfying a primary outage probability threshold. By letting the “best" CR relay assist the secondary transmission, the secondary outage probability can be considerably reduced. In [@Yener], secondary transmissions are assisted by a group of CR relay nodes located at different positions. The outage probability was investigated when all the relays forward simultaneously their received signals. The system achieves full diversity when the number of cooperating relay nodes is adequately selected. In [@Jaafar_Globecom], we proposed a new relaying scheme for CRNs where a CR relay node is able to assist simultaneously the primary and secondary transmissions. It has been shown that for certain relay’s position, assisting simultaneously both transmissions provides better outage performance than assisting only the primary or the secondary transmission. However, the proposed scheme is greedy on the relay’s transmit power. Even though, the previous works have investigated the utilization of user cooperation in CRNs for assisting the primary transmissions, the secondary transmissions or both, no work has investigated adaptive relaying schemes where the relay node decides independently when and which communication to assist. Consequently, we propose in this paper a novel opportunistic adaptive relaying scheme, where the CR relay node is able to decide when to cooperate or not, and in case of cooperation, whom to cooperate with (primary transmission or secondary transmission) depending on the channel states. We propose to extend the adaptive relaying scheme to the case where the relay node can also cooperate simultaneously with both transmissions. The paper is organized as follows. Section II presents the system model. In section III, we describe the two adaptive relaying schemes and we provide analytically the secondary outage probability for the first adaptive relaying scheme. Section IV shows and discusses the numerical and simulation results and a conclusion closes the paper in section V. System Model ============ We assume a CRN where one primary transmitter (PT) transmits data to a primary destination (PD) and a secondary transmitter (ST) communicates with a secondary destination (SD) over the same frequency band (Fig. \[Fig:Network\]). We assume a decode-and-forward secondary CR relay node (R) that can assist the primary or the secondary transmission in order to increase the secondary access to the licensed spectrum bands with respect to a certain primary outage probability threshold. In the extension, we assume that the relay is able to assist both transmissions. ![The Cognitive Radio Network[]{data-label="Fig:Network"}](Fig_Network1.eps){width="220pt"} We assume that PT and ST transmit their signals $x_p$ and $x_s$ (where $E\{|x_p|^2\}=E\{|x_s|^2\}=1$) with powers $P_{p}$ and $P_{s}$ respectively in order to achieve data rates $R_p$ and $R_s$ respectively. We assume also that R uses transmit power $P_r \leq P_r^{max}$, where $P_r^{max}$ is the maximal relay transmit power. We assume that the channels are submitted to Rayleigh fading and path loss attenuation and are stationary during a time-slot (time slot $=1^{st}$ + $2^{nd}$ sub-slots). Following Fig. \[Fig:Network\], the received signals during the first sub-slot are expressed by: $$\label{eq:received_PD_direct} y_{a}(1)=\sqrt{P_{p}}h_{pa}x_p+\sqrt{P_{s}}h_{sa}x_s+n_{a}, %\label{eq:received_PT_R} %y_{r}(1)&=&\sqrt{P_{p}}h_{pr}x_p+\sqrt{P_{s}}h_{sr}x_s+n_{r},\\ %\label{eq:received_SD_direct} %y_{s}(1)&=&\sqrt{P_{s}}h_{ss}x_s+\sqrt{P_{p}}h_{ps}x_p+n_{s},$$ where $a=p,s$ or $r$ ($p$, $s$ and $r$ denote primary, secondary and relay node resp.), $h_{ba}$ ($b=p$ or $s$) is the channel gain between nodes $b$ and $a$ having variance $\sigma_{ba}^{2}=d_{ba}^{-\beta}$ , $d_{ba}$ is the distance between $b$ and $a$, $\beta$ is the path-loss exponent, and where $n_{a}$ is the additive white gaussian noise with zero mean and variance $N_0$ received at $a$. We assume a fixed $P_p$ and that $P_s$ is calculated with respect to the primary outage probability threshold denoted $\varepsilon$. $P_s$ is given similarly to [@Zou] by: $$\label{eq:condition_gamma_ST} P_s=\frac{2 P_p \sigma_{pp}^{2}}{\Lambda_p \sigma_{sp}^{2}} \rho^{+},$$ where $\rho^{+}=max(0,\rho)$, $\rho=\frac{e^{-\frac{\Lambda_p}{2 {\gamma}_{p}\sigma_{pp}^{2}}}}{1-\varepsilon}-1$, $\Lambda_p=2^{2 R_p}-1$ and $\gamma_p = P_p/N_0$. ST calculates $P_s$ assuming that PT repeats the same signal over the two sub-slots with the same transmit power $P_p$. Adaptive Relaying Schemes ========================= Description of adaptive relaying scheme 1 ----------------------------------------- This novel scheme aims to exploit efficiently the relay position, the acquired information and the propagation environment conditions. We define by $a_0=\frac{\gamma_s |h_{ss}|^2}{\gamma_p |h_{ps}|^2+1}$, $a_p=\frac{\gamma_s |h_{ss}|^2}{\gamma_r^{(p)} |h_{rs}|^2+1}$, $a_s=\frac{\gamma_r^{(s)} |h_{rs}|^2}{\gamma_p |h_{ps}|^2+1}$ where $\gamma_a=P_a/N_0$ ($a=p$ or $s$) and $\gamma_{r}^{(i)}$ is the transmit power of the relay node when assisting transmission $i$. We also define $E_i=\left\{ a_i=max\left\{ a_p, a_s, a_0 \right\} \right\}$ the opportunism condition ($i=p, s$ or $0$). At the first sub-slot, R attempts to decode $x_p$ or $x_s$ and then, a relaying procedure is chosen depending on the value of $D$ defined by: $$\begin{aligned} \label{eq:condition_CR_helps_PU_SU} \nonumber &\mathrm{If }\;A_p \cap \left\{ \left\{\bar{A_s} \cap (a_p > a_0)\right\} \cup \left\{A_s \cap E_p \right\} \right\} \mathrm{, then} &D = 1,\\ \nonumber &\mathrm{If }\;A_s \cap \left\{ \left\{\bar{A_p} \cap (a_s > a_0)\right\} \cup \left\{A_p \cap E_s \right\} \right\} \mathrm{, then} &D = 2,\\ %\nonumber &\mathrm{If}\;\left\{ \left\{A_p \cap B_p \right\} \cup \left\{ A_s \cap B_s \right\} \right\} \cap E,\;\mathrm{then}&\;D = 3,\\ \nonumber &\mathrm{Otherwise} &D = 0,\\\end{aligned}$$ where $\bar{A}$ is the complement of $A$, and $$A_p = \left\{\frac{1}{2}{{\log }_2}\left( {1 + \frac{{\gamma _p}{{\left| {h_{pr}} \right|}^2}}{{\gamma _s}{{\left| {h_{sr}} \right|}^2}+1}} \right){\rm{ \geq }}{R_p}\right\},$$ $$A_s = \left\{\frac{1}{2}{{\log }_2}\left( {1 + \frac{{\gamma _s}{{\left| {h_{sr}} \right|}^2}}{{\gamma _p}{{\left| {h_{pr}} \right|}^2}+1}} \right){\rm{ \geq }}{R_s}\right\}.$$ The comparison of $(a_i: i = p, s$ or $0)$ indicates which relaying would improve better the secondary outage probability. The different cases are detailed below. ### R assists the primary transmission ($D=1$) This case occurs either when (i) R succeeds to decode the primary signal but not the secondary signal and when relaying the primary signal provides lower secondary outage probability than the repetition (i.e., $a_p > a_0$) or (ii) R succeeds to decode both the primary and secondary signals and assisting the primary transmission provides the lowest $P_{out_{sec}}$(i.e., $E_p$). Hence, when the relay is able to decode the primary signal and the best choice is to assist the primary transmission, then $D=1$. Consequently, the received signals at PD and SD, on the second sub-slot, are respectively given by: $$\label{eq:received_R_helpPU_D1} y_{a}(2|D=1)=\sqrt{P_{r}^{(p)}}h_{ra}x_p+\sqrt{P_{s}}h_{sa}x_s+n_{a}.$$ After normalizing the noise variances and combining the signals received in the two sub-slots (given by (\[eq:received\_PD\_direct\]) and (\[eq:received\_R\_helpPU\_D1\])) with Maximum Ratio Combining (MRC) [@Yang_2009], the Signal-to-Interference-plus-Noise-Ratio (SINR) at PD is: $$\label{eq:SINR_PD_D1} SINR_{p}(D=1)=\frac{\gamma_{p}|h_{pp}|^2}{\gamma_{s}|h_{sp}|^2+1}+\frac{\gamma_{r}^{(p)}|h_{rp}|^2}{\gamma_{s}|h_{sp}|^2+1},$$ while SINR$_{s}$ at SD is given by: $$\label{eq:SINR_SD_D1_CR_help_PU} SINR_{s}(D=1)=\frac{\gamma_{s}|h_{ss}|^2}{\gamma_{p}|h_{ps}|^2+1}+\frac{\gamma_{s}|h_{ss}|^2}{\gamma_{r}^{(p)}|h_{rs}|^2+1}.$$ ### R assists the secondary transmission ($D=2$) This second case occurs when (i) R succeeds to decode only the secondary signal and when relaying the secondary signal is beneficial to the secondary transmission (i.e., $a_s > a_0$) or (ii) R succeeds to decode both signals and assisting the secondary transmission provides the lowest $P_{out_{sec}}$(i.e., $E_s$). In this case, R assists the secondary transmission and $D=2$. The received SINR at PD and that at SD are expressed as (\[eq:SINR\_SD\_D1\_CR\_help\_PU\]) and (\[eq:SINR\_PD\_D1\]) respectively, where indexes $p$ and $s$ are inverted. ### R does not assist the transmissions ($D=0$) When the relay is not able to decode the signals or when relaying is not beneficial to $P_{out_{sec}}$, the relay does not participate in the transmissions. In this case, we assume that the primary and secondary transmitters retransmit the same signals. Accordingly, the received SINR at SD is given by (eq.(7), [@Jaafar_Globecom]) and that at PD by inverting indexes $p$ and $s$ in (eq.(7), [@Jaafar_Globecom]). Outage Probability Analysis --------------------------- In this scheme, any of three cases can happen. We start by calculating the probability of occurrence of each one. An upper bound on the probability of occurrence of $D=1$ is given by: $$\begin{aligned} \label{eq:occurrence_D4_1} \nonumber P\left(D=1\right)&=& \frac{\tilde{\gamma}_{ps}}{\tilde{\gamma}_{ps}+\tilde{\gamma}_{rs}^{(p)}} \times \frac{\tilde{\gamma}_{pr}e^{-\frac{\Lambda_p}{\tilde{\gamma}_{pr}}-\frac{\Lambda_s(1+\Lambda_p)}{(1-\Lambda_p\Lambda_s)\tilde{\gamma}_{sr}}}}{\tilde{\gamma}_{pr}+\Lambda_p \tilde{\gamma}_{sr}}\\ \nonumber &+& \frac{\tilde{\gamma}_{ps}}{\tilde{\gamma}_{ps}+\tilde{\gamma}_{rs}^{(p)}} \left(1-\frac{\tilde{\gamma}_{sr}e^{-\frac{\Lambda_s}{\tilde{\gamma}_{sr}}}}{\tilde{\gamma}_{sr}+\Lambda_s \tilde{\gamma}_{pr}} \right)\\ \nonumber &\times & \left( 1-e^{\frac{-\Lambda_s(1+\Lambda_p)}{(1-\Lambda_p \Lambda_s)\tilde{\gamma}_{sr}}} \right)-\phi_i\left( -\frac{1}{\tilde{\gamma}_{ps}}- \frac{\tilde{\gamma}_{rs}^{(s)}}{\tilde{\gamma}_{ss}\tilde{\gamma}_{ps}} \right)\\ \nonumber &\times & \frac{\tilde{\gamma}_{ps}}{\tilde{\gamma}_{ps}+\tilde{\gamma}_{rs}^{(p)}} \left( \frac{1+\tilde{\gamma}_{ps}}{\tilde{\gamma}_{ss}} \right)e^{ \frac{1}{\tilde{\gamma}_{ps}} + \frac{\tilde{\gamma}_{rs}^{(s)}}{\tilde{\gamma}_{ss}\tilde{\gamma}_{ps}} }\\ &\times & \frac{\tilde{\gamma}_{sr}e^{-\frac{\Lambda_s}{\tilde{\gamma}_{sr}}-\frac{\Lambda_p(1+\Lambda_s)}{(1-\Lambda_p\Lambda_s)\tilde{\gamma}_{pr}}}}{\tilde{\gamma}_{sr}+\Lambda_s \tilde{\gamma}_{pr}},\end{aligned}$$ where $\tilde{\gamma}_{ab}=\gamma_a \sigma_{ab}^2$, $\tilde{\gamma}_{rb}^{(i)}=\gamma_r^{(i)} \sigma_{rb}^2$ ($a=p,s$ or $r$ and $b=p,s$ or $r$) and $\phi_i(x)=\int_{-\infty}^{x}{\frac{e^t}{t}dt}$. See Appendix A. By following similar calculations, we can obtain $P(D=2)$. Finally, $P(D=0)=1-\sum\limits_{i=1}^{2}P(D=i)$. We next present the conditional primary and conditional secondary outage probabilities for each case: - $D=1$ The conditional primary outage probability is given by: $$\begin{aligned} \label{eq:outage_prob_CR_help_PU_D1_0} \nonumber P_{pri}(out.|D=1)&=&P\left( SINR_{p}(D=1)<\Lambda_p \right)\\ &=&P(\omega<\Lambda_p+\Lambda_p \omega_1 - \omega_2)\end{aligned}$$ where $\Lambda_a=2^{2R_a}-1$ ($a=p$ or $s$), $\omega=\gamma_{r}^{(p)}|h_{rp}|^2$, $\omega_1=\gamma_{s}|h_{sp}|^2$ and $\omega_2=\gamma_{p}|h_{pp}|^2$. We shall make use of the following Lemma. The exact closed-form expression of the conditional primary outage probability is expressed by: \[Lemma1\] $$\label{eq:outage_prob_CR_help_PU_D1} P_{pri}(out.|D=1)=\lambda_1+\lambda_2,$$ where $$\begin{aligned} \label{eq:d1} \nonumber \lambda_1&=&\frac{\tilde{{\gamma}}_{sp}^2\Lambda_p^2 +\tilde{{\gamma}}_{rp}^{(p)}\tilde{{\gamma}}_{sp}\Lambda_p\left( 1-e^{-\frac{\Lambda_p}{\tilde{\gamma}_{rp}^{(p)}}} \right) }{(\tilde{{\gamma}}_{pp}+\Lambda_p \tilde{{\gamma}}_{sp})(\tilde{{\gamma}}_{rp}^{(p)}+\Lambda_p \tilde{{\gamma}}_{sp})},\\ \\ \nonumber \lambda_2&=&\frac{\tilde{\gamma}_{pp}^2 \left( 1-e^{-\frac{\Lambda_p}{\tilde{\gamma}_{pp}}} \right)-\tilde{\gamma}_{pp} \tilde{\gamma}_{rp}^{(p)} \left( 1-e^{-\frac{\Lambda_p}{\tilde{\gamma}_{rp}^{(p)}}} \right) }{(\tilde{{\gamma}}_{pp}+\Lambda_p \tilde{{\gamma}}_{sp})(\tilde{{\gamma}}_{rp}^{(p)}-\tilde{{\gamma}}_{pp})},\\ \label{eq:d2} &\forall&\; \tilde{{\gamma}}_{pp} \neq \tilde{{\gamma}}_{rp}^{(p)}.\end{aligned}$$ See Appendix B. An upper bound on the conditional secondary outage probability is given by: \[Lemma2\] $$\begin{aligned} \label{eq:conditional_sec_D1} \nonumber P_{sec}(out.|D=1)&=& \left( 1-e^{-\frac{\Lambda_s}{\tilde{\gamma}_{ss}}}\left( 1+\frac{ln\left( 1+ \frac{\Lambda_s \tilde{\gamma}_{ps}}{\tilde{\gamma}_{ss}} \right)}{\tilde{\gamma}_{ps}} \right) \right)\\ &\times& \left( \tilde{\gamma}_{rs}^{(p)} +1 \right)=\varphi \times \left( \tilde{\gamma}_{rs}^{(p)} +1 \right).\end{aligned}$$ See Appendix C. By assisting the primary transmission, this relaying procedure aims to reduce the interference caused to the secondary transmission with respect to the primary outage threshold $\varepsilon$. For that purpose, R should control its transmit power $P_r$ to be as low as possible. This value of $P_r$, denoted $P_{r,num}^{(p)}$, is evaluated numerically by solving $P_{pri}(out.|D=1)=\varepsilon$. If $P_{r,num}^{(p)}>P_r^{max}$, then relaying is not beneficial and $D=0$. - $D=2$ Due to the similarity of the outage probability analysis of this relaying procedure to the first one, it is not given in details. However, by inverting indexes $p$ and $s$ and indexes $pri$ and $sec$ in equations (\[eq:outage\_prob\_CR\_help\_PU\_D1\_0\])-(\[eq:conditional\_sec\_D1\]), we obtain an accurate outage probability analysis. The relay transmit power should also be calculated such that $P_{pri}(out.|D=2)=\varepsilon$. The relay transmit power is given by $\gamma_{r,num}^{(s)}=\frac{\varepsilon/\varphi'-1}{\sigma_{rp}^2}$, where $\varphi'=\left( 1-e^{-\frac{\Lambda_p}{\tilde{\gamma}_{pp}}}\left( 1+\frac{ln\left( 1+ \frac{\Lambda_p \tilde{\gamma}_{sp}}{\tilde{\gamma}_{pp}} \right)}{\tilde{\gamma}_{sp}} \right) \right)$. Then, $\gamma_r^{(s)}=min\left( \gamma_{r,num}^{(s)}, \gamma_r^{max} \right)$. - $D=0$ The outage probability analysis of this case is presented in [@Jaafar_Globecom]. The conditional secondary outage probability is given by (eq.(16), [@Jaafar_Globecom]), while $P_{pri}(out.|D=0)$ is obtained by simply inverting indexes $p$ and $s$ in (eq.(16), [@Jaafar_Globecom]). Finally, upper bounds on the primary and secondary outage probabilities are given by: $$\label{eq:outage_adaptive} P_{out_{c}}= \sum_{i=0}^{2}{P(D=i)P_c(out.|D=i)},$$ where $c=pri$ or $sec$. The obtained expressions are upper bounds since some of the conditional outage probabilities calculated are upper bounds. Extension to “Adaptive relaying scheme 2" ----------------------------------------- In this extension, we assume that R is equipped with a SIC (Successive Interference Cancelation) receiver [@Tse], and hence it is able to decode both signals. We define $a_{ps}=\frac{(1-\alpha)\gamma_r^{(ps)} |h_{rs}|^2}{\alpha\gamma_r^{(ps)} |h_{rs}|^2+1}$, where $0 \leq \alpha \leq 1$. We also define $C_{i}=\left\{a_i=max\left\{ a_0, a_p, a_s, a_{ps} \right\} \right\}$ ($i=0,p,s$ or $ps$) and $E=\left\{A_p \cap B_p \right\} \cup \left\{ A_s \cap B_s \right\}$ the event of a successful successive decoding of both signals, where $B_i = \left\{\frac{1}{2} \log_2\left( {1 + {\gamma _{i}}{{\left| {{h_{ir}}} \right|}^2}} \right){\rm{ \geq }}{R_i}\right\}$. We call this extension “Adaptive relaying scheme 2". For this scheme, we distinguish four relaying procedures: $$\begin{aligned} \label{eq:condition_CR_helps_PU_SU} \nonumber &\mathrm{If}\; E \cap C_{ps},\;\mathrm{then}&\;D = 3,\\ \nonumber &\mathrm{If}\;\left\{A_s \cap \bar{B}_s \cap (a_s >a_0) \right\} \cup \left\{E \cap C_s \right\},\;\mathrm{then}&\;D = 2,\\ \nonumber &\mathrm{If}\;\left\{A_p \cap \bar{B}_p \cap (a_p >a_0) \right\} \cup \left\{E \cap C_p \right\},\;\mathrm{then}&\;D = 1,\\ \nonumber &\mathrm{Otherwise}\;&D = 0.\end{aligned}$$ The relaying procedures for $D=0,1$ or $2$ are identical to the first scheme. When $D=3$, a fraction of the relay transmit power $\alpha P_r^{(ps)}$ is used to send $x_p$ and the rest, i.e., $(1-\alpha) P_r^{(ps)}$ is used to transmit $x_s$. SINRs at PD and at SD are then given by (eq.(10), [@Jaafar_Globecom]) and (eq.(11), [@Jaafar_Globecom]) respectively. The parameter $\alpha$ is calculated using (eq.(33), [@Jaafar_Globecom]) where $\gamma_r^{(ps)}=\gamma_r^{max}$. ![Comparaison of different relaying schemes[]{data-label="Fig:Compare_schemes"}](Fig_Comparaison_NEW_PU_SU_PUSU_Rep_Rp08_Rs02_epsilon01_gammap20_3.eps){width="225pt"} Numerical and Simulation Results ================================ We consider the CRN presented in Fig. \[Fig:Network\] where the coordinates of PT, ST, PD and SD are given by (0,1.82), (0,0), (1,1.82) and (1,0) respectively (coordinates are in distance units). We assume that $R_p=0.8 bits/s/Hz$, $R_s=0.2 bits/s/Hz$, $\varepsilon=0.1$, $\beta=4$ and $\gamma_p=\gamma_r^{max}=20dB$. We measure the average of the secondary outage probability calculated for different random positions of the relay node in the plan of coordinates (X,Y) where $0.1 \leq X \leq 0.9$ and $0.1 \leq Y \leq 1.7$ unless otherwise is stated. In Fig. \[Fig:Compare\_schemes\], we compare the “Adaptive relaying scheme 1" to other transmission schemes presented in the literature [@Zou; @Jaafar_Globecom]. At low $\gamma_p$ ($\gamma_p \leq 8dB$), no secondary transmissions are allowed. When $\gamma_p$ is higher than the cutoff value, the “Adaptive relaying scheme 1" presents, as expected, the best outage performance since the proposed scheme chooses the most adequate relaying procedure. The “R assists secondary transmissions" and “R assists primary transmissions" schemes outperforms the non cooperative scheme. When R assists only the secondary transmissions, the performances are degraded by the fact that the relay transmit power is limited to $P_r^{max}$. ![“Adaptive relaying scheme" versus “Adaptive relaying scheme 2" (simulations only)[]{data-label="Fig:compare_adapt12"}](Figure_Comparaison_adapt_adapt2_data_rates_gammap20_copy4.eps){width="225pt"} ![Secondary Outage Probability vs. Relay position (simulations only)[]{data-label="Fig:3D"}](Fig_3D_Adaptive_relaying_gammap20_gammar30_Rp08_Rs02_epsilon01_NEW2.eps){width="225pt"} In Fig. \[Fig:compare\_adapt12\], we compare the secondary outage probabilities of the two adaptive relaying schemes for different $R_p$ values and where we assume $R_s=\frac{R_p}{2}$. For $R_p \leq 2.2 bits/s/Hz$, the second scheme outperforms the first one. Indeed, the proposed second scheme offers more relaying possibilities for R and hence improves the secondary performance. For $R_p \geq 2.2 bits/s/Hz$, no secondary or relaying transmissions are allowed due to the high primary outage probability requirement that blocks any interfering transmission. In Fig. \[Fig:3D\], we present the secondary outage performance using the “Adaptive relaying scheme 2" for different positions of the relay node on the plan (X,Y) where $-0.5 \leq X \leq 1.5$ and $0 \leq Y \leq 2$. When R is close to the secondary nodes, the secondary outage performance is improved. Indeed, when R is close to the primary nodes, the cases $D=1$ and $D=3$ are predominant and the outage probability gain comes from the interference reduction. As R gets closer to the secondary nodes, the cases $D=2$ and $D=3$ occur more often. Hence, the outage performance gain is issued from the interference reduction ($D=3$) and cooperation ($D=2$). Moreover, when R is in the middle zone, the secondary outage probability decreases. In this zone, the condition of the channels linked to the relay node favors successful decoding and efficient forwarding of the signals. Conclusion ========== In this paper, we proposed adaptive relaying schemes for cognitive radio networks, where a relay node is able to decide assisting the primary, the secondary or both transmissions depending on the channels condition. We showed by analysis and by simulation that the first adaptive relaying scheme, where the relay may help the primary or the secondary transmission, outperforms the non adaptive relaying schemes in terms of secondary outage probability, with respect to a primary outage probability threshold. Then, a second scheme has been proposed considering that the relay may help simultaneously both transmissions. Simulations show the performance improvement of the second scheme, specially at low data rates. Proof of Eq.(\[eq:occurrence\_D4\_1\]) ====================================== Due to the independency between events, we have: $$\begin{aligned} \label{eq:d_4_1_demo} \nonumber P(D=1)&=&P\left( A_p \cap \bar{A}_s \right) P\left(a_p > a_0 \right)\\ \nonumber &+& P(A_p|A_s)P(A_s)P(E_p)\\ \nonumber &=&P(\gamma_p |h_{pr}|^2 \geq max\{ \Lambda_p (1+\gamma_s |h_{sr}|^2) ,\\ \nonumber &{}& \frac{\gamma_s |h_{sr}|^2}{\Lambda_s}-1\})P(\gamma_r^{(p)} |h_{rs}|^2 \leq \gamma_p |h_{ps}|^2)\\ \nonumber &+& P(\gamma_p |h_{pr}|^2\geq \frac{\Lambda_p (1+ \Lambda_s)}{1-\Lambda_p \Lambda_s})P(a_p>a_0)\\ &{}& P(\frac{\gamma_s |h_{sr}|^2}{\gamma_p |h_{pr}|^2+1}\geq \Lambda_s)P(a_p>a_s). \label{eq:d_4_1_demo2}\end{aligned}$$ Since $\gamma_a |h_{ab}|^2$ and $\gamma_c |h_{cb}|^2$ are exponential distributed random variables with parameters $1/\tilde{\gamma}_{ab}$ and $1/\tilde{\gamma}_{cb}$ respectively ($a=p$ or $s$, $c=p$ or $s$, and $b=p,s$ or $r$), then $\forall x \in \mathbf{R}$: $$\begin{aligned} \label{eq:liste1} P(\gamma_a |h_{ab}|^2 \leq \gamma_c |h_{cb}|^2)&=&\frac{\tilde{\gamma}_{cb}}{\tilde{\gamma}_{ab}+\tilde{\gamma}_{cb}} ,\\ \label{eq:liste3} P(\gamma_a |h_{ab}|^2 \geq (\gamma_c |h_{cb}|^2+1) x)&=&\frac{{\tilde{\gamma}_{ab}}e^{-\frac{x}{\tilde{\gamma}_{ab}}}}{{\tilde{\gamma}_{ab}}+x {\tilde{\gamma}_{cb}}},\end{aligned}$$ and the probability density function (pdf) of $X_{ab}=\frac{\gamma_a |h_{as}|^2}{\gamma_b |h_{bs}|^2+1}$ ($a=s$ or $r$; $b=p$ or $r$) is given by: $$\label{eq:pdf_quotient} f_{X_{ab}}(x)=\frac{e^{-\frac{x}{\tilde{\gamma}_{as}}}}{\tilde{\gamma}_{as}+x \tilde{\gamma}_{bs}}\left( 1+ \frac{\tilde{\gamma}_{as}\tilde{\gamma}_{bs}}{\tilde{\gamma}_{as}+x \tilde{\gamma}_{bs}} \right),\; x \geq 0.$$ Since $\frac{1+\frac{\tilde{\gamma}_{rs}^{(s)}\tilde{\gamma}_{ps}}{\tilde{\gamma}_{rs}^{(s)}+y \tilde{\gamma}_{ps}}}{\tilde{\gamma}_{ss}+y \tilde{\gamma}_{rs}^{(p)}} \leq \frac{1+\tilde{\gamma}_{ps}}{\tilde{\gamma}_{ss}}$, $\forall y\geq 0$ and using (\[eq:pdf\_quotient\]), we get: $$\label{eq:upPEp} P(a_p>a_s)\leq -\phi_i\left( -\frac{1}{\tilde{\gamma}_{ps}}- \frac{\tilde{\gamma}_{rs}^{(s)}}{\tilde{\gamma}_{ss}\tilde{\gamma}_{ps}} \right)\left( \frac{1+\tilde{\gamma}_{ps}}{\tilde{\gamma}_{ss}} \right)e^{ \frac{1}{\tilde{\gamma}_{ps}} + \frac{\tilde{\gamma}_{rs}^{(s)}}{\tilde{\gamma}_{ss}\tilde{\gamma}_{ps}} }.$$ Using (\[eq:liste1\]),(\[eq:liste3\]) and (\[eq:upPEp\]) in (\[eq:d\_4\_1\_demo\]), we obtain (\[eq:occurrence\_D4\_1\]). Proof of Lemma.\[Lemma1\] ========================= From (\[eq:outage\_prob\_CR\_help\_PU\_D1\_0\]), $\omega$, $\omega_1$ and $\omega_2$ are exponential random variables with parameters $1/\tilde{\gamma}_{rp}$, $1/\tilde{\gamma}_{sp}$ and $1/\tilde{\gamma}_{pp}$ respectively. Thus, the pdfs of $\omega$ and $\phi=\omega_2-\Lambda_p \omega_1$ are: $$\nonumber f_\omega(\omega)=\frac{e^{-\frac{\omega}{\tilde{\gamma}_{rp}}}}{\tilde{\gamma}_{rp}},\omega \geq 0;\;\;\nonumber {{f_\phi}(\phi) = \left\{ {\begin{array}{*{20}{c}} {\frac{{{e^{ - \frac{\phi}{{{\tilde{ \gamma }_{pp}}}}}}}}{{{\tilde{ \gamma }_{pp}} + {\Lambda _p} \tilde{\gamma}_{{sp}}}},}&{\phi \geq 0} \\ {\frac{{{e^{\frac{\phi}{{{\Lambda _p}\tilde \gamma_{{sp}}}}}}}}{{{\tilde{ \gamma }_{pp}} + {\Lambda _p} \tilde{\gamma}_{{sp}}}},}&{\phi \leq 0.} \end{array}} \right.}$$ Finally, we obtain the conditional primary outage probability: $$\begin{aligned} \label{eq:p_out_demo_CR_help_PU} \nonumber P(z<\Lambda_p)& = & \int_{ - \infty }^{{\Lambda _p}} {{f_z}(z)dz}=\int_{ - \infty }^{{0}} {\frac{{{\Lambda _p}\tilde{\gamma} {_{sp}}{e^{\frac{z}{{{\Lambda _p}\tilde{\gamma} {_{sp}}}}}}}}{{{\beta _1}{\beta _2}}}}dz\\ \nonumber &+& \int_{0 }^{{\Lambda_p}} {\frac{{{\Lambda _p}\tilde{\gamma} {_{sp}}}e^{-\frac{z}{\tilde{\gamma}_{rp}^{(p)}}}}{{{\beta _1}{\beta _2}}} + \frac{{{\tilde{\gamma} _{pp}}\left( {{e^{ - \frac{z}{{{\tilde{\gamma} _{rp}^{(p)}}}}}} - {e^{ - \frac{z}{{{\tilde{\gamma} _{pp}}}}}}} \right)}}{{{\beta _1}\left( {{\tilde{\gamma} _{rp}^{(p)}} - {\tilde{\gamma} _{pp}}} \right)}} } dz\\ & = & \lambda_1+\lambda_2,\end{aligned}$$ where $z=\omega+\phi$, $f_z(z)$ is its pdf, $\beta_1=\tilde{\gamma}_{pp}+\Lambda_p \tilde{\gamma}_{sp}$, $\beta_2=\tilde{\gamma}_{rp}^{(p)}+\Lambda_p \tilde{\gamma}_{sp}$; $\lambda_1$ and $\lambda_2$ are given by (\[eq:d1\]) and (\[eq:d2\]) respectively. This completes the proof of Lemma.\[Lemma1\]. Proof of Lemma.\[Lemma2\] ========================= Using (\[eq:pdf\_quotient\]) for $X_{sb}$ ($b=p$ or $r$), we obtain: $$\begin{aligned} \label{eq:prob_sec} &P&\left( X_{sp} + X_{sr} < \Lambda_s \right)=\\ %\nonumber &=&\int_{0}^{\Lambda_s}{f_{X_r}(x)} \left( 1-\frac{\tilde{\gamma}_{ss}e^{-\frac{\Lambda_s-x}{\tilde{\gamma}_{ss}}}}{\tilde{\gamma}_{ss}+ (\Lambda_s-x)\tilde{\gamma}_{ps}} \right)dx\\ \nonumber &=& \int_{0}^{\Lambda_s}{\frac{\left( 1+ \frac{\tilde{\gamma}_{ss}\tilde{\gamma}_{rs}^{(p)}}{\tilde{\gamma}_{ss}+x \tilde{\gamma}_{rs}^{(p)}} \right)}{\tilde{\gamma}_{ss}+x \tilde{\gamma}_{rs}^{(p)}}} \left( e^{-\frac{x}{\tilde{\gamma}_{ss}}}- \frac{\tilde{\gamma}_{ss}e^{-\frac{\Lambda_s}{\tilde{\gamma}_{ss}}}}{\tilde{\gamma}_{ss}+ (\Lambda_s-x) \tilde{\gamma}_{ps}} \right)dx.\end{aligned}$$ Since ${\frac{\left( 1+ \frac{\tilde{\gamma}_{ss}\tilde{\gamma}_{rs}^{(p)}}{\tilde{\gamma}_{ss}+x \tilde{\gamma}_{rs}^{(p)}} \right)}{\tilde{\gamma}_{ss}+x \tilde{\gamma}_{rs}^{(p)}}}\leq \frac{\tilde{\gamma}_{rs}^{(p)}+1}{\tilde{\gamma}_{ss}}$, then using this upper bound in (\[eq:prob\_sec\]), we obtain (\[eq:conditional\_sec\_D1\]). This completes the proof of Lemma.\[Lemma2\].

--- author: - 'Yuan-Yuan Zhang' - 'Fu-Lin Zhang[^1]' bibliography: - 'LHSTStateBits.bib' title: 'Local-hidden-state models for T-states using finite [shared randomness]{} [^2]' --- Introduction ============ Nonclassical correlations in composite quantum systems and their hierarchy are fundamental issues in quantum information [@Book; @RevModPhys.81.865; @RMP2012Vedral; @RMP2014bell; @JPA2014LHV]. Many concepts of these correlations can be traced back to the early days of quantum mechanics, and play key roles in several quantum information processes. On the other hand, the tasks in quantum information also provide points of view to study the correlations. An important example is the work of Wiseman [*et al.*]{} [@PRL2007Steering], in which they define Bell nonlocality and Einstein-Podolsky-Rosen (EPR) steering according to two tasks, and prove that the former is a sufficient condition for the [latter]{} and entanglement is necessary for both of them. In the tasks of Wiseman [*et al.*]{} [@PRL2007Steering], two observers, Alice and Bob, share a bipartite entangled state. Alice can affect the postmeasured states left to Bob by choosing different measurements on her half. Such ability is termed *steering* by Schrödinger [@steer1935]. EPR steering from Alice to Bob exists when Alice can convince Bob that she has such ability, which is equivalent to the fact that unnormalized postmeasured states can not be described by a local-hidden-state (LHS) model. Further, their state is Bell nonlocal, when the two observers can convince Charlie, a third person, that the state is entangled. This is demonstrated by the inexistence of local-hidden-variable (LHV) model explaining correlations of outcomes of their joint local measurement. A LHS model is a particular case of a LHV model, of which the hidden variable is a single-particle state and one of the response functions is the probability of measurement on the state. Construction of local models, especially the optimal ones, provides a division between the quantum and classical worlds, in the sense of whether the nonclassical correlations exist. However, it is an extremely difficult problem to explicitly derive optimal models. Only a few models beyond Werner’s results [@Werner1989] have been reported, such as the ones in [@PRL2014oneway; @PRL2005LHVBit; @JOSAB2015Steering; @arxiv2015UnSteer], most of which are for states with high symmetries. Our recent work [@zhang2017LHS] shows the possibility of generating local models for states with a lower symmetry, from the ones with a high symmetry. Namely, we obtain the optimal [models for T-states (Bell diagonal states)]{}, given by Jevtic [*et al.*]{} [@JOSAB2015Steering] based on the steering ellipsoid [@PRL2014Ellips], by mapping the problem to the one of the Werner state. On the other hand, Bowles [*et al.*]{} [@PRL2005LHVBit] raise the issue of constructing local models using finite shared randomness. This comes from their consideration about the cost of classically, measured by classical bits encoding the local variable, simulating the correlations in an entangled state. They give a series of LHV models for Werner states using finite shared randomness, and prove the existence of the ones for entangled states admitting a LHV model. [ These results inspire a method for constructing LHV models for entangled states, in which the problem of finding a local model for an infinite set of measurements is mapped to the one of a finite set of measurements [@Arxiv2015LHV; @PRL2016Algorithmic; @RPP2017RevLHS]. In addition, the concepts of superlocality [@PRA2015SupLocal; @PRA2017SupLocal] and superunsteerability [@PRA2018SupLHS] stemm from the study of shared classical randomness required to simulate local correlations. ]{} In the present work, we [ study local models for T-states by extending our strategy in [@zhang2017LHS] to ]{} the case with finite shared randomness. [ They are LHS models, as the shared local variables are sets of discrete states on the Bloch sphere and Bob’s response function is his measurement probability on these states. Expressing the discrete distributions for Werner states in *Protocols 1* and *2* of [@PRL2005LHVBit] in terms of Dirac delta functions, we derive a family of LHS models for T-states by using the mapping in [@zhang2017LHS]. The one generated from the most economical LHV model for Werner state is discussed in detail, which provides an example to observe the continuously changing shared randomness with entanglement. Besides, we construct a LHS model, not belonging to the two protocols in [@PRL2005LHVBit], for the critical separable Werner state, by decomposing it into product states. It can be transformed into the LHS models for the critical separable T-states by a generalization of the original mapping in [@zhang2017LHS]. This ]{} shows the possibility of generating the separable boundary for a class of states with a low symmetry, and decomposing them into product states, from a higher symmetric case. Preliminaries {#LHSReview} ============= LHS model --------- We first give a brief review of the concepts of EPR steering and LHS model, under the context of two-qubit system and projective measurements. An arbitrary two-qubit state shared by Alice and Bob can be written as $$\label{rhoab} {\rho}_{AB}=\frac{1}{4} ({\mathbb{I}}\otimes {\mathbb{I}}+\vec{a} \cdot \vec{\sigma} \otimes {\mathbb{I}}+{\mathbb{I}}\otimes \vec{b} \cdot \vec{\sigma} + \sum_{ij} T_{ij} \vec{\sigma}_{i} \otimes \vec{\sigma}_{j} ),$$ where $ {\mathbb{I}}$ is the unit operator, $ \vec{a} $ and $ \vec{b} $ are the Bloch vectors for Alice and Bob’s qubit, $ \vec{\sigma} = ( \sigma_x,\sigma_y,\sigma_z ) $ is the vector of the Pauli operators, and $ T_{ij} $ is correlation matrix. We focus on the case in which Alice makes a projection measurement on her part. The measurement operator of Alice uniquely corresponds to a unit vector $\vec{x}$ and a outcome $a=\pm1$ as $$\Pi _a^{\vec{x}} = \frac{1}{2} ({{\mathbb{I}}+ a\vec{x}\cdot \vec{\sigma} } ) .$$ After the measurement, Bob’s state becomes $$\label{ assem} \begin{split} \rho _a^{\vec{x}} &= {\mbox{Tr}}( {\Pi _a^{\vec{x}} \otimes {\mathbb{I}}{\rho _{AB}}} ) \\ &= \frac{1}{4} [ { ( {1 + a\vec{a} \cdot \vec{x}} ) {\mathbb{I}}+ ( {\vec{b} + a{T^{\rm T}}\vec{x} } ) \cdot \vec{\sigma} } ], \end{split}$$ where $T^{\rm T}$ is transposed $T$. The set of $\rho _a^{\vec{x}} $ is called an assemblage. A LHS model is defined as $$\label{LHS} \rho^{\rm{LHS}} = \int \omega ( \vec{\lambda} ) p ( a| \vec{x},\vec{\lambda} ) \rho_{\vec{\lambda}} d \vec{\lambda} .$$ Here, $ \rho_{\vec{\lambda}} $ is a hidden state depending on the hidden variable $ \vec{\lambda} $ with the distribution function $ \omega ( \vec{\lambda} ) $. And, $ p ( a| \vec{x},\vec{\lambda} ) $ is a response function simulating the probability of Alice’s outcome, with $ p ( a| \vec{x},\vec{\lambda} ) \geqslant 0$ and $p ( 1| \vec{x},\vec{\lambda} ) + p ( -1| \vec{x},\vec{\lambda} ) = 1 $. If there exists a LHS model satisfying $$\begin{aligned} \label{AssEqLHS} \rho _a^{\vec{x}} = \rho^{\rm{LHS}}\end{aligned}$$ for all the measurements, the outcomes of Alice’s measurements and Bob’s collapsed state can be simulated by a LHS strategy without any entangled state [@PRL2007Steering]. On the contrary, if a LHS model satisfying (\[AssEqLHS\]) does not exist, $\rho_{AB}$ is termed steerable from Alice to Bob. [Without loss of generality, we may]{} take a hidden variable to the unit Bloch vectors and the local hidden states to be corresponding pure qubit states [[@arxiv2015UnSteer]]{} as $$\rho_{\vec{\lambda}} = |\vec{\lambda}\rangle \langle \vec{\lambda} |=\dfrac{1}{2} ( {\mathbb{I}}+ \vec{\lambda} \cdot \vec{\sigma} ) .$$ Then, $d \vec{\lambda}$ is the surface element on the Bloch sphere. We can take $$\label{PA} p ( a| \vec{x},\vec{\lambda} ) = \frac{1}{2}\bigr[ 1 + af(\vec{x},\vec{\lambda})\bigr] ,$$ with $ f(\vec{x},\vec{\lambda}) \in [-1,1] $. The LHS model can be rewritten as $${\rho ^{\rm LHS}} \!\!=\!\! \int \!\! { \omega ( \vec{\lambda} ) \! \frac{1}{4} \biggr[ {{\mathbb{I}}\! +\! \vec{\lambda} \! \cdot \! \vec{\sigma} \!+\! af(\vec{x},\! \vec{\lambda}) \!+\! a f(\vec{x},\! \vec{\lambda}) \vec{\lambda} \! \cdot \! \vec{\sigma} } \biggr] } d\vec{\lambda}.$$ Consequently, the equation (\[AssEqLHS\]) requires \[Reqs\] $$\begin{aligned} &\int \omega(\vec{\lambda}) d \vec{\lambda} =1 , \label{omega1}\\ &\int \omega(\vec{\lambda}) f (\vec{x},\vec{\lambda}) d \vec{\lambda} = \vec{a}\cdot\vec{x} , \label{a0} \\ &\int \omega(\vec{\lambda}) \vec{\lambda} d \vec{\lambda} =\vec{b} , \label{b0} \\ &\int \omega(\vec{\lambda}) f (\vec{x},\vec{\lambda}) \vec{\lambda} d \vec{\lambda} = T^{\rm T} \vec{x}. \label{Tx}\end{aligned}$$ The spin correlation matrix can always be diagonalized by local unitary operations, which preserve steerability or unsteerability. Hence, we consider the diagonalized $T$, that $T={\mbox{Diag}}\{T_x,T_y,T_z\}$, and omit its superscript ${\rm T}$ in the following parts of this article. [ Constructing ]{} a LHS model for a state $\rho_{AB}$ is equivalent to finding a pair of $ \omega ( \vec{\lambda} )$ and $ f ( \vec{x} ,\vec{\lambda} )$ fulfilling these requirements. T-states -------- The state (\[rhoab\]) is called a T-state, when the Bloch vectors, $\vec{a}$ and $\vec{b}$, vanish. In our recent work [@zhang2017LHS], we present an approach to derive the optimal LHS model for T-states. We first assume the correlation matrix on the EPR-steerable boundary being $T_0$ and $T= t T_0$ with $t\geq0$. That is, the T-state with $t>1$ is EPR-steerable, and the one with $0\leq t\leq 1$ admits a LHS model. The key step is multiplying both sides of Eq. (\[Tx\]) by $ T^{-1}_0 $ and defining the unit vector $ \vec{\lambda '} = {{T_0^{ - 1}\vec{\lambda} }}{ | {T_0^{ - 1}\vec{\lambda} } |^{-1} } $, where $ |\cdot| $ is the Euclidean vector norm. Then the condition (\[Tx\]) is rewritten as $$\label{Tx1} \int \omega' ( \vec{\lambda}' ) \frac{1}{ | T_0 \vec{\lambda}' | } f (\vec{x},\vec{\lambda}) \vec{\lambda}' d\vec{\lambda}' = t \vec{x},$$ where $ \omega' ( \vec{\lambda}' ) $ is the distribution function of the new defined hidden variable $\vec{\lambda}' $, and $ d\vec{\lambda}' $ is a surface element on its unit sphere. These variables are connected by a Jacobian determinant as $$d \vec{\lambda} = | \det T_0 | | {T_0^{ - 1}\vec{\lambda} } |^3 d \vec{\lambda}' , \ \ \ \omega ( \vec{\lambda} ) d\vec{\lambda} =\omega' ( \vec{\lambda}' ) d\vec{\lambda}'.$$ In the optimal LHS model of the critical Werner state [@PRL2007Steering; @Werner1989], with $T_0=- {\mbox{Diag}}[1/2,1/2,1/2]$, the functions in (\[Tx1\]) satisfy $$\omega' ( \vec{\lambda}' ) =\frac{1}{2 \pi} | T_0 \vec{\lambda}' |, \ \ \ f (\vec{x},\vec{\lambda}) ={\mbox{sgn}}(\vec{x}\cdot\vec{\lambda}').$$ We find that, these relations give exactly the optimal LHS model for an arbitrary T-state and leads to the critical condition [@zhang2017LHS; @JOSAB2015Steering; @EPL2016Tstate] $$\int \frac{1}{2 \pi} | T_0 \vec{\lambda}' | d\vec{\lambda}' = 1.$$ An explicit expression for this integral can be found in the work of Jevtic *et al.* [@JOSAB2015Steering]. LHS models for T-States with finite [shared randomness]{} ========================================================== We now generate the LHS models for [T-states]{} with finite [shared randomness]{}, using our approach reviewed above. The formulas in the above section are derived based on continuous local variables. To utilize these results, we represent the distribution of the finite hidden variables as delta functions. We mainly concentrate on the details of the two models corresponding to the most economical one of the Werner state and the one for the separable Werner state. LHS model on the icosahedron ---------------------------- In the most economical model simulating an entangled Werner state, Alice and Bob share $i =\{1,...,12\}$ uniformly distributed, corresponding to $12$ vertices of the icosahedron represented by the normalized vectors $\vec{v}_i$. That is, the distribution is given by $$\omega ( \vec{\lambda} ) = \sum_{i} \frac{1}{12} \delta ( \vec{\lambda} -\vec{v}_i ) .$$ The radius of a sphere inscribed inside the icosahedron is $l=\sqrt{(5+2\sqrt{5})/15}$. The icosahedron is symmetric under $\vec{v}_i \rightarrow -\vec{v}_i $, and its vertices satisfy the properties $\sum_{j} {\mbox{sgn}}(\vec{v}_j \cdot \vec{v}_i) \vec{v}_j= 2 \gamma \vec{v}_i$ with $\gamma=1+\sqrt{5}$. Then, the vector $l\vec{x}$ can always be represented as a convex decomposition $l\vec{x}=\sum_i \omega_i \vec{v}_i $ with $\omega_i \geq0$ and $\sum_i \omega_i =1$. Defining the function $$f (\vec{x},\vec{\lambda}) = - \sum_i \omega_i {\mbox{sgn}}( \vec{v}_i \cdot \vec{\lambda} ) ,$$ one can obtain $$\int \! \biggr[\! \sum_{j} \frac{1}{12} \delta ( \vec{\lambda} -\vec{v}_j ) \! \biggr]\biggr [\! - \! \sum_i \omega_i {\mbox{sgn}}( \vec{v}_i \cdot \vec{\lambda} ) \! \biggr] \! \vec{\lambda} d\vec{\lambda} = -\! \frac{\gamma l}{6} \vec{x},$$ which is the relation (\[Tx\]) for the Werner state. To fulfill the condition (\[Tx\]), equivalently the equation (\[Tx1\]), for T-states, an intuitive construction is given by $$\begin{aligned} && \omega' ( \vec{\lambda}' ) = \sum_{i} \frac{S}{12} \delta ( \vec{\lambda}' - \vec{v}_{i} ) | T_0 \vec{\lambda}' |, \\ && f (\vec{x},\vec{\lambda}) = \sum_i \omega_i {\mbox{sgn}}( \vec{v}_{i} \cdot \vec{\lambda}' ), \label{fT}\end{aligned}$$ where $S$ is a constant determined by the normalization condition (\[omega1\]). They [lead]{} to the visibility parameter in (\[Tx1\]) being $$\label{tLHS} t = S \frac{\gamma l}{6} =\frac{2 \gamma l }{ \sum_i | T_0 \vec{ v}_{i} | }.$$ Straightforward calculation gives the distribution of $\vec{\lambda}$ as $$\label{omegaT} \omega (\vec{\lambda}) = \sum_{i}\frac{S}{ 12 } \delta ( \vec{\lambda}- { T_0 \vec{v}_{i} }{|T_0 \vec{v}_{i}|^{-1}} ) |T_0 \vec{v}_{i}|.$$ Then, both the integrals in (\[a0\]) and (\[b0\]) can be easily checked to be zero, by using the symmetries $\omega (-\vec{\lambda}) =\omega (\vec{\lambda}) $ and $f (\vec{x},-\vec{\lambda}) =-f (\vec{x},\vec{\lambda}) $. Therefore, the functions (\[fT\]) and (\[omegaT\]) represent a LHS model for the T-state with the visibility parameter in (\[tLHS\]). And the extension to smaller amounts of $t$ is straightforward. Obviously, in our LHS models for T-states, the hidden variable $i=\{1,...,12\}$ distributes nonuniformly, whose probability is proportional to $|T_0 \vec{v}_{i}|$. The corresponding unit vectors $\vec{\lambda}'$ locate on $ \vec{v}_{i} $, and the Bloch vector of hidden states $\vec{\lambda}$ on $ { T_0 \vec{v}_{i} }{|T_0 \vec{v}_{i}|^{-1}}$. Both the distribution and visibility parameter, given in (\[tLHS\]), covered by the model, depend on the orientation of the icosahedron. A natural question is which orientation is optimal, in the sense of maximizing the parameter $t$, or equivalently $S$. Optimal icosahedron ------------------- Since it is a complex problem to perform general maximisations, we consider the special case with an axial symmetry that $|T_{0,x}|=|T_{0,y}|$ with the aid of numerical calculation. Then, the relation between $|T_{0,x}|$ and $|T_{0,z}|$ can be written as a simple formula [@JOSAB2015Steering]. And the orientation of the icosahedron can be represented by the intersection of Z-axis with the surface of the icosahedron. There are three types of special points on the surface, which are vertices, midpoints of edges and centre of faces. We suspect that the maximum of $S$ occurs at these special points. ![ (Color online) The icosahedron in the construction of LHS models in *Protocol 1*. Dashed blue lines show intersections of Z-axis with the surface of icosahedron during our rotation. Vertices, midpoints of edges and centre of faces on the dashed blue lines are marked as $A_i$, $B_j$ and $C_k$ respectively. []{data-label="icos"}](figIcos.eps){width="6cm"} ![ (Color online) The solid curve shows the maximum of $t$ in the LHS models based on the icosahedron, in company with the values for ten thousand random orientation, and the dashed line is for the value of the Werner state. []{data-label="figt"}](figt.eps){width="8.34cm"} ![ (Color online) The solid curve shows the shared randomness in the LHS model based on the icosahedron in optimal orientation, and the dashed line is for the value of the Werner state. []{data-label="figEntr"}](figEntr.eps){width="8cm"} ![ (Color online) The solid curve shows entanglement of T-states admitting the optimal LHS model based on the icosahedron, the dashed line is for the value of the Werner state, and the dot-dashed curve is for T-states on the EPR-steerable boundary. []{data-label="figEntangle"}](figEntangle.eps){width="8.35cm"} Choosing a trajectory of the intersection, as shown in Fig. \[icos\], consisting of an edge and two medians of faces, one can plot the values of $S$ versus the location of intersection (we omit the figures here). These curves indicate that the maximum of $S$ on the trajectory occurs at vertices when $|T_{0,z}|\leq 1/2$ , at the centre of faces when $1/2<|T_{0,z}| \lesssim 0.89$, or at midpoints of edges [when]{} $ |T_{0,z}| \gtrsim 0.89$. These maximums, in the same sequence, can be analytically expressed as $$\begin{aligned} && S_A = \frac{6}{ \sqrt{Z} + \sqrt{ 20 X + 5 Z } }, \\ && S_C = \frac{ \sqrt{30}} { \sqrt{X {\alpha_+} + Z {\beta_-} } + \sqrt{X {\alpha_- } + Z {\beta_+} }},\\ && S_B = \frac{3\sqrt{10}}{\sqrt{ 10X} + \sqrt{ X {\alpha_+}+ Z {\alpha_-} } + \sqrt{ X {\alpha_-} +Z {\alpha_+} } }, \ \ \ \ \ \\end{aligned}$$ where $X=T_{0,x}^2$, $Z=T_{0,z}^2$, $\alpha_{\pm}=5 \pm \sqrt{5}$ and $\beta_{\pm}=5/2 \pm \sqrt{5} $. We find that they are optimal among arbitrary orientations of the icosahedron, by comparing them with one hundred thousand randomly generated intersections. One-tenth of the random points are shown in Fig. \[figt\], in company with the corresponding maximums of $t$. Our construction provides a family of LHS models with a fixed dimensionality of the local variable. It is interesting to observe the continuously changing shared randomness, and its relation with the region of T-states admitting our models. We plot the maximums of $t$ in Fig. \[figt\], which measure how close our models get to the EPR-steerable boundary. The corresponding shared randomness, measured by the entropy of the distribution (\[omegaT\]) [@PRL2005LHVBit], [ $H=-\sum_i q_i \log_2 q_i$ with $q_i= |T_0\vec{v}_i|S/12$, ]{} is shown in Fig. \[figEntr\]. Obviously, the visibility parameter and entropy show two opposite trends. The anisotropy of the correlation matrix enhances the maximums of $t$, while it decreases the entropy. Among the family of T-states, the LHS model for the Werner state, with the maximum distance to the EPR-steerable boundary, requires the most shared randomness. This anomalous phenomenon prompts us to go back to the original point: the cost of classically simulating the correlations of entangled states [@PRL2005LHVBit]. It is direct to derive the entanglement, [measured]{} by concurrence [@Wootters98], for axial symmetric T-states as $\max \{0, (2t|T_{0,x}|+t|T_{0,z}|-1)/2\}$. As shown by the solid line in Fig. \[figEntangle\], the entanglement reveals a similar tend as the number of classical bits to [simulate]{} it. The degree of entanglement reaches its maximum at the point of Werner states, $T_{0,z}=1/2$, and decreases with the anisotropy. Comparing with the maximums of $t$, one can find that the points on the EPR-steerable boundary with small entanglement are easy to approach, in the sense of the cost of classically simulating the correlations of entangled states. In Fig. \[figEntangle\], a noteworthy point is the small interval with zero entanglement. This indicates that our LHS models are not the most economical ones, at least for the separable state in the small interval. This is because a $4$-dimensional local variable is sufficient to simulate a separable two-qubit state, while the least shared bits in our construction for T-states is $2.96$. We shall present more discussion about the LHS models for separable T-states below. Separable boundary ------------------ The above results can be straightforwardly extended to any LHS model for the Werner state in *Protocols 1 and 2* of [@PRL2005LHVBit] using a $3$-dimensional polyhedron with $D$ vertices. We omit these formulas for brevity. In this part, we focus on the case with a shared variable of dimension $D=4$. In the results for Werner state [@PRL2005LHVBit], the tetrahedron, with $4$ vertices, is without inversion symmetry and hence be excluded from *Protocols 1 and 2*. On the other hand, the maximum visibility parameter one can simulate with $D=4$ is the boundary of the separable Werner state [@PRL2005LHVBit]. Here our question is *whether one can generate the boundary of the separable T-states from the one of the Werner state, as we do in the study of EPR-steering [@zhang2017LHS]*. To answer the above question, we restrict the response function to the form $$f(\vec{x},\vec{\lambda})=\vec{x}\cdot \vec{\eta} ,$$ where $\vec{\eta} $ is Alice’s Bloch vector depending on $\vec{\lambda}$. We term the corresponding LHS model as a *LHS model for separable state*. The entanglement of the two-qubit state $\rho_{AB}$ is demonstrated by the inexistence of a LHS model with $ f(\vec{x},\vec{\lambda})$ in the above form. On can derive the solution for Werner states to the conditions (\[Reqs\]), by decomposing the critical separable Werner state into four product states. Let the Bell states $|\Psi_{\pm}\rangle=(|00\rangle\pm|11\rangle)/\sqrt{2}$ and $|\Phi_{\pm}\rangle=(|01\rangle\pm|10\rangle)/\sqrt{2}$. The critical separable Werner state is $\rho_{AB}^w=(3 |\Phi_-\rangle \langle\Phi_-|+|\Phi_+\rangle \langle\Phi_+|+|\Psi_+\rangle \langle\Psi_+|+|\Psi_-\rangle \langle\Psi_-| )/6$. We assume the normalized state $$|\phi_i\rangle \propto \sqrt{3}|\Phi_-\rangle + e^{i \theta_{i1}} |\Phi_+\rangle + e^{i \theta_{i2} }|\Psi_+\rangle + e^{i \theta_{i3}} |\Psi_-\rangle,$$ to be separable, and to satisfy $\rho_{AB}^w=\sum^4_i |\phi_i\rangle\langle \phi_i|/4$. There exist two solutions to these conditions, one of which is given by $$\begin{aligned} | \phi_1 \rangle \! =\! (\sin\! \frac{\alpha}{2}\! | 0 \rangle \!-\! \cos\! \frac{\alpha}{2}\!e^{i \beta}\!| 1 \rangle )\! \otimes \! (\cos\! \frac{\alpha}{2}\!| 0 \rangle \!+\! \sin\!\frac{\alpha}{2}\! e^{i\beta}\!| 1 \rangle),\end{aligned}$$ $| \phi_2 \rangle=\sigma_x\otimes\sigma_x | \phi_1 \rangle$, $| \phi_3 \rangle=\sigma_y\otimes\sigma_y | \phi_1 \rangle$, and $| \phi_4 \rangle=\sigma_z\otimes\sigma_z | \phi_1 \rangle$ , where $\alpha=\arccos (1/\sqrt{3})$ and $\beta=-{\pi}/{4}$. Alice’s measurements on the decomposition $\rho_{AB}^w=\sum^4_i |\phi_i\rangle\langle \phi_i|/4$ lead to a LHS model using the tetrahedron. Namely, it is defined by $$\begin{aligned} \omega(\vec{\lambda})=\sum_{i} \frac{1}{4} \delta(\vec{\lambda}-\vec{v}_i) ,\ \ \ \vec{\eta} =-\vec{\lambda},\end{aligned}$$ with the $4$ vertices of the tetrahedron $\vec{v}_1=(1,-1,1)/\sqrt{3}$, $\vec{v}_2=(1,1,-1)/\sqrt{3}$ , $\vec{v}_3=(-1,1,1)/\sqrt{3}$ and $\vec{v}_4=(-1,-1,-1)/\sqrt{3}$. They satisfy $$\begin{aligned} \label{SepW} \int \biggr[\sum_{i} \frac{1}{4} \delta(\vec{\lambda}-\vec{v}_i)\biggr] \biggr[ \vec{x}\cdot(-\vec{\lambda})\biggr] \vec{\lambda} d\vec{\lambda}=-\frac{1}{3}\vec{x}.\end{aligned}$$ The other solution leads to a model on the mirror image of the tetrahedron. We now turn to the T-states on the separable boundary. It is universal to consider a positive definite $T$, as any minus sign can be merged into $\vec{\eta}(\vec{\lambda})$. Here we perform $T^{-\frac{1}{2}}$ on the condition (\[Tx\]), and define the unit vector $ \vec{\lambda }'' = {{T^{-\frac{1}{2}}\vec{\lambda} }}{ | {T^{-\frac{1}{2}}\vec{\lambda} } |^{-1} } $ and its distribution $\omega'' ( \vec{\lambda}'') $. Then the condition (\[Tx\]) for a separable state is rewritten as $$\label{Tx3} \int \omega'' ( \vec{\lambda}'' ) \frac{1}{ | T^{\frac{1}{2}}\vec{\lambda}'' | }\bigr[(T^{-\frac{1}{2}} \vec{x})\cdot\vec{\eta} \bigr] \vec{\lambda}'' d\vec{\lambda}'' = \vec{x}.$$ Defining the unit vector $ \vec{\eta }'' = {{T^{-\frac{1}{2}}\vec{\eta} }}{ | {T^{-\frac{1}{2}}\vec{\eta} } |^{-1} } $, one can find that $ (T^{-\frac{1}{2}} \vec{x})\cdot\vec{\eta} = \vec{x} \cdot \vec{\eta }'' | T^{\frac{1}{2}}\vec{\eta}'' |^{-1}$. From the integral (\[SepW\]), it was easy to find a pair of $ \omega'' ( \vec{\lambda}'' ) $ and $\vec{\eta}''$ satisfying (\[Tx3\]) as $$\omega'' ( \vec{\lambda}'' ) =\sum_{i} \frac{3}{4} \delta(\vec{\lambda}''-\vec{v}_i)| T^{\frac{1}{2}}\vec{v}_i |^{2},\ \ \ \vec{\eta}''=\vec{\lambda}''.$$ The normalization condition (\[omega1\]) and the coordinates of [$\vec{v}_i$]{} lead to $$| T_x|+|T_y|+|T_z|=1,$$ which is nothing but the separable boundary of T-states [@horodecki1996information; @dakic2010necessary]. Then, in the space of $\lambda$, the distribution and Bloch vector of Alice are $$\label{omegaTS} \omega ( \vec{\lambda}) =\sum_{i} \frac{1}{4} \delta\bigr(\vec{\lambda}-\sqrt{3}T^{\frac{1}{2}} \vec{v}_i\bigr),\ \ \ \vec{\eta}=\vec{\lambda}.$$ Substituting them into the equations (\[a0\]) and (\[b0\]), one can confirm both the integrals to be zero. In the LHS models for separable T-states, defined by (\[omegaTS\]), the shared variables are encoded on $\sqrt{3}T^{\frac{1}{2}} \vec{v}_i$, and are uniformly distributed. The amount of shared randomness is $2$ bits, which is not affected by the anisotropy of the correlation matrix. The models are optimal in the sense of reaching the separable boundary. However, the question as to whether they are the most economical remains open. Summary ======= We study LHS models for T-states using finite shared randomness. The models are generated from the ones for Werner states, two of which are mainly discussed. The first is derived by using our recent approach [@zhang2017LHS] on the most economical model for the Werner state. It provides an example to observe the continuously changing shared randomness with an entangled state. With the increase of anisotropy, the amount of shared classical bits drops along with entanglement, although the model gets closer to the EPR-steerable boundary. The second one is restricted to simulate a separable state by a condition on Alice’s response function. It is derived from the one for the Werner state by a generalized generating approach, and reaches exactly the separable boundary of T-states. The cost of classical randomness in this model is $2$ bits, which is not affected by the anisotropy of the correlation matrix. It would be interesting to consider the open questions or extensions below. First, our approach to derive the LHS models for T-states on the separable boundary is actually to decompose them into product states. Geneneralizing this method may be a starting point to define T-states in higher-dimensional systems, which has been raised in our recent work [@zhang2017LHS]. Second, in what region our model using the icosahedron is the most economical one? Third, what is the minimal cost to classically simulate a separable state? This is a nontrivial question, as in LHS models on the separable boundary, the amount of bits is different from the entropy of states. This difference originates from the superposition of states in composite quantum systems, and may be interpreted as a kind of quantum correlation. This work is supported by the NSF of China (Grant No. 11675119, No. 11575125 and No. 11105097). [^1]: Corresponding author: flzhang@tju.edu.cn [^2]: EPL, 127 (2019) 20007

--- abstract: | We revisit the Anderson-Hasegawa double-exchange model and critically examine its exact solution when the core spins are treated quantum mechanically. We show that the quantum effects, in the presence of an additional superexchange interaction between the core spins, yield a term, the significance of which has been hitherto ignored. The quantum considerations further lead to new results when polaronic effects, believed to be ubiquitous in manganites due to electron-phonon coupling, are included. The consequence of these results for the magnetic phase diagrams and the thermal heat capacity is also carefully analysed. PACS No. 63.20. K, 71.38, 75.30. E\ author: - 'Manidipa Mitra, P. A. Sreeram and Sushanta Dattagupta' title: 'Quantum Treatment of the Anderson-Hasegawa Model – Effects of Superexchange and Polarons' --- 24 true pt Introduction ============ The emergence of manganites as a technologically important material due primarily to the occurrence of colossal magnetoresistance (CMR) in e.g. $La_{1-x}Ca_xMnO_3$ [@jin] has rekindled the interest of the condensed matter physics community in the double-exchange mechanism. The physics of double-exchange has successfully correlated ferromagnetism and metallicity in the doping range of $0.2 < x < 0.4$. The basic ingredient of double-exchange, proposed by Zener fifty years ago [@zen], is encapsulated within a simple two site model of Anderson and Hasegawa [@and; @degen; @kubo]. The latter has led to a plethora of theoretical and experimental investigations in recent years [@rev1; @rev2], which have gone on to add one or the other feature to the original model, often without abundant care, as we shall argue here. The undoped manganite system is characterised by the presence of an incomplete d-shell of the Manganese ions, which consists of 3 electrons in the $t_{2g}$ state and one electron in the $e_g$ state. The $t_{2g}$ spins are deep inside the d-level and are assumed to be unimportant in the process of charge transfer. However, these spins do show a tendency to align antiferromagnetically both in the parent compound and in the completely doped system. The Coulomb energy cost for the $e_g$ electrons to hop onto the adjacent Manganese ion in the absence of a hole is very large. The undoped system is thus an insulator. We will henceforth refer to the $e_g$ electron as the itinerant electron, the $t_{2g}$ electrons as the core electrons and the total spin in the $t_{2g}$ level as the core spin. Doping the system with a divalent atom like $Ca$ or $Sr$ leads to creation of holes in the $e_g$ level in a fraction of the Manganese ions. The itinerant electron from the occupied $e_g$ level of one Manganese site can then hop into its nearest neighbor Oxygen site, which facilitates hopping to the nearest neighbor unoccupied $e_g$ level of another Manganese ion, thus leading to a finite conductivity. This process, called the “double exchange mechanism", plus the presence of strong Hund’s rule coupling between the core spin and the itinerant electron, results in an indirect coupling between neighboring core spins . This in turn relates the magnetic order of the underlying lattice with the kinetic energy of the itinerant electron. The conclusions derived by Anderson and Hasegawa [@and] can be summarised as follows. In the limit when Hund’s coupling is infinitely strong, the itinerant electron would like to have its spin aligned with the local core spin. Additionally, if the core spin is treated , the appropriate axis of quantization is the direction of the core spin vector, as far as the itinerant electron is concerned. Now, as the latter hops, it has to readjust its spin to be realigned with the new core spin partner, amounting to a rotation of the quantization axis by an angle $\theta$, which is the polar angle between the core spins $\vec{S_1}$ and $\vec{S_2}$. From the property of the spin-1/2 rotation operator it follows that the hopping or the overlap matrix element $t$ will be renormalized to $t \cos(\theta/2)$, assuming azimuthal symmetry. It then follows that if $\theta$ equals $\pi$, the core spins have antiferromagnetic (AFM) coupling and hopping of the itinerant electron is totally inhibited. On the other hand, if $\theta$ equals zero, the core spins have ferromagnetic (FM) coupling and hopping is accentuated, thus synergising transport with ferromagnetism. This is the first result of Anderson and Hasegawa. The latter then proceeded to a quantum treatment of the core spins, but still operating within the infinite Hund’s coupling limit. Interestingly, it turns out that the energy eigenvalues are identical to the earlier classical case, provided $\cos(\theta/2)$ is identified as a which equals $(S_0+1/2) /(2S+1)$, where $S$ = $\mid \vec{S_1}\mid$ = $\mid \vec{S_2}\mid$ and $S_0$ = $\mid \vec{S_1} + \vec{S_2} + \vec{\sigma} \mid$, $\vec{\sigma}$ being the spin of the itinerant electron ($\mid \vec{\sigma} \mid$ = 1/2). Curiously, since $\vec{S_1}$, $\vec{S_2}$ and $\vec{\sigma}$ add up to $(2S+1/2)$ in the FM case for large Hund’s coupling, the parameter $\cos(\theta/2)$ would indeed be equal to unity, as in the classical case. But in the AFM case, the parameter reduces to $1/(2S+1)$, which goes to the classical value of zero only when $S\rightarrow \infty$, thus necessitating an additional constraint on the core spin, if the “classical" interpretation is to be taken seriously. Needless to say, in between FM and AFM cases, the parameter $\cos(\theta/2)$ would go through a set of discrete (and not continuous) values, thus pointing to the need of a more careful treatment of the core spins in the quantum case. One of the directions in which the Anderson-Hasegawa treatment has been extended is to recognize the importance of an additional superexchange term between the core spins proportional to $\vec{S_1}.\vec{S_2}$. It has been assumed in the literature till now that the superexchange term can be simply taken as an extra term to be added to the Hamiltonian and that the large Hund’s rule coupling affects only the process of charge transfer in these systems. In this paper we show inter alia that superexchange is itself modified in a nontrivial manner if the core spins are dealt with quantum mechanically, which leads to a correlated diagonal disorder in these systems, even in the cleanest samples. Our starting point therefore is the Hamiltonian for a two site one electron model, including the superexchange interaction, given by $$H=-t\sum_\tau (c_{1\tau }^{\dagger }c_{2\tau }+h.c.)-J_{H}\sum _{i=1}^{2} \vec{S_{i}}.\vec{\sigma _{i}} + J \vec{S_1}.\vec{S_2}.$$ Here $t$ is the hopping matrix element for the itinerant electron between the two sites, $c_{i\tau }^{\dagger }$ ($c_{i\tau }$) is the creation(annihilation) operator of the itinerant electron at site $i$ having spin projection $\tau$, $J_{H}$ is the Hund’s rule coupling strength, $\vec{S_{i}}$ is the core spin at site $i$ and $\vec{\sigma _{i}}$ is the spin of the itinerant electron at the site $i$. The parameter $J$ is the superexchange interaction strength between the core spins in the nearest neighbor sites. For our case we consider $\mid \vec S_i \mid = S$, i.e. the core spins on all the sites are taken to have the same value. With the preceding background, the motivation behind our work and the plan of this paper are as follows. We reiterate that the two site, single electron, double-exchange model is the simplest basic framework for interpreting a large number of fascinating properties of manganites. While Anderson and Hasegawa did provide a quantum solution to the model, especially in the limit of large Hund’s rule coupling, subsequent authors seem to have gone ahead in a somewhat cavalier fashion, in our opinion, about the classical limit of infinitely large core spins. This has led to some confusion about the interpretation of parameters in the model which needs to be cleared. As the value of the core spins in most studied CMR systems is indeed finite — three-halves for manganese — it is important to delineate the quantum versus classical effects, especially while considering additional phenomena, e.g., polaron-induced hopping and thermodynamics. With this aim in mind, we organize the paper as follows. We present in Sec. II, the exact quantum mechanical solution to the Anderson-Hasegawa model with an additional superexchange term, take the large Hund’s rule coupling limit and discuss the outcome of a hitherto ignored ‘site-disorder’ term. In Sec. III we reexamine the issue of polaron-assisted hopping in the light of our fully quantum calculation. The results in this section are then employed in Sec. IV for the computation of heat capacity and phase diagram, wherein we also specify the difference between our results and those which treat the core spins classically. Finally, in Sec. V, we present some concluding remarks. Exact solution for the Anderson-Hasegawa Model ============================================== In order to find the ground state of the system we follow the quantum mechanical calculation carried out by Anderson and Hasegawa [@and]. We first note that the Hund’s rule coupling term, proportional to $J_H$ is diagonal in the states given by $\mid \psi_1^{\pm}\rangle$ = $\mid S_1,\frac{1}{2},(S_1 \pm \frac{1}{2}),S_2;S_0,M>$ and $\mid \psi_2^{\pm}\rangle$ = $\mid S_1,\frac{1}{2},S_2,(S_2 \pm \frac{1}{2});S_0,M>$, while the hopping part of the Hamiltonian connects these two sets of states, corresponding as it does to a recoupling of the itinerant electron’s spin (1/2) from the site spin $S_1$ to $S_2$ and is thus given by the Wigner 6j (or Racah) coefficient (W)[@racah]. Here, $M$ = $S_0^z$. The superexchange term proportional to $J$, is off-diagonal in the basis states chosen above. However, it is diagonal in the states given by $\mid \phi(S^\prime) \rangle$ = $\mid \frac{1}{2},S_1,S_2,(S^\prime);S_0,M \rangle$, where $S^\prime$ = $\mid \vec{S_1}+\vec{S_2}\mid$. We can then relate the states $\mid \psi_{1,2}^{\pm} \rangle$ to the states $\mid \phi(S^\prime) \rangle$ through appropriate Racah coefficients again and we find, $$\begin{aligned} \mid \psi_{1}^{\pm} \rangle &=& \sum_{S^\prime} \sqrt{\left[2(S_1+\frac{1}{2}) +1\right]} \sqrt{(2S^\prime+1)} ~W\left(\frac{1}{2} S_1 S^\prime S_2;(S_1+ \frac{1}{2})S^\prime\right) \mid \phi(S^\prime)\rangle , \nonumber\\ \mid \psi_{2}^{\pm} \rangle &=& \sum_{S^\prime} \sqrt{\left[2(S_2+\frac{1}{2}) +1\right]} \sqrt{(2S^\prime+1)} ~W\left(\frac{1}{2} S_1 S^\prime S_2;(S_2+ \frac{1}{2})S^\prime\right) \mid \phi(S^\prime)\rangle. \label{trans-1} \end{aligned}$$ Clearly, since $S^\prime$ (the total core spin) must couple to the itinerant electron spin to give the total angular momentum $S_0$, the only values of $S^\prime$ to be summed over are $S^\prime$ = $S_0+\frac{1}{2}$ and $S_0- \frac{1}{2}$ . The particular Racah coefficients which occur (with $S_1=S_2=S$), have convenient closed expressions such as, $$\begin{aligned} \mid \psi_1^+ \rangle &=& \cos(\alpha/2) \mid \phi(S_0-\frac{1}{2})\rangle + \sin(\alpha/2) \mid \phi(S_0+\frac{1}{2}) \rangle , \nonumber \\ \mid \psi_1^- \rangle &=& -\sin(\alpha/2) \mid \phi(S_0-\frac{1}{2})\rangle + \cos(\alpha/2) \mid \phi(S_0+\frac{1}{2})\rangle , \nonumber \\ \label{transform-2}\end{aligned}$$ where, $$\cos(\alpha/2)= \left[\frac{\left(2S+S_0+\frac{3}{2}\right)} {2(2S+1)}\right]^{1/2}.$$ The relations between $\mid \psi_2^{\pm}\rangle$ and $\mid \phi(S_0\pm\frac{1} {2})\rangle$ states are the same as those between $\mid \psi_1^{\pm}\rangle$ and $\mid \phi(S_0\pm\frac{1}{2})\rangle$. Note that the expression for $\cos(\alpha/2)$ can be written in terms of $\cos(\theta/2)$, which actually gives the relation between $\alpha$ and $\theta$ as $\alpha = \theta/2$. Thus the Hamiltonian matrix in the space of $\mid \psi_1^\pm \rangle$ and $\mid \psi_2^\pm \rangle$ can be written as follows : $$\left ( \begin{array}{cccc} P_1 & P_2 & -t \cos(\theta/2) & -t \sin(\theta/2) \\ &&&\\ P_2 & P_3 & t \sin(\theta/2) & -t \cos(\theta/2) \\ &&&\\ -t \cos(\theta/2) & t \sin(\theta/2) & P_1 & P_2 \\ &&&\\ -t \sin(\theta/2) & -t \cos(\theta/2) & P_2 & P_3 \end{array} \right),$$ where, $$\begin{aligned} P_1 &=& \frac{J}{2}\left(R_1 \sin^2(\alpha/2) + R_2 \cos^2(\alpha/2) \right) -\frac{J_H}{2} S, \nonumber \\ P_2 &=& \frac{J}{2}(R_1-R_2) \cos(\alpha/2)\sin(\alpha/2),\nonumber \\ P_3 &=& \frac{J}{2}\left(R_1 \cos^2(\alpha/2) + R_2 \sin^2(\alpha/2)\right) +\frac{J_H}{2} (S+1), \end{aligned}$$ and, $$\begin{aligned} R_1 = \left ( S_0+\frac{1}{2}\right ) \left (S_0 + \frac{3}{2} \right )- 2 S (S+1), \nonumber \\ R_2 = \left ( S_0-\frac{1}{2}\right ) \left (S_0 + \frac{1}{2} \right )- 2 S (S+1). \end{aligned}$$ The eigenvalues (E) of the Hamiltonian matrix are obtained from, $$2 E= \frac{J_H}{2}+K_1(J) \pm \sqrt{4 t^2 + K_T^2(J) + K_3^2(J) \pm 4 t \cos(\theta/2)\sqrt{ K_T^2(J)+ K_3^2(J)}}, \label{exact-1}$$ where, $$\begin{aligned} K_1(J)&=& J\left[\left( S_0+\frac{1}{2}\right)^2-2S(S+1)\right] \nonumber \\ K_T(J)&=&\left[\frac{J_H(2S+1)}{2}+ K_2(J)\right] \\ K_2(J)&=& J\cos(\alpha) (S_0+\frac{1}{2}) \\ K_3(J)&=& J\sin(\alpha) (S_0+\frac{1}{2}) .\end{aligned}$$ While the exact result of Eq.(\[exact-1\]) may be of interest in its own right, we examine the limit of large Hund’s rule coupling, by expanding the square root term (upto $O(1/J_H)$). We find that the lowest energy eigenvalues are given by $$E_m = -\frac{J_H S}{2}-t\cos(\theta/2)+\frac{J}{2}\left[\bar{S}^\prime (\bar{S}^\prime+1)-2S(S+1)\right]+J\sin^2(\alpha/2)(S_0+\frac{1}{2}), \label{emin-1}$$ where $\bar{S}^\prime = S_0 - 1/2$. The first two terms in Eq. (\[emin-1\]) are the terms obtained by Anderson and Hasegawa. We emphasise once again that the parameters $\cos(\theta/2)$ and $\cos(\alpha/2)$ take values which depend on the quantum values of the core spins. The third term is the result of the superexchange interaction in the absence of the itinerant electron. The fourth term, a novel one, is purely due to the double-exchange mechanism in the presence of the itinerant electron. The explicit form of this term is given by, $$\Delta E_J = \frac{J}{2}\frac{2S-\bar{S}^\prime}{2S+1}(\bar{S}^\prime+1).$$ It is to be noted that in the ferromagnetic limit (i.e. $\bar S^\prime = 2S$) $\Delta E_J$ vanishes exactly. There are primarily three important points to be made about $\Delta E_J$ : \(a) This term is an on-site term : It does not involve physical transfer of the itinerant electron from one site to the other. \(b) This term vanishes in the absence of the itinerant electron. Thus, it exists only on the site at which the electron resides and hence, at any site $i$, it will be proportional to $n_i$, where $n_i$ is the number of itinerant electrons at the site $i$ and is taken to be either 1 or 0 in the absence of double occupancy. \(c) At any site $i$ this term depends on 3 spin values : the spin of the itinerant electron on the site $i$ ($\sigma_i$), the core spin at site $i$ ($S_i$) and the core spin on the neighboring site ($S_j$), $j$ being the neighbor of the site $i$. Thus, the extra energy term $\Delta E_J$ corresponds to a site energy term which is correlated with the core spins of the nearest neighbors. We may therefore propose an effective double-exchange Hamiltonian in the full lattice as $$H_{{\rm eff}}= \sum_i \epsilon_i n_i -t \sum_{<ij>} \cos(\theta_{ij}/2) (c_i^\dagger c_j + H.C.) + \sum_{<ij>} J_{ij} \vec{S_i}.\vec{S_j} ,$$ where $$\epsilon_i = \sum_j \frac{J_{ij}}{2}\frac{2S-S^\prime_{ij}}{2S+1} (S^\prime_{ij}+1),$$ and $$S^\prime_{ij} = \mid \vec{S_i} + \vec{S_j} \mid .$$ We have explicitly taken $J_{ij}$ in order to accomodate effects of anisotropic superexchange also. The classical limit of the extra term is given by $$\Delta E_J^{Cl} = \frac{J}{2} (2S+1)\left[1-\cos(\theta/2)\right]\cos(\theta/2) ,$$ which goes to zero in both the ferromagnetic as well as the antiferromagnetic limits. Again, we see a clear distinction between the classical and the quantum results in the antiferromagnetic limit. We emphasise that in taking the purely classical expression, one actually loses the effect of the quantum fluctuations which are present in these systems not only because of the fluctuating spins but also due to the on site disorder and the hopping, correlated with the spins on the lattice. A variety of interesting physical phenomena could be studied by taking into consideration the quantum Hamiltonian. One of these concerns the polaron effects, which are discussed below in Section III. Anderson-Hasegawa-Holstein Model ================================ Experiments in manganites - both thermodynamic and transport - seem to suggest the importance of polaron formation and the consequent localization of charge carriers [@millis95]. The minimal model which reflects such lattice carrier interaction on the double-exchange can be introduced by dovetailing the Holstein mechanism on the Anderson-Hasegawa Hamiltonian. Therefore, in view of the results presented in Sec. II, in the limit of large Hund’s rule coupling, we may write a two site Anderson-Hasegawa-Holstein Hamiltonian as, $$\begin{aligned} H &=& \epsilon \sum_{i=1}^2 \sum_{\sigma} n_{i \sigma} - \sum_{\sigma} t\frac{S_0 + \frac{1}{2}}{2 S + 1}(c_{1\sigma}^{\dag} c_{2 \sigma}+ c_{2 \sigma}^{\dag} c_{1 \sigma}) + g_1 \omega_0 \sum_{i=1}^2\sum_{\sigma} n_{i \sigma} (b_i + b_i^{\dag})\nonumber \\ &+& g_2 \omega_0 \sum_{\sigma} \left[ n_{1 \sigma} (b_{2} + b_{2}^{\dag}) +n_{2 \sigma} (b_1 + b_1^\dagger)\right] + \omega_0 \sum_{i=1}^2 b_i^{\dag} b_i + J \vec S_{1}.\vec S_{2} +\Delta E_J, \label{qdh-1}\end{aligned}$$ where, $g_{1}(g_{2})$ denotes the on-site (intersite) electron-phonon coupling strength and $\epsilon $ is the bare site energy. This site energy is to be contrasted with the site energy which was derived in the previous section. Here, since we are interested in doing a two-site model, the extra term in the Hamilonian due to the superexchange interaction, has been introduced simply as $\Delta E_J$. Note that we have considered a single phonon mode for interatomic vibrations of frequency $\omega_0$ for which $b_i$ and $b_{i}^{\dag}$ are the annihilation and creation operators. We separate out the in-phase mode and the out-of-phase mode by introducing new phonon operators $a=~(b_1+b_2)/ \sqrt 2$ and $d=~(b_1-b_2)/\sqrt 2 $ in the Hamiltonian. The in-phase mode does not couple to the electronic degrees of freedom whereas the out-of-phase mode does, leading to a Hamiltonian $H_d$, given by, $$H_d = \omega_0 d^\dagger d +\epsilon \sum_{i=1}^2 n_i - t\left(\frac{S_0+\frac{1}{2}}{2S+1}\right) (c_1^\dagger c_2 + h.c.) + g_-\omega_0(n_1-n_2)(d+d^\dagger) +J \vec S_1.\vec S_2 + \Delta E_J,$$ which represents an effective electron-phonon system. Following [@jayee] we use a Modified Lang-Firsov (MLF) transformation with variable phonon basis and obtain, $$\begin{aligned} \tilde{H_d}&=& e^R H_d e^{-R}\nonumber\\ &=&\omega_0 d^{\dag} d + \sum_{i} \epsilon_p n_{i} - t\frac{S_0 + \frac{1}{2}}{2 S + 1} ~ [c_{1}^{\dag} c_{2}~ \exp (2 \lambda (d^{\dag}-d)) + c_{2}^{\dag} c_{1}~\exp(-2 \lambda (d^{\dag}-d))] \nonumber \\ &+& \omega_0 (g_- - \lambda)(n_1 - n_2)(d + d^{\dag}) + J \sum_{<ij>} \vec S_{i}.\vec S_{j} + \Delta E_J , \label{qdh-2}\end{aligned}$$ where $R =\lambda (n_1-n_2) ( d^{\dag}-d)$, $\lambda$ is a variational parameter related to the displacement of the $d$ oscillator, $g_{-}=(g_1-g_2)/\sqrt 2$ and $\epsilon_p = \epsilon - \omega_0 ( 2 g_{-} - \lambda) \lambda$. The basis set is given by $|\pm,N \rangle = \frac{1}{\sqrt 2} (c_{1}^{\dag} \pm c_{2}^{\dag})$ $|0\rangle_e |N\rangle$, where $|+\rangle$ and $|-\rangle$ are the bonding and the antibonding electronic states and $|N\rangle$ denotes the $N$th excited oscillator state within the MLF phonon basis. The diagonal part of the Hamiltonian $\tilde{H_d}$ in the chosen basis is treated as the unperturbed Hamiltonian ($H_0$) and the remaining part of the Hamiltonian $H_{1}= \tilde{H_{d}}-H_0$, as the perturbation. The unperturbed energy of the state $| \pm,N\rangle$ is given by $$\begin{aligned} E_{\pm,N}^{(0)}&=& \langle N,\pm|H_0|\pm, N \rangle\nonumber\\ &=& N \omega_0 + \epsilon_p \mp t_{eff} \left[ \sum_{i=0}^{N} \frac{(2\lambda)^{2i}}{i!} (-1)^i N_{C_i}\right] +J\vec S_1 . \vec S_2 +\Delta E_J\end{aligned}$$ where $t_{eff}=t~\frac{S_0 + \frac{1}{2}}{2 S + 1} ~\exp{(-2\lambda^2)}$, $N_{C_i}=\frac{N!}{(N-i)!~~ i!}$. The general off-diagonal matrix elements of $H_1$ between the two states $|\pm,N \rangle$ and $|\pm,M \rangle$ may be calculated for $(N-M)>0$ as in Ref. [@jayee]. The unperturbed ground state is the $|+\rangle|0\rangle$ state and the unperturbed energy, $ E_0^{(0)}=\epsilon_p - t_{eff} + J \vec S_1.\vec S_2 + \Delta E_J $. However, in this exact quantum limit of core spins, for given values of $g_-$ and $J$, $E_0^{(0)}$ can have four values corresponding to ferromagnetic (FM), canted 1 (CA1), canted 2 (CA2) and antiferromagnetic (AFM) orientation of the two spins for $\mid \vec S_{12} \mid = \mid \vec S_1+ \vec S_2 \mid = 3,2,1,0$ respectively. Minimizing the unperturbed ground state energy $\lambda $ is calculated and is given by $$\begin{aligned} \lambda&=&\frac {\omega_0g_{-}}{\omega_0+2t_{eff} } .\end{aligned}$$ We have evaluated the perturbation correction to the energy upto the sixth order and the wave function upto the fifth order. The convergence of the perturbation series is very good for $t/\omega_0 \le 1$. To obtain the ground state spin order of the core spins we calculate the energy for each set of values of $g_{-}$ and $J$ with four possible $\vec S_{12}$ and find out the combination for which the energy is the minimum. Further, to study the effect of an external magnetic field ($\vec h$) we include a term $- \tilde g \mu_B (\vec S_1 + \vec S_2). \vec h$ to the Hamiltonian in equation (\[qdh-1\]), $\tilde g$ being the Lande g factor. We assume that the external magnetic field is along the direction of $\vec S_{12}$ and is expressed in units of $\mu_{eff}(=\tilde g \mu_B)$=1. It is expected that the charge transfer from site ‘1’ to ‘2’ depends on the spin order of the core spins as well as the electron-phonon interaction. In the double-exchange model, the effective hopping reaches its maximum value in the ferromagnetic state and decreases as it approaches the antiferromagnetic limit. Moreover, in a lattice, the electron produces lattice deformations and which in turn localize the electron for strong electron-phonon coupling. To study the polaronic character one calculates the static correlation functions $\langle n_1 u_{1}\rangle_{0}$ and $\langle n_1 u_{2}\rangle_{0}$, where $u_1$ and $u_2$ are the lattice deformations at sites 1 and 2 respectively, produced by an electron at site 1 [@jayee; @euro]. In the present report with a two-site one electron model, following [@euro], we calculate $- <n_1(u_1-u_2)>_0/g_{-}=\frac{\lambda^{corr}}{g_{-}}$ and study the nature of the polaron crossover for different ranges of $g_{-}$ and $J$. In the ‘large’ polaron limit this parameter takes a small value, while with increasing electron-phonon coupling it tends to unity, showing a distinct crossover from ‘large’ to ‘small’ polaron behavior. The measure of delocalization of the electron for various ranges of $g_{-}$ as well as $J$ will be evident from the kinetic energy. So we have also calculated the kinetic energy, given by, $$t_{eff}^{KE}=-E_{Kin}=<\psi_G| t \frac{S_0 + \frac{1}{2}}{2 S + 1} ~ [c_{1}^{\dag} c_{2}~ \exp (2 \lambda (d^{\dag}-d)) + c_{2}^{\dag} c_{1}~\exp(-2 \lambda (d^{\dag}-d)) ~] |\psi_G>,$$ where $\psi_{G}$ is the ground state wave-function, is evaluated upto the fifth order in the perturbation. The numerical evaluation of $E_{Kin}$ will be presented below in Sec. IV. Phase diagrams and Specific Heat ================================ Recently, there have been many experimental reports on manganites at low doping and low temperatures with and without an external magnetic field [@cv; @cv1; @cv2]. Okuda et al have estimated the electronic specific heat for $La_{1-x}Sr_xMnO_3$ in the ferromagnetic regime and concluded that the carrier mass-renormalization near the metal-insulator transition at $x=0.16$ is minimal. They have also observed a decrease in the low temperature specific heat in the presence of a magnetic field. Motivated by these observations, we have carried out a calculation of the specific heat, based on the partition function of the system which, from a cumulant expansion upto the 2nd order, is given by [@sd], $$Z(\beta) = Z_0(\beta) exp{(-\int^{\beta}_{0} d\beta^\prime \int^{\beta^{\prime}}_{0} d\beta^{\prime \prime}\langle \tilde H_1(\beta^\prime)\tilde H_1(\beta^{\prime \prime})\rangle)} ,$$ where $Z_0(\beta) = Tr (e^{-\beta H_0})$ ; $\tilde H_{1} (\beta) = e^{\beta H_0} H_1 e^{-\beta H_0}$, and $\beta = \frac{1}{K_B T}$. The expression $\langle \rangle$ denotes the usual canonical averaging. The specific heat is then calculated (in arbitrary units) from the well known relation: $$C_V=-\frac{d}{dT}(\frac{d}{d\beta}lnZ(\beta)),$$ and in the low temperature regime, to which only the zero-and one-phonon states contribute. If the localized core spin at each site is $\frac{3}{2}$ then the possible values of $\mid \vec S_1 + \vec S_2 \mid = S_{12}$ are 3, 2, 1 and 0. The ferromagnetic (FM) and antiferromagnetic (AFM) orders are obviously related to $S_{12}=$ 3 and 0, whereas $S_{12}=$ 2 and 1 are referred as canted 1 (CA1) and canted 2 (CA2) states respectively. The Fig. 1 shows the phase diagram for the four possible spin orders for our system, in the $g_{-}$ vs $J$ plane. For small values of $g_{-}$ and $J$, the FM state is the most stable one, and with increasing $J$, the ground state first becomes CA1 and then CA2. For a very large value of the superexchange interaction $J$, the system is in an AFM order for any value of $g_{-}$. However, with increasing electron-phonon interaction $g_{-}$, the CA1 and CA2 phase become narrower. Indeed, for larger values of $g_{-}$ the FM state appears for very low $J$ but with a small increase of $J$ the system transits to the AFM phase. The CA1 and CA2 phases in fact do not appear at all as the phase changes from the FM to AFM state with increasing superexchange interaction $J$, for large values of $g_{-}$. It can be further shown that for a very large value of $g_{-}$ the ground state is AFM for any value of $J$. It is evident from the phase diagram that for a particular value of $J$ the ground state changes as the electron-phonon coupling ($g_-$) increases (Fig. 1). But the change of phase from one to another is not continuous with $g_{-}$, for the quantum consideration of the core spins. This is shown in Fig. 2. For small values of $J=$ 0.01, the FM state exists even for a large value of $g_{-}$ and then it sharply changes to the AFM state. On the other hand, for larger values of $J=$ 0.04, 0.09, the system passes sharply to the canted phases (CA1 and CA2) and then the AFM state, with increasing $g_{-}$. For $J=$0.04 the CA1 and CA2 regimes are very narrow and for $J=$0.09 the CA1 and CA2 orders persist for a wider range of $g_-$. This is to be contrasted with the classical core spin model in which a similar study shows that the transitions to different core spin orientations are continuous for the same range of values for $g_{-}$ and $J$ [@euro]. In the classical case only three phases(FM, AFM and Canted) are present. The relative angle $\theta$ between classical core spins can take any value from $0$ to $\pi$, so any spin orientation other than FM ($\theta=0$) and AFM ($\theta = \pi$) yields a canted phase. Hence, in the classical limit of the core spins, for certain values of $J$, the FM-AFM transition is a smooth and continuous transition with $g_{-}$, whereas for spin $\frac{3}{2}$, the FM-AFM transition with $g_{-}$ is never continuous for any $J$. The probability of hopping of the itinerant electron from site to site is a maximum in the FM state as would be expected from the double-exchange mechanism. But, for a very strong electron-phonon coupling $g_{-}$, the electron may be localized forming a small polaron. For low values of $J$ we find both small and large polaron ground states in the FM phase (Fig. 3). The large to small polaron crossover is indicated by the relative deformation of the two lattice sites which is measured by the static correlation function $\frac {\lambda_{corr}}{g_{-}}$. In Fig. 3 the kinetic energy $t_{eff}^{KE}$ is large for small values of $g_{-}$, where the polaron is large, and for large $g_{-}$, the kinetic energy reduces rapidly, while $\frac {\lambda_{corr}}{g_{-}}$ rises, showing a smooth crossover to the small polaron regime. The classical and quantum formulations of the double-exchange model turn out to be the same in the FM limit of the core spins. So the nature of the kinetic energy and the polaron crossover in the FM state, as shown in Fig. 3, will be unaltered in the classical limit of core spins. However, in the AFM limit the two approaches (classical and quantum) are not equivalent, as has been argued earlier also. In the $S \rightarrow \infty$ limit the hopping probability is zero for the AFM case, while for $S=\frac{3}{2}$ the parameter modifying the hopping probability $t$, takes a finite value $0.25$ resulting in a finite charge transfer, even in the AFM limit. In Fig. 4 we show the nature of variation of the kinetic energy as well as the polaron crossover in different magnetic ground states. For $J=0.09$ the ground state is FM for low $g_{-}$, and with increasing $g_{-}$, the ground state changes sharply to CA1, CA2 and lastly to the AFM state. Since at each transition (from FM $\rightarrow $ CA1 $\rightarrow $ CA2 $\rightarrow $ AFM) the effective hopping reduces due to the double-exchange interaction, it is obvious that the kinetic energy will show a sharp drop at each transition point. It is expected that the polaron crossover will also show concomitant sharp jumps at each magnetic transition and the crossover to small polaron behavior will occur at lower value of $g_{-}$ than in the FM limit. This is shown clearly in Fig. 4. It is further evident that in a double-exchange system, both the magnetic transitions and the electron-phonon coupling localize the electron, when the polaron crossover and magnetic transitions are overlapping. The locations of the large polaron region (A) and the small polaron region (B) are indicated in the $g_-$ vs $J$ phase diagram (Fig. 1). In Fig. 1, for large values of $g_-$, the CA1 and CA2 phases are very narrow and appear as a single phase boundary of FM-AFM region. So the line of separation of polaronic regimes (A and B) appears as a point for CA1 and CA2 in Fig. 1. Thus the results presented in Fig. 3 and Fig. 4 have a bearing on the transport behavior of our model. Having discussed transport we now redirect our attention to thermodynamic properties. With this in mind we show in Fig. 5 the variation of the specific heat in the low temperature region in the FM state with zero and one phonon states. With application of an external magnetic field $\vec h$, $C_V$ takes lower values than for $\vec h= 0$ which is expected, as the average energy decreases with application of $\vec h$ in the FM state. For CA1($\mid \vec S_{12}\mid=2$), CA2($\mid \vec S_{12}\mid=1$) and AFM ($\mid \vec S_{12} \mid = 0$) states the external magnetic field will tend to align the core spins to feromagnetic order($\mid \vec S_{12}\mid = 3$). For CA1, CA2 and AFM states at low field and low temperatures it can be shown from the present calculation that $C_V$ does not change much from the $\vec h=0$ limit as long as $\vec h$ does not shift $\mid \vec S_{12}\mid$ to higher values. For larger $\vec h$, as the ground state changes from lower $\mid \vec S_{12}\mid $ to a higher one, $C_V$ decreases in the low temperature region. For CMR materials, there are some reports on measurements of $C_V$ but these are measured in the FM state [@cv; @cv1; @cv2]. In such cases, it was found that for low doping regions, $C_V$ decreases with an increasing magnetic field. Our present calculation of $C_V$ seemingly agrees with these experimental findings. The difference in the quantum and classical cases for specific heat, as far as the core spins are concerned, is exemplified in Fig. 6 and Fig. 7 for FM and AFM cases respectively. The quantum results evidently yields the correct low-temperature limit. Conclusions =========== The Anderson-Hasegawa model, though almost fifty years old, is able to capture all the crucial features of the double-exchange mechanism, originally proposed by Zener. The model is restricted to just two sites but the limitation should not be too serious when the electron hopping, influenced by thermal fluctuations, lattice distortions, phonon effects and other interactions, is expected to be incoherent. Incoherent hopping, albeit quantum in nature, is quite distinct from coherent band-like propagation, and approximately follows a Markovian process as far as the quantum diffusion of the electron is concerned. For a Markovian process only the pre-hopping and post-hopping sites matter. Therefore, the two site abstraction of the underlying three dimensional lattice provides the simplest paradigm which can be exploited for analyzing a variety of phenomena which are of current interest in manganites. With this in mind, we have felt the need of carefully reexamining the exact quantum solution of the Anderson-Hasegawa model, for realistic values of the core spins. We have further used this model as the basic building block, in order to incorporate one or the other phenomena of relevance to manganites. These include superexchange and polarons, which have been the focus of our attention here. Indeed, we have found that Superexchange, when properly treated in conjunction with quantum spin dynamics of the core spins, leads to additional terms in the Anderson-Hasegawa Hamiltonian, which are absent in the classical approximation of the core spins. A similar effect can also be observed with the exact solution of the Anderson-Hasegawa model, with the addition of phonon coupling. This may lead to the enhancement of the site energy term, which will have important consequences in these systems, especially in the paramagnetic state of the manganite systems. Moreover, the discreteness associated with the effective hopping in this quantum case was shown to have further consequence for thermodynamic and transport properties. In conclusion, therefore, we find that the Anderson-Hasegawa model continues to remain relevant for the understanding of topically important issues in manganites. Acknowledgement =============== The authors are deeply grateful to Prof. B. Dutta Roy for very helpful discussions on the Racah algebra for the Anderson-Hasegawa model. Discussions with Prof. S. Satpathy on the polaronic mechanism have been quite valuable. SD wishes to thank Prof. S. D. Mahanti for generating initial interest in manganites. [999]{} S. Jin, T. Tiefel, M. McCormack, R. Fastnacht, R. Ramesh and L. Chen, Science [**264**]{}, 413 (1994); S. Jin, M. McCormack, T. Tiefel and R. Ramesh, Jour. Appl. Phys. [**76**]{}, 6929 (1994). C. Zener, Phys. Rev. [**82**]{}, 403 (1951) P. W. Anderson and H. Hasegawa, Phys. Rev, [**100**]{}, 675 (1955). P. G. de Gennes, Phys. Rev. [**118**]{}, 141 (1960). K. Kubo and N. Ohata, Jour. Phys. Soc. Japan, [**33**]{}, 21 (1972). Myron B. Salamon and Marcelo Jaime, Rev. Mod. Phys. [**73**]{}, 583 (2001). J. M. D. Coey, M. Viret, S. von Molnar, Adv. in Phys. [**48**]{}, 167(1999). L. C. Biedenharn, J. M. Blatt and M. E. Rose, Rev. Mod. Phys. [**24**]{}, 249 (1952) A. J. Millis, P. B. Littlewood and B. I. Shraiman, Phys. Rev. Lett. [**74**]{}, 5144(1995). A. N. Das and Jayita Chatterjee, Int. Jour. Mod. Phys., [**13**]{}, 3903 (1999); Jayita Chaterjee and A. N. Das, Phys. Rev. B [**61**]{}, 4592 (2000). J. Chatterjee, M. Mitra and A. N. Das, Euro. Phys. J. B [ **18**]{} , 573(2000). T. Okuda. A. Asamitsu, Y. Tomioka, T. Kimura, Y. Taguchi and Y. Tokura, Phys. Rev. Lett. [**81**]{}, 3203(1998). T. Okuda, T. Kimura and Y. Tokura, Phys. Rev. B [**60**]{}, 3370(1999). M. Roy, J. F. Mitchell, A. P. Ramirez, P. Schiffer, cond-mat/0101223. see, for example, L. P. Kadanoff and G. Baym, (Benjamin, N. Y., 1962). 1.8in [**Figure Captions :** ]{} FIG. 1. The $g_{-}$ vs $J$ phase diagram ($\vec h = 0$) for $\mid \vec S_1 \mid = \mid \vec S_2 \mid = \frac{3}{2}$ and $t=1$. ([**A**]{}) and ([**B**]{}) denote large polaron and small polaron region respectively. 0.5cm FIG. 2. Variations of ground state spin configuration $\mid \vec S_1 + \vec S_2 \mid$ with $g_{-}$ for $t=1$ and $J=$ 0.01, 0.04 and 0.09, $h=0$ (in units of $\omega_0 = 1$). 0.5cm FIG. 3. Variations of $t_{eff}^{KE}$ and $\lambda_{corr}/g_-$ with $g_-$ for $t=1.0$, $J=0.01$ and $h=0$. 0.5cm FIG. 4. Variations of effective kinetic energy $t_{eff}^{KE}$ (dashed line) and polaron crossover $\lambda_{corr}/g_-$ (solid line) with $g_-$, for $t=1.0$, $J=0.09$ and $h=0$. The sharp jumps in $t_{eff}^{KE}$ and $\lambda_{corr}/g_-$ occur at values of $g_-$ where the magnetic transitions take place (see Fig. 2). 0.5cm FIG. 5. Variations of $C_V$ (in arbitrary units) for $g_-=0.6$, $J=0.01$ and $t=1$, for different values of the magnetic field $h=$ 0, 0.01, 0.05. FIG. 6. Variations of $C_V$ (in arbitrary units) for $g_-=0.2$, $J=0.02$, $h=0$ and $t=1$, in classical (solid line) and quantum (dashed line) formulation of the core spins. The ground state is FM. FIG. 7. Variations of $C_V$ (in arbitrary units) for $g_-=0.9$, $J=0.2$, $h=0$ and $t=1$, in classical (solid line) and quantum (dashed line) formulation of the core spins. The ground state is AFM.

--- abstract: 'If two probability density functions (PDFs) have values for their first $n$ moments which are quite close to each other (upper bounds of their differences are known), can it be expected that the PDFs themselves are very similar? Shown below is an algorithm to quantitatively estimate this “similarity” between the given PDFs, depending on how many moments one has information about. This method involves the concept of functions behaving “similarly” at certain “length scales”, which is also precisely defined. This technique could find use in data analysis, to compare a data set with a PDF or another data set, without having to fit a functional form to the data.' address: - 'Department of Physics, University of Maryland, College Park, MD 20740, USA' - 'Department of Mathematics, University of Maryland, College Park, MD 20740, USA' author: - Pranava Chaitanya Jayanti - Konstantina Trivisa bibliography: - 'mendeley\_v2.bib' title: Bounded Statistics --- July 2018 [*Keywords*]{}: moment problem, probability density functions, inverse problems, data analysis, characteristic functions\ Introduction ============ Modern scientific efforts often involve collecting, filtering, analysing and interpreting extremely large amounts of raw data. Famous examples from fundamental physics include the experiments at the Large Hadron Collider (LHC) and the Laser Interferometer Gravitational-Wave Observatory (LIGO). In the more applied sciences, one can think of complicated weather models that are tested for accuracy by comparison with meteorological data. Regression techniques are routinely used by engineers to fit very sophisticated equations of state to pressure-volume-temperature data of complex cryogenic mixtures. In all these cases, it would be of great use to develop a preliminary idea of the nature of a given data set or probability density function (PDF). This could help optimise computational costs. To this end, one may ask: how much information regarding the underlying PDF can be extracted from the moments of a distribution? This is famously known as the moment problem [@Shohat1970TheMoments] and is a highly non-trivial question. It is a very important inverse problem with wide applications in data analysis. The question that is addressed in this article is a slight variant of the moment problem: is it possible to compare different distributions based on their moments alone, without having to explicitly construct their PDFs? More precisely, the moment problem can be stated as follows: given a sequence of real numbers $\{\mu_k\}_{k=1,2,\dots}$, is it possible to find a PDF $f(x)$ such that $\{\mu_k\}$ are its moments, i.e., $\mu_k=\int_a^bx^k f(x) dx$? Depending on the domain of integration, the moment problem is classified into three categories: the Hausdorff problem which has a finite domain (usually taken to be $[0,1]$ with no loss in generality), the Stieltjes problem with a half-infinite domain $[0,\infty)$, and the Hamburger problem which spans the whole real line $(-\infty,\infty)$. These cases have been well-studied and conditions for existence and/or uniqueness of solutions have been formulated [@Shohat1970TheMoments]. However, in many cases, an explicit construction of the PDF [@Talenti1987RecoveringMoments] may not be the goal, but it is done anyway. For instance, consider the ubiquitous situation of comparing two data sets. One way to do this is to fit PDFs using some regression techniques to the two data sets, and then compare the two PDFs. For the Hausdorff moment problem, this procedure of constructing the PDF from a finite set of moments is ill-defined in the sense of Hadamard, i.e., it is highly sensitive to the values of the moments due to the (exponentially) large condition number of the corresponding Hankel matrix [@Talenti1987RecoveringMoments]. It is also computationally very expensive, especially for large data sets. The computational cost can be decreased if an approximate form for the PDF can be deduced a priori, and this is the focus of this article. Given the first $n$ moments of each distribution/data set, or even just how far apart they are from each other, it is shown that the closeness (or similarity) of the underlying distributions can be commented upon in a very quantitative manner. The moments of a distribution are directly related to the derivatives of the characteristic function of the PDF. Hence, the closeness of the moments of two distributions directly translates to the similarity of the derivatives (at $k=0$) of their characteristic functions. This fact is used, along with Taylor series expansions of the characteristic functions, to precisely characterize the “similarity” of the two distributions. In the case of a Hausdorff moment problem, uniqueness is guaranteed, i.e., *if* there exists a function that solves the problem, it is unique [@Shohat1970TheMoments; @Talenti1987RecoveringMoments]. In other words, for the case of a finite domain, the moments characterize the PDF completely. Thus, the two PDFs can be expected to be the same if and only if they have exactly the same moments, which is a trivial case. In the case that they are different, the PDFs may be expected to behave similarly only on certain “large” cut-off length scales, and differ significantly on the finer scales. This concept of “scale-wise similarity” is also precisely defined in the article. The result of this algorithm is the estimation of a cut-off length scale over which the two PDFs are similar (as defined by a user-specified tolerance), given how far apart the first few moments are. The outline of this article is as follows: Section \[definitions\] of this article defines some of the concepts used in the algorithm and Section \[estimates\] establishes the estimates used in this work. Section \[existence & uniqueness\] describes the algorithm, while Section \[infinite moments\] consists of a brief discussion on the cases of bounded and unbounded support. Then, we illustrate the algorithm with some numerical examples in Section \[numerical examples\], following which the method is extended to higher dimensions in Section 7. Conclusions and scope for future work are presented in Sections 8 and 9, respectively. Definitions =========== Distance metric {#distance metric} --------------- Consider two absolutely integrable functions over a compact support, $f,g : [a,b]\rightarrow \mathbb{R}$. Define a distance metric between the two functions as: $$d(f,g) = \frac{\int_a^b |f(x)-g(x)| dx}{\int_a^b dx}$$ Notice that this definition conforms to all the requirements of a distance metric. Characteristic function {#characteristic function} ----------------------- The characteristic function (denoted by a capital letter) is the Fourier transform of a PDF. Here, and in what follows, the Fourier transform and its inverse are given by: $$\label{eq:characteristic function} F(k) = \int_a^b e^{ikx}f(x) dx \ , \ f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-ikx}F(k) dk$$ Assuming that the derivatives of the characteristic function exist, and that they all vanish at infinity, it can be easily shown that for a normalized PDF of a random variable $x$, the $n^{\text{th}}$ moment can be expressed as: $$\langle x^n \rangle = \int_a^b x^n f(x) dx = (-i)^n F^{(n)}(0)$$ where $F^{(n)}(0) = \frac{d^n}{dk^n}F(k) \vert_{k=0}$. Let the $n^{\text{th}}$ moments of two normalized PDFs $f(x)$ and $g(x)$ be denoted by $\langle x^n \rangle_f$ and $\langle x^n \rangle_g$, respectively. Define the bound on the difference of the $n^{\text{th}}$ moments by: $$M_n \ge \left\vert \langle x^n \rangle_f - \langle x^n \rangle_g \right\vert = \left\vert F^{(n)}(0) - G^{(n)}(0) \right\vert$$ Low-pass filtering {#low-pass filtering} ------------------ In what follows, the notation in [@Frisch1995Turbulence:Kolmogorov] has been used to describe the large scale behaviour of functions. Filtering operators, one of which is extensively used in this article, are defined below. The action of the low-pass filtering operator $P_K$, associated with inverse length scale $K$, on an integrable function $f(x)$ is given by: $$\label{eq:low-pass filter} P_K[f(x)] = f^<(x) = \frac{1}{2\pi}\int_{-K}^{K} e^{-ikx}F(k) dk$$ where $F(k)$ is the Fourier transform of $f(x)$. \ The low-pass filtering operator has a “smoothing effect” in the sense that it removes any fluctuations of length scale less than $Κ^{-1}$. It is trivial to show that $P_K$ is a projection operator. The complement of the low-pass filter is called the high-pass filter, which removes the large scale behaviour and returns the small-scale features of the function. The action of the high-pass filtering operator $Q_K$, associated with inverse length scale $K$, on an integrable function $f(x)$ is given by: $$Q_K[f(x)] = (\mathbb{I} - P_K)[f(x)] = f^>(x) = \frac{1}{2\pi}\int_{|k|>K} e^{-ikx}F(k) dk$$ \ As an illustration, the action of these filters on a test function is shown in Figure \[fig:figure1\]. Estimates of the distance {#estimates} ========================= As stated in the introduction, we will use the characteristic relation between the moments and the derivatives of the characteristic functions to construct the required algorithm. Consider two PDFs $f(x)$ and $g(x)$ over the compact support $[0,1]$. Let their respective characteristic functions be $F(k)$ and $G(k)$. Assuming we have knowledge of the first $n$ moments (starting from the zeroth moment), we expand the characteristic functions in a Taylor series about $k=0$. $$\begin{split} F(k) &= F^{(0)}(0) + \frac{F^{(1)}(0)}{1!}k + \frac{F^{(2)}(0)}{2!}k^2 + \dots + \frac{F^{(n-1)}(0)}{(n-1)!}k^{n-1} + \frac{F^{(n)}(t_Fk)}{n!}k^n\\ G(k) &= G^{(0)}(0) + \frac{G^{(1)}(0)}{1!}k + \frac{G^{(2)}(0)}{2!}k^2 + \dots + \frac{G^{(n-1)}(0)}{(n-1)!}k^{n-1} + \frac{G^{(n)}(t_Gk)}{n!}k^n \end{split}$$ where $0\le t_F,t_G \le 1$. The last term in the expansion is the Lagrange form of the remainder in Taylor’s theorem, which is applicable when the function being expanded as a series belongs to $C^{n-1}$ over the entire domain and is $n$-times differentiable. (For a rather straightforward proof that the characteristic function satisfies these conditions, see \[appendix\_smoothness\_characteristic function\]). Using these Taylor expansions, and the bounds on the difference of the moments (Section \[characteristic function\]), $$\begin{split} |F(k) - G(k)| &\le \left\vert F^{(0)}(0) - G^{(0)}(0)\right\vert + \left\vert(F^{(1)}(0)-G^{(1)}(0))\frac{k}{1!}\right\vert + \left\vert(F^{(2)}(0)-G^{(2)}(0))\frac{k^2}{2!}\right\vert\\ &+ \dots + \left\vert(F^{(n-1)}(0)-G^{(n-1)}(0))\frac{k^{n-1}}{(n-1)!}\right\vert + \left\vert(F^{(n)}(t_Fk)-G^{(n)}(t_Gk))\frac{k^n}{n!}\right\vert\\ \end{split}$$ $$\begin{split} \Rightarrow |F(k) - G(k)| &\le M_0 + \frac{M_1}{1!}|k| + \frac{M_2}{2!}|k|^2 + \dots + \frac{M_{n-1}}{(n-1)!}|k|^{n-1}\\ &\qquad \qquad \qquad + \frac{\left\vert F^{(n)}(t_Fk)-G^{(n)}(t_Gk)\right\vert}{n!}|k|^n\\ \end{split}$$ From , we have the following inequality: $$|f^<(x)-g^<(x)|=\left\vert \frac{1}{2\pi} \int_{-K}^{K} e^{-ikx}\left( F(k)-G(k) \right) dk\right\vert \le \frac{1}{2\pi}\int_{-K}^{K} |F(k)-G(k)|dk$$ By the definition of the distance metric and using this inequality, $$\begin{split} d(f^<,g^<) &= \frac{\int_0^1 |f^<(x)-g^<(x)|dx}{\int_0^1 dx} \le \frac{\int_0^1 \frac{1}{2\pi}\int_{-K}^{K} |F(k)-G(k)|dk dx}{\int_0^1 dx}\\ &\le \frac{1}{2\pi}\int_{-K}^{K} \Big\{ M_0 + \frac{M_1}{1!}|k| + \frac{M_2}{2!}|k|^2 + \dots + \frac{M_{n-1}}{(n-1)!}|k|^{n-1}\\ &\qquad \qquad \qquad \qquad + \frac{\left\vert F^{(n)}(t_Fk)-G^{(n)}(t_Gk)\right\vert}{n!}|k|^n \Big\} dk \\ \end{split}$$ Recalling , we have the following bound: $$F^{(n)}(t_Fk) = i^n\int_0^1 x^n f(x) e^{it_fkx} dx \Rightarrow \left\vert F^{(n)}(t_Fk) \right\vert \le \int_0^1 x^n f(x) dx = \langle x^n \rangle_f$$ Similarly, $\left\vert G^{(n)}(t_Gk) \right\vert \le \langle x^n \rangle_g$. Combining these results, $$\label{eq:preliminary distance} d(f^<,g^<) \le \frac{1}{\pi} \left\{ M_0K + \frac{M_1}{2!}K^2 + \dots + \frac{M_{n-1}}{n!}K^n + \frac{\langle x^n \rangle_f + \langle x^n \rangle_g}{(n+1)!}K^{n+1} \right\}$$ At this stage, all that remains is for us to estimate the unknown $\langle x^n \rangle_f$ (and $\langle x^n \rangle_g$) in terms of the lower moments, which are known. Depending on how much a priori information we have about the PDF $f(x)$, we may be able to estimate these unknown higher moments in various ways (see \[bounds on higher moments\]). At this stage, we will consider the simplest case, where we know nothing about $f(x)$ except that it is a PDF, i.e., it is integrable. Since $0\le x\le 1$, using Hölder’s inequality, it is easy to establish that $$\langle x^n \rangle_f \le \langle x^m \rangle_f \ \forall \ 0\le m<n$$ Thus, the best (smallest) upper bound is when $m=n-1$. From and this estimate, $$\label{eq:final distance} d(f^<,g^<) \le \frac{1}{\pi} \left\{ M_0K + \frac{M_1}{2!}K^2 + \dots + \frac{M_{n-1}}{n!}K^n + \frac{R_n}{(n+1)!}K^{n+1} \right\}$$ where $R_n = \langle x^{n-1} \rangle_f + \langle x^{n-1} \rangle_g$.\ This means that the two PDFs have “similar” behaviour over length scales greater than 1/Κ, if the distance between the smoothed functions is less than some specified tolerance. The functions may, however, differ from each other significantly over small scales, which is expected since their moments are not exactly equal, but are only similar in the sense of their difference being bounded above. Existence and uniqueness of solutions {#existence & uniqueness} ===================================== Given a PDF, its closeness (at a certain length scale) to any other PDF can be estimated by specifying a tolerance for considering two functions to “behave similarly” and checking if there exists a length scale over which this is achieved. The question to be posed is the following: *For any $\varepsilon>0$, does there exist an inverse length scale $K$ such that $d(f^<,g^< )≤\varepsilon$, where $f^<$ and $g^<$ are the low-pass filtered functions as defined in ?* This is answered by looking at the polynomial equation (in the variable $K$): $$\label{eq:polynomial in K} \frac{1}{\pi} \left\{ M_0K + \frac{M_1}{2!}K^2 + \frac{M_2}{3!}K^3 + \dots + \frac{M_{n-1}}{n!}K^n + \frac{R_n}{(n+1)!}K^{n+1}\right\} - \varepsilon = 0$$ and calculating the inverse length scale $K$ which is a root of this equation. is a polynomial equation of the $(n+1)^{\text{th}}$ degree, leading to the existence of $n+1$ solutions. Of these, the sought-after one is a real and positive value of $K$, preferably a unique root (to prevent further complications of having to choose between multiple solutions). There exists a unique positive solution to , which characterizes the desired cut-off scale. *Proof*: is a polynomial equation of degree $n+1$ with all positive coefficients, except the constant term. Using Descartes’ rule of signs, it can be concluded that there is exactly one positive root, since exactly one sign change occurs throughout the polynomial.\ Infinite/non-existent moments {#infinite moments} ============================= Bounded support {#bounded support} --------------- For distributions defined on a bounded support, all moments exist regardless of the PDF. This can be shown as follows. (Since any bounded domain $[a,b]$ can be mapped to the compact set $[0,1]$, only the latter shall be considered, without any loss of generality.) $$\langle x^n \rangle = \int_0^1 x^n f(x) dx \le \int_0^1 f(x) dx = 1$$ where the last equality follows from normalization, which is valid for all PDFs since they are Lebesgue-integrable (by definition). Thus, any PDF on a bounded support has all moments and they are all finite. Hence, the method described in this report is certainly applicable to PDFs with bounded support. Unbounded support {#unbounded support} ----------------- It is not necessary that all the moments, or even any moments, exist for a PDF with an unbounded support. For instance, while the normal distribution possesses all moments, the Cauchy distribution has none. If the $n^{\text{th}}$ moment of a distribution does not exist, it means that the $n^{\text{th}}$ derivative of the characteristic function does not exist as a limit at $k=0$. Similarly, the $n^{\text{th}}$ derivative of the characteristic function could blow up at $k=0$, signifying the blow up of the $n^{\text{th}}$ moment. In the latter case, the limit exists and is equal to $\infty$.\ *Note: However, it can be easily shown that for a PDF with an unbounded support, the existence of a finite $n^{\text{th}}$ moment implies the existence and finiteness of all moments that are lesser than $n$. For instance, $\forall \ 0\le m<n$,* $$\begin{aligned} \langle x^n \rangle = \int_0^{\infty} x^n f(x) dx &= \int_0^1 x^n f(x) dx + \int_1^{\infty} x^n f(x) dx\\ &\ge \int_0^1 x^n f(x) dx + \int_1^{\infty} x^m f(x) dx\\ &= \int_0^1 (x^n - x^m) f(x) dx + \langle x^m \rangle\end{aligned}$$ *The first term in the last step is clearly finite.*\ If the higher derivatives of the characteristic functions become arbitrarily large, this corresponds to very large fine-scale fluctuations (this interpretation follows from the relation between the moments and the derivatives of the characteristic function). These fluctuations are not physical in the sense that they will prevent any kind of macroscopic equilibration from occurring, and hence characterize far-from-equilibrium processes in their transient states[^1]. Such states of these processes may not even have a well-defined PDF to begin with, and the method of this article is irrelevant to such extreme cases. Numerical examples {#numerical examples} ================== In this section, the above method is applied to some PDFs with compact support $[0,1]$. Two cases are considered: a “structured” PDF (with a closed-form expression) and an “unstructured" PDF (generated with random numbers). In both cases, various realisations of the PDFs are compared to deduce their similarity, given the first few moments. The calculations were performed using MATLAB for different illustrative values of the parameters. In Figures 2-6 and 8-12 below, the two original PDFs are denoted by solid (red and blue) lines, while the smoothed functions are shown as dashed (red and blue) lines. Case 1: Normal random variables {#normal random variables} ------------------------------- For the first illustration, we consider a normally-distributed random variable $N(\mu,\sigma^2)$ over the compact support $[0,1]$. Its PDF is of the form: $$\begin{split} f(x) &= \frac{1}{N_f}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\\ N_f = \int_0^1 e^{-\frac{(x-\mu)^2}{2\sigma^2}} dx &= \sigma \sqrt{\frac{\pi}{2}} \left[ \text{erf}\left( \frac{1-\mu}{\sqrt{2}\sigma} \right) - \text{erf}\left( \frac{-\mu}{\sqrt{2}\sigma} \right) \right]\\ \end{split}$$ where $N_f$ is the normalization factor, and $\text{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-y^2} dy$ is the error function. Using , we find the characteristic function corresponding to this PDF: $$\label{eq:complicated characteristic function} F(k) = e^{-\frac{\mu^2}{2\sigma^2}}e^{-\frac{\sigma^2}{2}\left( k-\frac{i\mu}{\sigma^2} \right)^2} \frac{\left[ \text{erf}\left( \frac{1-(\mu+i\sigma^2 k)}{\sqrt{2}\sigma} \right) - \text{erf}\left( \frac{-(\mu+i\sigma^2 k)}{\sqrt{2}\sigma} \right) \right]}{\left[ \text{erf}\left( \frac{1-\mu}{\sqrt{2}\sigma} \right) - \text{erf}\left( \frac{-x}{\sqrt{2}\sigma} \right) \right]}$$ From and , and a few variable substitutions, we arrive at an integral expression for the smoothed PDF: $$\label{eq:smoothed function_normal random} f^<(x) = f(x) \times \frac{i}{2\sqrt{\pi}} \int_{\frac{\sigma}{\sqrt{2}}\left( \frac{x-\mu}{\sigma^2}+iK \right)}^{\frac{\sigma}{\sqrt{2}}\left( \frac{x-\mu}{\sigma^2}-iK \right)} e^{k^2}\left[ \text{erf}\left( k+\frac{1-x}{\sqrt{2}\sigma} \right) - \text{erf}\left( k+\frac{-x}{\sqrt{2}\sigma} \right) \right] dk$$ was numerically integrated on MATLAB, and the algorithm was applied to two such PDFs and the cut-off scales (and other details) were evaluated. The results are shown in Table \[tab:table\_normal\] and Figures \[fig:figure2\] - \[fig:figure6\]. The following observations can be made: 1. The cut-off scale is higher (smaller length scales are probed) in the case where $\mu_1=\mu_2,\sigma_1\neq \sigma_2$, when compared to the case of $\mu_1\neq \mu_2,\sigma_1=\sigma_2$. This is expected since it is harder to “match” two PDFs whose peaks are separated, as opposed to when one is simply broader than the other. 2. Increasing the number of moments increases the cut-off (wavenumber) scale. This is because smaller length scales can be probed with more information (moments). 3. Increasing the number of moments also increases the distance between the smoothed functions. This could perhaps be due to the moments acting as constraints that are to be adhered to while comparing the functions. (It is to be noted that this trend is not observed in the case of the “less-structured” PDFs considered in the next section.) 4. Reducing the tolerance reduces the cut-off scale. Once again, this is expected since a tighter tolerance may be achieved only by sufficient smoothing of the functions. Case 2: Scale-separated PDFs ---------------------------- Scale-separation is commonly encountered in nature, when the dynamics of a physical system can be separated into two (usually narrow) intervals of length/time scales (see Figure \[fig:figure1\]). For instance, in certain models of combustion, the assumption of “fast chemistry” is invoked to simplify the analysis. This means that some of the reactions at equilibrium are much faster than others, so that they may be considered instantaneous, and this is a manifestation of temporal scale-separation. As an example of spatial scale-separation, one could consider fluid flow that is not fully turbulent. In such a flow, it is possible to clearly discern the large-scale flow features from the small-scale fluctuations. In summary, scale-separation refers to a case where a range of intermediate (length/time) scales are absent. Due to their ubiquity, it is useful to see an illustration of the method applied to a scale-separated PDF. For this purpose, a spectrum was created by generating normally-distributed random numbers (with zero mean) on MATLAB and arranging them in decreasing order of magnitude. The rearrangement of the numbers was done to imitate the spectrum of a system in a statistically steady state, where fluctuations decrease with decreasing scale size. Further, the spectrum was set to zero in some intermediate wavenumbers to mimic scale-separation (Figure \[fig:figure7\]). Two such PDFs were constructed as different realizations of the random spectrum and the algorithm described in Section \[existence & uniqueness\] was used to analyse their similarity and determine the cut-off scale. Using the same random spectrum, the effect of varying parameters is discussed. (The random spectra used to construct Figure \[fig:figure12\] are different from the ones used for Figures \[fig:figure8\] - \[fig:figure11\].) Integrations were performed numerically and the results are shown in Table \[tab:table\_scale-separated\].\ The following observations, similar to those in Section \[normal random variables\], can be made: 1. Comparing the results for Figures \[fig:figure8\] and \[fig:figure12\], it is seen that the cut-off scale is higher (smaller length scales are probed) in the former. Observing the graphs of the two cases reveals that in Figure \[fig:figure12\], the original PDFs are very much “out of phase”, which means they are less similar to begin with than in Figure \[fig:figure8\]. 2. Increasing the number of moments increases the cut-off (wavenumber) scale. 3. Reducing the tolerance reduces the cut-off scale. Extension to higher dimensions {#extension to higher dimensions} ============================== All of the above analysis was done for PDFs that were functions of just one variable. Can it be extended to a multivariate PDF (i.e., to higher dimensions, say $d$)? It is possible, as will be shown below. The steps are similar to the ones in the 1-dimensional case, but there is a subtle difference in the end. Let $x = (x_1,x_2,\dots,x_d) \in \mathbb{R}^d$ and let $f(x)$ be a PDF supported over the unit ball in $\mathbb{R}^d$. The characteristic function in $d$ dimensions is given by: $$F(k) = \int_a^b e^{ikx}f(x) dx \ , \ f(x) = \frac{1}{(2\pi)^d}\int_{-\infty}^{\infty} e^{-ikx}F(k) dk$$ where $k=(k_1,k_2,\dots,k_d) \in \mathbb{R}^d$ is the wavevector. Let $\alpha = (\alpha_1,\alpha_2,\dots,\alpha_d)$ and $|\alpha| = \alpha_1+\alpha_2+\dots+\alpha_d$, where each $\alpha_i \in \{0,1,2,\dots\}$. Then the moments of the PDF are defined as follows: $$\langle x^{\alpha} \rangle := \langle x_1^{\alpha_1}x_2^{\alpha_2}\dots x_d^{\alpha_d}\rangle = \int x_1^{\alpha_1}x_2^{\alpha_2}\dots x_d^{\alpha_d}f(x) dx = (-i)^{|\alpha|}D^{\alpha}F(0)$$ where $D^{\alpha} \equiv \frac{\partial^{\alpha}}{\partial k_1^{\alpha 1} \partial k_2^{\alpha 2} \dots \partial k_d^{\alpha d}}$. The upper bounds for the moments are defined as: $$M_{\alpha} \ge |\langle x^{\alpha} \rangle_f - \langle x^{\alpha} \rangle_g| = |D^{\alpha}F(0) - D^{\alpha}G(0)|$$ The low-pass filtering operator is defined for a $d$-dimensional cut-off scale $K = (K_1,K_2,\dots,K_d)$: $$P_K[f(x)] = f^<(x) = \frac{1}{(2\pi)^d}\int_{-K_d}^{K_d}\dots \int_{-K_2}^{K_2}\int_{-K_1}^{K_1} e^{-ikx}F(k) dk_1dk_2\dots dk_d$$ Expanding the characteristic function in a Taylor series about the origin, $$F(k) = \sum_{|\alpha|\le n-1} \frac{D^{\alpha}F(0)}{\alpha !}k^{\alpha} + \sum_{|\alpha|= n} \frac{D^{\alpha}F(tk)}{\alpha !}k^{\alpha}$$ where $t\in (0,1)$ and $\alpha ! = \alpha_1 ! \alpha_2 !\dots \alpha_d !$. As in the 1-dimensional case, the remainder term in the Taylor series can be bounded as: $$|D^{\alpha}F(tk)| \le \langle x^{\alpha} \rangle \le \langle x^{\beta} \rangle$$ $\beta$ is obtained by reducing any one non-zero component of $\alpha$ by $1$. Finally, the distance metric in $d$-dimension is: $$d(f,g) = \frac{\int_{B(0,1)}|f(x)-g(x)|dx}{\int_{B(0,1)}dx}$$ Following the same sequence of inequalities as before, the distance inequality for the smoothed functions becomes: $$d(f^<,g^<) \le \frac{1}{(2\pi)^d} \int_{-K_d}^{K_d}\dots \int_{-K_2}^{K_2}\int_{-K_1}^{K_1} \left\{ \sum_{|\alpha| \le n-1} \frac{M_{\alpha}}{\alpha!}|k|^{\alpha} + \sum_{|\alpha| = n} \frac{R_{\alpha}}{\alpha!}|k|^{\alpha} \right\} dk_1dk_2\dots dk_d$$ where $R_{\alpha}=\langle x^{\beta_f} \rangle_f + \langle x^{\beta_g} \rangle_g$ is the remainder term. Here, $\beta_f$ and $\beta_g$ are chosen so that the remainder term is the least possible (among all the moments of that order). Note that $|k|^{\alpha} = |k_1|^{\alpha_1}|k_2|^{\alpha_2}\dots |k_d|^{\alpha_d}$. $$\therefore d(f^<,g^<) \le \frac{1}{\pi^d} \left\{ \sum_{|\alpha|\le n-1} \frac{M_{\alpha}}{(\alpha + \mathbb{I})!}K^{\alpha + \mathbb{I}} + \sum_{|\alpha|=n} \frac{R_{\alpha}}{(\alpha + \mathbb{I})!}K^{\alpha + \mathbb{I}} \right\} = \varepsilon$$ where $\mathbb{I} = (1,1,\dots,1)$. The difference in the higher dimensional case $(d>1)$ is that the solution $(K_1,K_2,…,K_d)$ is not unique. However, a unique (and conservative) value for the cut-off scale, that is common over all the dimensions, can be determined. If $\max_{i} \{K_i \}=\kappa$, then setting all the $K_i=\kappa$, the resulting equation is of the form: $$\frac{\kappa^d}{\pi^d} \left\{ \sum_{j=0}^{n} a_j \kappa_j \right\} = \varepsilon$$ where the $a_j$ are the sum of various moments (divided by the appropriate factorials) and the remainder term. The existence and uniqueness of this $\kappa$ can be proved just as in the 1-dimensional case. Conclusions =========== A method has been proposed to estimate a length scale to which given PDFs must be smoothed for their (normalized) $L^1$ distance to be less than a user-specified tolerance, given the first few moments of the PDFs. It has been shown that such a cut-off scale indeed exists and is unique. Two numerical examples were used as proof of concept, as well as to illustrate the working of the algorithm. An extension to higher dimensions has been outlined. This scheme is hoped to find use in data analysis for comparing large data sets, as calculating moments is less computationally-intensive than having to reverse-engineer the PDFs of the data sets in order to compare them. Scope for future work {#scope for future work} ===================== The calculations in Section \[estimates\] involves a series of inequalities, which lead to a very conservative estimate of the distance between the smoothed functions and the cut-off scale, as seen in Tables \[tab:table\_normal\] and \[tab:table\_scale-separated\]. A useful direction for future research is to sharpen the estimates with more accurate inequalities. Moreover, the method described in this article only deals with PDFs having bounded support. As discussed in Section \[unbounded support\], unbounded domains pose major problems in the existence or the finiteness of moments. Extending this scheme, or formulating an entirely new one, for unbounded domains could be another meaningful and interesting research problem. PCJ wishes to thank Andy Sebastian and Andrew Corson for helpful discussions in the preliminary stages of this research and acknowledge Joydeep Singha’s suggestion to extend this algorithm to multiple dimensions. The authors are also grateful to Dr. Chris Jarzynski and Dr. Venkatarathnam Gadhiraju for their valuable suggestions to improve the manuscript. Bounds on higher moments {#bounds on higher moments} ======================== Consider a PDF $f(x)$ with compact support $[0,1]$. Then, its $n^{\text{th}}$ moment can be bounded from above by a lower moment $m$ and/or other terms, depending on the smoothness of the PDF. This is useful in bounding the remainder term in the algorithm discussed above. PDF is bounded: $f(x) \in L^{\infty}([0,1])$ -------------------------------------------- For all $m,n \in \mathbb{N}$ with $m<n$, we use the Cauchy-Schwartz inequality to obtain: $$\begin{aligned} \langle x^n \rangle = \int_0^1 x^n f(x) dx &\le \left( \int_0^1 x^{2(n-m)} dx \right)^{\frac{1}{2}} \left( \int_0^1 x^{2m} f^2(x) dx \right)^{\frac{1}{2}}\\ &\le \left( \frac{1}{2(n-m)+1} \right)^{\frac{1}{2}} \left( \left\Vert f \right\Vert_{L^{\infty}} \int_0^1 x^{2m} f(x) dx \right)^{\frac{1}{2}}\end{aligned}$$ $$\label{appendix_bounded} \langle x^n \rangle \le \left( \frac{\left\Vert f \right\Vert_{L^{\infty}} \langle x^{2m} \rangle}{2(n-m)+1} \right)^{\frac{1}{2}}$$ Three special cases of may be of interest. 1. $m=0$ $$\Rightarrow \langle x^n \rangle \le \left( \frac{\left\Vert f \right\Vert_{L^{\infty}}}{2n+1} \right)^{\frac{1}{2}}$$ 2. $n$ is even and $m=\frac{n}{2}$ $$\Rightarrow \langle x^n \rangle \le \frac{\left\Vert f \right\Vert_{L^{\infty}}}{n+1}$$ 3. $n$ is odd and $m=\frac{n-1}{2}$ $$\Rightarrow \langle x^n \rangle \le \left( \frac{\left\Vert f \right\Vert_{L^{\infty}} \langle x^{n-1} \rangle}{n+2} \right)^{\frac{1}{2}}$$ PDF is absolutely continuous ---------------------------- In this case, for some $a \in [0,1]$, $$|f(x)| \le |f(a)| + \left\vert \int_a^x |f'(y)|dy \right\vert \le |f(a)| + \int_0^1 |f'(y)|dy$$ $$\label{appendix_absolutely continuous} \Rightarrow \left\Vert f \right\Vert_{L^{\infty}} \le |f(a)| + \left\Vert f' \right\Vert_{L^1} \le |f(a)| + \left\Vert f' \right\Vert_{L^p}$$ for $1\le p\le \infty$. This bound may be used in . $f(1), f^{(1)}(1), \left\Vert f^{(2)} \right\Vert_{L^{\infty}}$ are finite and known ------------------------------------------------------------------------------------ $$\begin{aligned} \langle x^n \rangle &= \int_0^1 x^n f(x) dx \xrightarrow[\text{parts, twice}]{\text{integrating by}} \\ &= \frac{f(1)}{n+1} - \frac{f^{(1)}(1)}{(n+1)(n+2)} + \frac{\int_0^1 x^{n+2}f^{(2)}(x) dx}{(n+1)(n+2)}\\\end{aligned}$$ $$\Rightarrow \langle x^n \rangle \le \frac{f(1)}{n+1} - \frac{f^{(1)}(1)}{(n+1)(n+2)} + \frac{\left\Vert f^{(2)} \right\Vert_{L^{\infty}}}{(n+1)(n+2)(n+3)}$$ This can, of course, be extended to higher derivatives by integrating by parts repeatedly, depending on how much information one already has about the PDF. Smoothness of the characteristic function {#appendix_smoothness_characteristic function} ========================================= If a random variable has moments up to the order $n$, then the corresponding characteristic function belongs to $C^n(\mathbb{R})$. *Proof*: (Adapted from [@Lukacs1970CharacteristicFunctions]) From , we know that for a random variable with a PDF given by $f(x)$, $$F^{(n)}(k) = i^n \int_0^1 x^n f(x) e^{ikx} dx$$ In the case that a PDF does not exist, this can be rewritten in terms of the cumulative distribution function as: $$F^{(n)}(k) = i^n \int_0^1 x^n e^{ikx} d\Phi(x)$$ Thus, if the PDF exists, then $f(x) dx = d\Phi(x)$. Now, consider $$\left\vert F^{(n)}(k+h) - F^{(n)}(k) \right\vert \le \int_0^1 x^n \left\vert e^{ihx} -1 \right\vert d\Phi(x) = 2\int_0^1 x^n \left\vert \sin{\frac{hx}{2}} \right\vert d\Phi(x)$$ The RHS is independent of $k$, and less than $2\int_0^1 x^n d\Phi(x)$ which is twice the $n^{\text{th}}$ moment. Also, the RHS can be made arbitrarily small by taking the limit $h \rightarrow 0$. From all these observations, we conclude that $F^{(n)}(k)$ is uniformly continuous on the entire real line. Thus, $F(k) \in C^n(\mathbb{R})$. \ From the discussion in Section \[bounded support\], we know that a random variable over a compact support has all moments. Combining this with the above theorem gives us the following corollary. For a random variable over a compact support, the characteristic function belongs to $C^{\infty}(\mathbb{R})$. References {#references .unnumbered} ========== to 1.0 Figure number & Number of moments $(n)$ & Tolerance $(\varepsilon)$ & Cut-off scale $(K)$ & Original distance $d(f,g)$ & Smoothed distance $d(f^<,g^<)$ & Parameters of PDFs $(\mu_1, \sigma_1)$ $(\mu_2, \sigma_2)$\ Figure \[fig:figure2\] & 3 & 0.1 & 1.396927 & 0.561495 & 0.010545 & $(0.4,0.25)$ $(0.6,0.25)$\ Figure \[fig:figure3\] & 3 & 0.1 & 1.854235 & 0.289393 & 0.005971 & $(0.5,0.2)$ $(0.5,0.3)$\ Figure \[fig:figure4\] & 6 & 0.1 & 1.537326 & 0.561495 & 0.013916 & $(0.4,0.25)$ $(0.6,0.25)$\ Figure \[fig:figure5\] & 6 & 0.1 & 2.863850 & 0.289393 & 0.019028 & $(0.5,0.2)$ $(0.5,0.3)$\ Figure \[fig:figure6\] & 3 & 0.01 & 0.564254 & 0.561495 & 0.000723 & $(0.4,0.25)$ $(0.6,0.25)$\ \[tab:table\_normal\] to Figure number & Number of moments $(n)$ & Tolerance $(\varepsilon)$ & Cut-off scale $(K)$ & Original distance $d(f,g)$ & Smoothed distance $d(f^<,g^<)$\ Figure \[fig:figure8\] & 20 & 1 & 6.949457 & 0.048280 & 0.024368\ Figure \[fig:figure9\] & 10 & 1 & 6.042744 & 0.048280 & 0.057852\ Figure \[fig:figure10\] & 5 & 1 & 4.178820 & 0.048280 & 0.029120\ Figure \[fig:figure11\] & 5 & 0.1 & 2.816885 & 0.048280 & 0.009402\ Figure \[fig:figure12\] & 20 & 1 & 6.364578 & 0.135750 & 0.086696\ \[tab:table\_scale-separated\] ![(a) Test function and action of (b) low-pass and (c) high-pass filters. (Reproduced with permission from Chapter 2 of [@Frisch1995Turbulence:Kolmogorov])[]{data-label="fig:figure1"}](figure1.eps){width="\linewidth"} to {width=".75\linewidth"}\ (a)\ {width=".75\linewidth"}\ (b)\ to {width=".75\linewidth"}\ (a)\ {width=".75\linewidth"}\ (b)\ to {width=".75\linewidth"}\ (a)\ {width=".75\linewidth"}\ (b)\ to {width=".75\linewidth"}\ (a)\ {width=".75\linewidth"}\ (b)\ to {width=".75\linewidth"}\ (a)\ {width=".75\linewidth"}\ (b)\ ![Amplitudes of the sine-component of one of the functions in Figures \[fig:figure8\] - \[fig:figure11\][]{data-label="fig:figure7"}](figure7.eps){width="\linewidth"} ![Scale-separated PDFs constructed from random spectra.\ $n=20$, $\varepsilon=1$[]{data-label="fig:figure8"}](figure8.eps){width="\linewidth"} ![Scale-separated PDFs constructed from random spectra.\ $n=10$, $\varepsilon=1$[]{data-label="fig:figure9"}](figure9.eps){width="\linewidth"} ![Scale-separated PDFs constructed from random spectra.\ $n=5$, $\varepsilon=1$[]{data-label="fig:figure10"}](figure10.eps){width="\linewidth"} ![Scale-separated PDFs constructed from random spectra.\ $n=5$, $\varepsilon=0.1$[]{data-label="fig:figure11"}](figure11.eps){width="\linewidth"} ![Scale-separated PDFs constructed from random spectra.\ $n=20$, $\varepsilon=1$[]{data-label="fig:figure12"}](figure12.eps){width="\linewidth"} [^1]: Note that while turbulent flow is also (strictly speaking) far-from-equilibrium, there is a difference between the stationary and transient states of turbulent flow. In the former, there is statistical equilibrium, and the Kolmogorov spectrum is evidence of fluctuations decreasing with reducing length scales.

--- abstract: 'The set of realizable refractive indices as a function of frequency is considered. For passive media we give bounds for the refractive index variation in a finite bandwidth. Special attention is given to the loss and index variation in the case of left-handed materials.' address: - 'Department of Electronics and Telecommunications, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway' - 'Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway' author: - Johannes Skaar - Kristian Seip title: Bounds for the refractive indices of metamaterials --- Introduction ============ During recent years several new types of artificial materials or metamaterials with sophisticated electromagnetic properties have been designed. The fabrication of custom structures with dimensions much smaller than the relevant wavelength has made it possible to tailor the effective electric permittivity $\epsilon$ and the magnetic permeability $\mu$. For example, materials with thin metal wires simulate the response of a low-density plasma so that ${\rm{Re}\,}\epsilon$ may become negative in the microwave range [@pendry1996]. Similarly, with the help of a split-ring structure, a strong magnetic resonance is achieved so that ${\rm{Re}\,}\mu$ may be negative [@pendry1999]. Passive media with ${\rm{Re}\,}\epsilon$ and ${\rm{Re}\,}\mu$ simultaneously negative, first realized by Smith [*et al.*]{} [@smith], are particularly interesting. Such materials are often referred to as left-handed, since, for negative $\epsilon$ and $\mu$, the electric field, the magnetic field, and the wave vector form a left-handed set of vectors. As the Poynting vector and the wave vector point in opposite directions, the refraction at the boundary to a regular medium is negative. The concept of negative refraction, introduced by Veselago already in 1968 [@veselago], has opened a new horizon of applications in electromagnetics and optics. In particular the possibility of manipulating the near-field may have considerable potential, enabling imaging with no limit on the resolution [@pendry2000]. Materials with negative $\epsilon$ and $\mu$ are necessarily dispersive [@landau_lifshitz_edcm; @veselago], and loss is unavoidable. Loss has serious consequences for the performance of certain components; for example, it has been shown that the resolution associated with the Veselago–Pendry lens is strongly dependent on the loss of the material [@ramakrishna2002; @nieto-vesperinas]. Therefore, it is important to look for metamaterial designs with negative real part of the refractive index while the loss is low. In this paper, instead of performing a search in an infinite, complex design space, we will find ultimate, theoretical bounds based on causality. We will also find optimal $\epsilon(\omega)$ and $\mu(\omega)$ functions. For example, suppose our goal is refractive index close to –1 while loss is negligible in a limited bandwidth $\Omega\equiv[\omega_1,\omega_2]$. What is then the minimal variation of the refractive index in $\Omega$, given that the medium is passive? If we force the real part of the refractive index to be exactly –1 in $\Omega$, what will then be the minimal loss there? It is common to assume that the medium can be described by some specific model, such as, for example, single or multiple Lorentzian resonances. While this permits a straightforward analysis, it is not clear if a more general medium would give a more optimal response in some sense. We will therefore not use a spesific model, but rather assume only causality. Realizable electromagnetic parameters ===================================== Any electromagnetic medium must be causal in the microscopic sense; the polarization and magnetization cannot precede the electric and magnetic fields, respectively. This means that $\epsilon(\omega)$ and $\mu(\omega)$ obey the Kramers–Kronig relations. In terms of the susceptibilities $\chi=\epsilon-1$ or $\chi=\mu-1$, these relation can be written $$\begin{aligned} {\rm{Im}\,}\chi=\mathcal H\,{\rm{Re}\,}\chi,\label{KKchi1}\\ {\rm{Re}\,}\chi=-\mathcal H\, {\rm{Im}\,}\chi,\label{KKchi2}\end{aligned}$$ where $\mathcal H$ denotes the Hilbert transform [@nussenzveig]. These conditions are equivalent to the fact that $\chi$ is analytic in the upper half-plane (${\rm{Im}\,}\omega>0$), and uniformly square integrable in the closed upper half-plane [^1]. The susceptibilities are defined for negative frequencies by the symmetry relation $$\chi(-\omega)=\chi^*(\omega), \label{sym}$$ so that their inverse Fourier transforms are real. For passive media, in addition to (\[KKchi1\])-(\[sym\]) we have: $${\rm{Im}\,}\chi(\omega)>0 \qquad\rm{for}\ \, \omega>0. \label{loss}$$ The losses, as given by the imaginary parts of the susceptibilities, can be vanishingly small; however they are always present unless we are considering vacuum [@landau_lifshitz_edcm]. Eqs. (\[KKchi1\])-(\[loss\]) imply that $1+\chi$ is zero-free in the upper half-plane [@landau_lifshitz_edcm]. Thus the refractive index $n=\sqrt{\epsilon}\sqrt{\mu}$ can always be chosen as an analytic function in the upper half-plane. With the additional choice that $n\to +1$ as $\omega\to\infty$, $n$ is determined uniquely, and it is easy to see that (\[KKchi1\])-(\[loss\]) hold for the substitution $\chi\to n-1$. While any refractive index with positive imaginary part can be realized at a single frequency, the conditions (\[KKchi1\])-(\[loss\]) put serious limitations on what is possible to realize in a finite bandwidth. First we will investigate the possibility of designing materials with the real part of the refractive index less than unity. In particular we will analyze to what extent it is possible in a limited bandwidth to have a constant index below unity (or even below zero) while the loss is small. We set $n-1=u+iv$ (or $\chi=u+iv$), where $u$ and $v$ are the real and imaginary parts of $n-1$ (or $\chi$), respectively. To begin with, we set $v(\omega)=0$ in the interval $\Omega=[\omega_1,\omega_2]$. (The case with a small imaginary part will be treated later.) By writing out the Hilbert transform and using (\[sym\]), we find $$\label{uint} u(\omega)=-\frac{2}{\pi}\int_0^{\omega_1} \frac{v(\omega')\omega'\rmd\omega'}{\omega^2-\omega'^2} +\frac{2}{\pi}\int_{\omega_2}^\infty \frac{v(\omega')\omega'\rmd\omega'}{\omega'^2-\omega^2}$$ for $\omega\in\Omega$. Note that both terms in (\[uint\]) are increasing functions of $\omega$. Since the goal is a constant, negative $u(\omega)$ in $\Omega$, the second term should be as small as possible. In the limit where the second term is zero, we obtain $$u(\omega)-u(\omega_1)=\frac{2}{\pi}\int_0^{\omega_1} \frac{v(\omega')\omega'\rmd\omega'}{\omega_1^2-\omega'^2}\frac{\omega^2-\omega_1^2}{\omega^2-\omega'^2},$$ and therefore $$\label{variationbound} u(\omega)-u(\omega_1)>|u(\omega_1)|\frac{\omega^2-\omega_1^2}{\omega^2}, \quad \omega\in\Omega,$$ provided $u(\omega_1)$ is negative. In particular, the largest variation in the interval is $$\label{variationboundOmega} u(\omega_2)-u(\omega_1) >|u(\omega_1)|(2\Delta-\Delta^2).$$ Here we have defined the normalized bandwidth $\Delta=(\omega_2-\omega_1)/\omega_2$. These bounds are realistic in the sense that equality is obtained asymptotically when $v(\omega)$ approaches a delta function in $\omega=0^+$. In this limit $u(\omega)=u(\omega_1)\omega_1^2/\omega^2$. It is interesting to estimate how much loss we must allow in the interval to wash out the variation (\[variationbound\]). Letting $v(\omega)$ approach a delta function in $\omega=0^+$, and adding the positive function $|u(\omega_1)|\sqrt{\omega_2^2-\omega^2}\sqrt{\omega^2-\omega_1^2}/\omega^2$ for $\omega\in\Omega$ correspond to a constant $u(\omega)=u(\omega_1)$ in $\Omega$. Furthermore, it can be shown that this particular $v$ corresponds to the minimal possible loss in the interval. The proof for this claim is given elsewhere [@seip2005]. Thus the maximal value of the (minimal) loss in the interval satisfies $$\label{maxloss} v_{\max}>|u(\omega_1)|\frac{\omega_2^2-\omega_1^2}{2\omega_1\omega_2}=|u(\omega_1)|\Delta+O(\Delta^2).$$ By a superposition of the optimal solutions associated with the bounds (\[variationbound\]) and (\[maxloss\]), we obtain a bound for the loss when a certain fraction $1-\alpha$ ($0\leq\alpha\leq 1$) of the variation (\[variationbound\]) remains: $$\label{maxlossvar} v_{\max} > \alpha|u(\omega_1)|\frac{\omega_2^2-\omega_1^2}{2\omega_1\omega_2}.$$ As an example, consider the case where the goal is refractive index close to –1 in an interval $\Omega$ with $\Delta\ll 1$. In the limit of zero imaginary index, (\[variationboundOmega\]) gives that the variation of the real index in the interval is larger than $4\Delta$. It is interesting that the minimal variation is obtained approximately if the medium has sharp Lorentzian resonances for a low frequency. For example, let $\epsilon(\omega)=\mu(\omega)=1+\chi(\omega)$, where $$\label{lorentz} \chi(\omega)=\frac{F\omega_{0}^2}{\omega_{0}^2-\omega^2-i\omega\Gamma}.$$ Here, $F$, $\omega_{0}$, and $\Gamma$ are positive parameters. If the bandwidth $\Gamma$ and center frequency $\omega_{0}$ are much smaller than $\omega_1$, ${\rm{Re}\,}n(\omega)\approx 1-F(\omega_{0}/\omega)^2$ and ${\rm{Im}\,}n(\omega)\approx F\omega_{0}^2\Gamma/\omega^3$ for $\omega\geq\omega_1$. If we require ${\rm{Re}\,}n(\omega_1)=-1$, we obtain ${\rm{Im}\,}n(\omega_1)\approx 2\Gamma/\omega_1$ and ${\rm{Re}\,}n(\omega_2)-{\rm{Re}\,}n(\omega_1)\approx 4\Delta$. When $\Gamma/\omega_1\to 0$, this corresponds to the optimal refractive index function associated with the bound (\[variationboundOmega\]). Furthermore, it is worth noting that if we want the real index variation to be zero in $\Omega$, the maximum imaginary part of the refractive index in $\Omega$ must be larger than $2\Delta$. The required imaginary part in $\Omega$ can roughly be approximated by weak resonances at $(\omega_1+\omega_2)/2$, see Fig. 1. ![The real (solid line) and imaginary (dashed line) refractive index associated with a Lorentzian resonance at $\omega=\omega_{0}$. The two figures represent the same functions but the scale is different. Also shown are the real part and imaginary part for the case where the real part is –1 in $\Omega$. In general this refractive index function can be found using the approach in [@seip2005]; however, when $\omega_1$ is much larger than the resonance frequency and bandwidth, the required $v$ in $\Omega$ is $2\sqrt{\omega_2^2-\omega^2}\sqrt{\omega^2-\omega_1^2}/\omega^2$. The parameters used are $\omega_1=2\omega_{0}$, $\omega_2=2.5\omega_{0}$, $\Gamma=0.1\omega_{0}$, and $F=8$.[]{data-label="fig:resonance"}](resonance.eps){height="6.7cm" width="8.1cm"} So far we have considered the case where the goal is a constant $u(\omega)<0$ in $\Omega$. If the goal is $u(\omega)>0$ in $\Omega$, it is the last term in (\[uint\]) that comes to rescue. Inspired by the result (\[variationbound\]), we may let $u(\omega)$ approach a delta function at a frequency much larger than $\omega$. Indeed, in the limit where this resonance frequency approaches infinity, the function $u(\omega)$ is constant and positive in $\Omega$ while $v(\omega)$ is zero. Of course this limit is not realistic; in practice the resonance frequency is limited to, say $\omega_{\max}$, where $\omega_{\max}>\omega_2$. The associated bounds are easily deduced along the same lines as above. For example, (\[variationboundOmega\]) becomes $$\label{variationboundOmegaomegamax} u(\omega_2)-u(\omega_1) >u(\omega_1)\frac{\omega_2^2-\omega_1^2}{\omega_{\max}^2-\omega_2^2}.$$ Similarly, there may be a lower bound $\omega_{\min}$, where $0<\omega_{\min}<\omega_1$, on the resonance frequency. The stricter inequalities in this case, corresponding to (\[variationbound\])-(\[maxlossvar\]), can be found in a similar fashion. Eq. (\[variationboundOmega\]) and (\[variationboundOmegaomegamax\]) have another interesting consequence. If the loss is zero in an infinitesimal bandwidth around $\omega$, we immediately find that the derivative $\rmd u/\rmd\omega$ is bounded from below: $$\label{dnbound} \frac{\rmd u}{\rmd\omega} > \cases{2|u(\omega)|/\omega & for $u(\omega)<0$, \\ 0 & for $u(\omega)\geq 0$.}$$ For the case $u(\omega)\geq 0$ we have set $\omega_{\max}=\infty$. Note that also this bound is tight. A similar bound was obtained previously for $\epsilon(\omega)$ and $\mu(\omega)$ [@landau_lifshitz_edcm]. Eqs. (\[dnbound\]) should also be compared to the weaker bound $\rmd n/\rmd\omega>|u(\omega)|/\omega$ which was obtained recently [@smith_kroll]. While the latter bound means that the group velocities of transparent, passive media are bounded by $c$, (\[dnbound\]) implies the maximum group velocity $c/(2-n)$ for $n<1$ (and trivially $c/n$ for $n \geq 1$). Here $c$ is the vacuum light velocity. When the loss in a bandwidth $\Omega$ is at most $v_{\max}$, (\[dnbound\]) becomes $$\label{dnboundloss} \frac{\rmd u}{\rmd\omega} > \cases{\left(\frac{2|u(\omega)|}{\omega}-\frac{4v_{\max}}{\pi\omega\Delta}\right) & for $u(\omega)<0$, \\ -\frac{4v_{\max}}{\pi\omega\Delta} & for $u(\omega)\geq 0$,}$$ to lowest order in $\Delta$, for $\omega$ close to $(\omega_1+\omega_2)/2$. In obtaining (\[dnboundloss\]) we have assumed that $v(\omega)$ is approximately constant in $\Omega$, and calculated the corresponding contribution to $\rmd u/\rmd\omega$. A similar bound can be derived when $v(\omega)$ varies slowly in $\Omega$; for example, if $v(\omega)$ is the imaginary part of a Lorentzian with $\Gamma=\omega_2-\omega_1$, the inequality holds with the replacement $4/\pi\to 2$. Note that without an assumption on the variation of $v(\omega)$, $\rmd u/\rmd\omega$ can take any value. A similar method as that leading to (\[variationboundOmega\]) can be used to find bounds for the variation of derivatives in $\Omega$, in the limit of no loss. For the first order derivative, the variation can be arbitrarily small to first order in $\Delta$, for any positive $\rmd u(\omega_1)/\rmd\omega$. For negative second order derivative $D\equiv \rmd^2 u/\rmd\omega^2$ (first order dispersion coefficient) we obtain $$\label{DvariationboundOmega} D(\omega_2)-D(\omega_1) > |D(\omega_1)|4\Delta+O(\Delta^2).$$ Discussion and conclusion ========================= We have considered the set of realizable permittivities, permeabilities and refractive indices. For passive media we have used (\[KKchi1\])-(\[loss\]) to prove ultimate bounds for the loss and variation of the real part of the permittivity, permeability, and refractive index. While the notation has indicated an isotropic medium, the bounds in this paper are valid for the effective index of the normal modes of anisotropic media as well. In this case the identification of associated $\epsilon$ and $\mu$ tensors from the effective index is more complicated, but nevertheless feasible. More generally, the bounds are valid for the effective index of any electromagnetic mode that can be excited separately and causally, provided the effective index is independent of the longitudinal coordinate. On the basis of causality, it is clear that the susceptibilities of active media also satisfy (\[KKchi1\])-(\[sym\]). However, (\[loss\]) is certainly not valid. Kre[ĭ]{}n and Nudel’man have shown how to approximate a square integrable function in a finite bandwidth by a function satisfying (\[KKchi1\])-(\[sym\]) [@nudelman; @krein2]. The approximation can be done with arbitrary precision; however, there is generally a trade-off between precision and the energy of $\chi$ outside the interval [@skaar2003]. Once a valid susceptibility has been found, a possible refractive index can be found e.g. by setting $n=\epsilon=\mu=1+\chi$. Hence, in principle, for active media $n$ can approximate any square integrable function in a limited bandwidth. References {#references .unnumbered} ========== [10]{} J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs. Extremely low frequency plasmons in metallic mesostructures. , 76:4773–4776, 1996. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart. Magnetism from conductors and enhanced nonlinear phenomena. , 47:2075–2084, 1999. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz. Composite medium with simultaneously negative permeability and permittivity. , 84:4184, 2000. V. G. Veselago. The electrodynamics of substances with simultaneously negative values of $\epsilon$ and $\mu$. , 10:509, 1968. J. B. Pendry. Negative refraction makes a perfect lens. , 85:3966–3969, 2000. L. D. Landau and E. M. Lifshitz. . Pergamon Press, New York and London, Chap. 9, 1960. S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz. The asymmetric lossy near-perfect lens. , 49:1747–1762, 2002. M. Nieto-Vesperinas. Problem of image superresolution with a negative-refractive-index slab. , 21:491–498, 2004. H. M. Nussenzveig. . Academic Press, New York and London, Chap. 1, 1972. K. Seip and J. Skaar. An extremal problem related to negative refraction, arxiv.org/math/0506620. , to appear, 2005. D. R. Smith and N. Kroll. Negative refractive index in left-handed materials. , 85:2933–2936, 2000. P. [Ya.]{} Nudel’man. Some limit approximation theorem for the synthesis of networks and signals. , 26:49–56, 1971. M. G. Kre[ĭ]{}n and P. [Ya.]{} Nudel’man. Approximation of functions in [$L\sb{2}(\omega \sb{1},\omega \sb{2})$]{} by transmission functions of linear systems with minimal energy. , 11(2):37–60, 1975. J. Skaar. A numerical algorithm for extrapolation of transfer functions. , 83:1213–1221, 2003. [^1]: If the medium is conducting at zero frequency, the electric $\chi$ is singular at $\omega=0$. Although $\chi$ is not square integrable in this case, similar relations as (\[KKchi1\])-(\[KKchi2\]) can be derived [@landau_lifshitz_edcm].

--- abstract: 'MAGIC, a system of two imaging atmospheric Cherenkov telescopes, achieves its best performance under dark conditions, i.e. in absence of moonlight or twilight. Since operating the telescopes only during dark time would severely limit the duty cycle, observations are also performed when the Moon is present in the sky. Here we develop a dedicated Moon-adapted analysis to characterize the performance of MAGIC under moonlight. We evaluate energy threshold, angular resolution and sensitivity of MAGIC under different background light levels, based on Crab Nebula observations and tuned Monte Carlo simulations. This study includes observations taken under non-standard hardware configurations, such as reducing the camera photomultiplier tubes gain by a factor $\sim$1.7 (reduced HV settings) with respect to standard settings (nominal HV) or using UV-pass filters to strongly reduce the amount of moonlight reaching the cameras of the telescopes. The Crab Nebula spectrum is correctly reconstructed in all the studied illumination levels, that reach up to 30 times brighter than under dark conditions. The main effect of moonlight is an increase in the analysis energy threshold and in the systematic uncertainties on the flux normalization. The sensitivity degradation is constrained to be below 10%, within 15-30% and between 60 and 80% for nominal HV, reduced HV and UV-pass filter observations, respectively. No worsening of the angular resolution was found. Thanks to observations during moonlight, the maximal duty cycle of MAGIC can be increased from $\sim$18%, under dark nights only, to up to $\sim$40% in total with only moderate performance degradation.' address: - 'ETH Zurich, CH-8093 Zurich, Switzerland' - 'Università di Udine, and INFN Trieste, I-33100 Udine, Italy' - 'INAF - National Institute for Astrophysics, viale del Parco Mellini, 84, I-00136 Rome, Italy' - 'Università di Padova and INFN, I-35131 Padova, Italy' - 'Croatian MAGIC Consortium, Rudjer Boskovic Institute, University of Rijeka, University of Split - FESB, University of Zagreb - FER, University of Osijek,Croatia' - 'Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Salt Lake, Sector-1, Kolkata 700064, India' - 'Max-Planck-Institut für Physik, D-80805 München, Germany' - 'Universidad Complutense, E-28040 Madrid, Spain' - 'Inst. de Astrofísica de Canarias, E-38200 La Laguna, Tenerife, Spain' - 'Universidad de La Laguna, Dpto. Astrofísica, E-38206 La Laguna, Tenerife, Spain' - 'University of Łódź, PL-90236 Lodz, Poland' - 'Deutsches Elektronen-Synchrotron (DESY), D-15738 Zeuthen, Germany' - 'Institut de Fisica d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain' - 'Università di Siena, and INFN Pisa, I-53100 Siena, Italy' - 'Institute for Space Sciences (CSIC/IEEC), E-08193 Barcelona, Spain' - 'Technische Universität Dortmund, D-44221 Dortmund, Germany' - 'Universität Würzburg, D-97074 Würzburg, Germany' - 'Finnish MAGIC Consortium, Tuorla Observatory, University of Turku and Astronomy Division, University of Oulu, Finland' - 'Unitat de Física de les Radiacions, Departament de Física, and CERES-IEEC, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain' - 'Universitat de Barcelona, ICC, IEEC-UB, E-08028 Barcelona, Spain' - 'Japanese MAGIC Consortium, ICRR, The University of Tokyo, Department of Physics and Hakubi Center, Kyoto University, Tokai University, The University of Tokushima, Japan' - 'Inst. for Nucl. Research and Nucl. Energy, BG-1784 Sofia, Bulgaria' - 'Università di Pisa, and INFN Pisa, I-56126 Pisa, Italy' - 'ICREA and Institute for Space Sciences (CSIC/IEEC), E-08193 Barcelona, Spain' - 'now at Centro Brasileiro de Pesquisas Físicas (CBPF/MCTI), R. Dr. Xavier Sigaud, 150 - Urca, Rio de Janeiro - RJ, 22290-180, Brazil' - 'Humboldt University of Berlin, Institut für Physik Newtonstr. 15, 12489 Berlin Germany' - also at University of Trieste - 'now at Finnish Centre for Astronomy with ESO (FINCA), Turku, Finland' author: - 'M. L. Ahnen' - 'S. Ansoldi' - 'L. A. Antonelli' - 'C. Arcaro' - 'A. Babić' - 'B. Banerjee' - 'P. Bangale' - 'U. Barres de Almeida' - 'J. A. Barrio' - 'J. Becerra González' - 'W. Bednarek' - 'E. Bernardini' - 'A. Berti' - 'W. Bhattacharyya' - 'B. Biasuzzi' - 'A. Biland' - 'O. Blanch' - 'S. Bonnefoy' - 'G. Bonnoli' - 'R. Carosi' - 'A. Carosi' - 'A. Chatterjee' - 'P. Colin' - 'E. Colombo' - 'J. L. Contreras' - 'J. Cortina' - 'S. Covino' - 'P. Cumani' - 'P. Da Vela' - 'F. Dazzi' - 'A. De Angelis' - 'B. De Lotto' - 'E. de Oña Wilhelmi' - 'F. Di Pierro' - 'M. Doert' - 'A. Domínguez' - 'D. Dominis Prester' - 'D. Dorner' - 'M. Doro' - 'S. Einecke' - 'D. Eisenacher Glawion' - 'D. Elsaesser' - 'M. Engelkemeier' - 'V. Fallah Ramazani' - 'A. Fernández-Barral' - 'D. Fidalgo' - 'M. V. Fonseca' - 'L. Font' - 'C. Fruck' - 'D. Galindo' - 'R. J. García López' - 'M. Garczarczyk' - 'M. Gaug' - 'P. Giammaria' - 'N. Godinović' - 'D. Gora' - 'S. Griffiths' - 'D. Guberman' - 'D. Hadasch' - 'A. Hahn' - 'T. Hassan' - 'M. Hayashida' - 'J. Herrera' - 'J. Hose' - 'D. Hrupec' - 'G. Hughes' - 'K. Ishio' - 'Y. Konno' - 'H. Kubo' - 'J. Kushida' - 'D. Kuveždić' - 'D. Lelas' - 'E. Lindfors' - 'S. Lombardi' - 'F. Longo' - 'M. López' - 'C. Maggio' - 'P. Majumdar' - 'M. Makariev' - 'G. Maneva' - 'M. Manganaro' - 'K. Mannheim' - 'L. Maraschi' - 'M. Mariotti' - 'M. Martínez' - 'D. Mazin' - 'U. Menzel' - 'M. Minev' - 'R. Mirzoyan' - 'A. Moralejo' - 'V. Moreno' - 'E. Moretti' - 'V. Neustroev' - 'A. Niedzwiecki' - 'M. Nievas Rosillo' - 'K. Nilsson' - 'D. Ninci' - 'K. Nishijima' - 'K. Noda' - 'L. Nogués' - 'S. Paiano' - 'J. Palacio' - 'D. Paneque' - 'R. Paoletti' - 'J. M. Paredes' - 'X. Paredes-Fortuny' - 'G. Pedaletti' - 'M. Peresano' - 'L. Perri' - 'M. Persic' - 'P. G. Prada Moroni' - 'E. Prandini' - 'I. Puljak' - 'J. R. Garcia' - 'I. Reichardt' - 'W. Rhode' - 'M. Ribó' - 'J. Rico' - 'A. Rugliancich' - 'T. Saito' - 'K. Satalecka' - 'S. Schroeder' - 'T. Schweizer' - 'A. Sillanpää' - 'J. Sitarek' - 'I. Šnidarić' - 'D. Sobczynska' - 'A. Stamerra' - 'M. Strzys' - 'T. Surić' - 'L. Takalo' - 'F. Tavecchio' - 'P. Temnikov' - 'T. Terzić' - 'D. Tescaro' - 'M. Teshima' - 'D. F. Torres' - 'N. Torres-Albà' - 'A. Treves' - 'G. Vanzo' - 'M. Vazquez Acosta' - 'I. Vovk' - 'J. E. Ward' - 'M. Will' - 'D. Zarić' bibliography: - 'mybibfile.bib' title: Performance of the MAGIC telescopes under moonlight --- Gamma-ray astronomy,Cherenkov telescopes,Crab Nebula Introduction ============ In the last decades the Imaging Atmospheric Cherenkov Technique (IACT) opened a new astronomical window to observe the $\gamma$-ray sky at Very High Energy (VHE, E$>$50 GeV). After the pioneering instruments of the last century, the three most sensitive currently operating instruments, VERITAS [@VERITAS2008], H.E.S.S.[@HESS2006] and MAGIC [@upgrade1], have discovered more than a hundred sources, comprised of a large variety of astronomical objects (see [@DeNaurois2015] for a recent review). The IACT uses one or several optical telescopes that image the air showers induced by cosmic $\gamma$ rays in the atmosphere, through the Cherenkov radiation produced by the ultra-relativistic charged particles of the showers. The air-shower Cherenkov light peaks in the optical/near-UV band. This faint light flash can be detected above the ambient optical light background using fast photodetectors. The IACT works only by night and preferentially during dark moonless conditions. IACT telescope arrays are usually optimized for dark nights, using as photodetectors UV-sensitive fast-responding photomultiplier tubes (PMTs), ideal to detect the nanosecond Cherenkov flash produced by an air shower. PMTs can age (gain degradation with time) quickly in a too bright environment, which restricts observations to relatively dark conditions. When IACT instruments operate only during moonless astronomical nights, their duty cycle is limited to 18% ($\sim$1500h/year), without including the observation time loss due to bad weather or technical issues. Every month around the full Moon, the observations are generally fully stopped for several nights in a row. Operating IACT telescopes during moonlight and twilight time would allow increasing the duty cycle up to $\sim$40%. This is interesting for many science programs, to obtain a larger amount of data and a better time coverage without full-Moon breaks. It may also be crucial for the study of transient events (active galaxy nucleus flares, $\gamma$-ray bursts, cosmic neutrino or gravitational wave detection follow-ups, etc.) that occur during moonlight time. With moonlight observation, the IACT can be more reactive to the variable and unpredictable $\gamma$-ray sky. Moreover, operation under bright background light offers the possibility to observe very close to the Moon to study for instance the cosmic-ray Moon shadow to probe the antiproton and positron fractions [@ElectronMoonShadow; @FirstMoonShadow] or the lunar occultation of a bright $\gamma$-ray source, which was used e.g. in hard X-ray for source morphology studies [@1975natur]. Different hardware approaches have been developed by IACT experiments to extend their duty cycle into moonlight time. One possibility is to restrict the camera sensitivity to wavelengths below 350nm, where the moonlight is absorbed by the ozone layer. This idea was applied to the Whipple 10m telescope, which was equipped with the dedicated UV-sensitive camera ARTEMIS [@ArtemisFilter], or with a simple UV-pass filter in front of the standard camera [@WhippleFilter]. The drawback of this technique is the dramatic increase of the energy threshold (a factor $\sim$4) due to the reduction of the collected Cherenkov light. The CLUE experiment [@CLUE] was a similar attempt with an array of 1.8m telescopes sensitive in the background-free UV range 190-230nm. More recently, the VERITAS collaboration also developed UV-pass filters to extend the operation during moonlight time [@VeritasFilter]. Another approach, developed first by the HEGRA collaboration [@HegraMoon], is to reduce the High Voltage (HV) applied to the PMTs (reducing the gain) to limit the anode current that can damage the PMTs. This, however, only allows observations at large angular distances from a partially illuminated Moon. An alternative way to safely operate IACT arrays under moonlight would be to use, instead of PMTs, silicon photomultiplier detectors, which are robust devices that can be exposed to high illumination levels without risk of damages. This was successfully demonstrated with the FACT camera [@FactMoon], which can operate with the full Moon inside its field of view (FOV). The use of a silicon photomultiplier camera is actually under consideration for the new generation of IACT instruments [@SiPM-SST1M; @SiPM-SCT; @SiPM-LST; @SiPM-ASTRI; @Light-trap]. The cameras of the MAGIC telescopes, which are equipped with low-gain PMTs, were designed from the beginning to allow observations during moderate moonlight [@OldMoon; @MagicMoon-ICRC2009]. The use of reduced HV [@MagicMoonShadow] and UV-pass filters [@Filters] were introduced later to extend the observations to all the possible Night Sky Background (NSB) levels, up to few degrees from a full Moon. IACT observations under moonlight are becoming more and more standard, and are routinely performed with the MAGIC and VERITAS telescopes. The performance of VERITAS under moonlight with different hardware settings at a given NSB level has been recently reported [@VeritasNew]. In this paper, we present a more complete study on how the performance of an IACT instrument is affected by moonlight and how it degrades as a function of the NSB. Our study is based on extensive observations of the Crab Nebula, adapted data reduction and tuned Monte Carlo (MC) simulations. The observations, carried out from October 2013 to March 2016 by MAGIC with nominal HV, reduced HV and UV-pass filters, cover the full range of NSB levels that are typically encountered during moonlight nights. The MAGIC telescopes under moonlight {#sec:MAGIC} ==================================== MAGIC (Major Atmospheric Gamma-ray Imaging Cherenkov) is a system of two 17m-diameter imaging atmospheric Cherenkov telescopes located at the Roque de los Muchachos Observatory on the Canary Island of La Palma, Spain, at an altitude of 2200m a.s.l. The telescopes achieve their best performance for VHE $\gamma$-ray observation in the absence of moonlight. Under such conditions, and for zenith angles below $30^\circ$, MAGIC reaches an energy threshold of $\sim50$GeV at trigger level, and a sensitivity above 220GeV of $0.67 \pm 0.04 \% $ of the Crab Nebula flux (Crab Units, C.U.) in 50 hours of observation [@upgrade2]. MAGIC is also designed to observe under low and moderate moonlight. Each camera consists of 1039 6-dynode PMTs, that are operated at a relatively low gain, typically of 3-4 $ \times 10^4$. This configuration was set specifically to decrease the amount of charge that hits the last PMT dynode (anode) during bright sky observations due to the Moon, preventing fast aging (see more details in Section 3.10 of [@upgrade1]). With the same criteria, there are established safety limits for the current generated in the PMTs. Individual pixels (PMT) are automatically switched off if their anode currents (DCs) are higher than 47$\mu$A and the telescopes are typically not operated if the median current in one of the cameras is above 15$\mu$A (as a reference, during dark time the median current is about 1$\mu$A). A detailed study on the gain drop of the MAGIC PMTs when exposed to high illumination levels was reported in [@OldMoon], which shows that while the detectors are operated at low gain and within the imposed safety limits no significant degradation is expected in the lifetime of MAGIC. The MAGIC trigger system {#sec:trigger} ------------------------ The standard MAGIC trigger has three levels. The first one (L0) is an amplitude discriminator that operates individually on every pixel of the camera trigger area. All the L0 signals are sent to the second level (L1), a digital system that operates independently on each telescope, looking for time-coincident L0 triggers in a minimum number of neighboring pixels (typically three). Finally, the third level (L3) looks for time coincidence of the L1 triggers of each telescope [@upgrade1]. The trigger rates depend on the discriminator threshold (DT) set on each PMT at the L0 level. The DTs are controlled by the Individual Pixel Rate Control (IPRC) software, which aims to keep stable the L0 rates of every pixel within certain desired limits. These limits are optimized to provide the lowest possible energy threshold while keeping accidental rates at a low level which can be handled by the data acquisition system (DAQ) without incurring a significant additional dead time. The accidental L0 triggers are dominated by NSB fluctuations. As they can vary significantly during observations, the DTs are constantly changed by the IPRC. If the L0 rate of one pixel moves temporary outside the imposed limits, as it could happen if, e.g., a bright star is in the FOV, the IPRC adjusts its DT until the rate is back within the desired levels (for more details see Section 5.3.4 of [@upgrade1]). Noise fluctuations are higher in a region with high density of bright stars, like the galactic plane, than in an extragalactic one. During relatively bright moonlight observations the main contribution to NSB comes from the Moon itself. Unlike stars, that only affect a few pixels, the moonlight scattered by the atmosphere affects the whole camera almost uniformly (with the exception of the region within a few degrees of the Moon). The induced noise depends on zenith angle, the angular distance between the pointing direction and the Moon, its phase, its position in the sky and its distance to the Earth [@Britzger]. Essentially, accidental L0 rates get higher during moonlight observations and IPRC reacts increasing the DTs, resulting in a higher trigger-level energy threshold. Moonlight observations ---------------------- In this work, the performance of MAGIC is studied for different NSB conditions. During the observations we do not measure directly the NSB spectrum, but just monitor the DC in every camera pixel. We infer the NSB level by comparing the measured median DC in the camera of one of the telescopes, MAGIC 1, with a reference average median DC that is obtained in a well-defined set of observation conditions. Here we use as reference the telescopes pointing toward the Crab Nebula at low zenith angle during astronomical night, with no Moon in the sky or near the horizon, and good weather (no clouds or dust layer). We shall refer to these conditions as $\textit{NSB}_{\text{Dark}}$. The median DC in MAGIC 1 during Crab dark observations is affected by hardware interventions: it depends on the PMTs HV and as so it can change after a camera flat-fielding. For the whole studied period Crab median DC during dark observations with nominal HV lies between 1.1 and 1.3$\mu$A[^1]. Due to the constraints imposed by the DC safety limits described in Section \[sec:trigger\], observations are possible up to a brightness of about 12$\times \,\textit{NSB}_{\text{Dark}}$ using the standard HV settings (nominal HV). Observations can be extended up to about 20$\times \, \textit{NSB}_{\text{Dark}}$ by reducing the gain of the PMTs by a factor $\sim$1.7 (reduced HV settings). When the HV is reduced there is less amplification in the dynodes and so fewer electrons hit the anode. However, the PMT gains cannot be reduced by an arbitrary large factor because the performance would significantly degrade, resulting in lower collection efficiency[^2], slower time response, larger pulse-to-pulse gain fluctuations and an intrinsically worse signal-to-noise ratio [@Photonis]. Even when the telescopes are operated with reduced HV, observations are severely limited or cannot be performed if the Moon phase is above $90\%$. Observations can, however, be extended up to about 100$\times \, \textit{NSB}_{\text{Dark}}$ with the use of UV-pass filters. This limit is achievable if the filters are installed and at the same time PMTs are operated with reduced HV. This is done only in extreme situations ($>$50$\times \, \textit{NSB}_{\text{Dark}}$). All the UV-pass filter data included in this work were taken with nominal PMT gain. In practice, observations can be performed in conditions that are safe for the PMTs as close as a few degrees away from a full Moon. The telescopes can be pointed almost at any position in the sky, regardless the Moon phase, and, as a result, they can be operated continuously without full Moon breaks [@Filters]. The characteristics of the filters are explained in Section \[sec:filters\]. As a first approximation, the brightness of the whole sky strongly depends on the Moon phase and its zenith angle. Figure \[fig:NSBvsDist\] shows the brightness of a Crab-like FOV, seen by MAGIC, as a function of the angular distance to the Moon for different Moon phases. The brightness values were simulated with the code described in [@Britzger], for a Moon zenith angle of 45$^\circ$. While the Moon phase is lower than $50\%$, the brightness is below 5$\times \, \textit{NSB}_{\text{Dark}}$ in at least $80\%$ of the visible sky and then in general operations can be safely performed with nominal HV. For phases larger than $80\%$, the brightness is typically above 10$\times \, \textit{NSB}_{\text{Dark}}$ in most of the sky when the Moon is well above the horizon, and the observations are usually only possible with reduced HV. When the Moon phase is close to 100%, observations are practically impossible without the use of UV-pass filters. Combining nominal HV, reduced HV and UV-pass filter observations, MAGIC could increase its duty cycle to $\sim$40%. ![Crab FOV brightness, simulated with the code described in [@Britzger], as a function of the angular distance to the Moon for different Moon phases (gray solid lines). Moon zenith angle was fixed at 45$^\circ$. In blue, green and red the maximum NSB levels that can be reached using nominal HV, reduced HV and UV-pass filters are shown, respectively.[]{data-label="fig:NSBvsDist"}](NSBSepPhase_HWlimits_20170213){width="\columnwidth"} UV-pass filters {#sec:filters} --------------- ![The blue curve shows the typical Cherenkov light spectrum for a vertical shower initiated by a 1TeV $\gamma$ ray, detected at 2200m a.s.l [@1TeVshower]. In green, the emission spectrum of the NSB in the absence of moonlight measured in La Palma [@Benn98]. The dotted curves show the shape of direct moonlight spectrum (black) and Reyleigh-scattered diffuse moonlight (grey) [@SMARTS1; @SMARTS2]. The four curves are scaled by arbitrary normalization factors. The filter transmission curve is plotted in red. As a reference, the quantum efficiency of a MAGIC PMT is plotted in orange (using the right-hand axis).[]{data-label="fig:filters_transmission"}](transmision1TeVnPMTqe_20150524){width="\columnwidth"} ![On the left, the UV-pass filters installed on the camera of one of the MAGIC telescopes. On the right, design of the frame that holds the filters. The outer aluminium ring is screwed to the camera.[]{data-label="fig:frame"}](frame.eps){width="\columnwidth"} Camera filters are used to reduce strongly the NSB light, while preserving a large fraction of the Cherenkov radiation that peaks at $\sim$330nm. The filter transmission must be high in UV and cut the longer wavelengths. They were selected to maximize the signal-to-noise ratio that scales as $T_{\text{Cher}}/\sqrt{T_{\text{Moon}}}$, with $T_{\text{Cher}}$ and $T_{\text{Moon}}$ the Cherenkov-light and the moonlight transmission of the filters, respectively. An additional constraint was imposed by the MAGIC calibration laser, which has a wavelength of 355nm. $T_{\text{Moon}}$ depends on the spectral shape of the scattered moonlight, which depends on the angular distance to the Moon. Far from it (tens of degrees away) the NSB is dominated by Rayleigh-scattered moonlight that peaks at $\sim$470nm. Close to the Moon, Mie scattering of moonlight dominates; its spectrum peaks at higher wavelengths and resembles more the spectrum of the light coming directly from the Moon (“direct moonlight”). The spectral shape of the NSB is also affected by the aerosol content and distribution, and by the zenith angle of the Moon. Typical spectra for Rayleigh-scattered and direct moonlight were computed using the code SMARTS [@SMARTS1; @SMARTS2], adding the effect of the Moon albedo. They can be seen in Figure \[fig:filters\_transmission\], together with the spectrum of the Cherenkov light from a vertical shower initiated by a 1TeV $\gamma$ ray, at 2200m a.s.l. [@1TeVshower]. Taking the spectral information of Cherenkov light and diffuse moonlight into account, we selected commercial inexpensive UV-pass filters produced by Subei[^3] (model ZWB3) with a thickness of 3mm and a wavelength cut at 420nm. The filter transmission curve is also shown in Figure \[fig:filters\_transmission\]. The transmission of the filters for Cherenkov light from air showers were measured by installing a filter in only one of the two telescopes, selecting image of showers with similar impact parameters (defined as the distance of the shower axis to the telescope center) for both telescopes, and comparing the integrated charge in both images. The measured Cherenkov-light transmission at 30$^\circ$ from zenith is $T_{\text{Cher}}=(47 \pm 5)\%$ . The transmission for the NSB goes from $\sim$20%, when pointing close to the Moon, to $\sim$33%, when background light is dominated by either Rayleigh-scattered moonlight or the dark NSB. Other parameters such as the Moon phase and zenith angle also affect the NSB transmission. The conversion from DC to NSB level could then be different depending on the observation conditions. For the performance study in this work we adopted a “mean scenario”, corresponding to an NSB transmission of 25%. The filters were bought in tiles of 20cm$\times$30cm, and mounted on a light-weight frame. This frame consists of an outer aluminum ring that is screwed to the camera and steel 6mm$\times$6mm section ribs that are placed between the filter tiles (see Figure \[fig:frame\]). The filter tiles are fixed to the ribs by plastic pieces and the space between tiles and ribs is filled with silicon. This gives mechanical stability to the system and prevents light leaks. Two people can mount, or dismount, the UV-pass filter on a MAGIC camera in about 15 minutes. Data sample and analysis methods {#sec:DatAna} ================================ To characterize the performance of MAGIC under moonlight we used 174 hours of Crab Nebula observations taken between October 2013 and January 2016, under NSB conditions going from 1 (dark) up to $30 \times \, \textit{NSB}_{\text{Dark}}$[^4]. Observations were carried out in the so-called wobble mode [@Fomin], with a standard wobble offset of $0.4^\circ$. All the data correspond to zenith angles between 5$^\circ$ and 50$^\circ$. For this study we selected samples that were recorded during clear nights, for which the application of the MC corrections described in [@fruck2013] are not required. Data were divided into different samples according to their NSB level and the hardware settings in which observations were performed (nominal HV, reduced HV or UV-pass filters), as summarized in Table \[tabTime\]. When dividing the data we aimed to have rather narrow NSB bins while keeping sufficient statistics in each of them ($\sim10$ hours per bin). Bins are slightly wider in the case of the UV-pass filter data to fulfill that requirement. ---------------------------------- ------------------- ------- Sky Brightness Hardware Settings Time \[$\textit{NSB}_{\text{Dark}}$\] \[h\] 1 (Dark) nominal HV 53.5 1-2 nominal HV 18.9 2-3 nominal HV 13.2 3-5 nominal HV 17.0 5-8 nominal HV 9.8 5-8 reduced HV 10.8 8-12 reduced HV 13.3 12-18 reduced HV 19.4 8-15 UV-pass filters 9.5 15-30 UV-pass filters 8.3 ---------------------------------- ------------------- ------- : Effective observation time of the Crab Nebula subsamples in each of the NSB/hardware bins.[]{data-label="tabTime"} ![Distributions of the pixel charge extracted with a sliding window for pedestal events (i.e., without signal) for different NSB/hardware conditions.[]{data-label="fig:pedchargedistr"}](PedestalChargeDistribution_BWcompatible_legUpdate){width="\columnwidth"} Analysis {#sec:analysis} -------- In this section we describe how moonlight affects the MAGIC data and how the analysis chain and MC simulations have been adapted. The data have been analyzed using the standard MAGIC Analysis and Reconstruction Software (MARS, [@TrueMARS]) following the standard analysis chain described in [@upgrade2], besides some modifications that were implemented to account for the different observation conditions. ---------------------------------- ------------------- ---------------- --------------------------------- ---------- Sky Brightness Hardware Settings Pedestal Distr Cleaning Level factors Size Cut mean / rms $\text{Lvl}_1$ / $\text{Lvl}_2$ \[$\textit{NSB}_{\text{Dark}}$\] \[phe\] \[phe\] \[phe\] 1 (Dark) nominal HV 2.0 / 1.0 6.0 / 3.5 50 1-2 nominal HV 2.5 / 1.2 6.0 / 3.5 60 2-3 nominal HV 3.0 / 1.3 7.0 / 4.5 80 3-5 nominal HV 3.6 / 1.5 8.0 / 5.0 110 5-8 nominal HV 4.2 / 1.7 9.0 / 5.5 150 5-8 reduced HV 4.8 / 2.0 11.0 / 7.0 135 8-12 reduced HV 5.8 / 2.3 13.0 / 8.0 170 12-18 reduced HV 6.6 / 2.6 14.0 / 9.0 220 8-15 UV-pass filters 3.7 / 1.6 8.0 / 5.0 100 15-30 UV-pass filters 4.3 / 1.8 9.0 / 5.5 135 ---------------------------------- ------------------- ---------------- --------------------------------- ---------- ### Moonlight effect on calibrated data {#sec:calib} After the trigger conditions are fulfilled, the signal of each pixel is recorded into a 30ns waveform. Then an algorithm looks over that waveform for the largest integrated charge in a sliding window of 3ns width, which is saved and later calibrated [@upgrade2]. In the absence of signal, the sliding window picks up the largest noise fluctuation of the waveform. The main sources of noise are the statistical fluctuations due to NSB photons, the PMT after pulses and the electronic noise. The noise due to background light fluctuations scales as the square root of the NSB (Poisson statistics). The after pulse rate is proportional to the PMT current, which increases linearly with the NSB. When the PMTs are operated under nominal HV, electronic noise has a similar level to the NSB fluctuation induced by a dark extragalactic FOV, which has no bright stars [@upgrade1]. For Crab dark observations, the brightness of the FOV ($\textit{NSB}_{\text{Dark}}$) is about 70% higher than dark extragalactic FOV, and the NSB-related noise already dominates. Figure \[fig:pedchargedistr\] shows the distribution of extracted charge in photoelectrons (phe) for pedestal events (triggered randomly without signal) under four different observation conditions. During observations of the Crab Nebula under dark conditions the pedestal distribution has an RMS of $\sim $1phe and a mean bias of $\sim2$phe. The distribution is asymmetric with larger probability of upward fluctuation (induced by the sliding window method) and an extra tail at large signals ($>$8phe) produced by the PMT after pulses. During moonlight observations, the noise induced by the NSB increases while the electronic noise remains constant (as long as the hardware settings remain unchanged). In fact, the electronic noise in terms of photoelectrons is proportional to the calibration constant, which depends on the hardware configuration of the observations. With reduced HV, all gains are lower, and hence the calibration constants increase resulting in higher electronic noise level in phe ($\sim$1.7) and, as a consequence, worse signal-to-noise ratio of integrated pulses. The transient time in PMTs also increases when the gain is lowered, but the delay in arrival time of pulses is $\sim$1 ns. The signal pulse is always well within the 30ns window and then the peak search method is not affected. During UV-pass filter observations PMTs are operated with nominal HV but some pixels are partially shadowed by the filter frame[^5]. The camera flat-fielding, which makes all pixels respond similarly to the same sky light input, gives higher calibration constants to the shadowed pixels. Thus, electronic noise on those pixels is larger, while in contrast the NSB noise is strongly reduced by the filters. The relative contribution of the electronic to the total noise is then also higher during UV-pass filter observations. Table \[tabNoise\] shows the typical pedestal distribution mean and RMS for all the NSB/hardware bins. The broader pedestal charge distribution has a double effect on the extraction of a real signal (Cherenkov light). If the signal is weak, the maximal waveform fluctuation may be larger than the Cherenkov pulse and the sliding window could select the wrong section. Then, the reconstructed pulse time is random and the signal is lost. If the signal is strong enough, the sliding window selects the correct region, the time and amplitude of the signal is just less precise (NSB does not induce a significant bias). Strong signals are almost not affected as their charge resolution is dominated by close to Poissonian fluctuations of the number of recorded phe. ### Moonlight-adapted image cleaning After the calibration of the acquired data, charge and timing information of each pixel is recorded. Most pixel signals contain only noise. The so-called sum-image cleaning [@upgrade2] is then performed to remove those pixels. In this procedure we search for groups of 4, 3 and 2 neighboring (4NN, 3NN, 2NN) pixels with a summed charge above a given level, within a given time window. The charge thresholds for 4NN-, 3NN-, 2NN-charge thresholds are set to $4 \times \text{Lvl}_1$, $3 \times 1.3 \times \text{Lvl}_1$, $2 \times 1.8 \times \text{Lvl}_1$, respectively, where $\text{Lvl}_1$ is a global factor adapted to the noise level of the observations. The time windows are kept fixed at 1.1ns, 0.7ns and 0.5ns, respectively, independent on the NSB level. Pixels belonging to those groups are identified as core pixels. Then all the pixels neighboring a core pixel that have a charge higher than a given threshold ($\text{Lvl}_2$) and an arrival time within 1.5ns with respect to that core pixel, are included in the image. In the MAGIC standard analysis [@upgrade2] the cleaning levels are set to $\text{Lvl}_1 = 6$phe and $\text{Lvl}_2=3.5$phe, which provide good image cleaning for any moonless-night observation. Higher cleaning levels would result in a higher energy threshold at the analysis level. In contrast, lower cleaning levels can also be used for dark extragalactic observation to push the analysis threshold as low as possible [@B0218]. The standard-analysis cleaning levels are then a compromise between robustness and performance, optimized to be used for any FOV, galactic or extragalactic, under dark and dim moonlight conditions. During moonlight observations the background fluctuations are higher and the cleaning levels must be increased accordingly. Those levels were modified to ensure that the fraction of pedestal events that contain only noise and survive the image cleaning is lower than 10%. They were optimized for every NSB/hardware bin independently to get the lowest possible analysis threshold for every bin. The optimized cleaning levels for each bin are shown in Table \[tabNoise\]. The time window widths were not modified for reduced HV observations, because the variations in the PMTs response are expected to be very small. We do not use variable cleaning levels that would automatically scale as a function of the noise because the MAGIC data reconstruction is based on comparison with MC simulations, which must have exactly the same cleaning levels as the data. During moonlight observations, the noise level is continuously changing, so it is not realistic to fine tune our MC for every observation. Instead we create a set of MC simulations for every NSB/hardware bin with fixed noise and cleaning levels. ### Moon-adapted Monte Carlo simulations {#sec:MC} {width="1.5\columnwidth"} MC simulations have mainly two functions in the MAGIC data analysis chain. A first sample (train sample) is used to build look-up tables and multivariate decision trees (random forest), which are employed for the energy and direction reconstruction and gamma/hadron separation [@upgrade2]. A second, independent sample (test sample) is used for the telescope response estimation during the source flux/spectrum reconstruction. We prepared MC samples adapted for every NSB/hardware bin. For nominal and reduced HV settings, we used the standard MAGIC MC simulation chain with additional noise to mimic the effect of moonlight (and reduced HV). The noise is injected after the calibration at the pixel signal level. First we model the noise distribution in a given integration window of 3ns that would produce the same pedestal charge distribution than the one obtained during observations (see Figure \[fig:pedchargedistr\]) using the sliding window search method described in Section \[sec:calib\]. We then extract a random value from the modeled noise distribution and add it to the extracted signal of the MC event. If the modified signal is larger than a random number following the pedestal charge distribution, this new value becomes the new charge and a random jitter is added to the arrival time (depending on the new signal/noise ratio). If the random pedestal signal is larger it means that the sliding window caught a spurious bump larger than the signal itself, then the pixel charge is set to this fake signal and the arrival time is chosen randomly according to the pedestal time distribution. This method allows us to adapt our MC to any given NSB without reprocessing the full telescope simulation and data calibration. In the case of the UV-pass filter observations, additional modifications on the simulation chain were implemented to include the filter transmission and the shadowing produced by the frame ribs. We did not simulate the effect of the moonlight on the trigger because it is very difficult to reproduce the behavior of the IPRC, which control the pixel DTs (see section \[sec:trigger\]). Instead, simulations were performed using the standard dark DTs and we later applied cuts on the sum of charge of pixels surviving the image cleaning (image size) on each telescope. This size cut acts as a software threshold and it is optimized bin-wise as the minimal size for which the data and MC distributions are matching. Even in the absence of moonlight a minimum cut in the total charge of the images is applied, as potential $\gamma$-ray events with lower sizes are either harder to reconstruct or to distinguish from hadron-induced showers [@upgrade2]. The used size cuts are given Table \[tabNoise\]. Figure \[fig:SizeM1\] compares size distributions of MC $\gamma$-ray events (simulated with the spectrum of the Crab Nebula reported in @upgrade2) with those of the observed excess events within a 0.14$^\circ$ circle from the Crab Nebula. Performance =========== In this section we evaluate how moonlight and the use of different hardware configurations affect the main performance parameters of the MAGIC telescopes. Energy threshold ---------------- ![Effective collection area at reconstruction level for zenith angles below 30$^\circ$ for four different observation conditions: Dark conditions with nominal HV (black), 3-5 $\times \textit{NSB}_{\text{Dark}}$ with nominal HV (blue), 5-8 $\times \, \textit{NSB}_{\text{Dark}}$ with reduced HV (green) and 8-15 $\times \textit{NSB}_{\text{Dark}}$ with UV-pass filters (red). The optimized cleaning levels and size cuts from Table \[tabNoise\] were used to produce these plots.[]{data-label="fig:CollAreaExample"}](CollAreavsNSB_Reconstruction_20161213){width="\columnwidth"} The energy threshold of IACT telescopes is commonly defined as the peak of the differential event rate distribution as a function of energy. It is estimated from the effective collection area as a function of the energy, obtained from $\gamma$-ray MC simulations, multiplied by the expected $\gamma$-ray spectrum, which is typically (and also in this work) assumed to be a power-law with a spectral index of $-2.6$. It can be evaluated at different stages of the analysis. The lowest threshold corresponds to the trigger level, which reaches $\sim50$ GeV during MAGIC observations in moonless nights at zenith angles below 30$^\circ$ [@upgrade2]. It naturally increases during moonlight observations, as the DTs are automatically raised by the IPRC (see Section \[sec:trigger\]). As explained in section \[sec:analysis\], our MC simulations do not reproduce the complex behavior of the trigger during such observations. Here we evaluate then the energy threshold at a later stage, after image cleaning, event reconstruction and size cuts (reconstruction level), for which a good matching between real data and MC is achieved. The effective collection area at the reconstruction level as a function of the energy for four different NSB/hardware situations are shown in Figure \[fig:CollAreaExample\]. In all four curves two regimes can be identified: one, at low energies, which is rapidly increasing with the energy and another, towards high energies, which is close to a plateau. As expected, the dark-sample analysis presents the largest effective area along the full energy range. The degradation due to moonlight is more important at the lowest energies, where the Cherenkov images are small and dim. The higher the size cuts and cleaning levels, the higher the energy at which the plateau is achieved. In the case of UV-pass filter observations, the used cleaning levels and size cuts are lower (in units of phe) than the ones applied during reduced HV data analysis, but due to the filter transmission, the plateau is reached at even higher energies. Above $\sim$1TeV the effective area is almost flat for the four studied samples and the effect of Moon analysis is very small (below $\sim$10%). The degradation of the effective area at low energies is directly translated into an increase of the energy threshold, as can be seen in Figure \[fig:EthExample\], where the differential rate plots for the same four NSB/hardware cases are shown. The energy threshold at reconstruction level is estimated by fitting a Gaussian distribution in a narrow range around the peak of these distributions[^6]. In Figure \[fig:EthNSB\] we show the obtained energy threshold as a function of the sky brightness for different hardware configurations at low ($<30^\circ$) and medium ($30^\circ - 45^\circ$) zenith angles[^7]. For low zenith angles it goes from $\sim$70GeV in the absence of moonlight to $\sim$300GeV in the brightest scenario considered. For medium zenith angles, the degradation is similar from $\sim$110GeV to $\sim$500GeV. The degradation of the energy threshold $E_{\text{th}}$ as a function of the NSB level can be roughly approximated, for nominal HV and reduced HV data, by $$\label{eq:ThrFit} E_{\text{th}}(\textit{NSB}) = E^{\text{Dark}}_{\text{th}} \times \left(\dfrac{\textit{NSB}}{\textit{NSB}_{\text{Dark}}}\right)^{0.4}$$ Where $E^{\text{Dark}}_{\text{th}}$ is the energy threshold during dark Crab Nebula observations. At the same NSB level, reduced HV data have a slightly higher energy threshold than nominal HV data due to higher electronic noise in phe units, while the UV-pass-filter energy threshold is significantly higher ($\sim$40%) than the one of reduced HV data without filters. The energy threshold increase with filters is due to the lower photon statistic (the same shower produces less phe). This degradation is reduced at higher NSBs (i.e. higher energies), where larger image sizes make the photon statistic less important than the signal-to-noise ratio in the energy threshold determination. ![Rate of MC $\gamma$-ray events that survived the image cleaning and a given quality size cut for an hypothetical source with an spectral index of $-2.6$ observed at zenith angles below 30$^\circ$. The four curves correspond to different observation conditions: Dark conditions with nominal HV (black), 3-5 $\times \textit{NSB}_{\text{Dark}}$ with nominal HV (blue), 5-8 $\times \, \textit{NSB}_{\text{Dark}}$ with reduced HV (green) and 8-15 $\times \textit{NSB}_{\text{Dark}}$ with UV-pass filters (red). Dashed lines show the gaussian fit applied to calculate the energy threshold on each sample.[]{data-label="fig:EthExample"}](EnergyThreshold_4NSBs_20170130_fits_slash){width="\columnwidth"} ![Energy threshold at the event reconstruction level as a function of the sky brightness for observations with nominal HV (black), reduced HV (green) and UV-pass filters (red) at zenith angles below 30$^\circ$ (filled circles, solid lines) and between 30$^\circ$ and 45$^\circ$ (empty squares, dashed lines). Gray lines represent the approximation given by equation \[eq:ThrFit\] for zenith angles below 30$^\circ$ (solid) and between 30$^\circ$ and 45$^\circ$ (dashed).[]{data-label="fig:EthNSB"}](EnergyThresholdvsNSB_30-45_20170130){width="\columnwidth"} Reconstruction of the Crab Nebula spectrum {#sec:CrabSpectra} ------------------------------------------ ### Standard cleaning MAGIC data are automatically calibrated with the standard analysis chain optimized for dark observations. Most of the analyses start from high level data, after image cleaning and event reconstruction. When dealing with moonlight data an adapted analysis is in principle required, as described in Section \[sec:analysis\]. However, the effect of weak moonlight can be almost negligible and the data can be processed following the standard chain. Here we want to determine which is the highest NSB level for which the standard analysis provides consistent results, within reasonable systematic uncertainties, with respect to those obtained with the dark reference sample. To answer this question we attempted to reproduce the Crab Nebula spectrum by applying the standard analysis, including standard dark MC for the train and test samples, to our moonlight data taken with nominal HV. To minimize systematic uncertainties we use typical selection cuts with 90% $\gamma$-ray efficiency for the $\gamma$-ray/hadron separation and sky signal region radius [@upgrade2]. The obtained Crab Nebula spectral energy distributions (SEDs) are shown in figure \[fig:CrabSEDStd\] for 1-8 $\times \, \textit{NSB}_{\text{Dark}}$. The image size cuts described in Section \[sec:MC\] were applied to produce these spectra. The SED obtained using data with 1-2 $\times \, \textit{NSB}_{\text{Dark}}$ is compatible, within errors, with the one obtained with dark data. This shows that the standard analysis is perfectly suitable for this illumination level. For brighter NSB conditions the reconstructed spectra are underestimated. With 2-3 $\times \, \textit{NSB}_{\text{Dark}}$, the data-point errors above $\sim$130GeV are below $\sim$20% while with 5-8 $\times \, \textit{NSB}_{\text{Dark}}$ the reconstructed flux falls below $\sim$50% at all energies. Thus, the standard analysis chain can be still used for weak moonlight at the price of additional systematic bias (10% for 1-2 $\times \, \textit{NSB}_{\text{Dark}}$ and 20% for 2-3 $\times \, \textit{NSB}_{\text{Dark}}$) but for higher NSB levels a dedicated Moon analysis is mandatory. ### Custom analysis Figure \[fig:CrabSEDMoon\] shows the spectra of the Crab Nebula obtained after applying the dedicated Moon analysis (dedicated MC, cleaning levels and size cuts) described in Section \[sec:analysis\] to each data set. In almost all the cases the fluxes obtained are consistent within $\pm$20% with the one obtained under dark conditions, at least up to 4TeV. The only exception is the brightest NSB bin (UV-pass filters data up to 30 $\times \textit{NSB}_{\text{Dark}}$) where the ratio of the flux to the dark flux gets slightly above $\sim$30% at energies between about 400 and 800GeV. It is also interesting to notice how the spectrum reconstruction improves when the dedicated moon analysis is performed by comparing the spectra obtained for the nominal HV samples in Figures \[fig:CrabSEDStd\] and \[fig:CrabSEDMoon\]. Angular resolution {#sec:AngRes} ------------------ {width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"} The reconstruction of the $\gamma$-ray arrival direction could be affected in two ways by moonlight. Firstly, as already discussed, it induces more background noise that affects the quality of the recorded images. Secondly the moonlight can disturb the tracking monitor of the telescope, which is based on a star-guiding system [@Riegel2005ICRC]. An eventual mispointing is ruled out by checking that for every NSB/hardware bin the center of the 2D-skymap event excess distribution (obtained with a Gaussian fit) is well within a 0.02$^\circ$ circle around the actual Crab Nebula position as expected from the pointing accuracy of MAGIC [@upgrade2]. To study the possible degradation of the point spread function (PSF), we compare the $\theta^2$ distribution obtained for Crab data taken under moonlight and under dark conditions, $\theta$ being the angular distance between the Crab Nebula position and the reconstructed event arrival direction. As explained in [@upgrade2], this distribution can be well fitted by a double exponential function. Figure \[fig:angRes\] shows the $\theta^2$ distribution of events with estimated energy above 300GeV and $\gamma$-ray/hadron separation cut corresponding to 90% $\gamma$-ray efficiency for four representative NSB/hardware bins. For all the NSB/hardware bins the $\theta^2$ distribution above the energy threshold is in good agreement with the PSF obtained under dark conditions. The angular resolution does not seem to be significantly affected by moonlight. Sensitivity ----------- As shown in previous sub-sections, moonlight observations are perfectly apt for bright $\gamma$-ray sources such as the Crab Nebula, whose spectrum and direction can be well reconstructed, with the only drawback being a higher energy threshold with respect to the one obtained in dark observations. However, one may wonder how the performance for the detection of weak sources is affected by moonlight, which may degrade the $\gamma$-ray/hadron separation power. To study this potential effect, we computed the minimal $\gamma$-ray flux that MAGIC can detect in 50h of observation, from $\gamma$-ray and background event rates obtained with the Crab Nebula samples analyzed in this work, following the method described in [@upgrade2] [^8]. For each NSB/hardware bin, the $\gamma$-ray and background rates are obtained for several analyses achieving different energy thresholds. Each analysis corresponds to a set of cuts in the image size and reconstructed energy as well as previously optimized $\gamma$-ray/hadron separation cuts. The analysis-level energy threshold is estimated by applying the same set of cuts to a $\gamma$-ray MC sample simulated with the same energy spectrum as the Crab Nebula and re-weighted to reproduce the same zenith-angle distribution as for the observations. To accumulate enough data in every NSB/hardware bin, we use data from a large zenith angle range going from 5$^\circ$ to 45$^\circ$. As the sensitivity and energy threshold depend strongly on the zenith angle and data sub-samples have different zenith angle distributions, the performances are corrected to correspond to the same reference zenith-angle distribution (average of all the data). To visualize the degradation caused by moonlight, the integral sensitivity computed for each NSB/hardware bin is divided by the one obtained under dark conditions at the same analysis-level energy threshold. The obtained sensitivity ratios are shown in Figure \[fig:Sens\] as a function of the energy threshold. The Moon data taken with nominal HV provide a sensitivity only slightly worse than the one obtained using dark data. The sensitivity degradation is constrained to be less than 10% below 1TeV and all the curves are compatible within error bars above $\sim$300GeV. Error bars increase with the energy because the event statistic decreases dramatically. These error bars are not independent as the data corresponding to a given energy threshold are included in the lower energy analysis. The only visible degradation is near the reconstruction-level energy threshold ($<$200GeV), where the sensitivity is 5-10% worse. For Moon data taken with reduced HV, the sensitivity degradation lies between 15% and 30%. It seems to increase with the NSB level, although above 400GeV the three curves are compatible within statistical errors. This degradation is caused by a combination of a higher extracted-signal noise (see section \[sec:analysis\]) and a smaller effective area. The degradation is even clearer in the UV-pass filter data, where the sensitivity is 60-80% worse than the standard one. Such a degradation is expected, especially due to the fact that the filters reject more than 50% of the Cherenkov light. Besides, sensitivity could also be affected by a poorer reconstruction of the images, especially in the pixels that are partially obscured by the filter frame ribs. At the highest energies ($>$2TeV) sensitivity seems to improve. This could be expected for bright images, that are less affected by noise, but higher statistics at those energies would be needed to derive further conclusions. ![Ratio of the integral sensitivity under moonlight to the dark sensitivity as a function of the analysis energy threshold, for nominal HV (top), reduced HV (middle) and UV-pass filter (bottom) data. The NSB levels are given in unit of $\textit{NSB}_{\text{Dark}}$[]{data-label="fig:Sens"}](Sensitivity_vs_Eth_MoonA "fig:"){width="\columnwidth"} ![Ratio of the integral sensitivity under moonlight to the dark sensitivity as a function of the analysis energy threshold, for nominal HV (top), reduced HV (middle) and UV-pass filter (bottom) data. The NSB levels are given in unit of $\textit{NSB}_{\text{Dark}}$[]{data-label="fig:Sens"}](Sensitivity_vs_Eth_MoonA_RedHV "fig:"){width="\columnwidth"} ![Ratio of the integral sensitivity under moonlight to the dark sensitivity as a function of the analysis energy threshold, for nominal HV (top), reduced HV (middle) and UV-pass filter (bottom) data. The NSB levels are given in unit of $\textit{NSB}_{\text{Dark}}$[]{data-label="fig:Sens"}](Sensitivity_vs_Eth_MoonA_Filters "fig:"){width="\columnwidth"} ![Daily light curve of the Crab Nebula above 300GeV for observation under different sky brightness with nominal HV (top), reduced HV (middle) and above 500GeV for UV-pass filters (bottom). Horizontal lines correspond to the constant flux fit of the different NSB bins. For comparison, the LC and constant fit of the dark observation are reproduced in every panel.[]{data-label="fig:LC"}](CrabLC_MoonA "fig:"){width="\columnwidth"} ![Daily light curve of the Crab Nebula above 300GeV for observation under different sky brightness with nominal HV (top), reduced HV (middle) and above 500GeV for UV-pass filters (bottom). Horizontal lines correspond to the constant flux fit of the different NSB bins. For comparison, the LC and constant fit of the dark observation are reproduced in every panel.[]{data-label="fig:LC"}](CrabLC_MoonA_redHV "fig:"){width="\columnwidth"} ![Daily light curve of the Crab Nebula above 300GeV for observation under different sky brightness with nominal HV (top), reduced HV (middle) and above 500GeV for UV-pass filters (bottom). Horizontal lines correspond to the constant flux fit of the different NSB bins. For comparison, the LC and constant fit of the dark observation are reproduced in every panel.[]{data-label="fig:LC"}](CrabLC_MoonA_Filter_500GeV "fig:"){width="\columnwidth"} Systematics ----------- During moonlight observations many instrumental parameters are more variable than during dark observations, in particular the trigger DTs and the extracted signal noise, and these variations induce larger MC/data mismatches and then larger systematic uncertainties. As shown in Section \[sec:CrabSpectra\], the Crab Nebula spectrum can be well reconstructed in every NSB/hardware bin. The reconstructed flux above the energy threshold of every NSB bin is within a 10%, 15%, 30% error band around the flux obtained under dark conditions for nominal HV, reduced HV and UV-pass filter observations, respectively. The spectral shape is particularly well reproduced in all hardware configurations. The dark-Moon flux ratios vary less than 10% over an order of magnitude in energy, corresponding to an additional systematic on the power-law spectral index below 0.05. The overall flux may mask large day-to-day fluctuations due to different sky brightness. To estimate this additional day-to-day systematic, we show in figure \[fig:LC\] the daily light curve (LC) of the Crab Nebula flux above 300GeV from October 2013 to March 2016 for every NSB level observed without UV-pass filters and the LC above 500GeV from January to October 2015 for the two NSB bins with UV-pass filters[^9]. Taking into account only statistical fluctuations, the $\chi^2$ test indicates that a constant flux is incompatible for every LC (even for dark observations). Assuming conservatively that the additional fluctuations are only due to systematic uncertainties (i.e., the Crab Nebula flux is constant), we estimate these systematic uncertainties by adding errors quadratically to the statistical errors in every data point until the constant-fit $\chi^2$ equals the number of degrees of freedom $k$ plus or minus $\sqrt{2k}$ (standard deviation of the $\chi^2$ distribution). In order to constrain strongly the constant fit we include data points of several NSB bins for the fit of moderate moonlight with nominal HV ($1-8 \times \, \textit{NSB}_{\text{Dark}}$), moonlight with reduced HV ($5-18 \times \, \textit{NSB}_{\text{Dark}}$) and strong moonlight with UV-pass filter ($8-30 \times \, \textit{NSB}_{\text{Dark}}$). Table \[tabSyst\] gives the day-to-day systematic errors obtained for these three hardware/NSB conditions as well as for dark observation with nominal HV. Sky Brightness Hardware Settings Day-to-day Systematics ----------------------------------------- ------------------- ------------------------ Dark ($\textit{NSB}_{\text{Dark}} = 1$) nominal HV $(7.6\pm1.2)$% 1-8 $\textit{NSB}_{\text{Dark}}$ nominal HV $(9.6\pm1.2)$% 5-18 $\textit{NSB}_{\text{Dark}}$ reduced HV $(15.4\pm3.2)$% 8-30 $\textit{NSB}_{\text{Dark}}$ UV-pass filters $(13.2\pm3.4)$% For dark observations, the obtained day-to-day systematic uncertainty is $(7.6\pm1.2)$%. This result is below the previous study based on Crab Nebula LC that reports a day-to-day systematic uncertainty of $\sim$12% for the period from November 2009 to January 2011 [@Perf2012] and from October 2009 to April 2011 [@Crab2015JHEAp]. This is consistent with the result after the telescope upgrade reported in [@upgrade2], which claims day-to-day systematic uncertainty below 11%. For observation under moonlight with nominal HV (NSB $<8 \times \, \textit{NSB}_{\text{Dark}}$), the obtained day-to-day systematic is $(9.6\pm1.2)$%, still below the 11%. The additional systematic due to the moonlight is marginal and can be only constrained to be below 9%. For brighter moonlight that requires hardware modifications, the systematic errors get larger. A few data points show a flux much lower than expected (down to $\sim$50%). The overall day-to-day systematic is estimated at $(15.4\pm3.2)$% for reduced HV and $(13.2\pm3.4)$% for UV-pass filters, corresponding to an additional systematic on top of the dark nominal HV systematic errors laying between 6% and 18%. For every hardware configuration, the additional day-to-day systematic errors is of the same order, or below, the systematic errors found for the overall flux. To summarize, the additional systematic uncertainties of MAGIC during Moon time depend on the hardware configuration and the NSB level. For moderate moonlight (NSB $<8 \times \, \textit{NSB}_{\text{Dark}}$) observations with nominal HV, the additional systematic errors on the flux is below 10%, raising the flux-normalization uncertainty (at a few hundred GeV) from 11% [@upgrade2] to 15%. For observations with reduced HV (NSB $<18 \times \, \textit{NSB}_{\text{Dark}}$) the additional systematic errors on the flux is $\sim$15%, corresponding to a full flux-normalization uncertainty of 19% after a quadratic addition. For UV-pass filter observations, the flux-normalization uncertainty increases to 30%. The additional systematic on the reconstructed spectral index is negligible ($\pm$0.04) and the overall uncertainty is still $\pm$0.15 for all hardware/NSB configurations. The uncertainty of the energy scale is not affected by the moonlight. It may increase for reduced HV and UV-pass filter observations but this effect is included in the flux-normalization uncertainty increase[^10]. Concerning the pointing accuracy, as discussed in Section \[sec:AngRes\], no additional systematic uncertainties have been found. Conclusions =========== For the first time the performance under moonlight of an IACT system is studied in detail with an analysis dedicated for such observations, including moonlight-adapted MC simulations. This study includes data taken with three different hardware settings: nominal HV, reduced HV and UV-pass filters. During moonlight, the additional noise results in a higher energy threshold increasing with the NSB level, which for zenith angles below 30$^{\circ}$ goes from $\sim$70GeV (at the reconstruction level) under dark conditions up to $\sim$300GeV in the brightest scenario studied (15-30 $\times \, \textit{NSB}_{\text{Dark}}$). With a dedicated moonlight-adapted analysis, we are able to reconstruct the Crab Nebula spectrum in all the NSB/hardware bins considered. The flux obtained is compatible within 10%, 15% and 30% with the one obtained under dark conditions for nominal HV, reduced HV and UV-pass filter observations, respectively. The systematic uncertainty on the flux-normalization, 11% for standard dark observation, increases to 15% for nominal HV moonlight observations with NSB $< 8\times \, \textit{NSB}_{\text{Dark}}$, 19% for reduced HV observations between 5 and 18 $\times \, \textit{NSB}_{\text{Dark}}$ and 30% for UV-pass filter observations between 8 and 30 $\text{NSB}_{\text{Dark}}$. No significant additional systematic on the spectral slope was found, and the overall uncertainty is still $\pm$0.15 as reported in [@upgrade2]. An eventual degradation in the sensitivity is constrained to be below 10% while observing with nominal HV under illumination levels $<8 \times \, \textit{NSB}_{\text{Dark}}$. The sensitivity degrades by 15 to 30% when observing with reduced HV and by 60 to 80% when observing with UV-pass filters. No significant worsening on the angular resolution above 300GeV was observed. The main benefit of operating the telescopes under moonlight is that duty cycle can be doubled, suppressing the need to stop observations around full Moon. Depending on the needed energy threshold, many projects can profit from this additional time. Already moderate moonlight observations lead to the discovery of several active galactic nuclei, such as PKS 1222+21 [@PKS1222], 1ES 1727+502 [@MAGIC1ES1727; @Veritas1ES], B3 2247+381 [@B322]. They are also used to study light curves of variable sources with better sampling, for instance the binary systems LSI +61 303 [@LSI] and HESS J0632+057 [@HESSj0632] and the active galactic nuclei PG1553+13 [@PG1553], or to accumulate large amount of data as for deep observations of the Perseus cluster [@PerseusCR]. The present study shows that, except for the energy threshold, the performance of IACT arrays is only moderately affected by moonlight. Hardware modifications to tolerate a strong sky brightness (reduced HV, UV-pass filters) seem to have more effect than the noise increase. The use of robust photodetectors, e.g. silicon photomultipliers, in the future should improve the performance under these bright conditions. The bright moonlight observations are particularly useful for projects in which the relevant physics lie above a few hundred GeV, such as long monitoring campaigns of VHE sources with hard spectrum or deep observation of supernova remnants for PeVatron studies. The eventual loss in sensitivity can be compensated with the possibility of much longer observation time in a less demanded observation period (currently often even used for technical work). In addition, observations under extreme NSB conditions are sometimes unavoidable, as in the case of the observation of the shadowing of cosmic rays by the Moon[^11]. Observations under moonlight open many possibilities that should be more and more used with the current flourish of the VHE $\gamma$-ray astronomy using the IACT. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank the Instituto de Astrofísica de Canarias for the excellent working conditions at the Observatorio del Roque de los Muchachos in La Palma. The financial support of the German BMBF and MPG, the Italian INFN and INAF, the Swiss National Fund SNF, the ERDF under the Spanish MINECO (FPA2015-69818-P, FPA2012-36668, FPA2015-68378-P, FPA2015-69210-C6-2-R, FPA2015-69210-C6-4-R, FPA2015-69210-C6-6-R, AYA2015-71042-P, AYA2016-76012-C3-1-P, ESP2015-71662-C2-2-P, CSD2009-00064), and the Japanese JSPS and MEXT is gratefully acknowledged. This work was also supported by the Spanish Centro de Excelencia “Severo Ochoa” SEV-2012-0234 and SEV-2015-0548, and Unidad de Excelencia “María de Maeztu” MDM-2014-0369, by the Croatian Science Foundation (HrZZ) Project 09/176 and the University of Rijeka Project 13.12.1.3.02, by the DFG Collaborative Research Centers SFB823/C4 and SFB876/C3, and by the Polish MNiSzW grant 2016/22/M/ST9/00382. References {#references .unnumbered} ========== [^1]: As the Crab Nebula is in the galactic plane, the NSB is lower by 30-40% for a large fraction of MAGIC observations, which point to extragalactic regions of the sky. During reduced HV and UV-pass filter observations the measured DC is lower than what would be obtained if observing under the same NSB conditions and nominal HV. Correction factors are applied to properly convert from DC to NSB level based on the gain reduction factor of the PMTs and on the moonlight transmission of the filters. [^2]: In MAGIC the HV divider chain is fixed for all dynodes and the voltage is also reduced at the first dynode. [^3]: [http://www.globalsources.com/sbgx.co]{} [^4]: Observations are possible at higher illumination levels, but it is hard to get Crab data under such occasions. In fact, only on rare situations MAGIC targets are found under higher NSB levels than the ones analyzed in this work. [^5]: The shadowing of the frame is important (blocking more than 40% of the incoming light) for $\sim$7% of the pixels. [^6]: Note that in those distributions the peak is broad, which means that it is possible to obtain scientific results with the telescopes below the defined threshold. [^7]: Here we compute an average over a relatively wide zenith range, but energy threshold dependence with the zenith angle is stronger for medium zenith angles (see Figure 6 in [@upgrade2]) [^8]: The sensitivity is defined as the integral flux above an energy threshold giving $N_{\text{excess}} / \sqrt{N_{\text{bgd}}} = 5$, where $N_{\text{excess}}$ is the number of excess events and $N_{\text{bgd}}$ the number of background events, with additional constraints: $N_{\text{excess}} > 10$ and $N_{\text{excess}} > 0.05 N_{bgd}$. [^9]: UV-pass filter observation started only in January 2015. We use higher cut in energy for the UV-pass filter LC because the last bin (NSB:15-30$ \times \, \textit{NSB}_{\text{Dark}}$) has an energy threshold above 300GeV at the observed zenith angles. [^10]: It is difficult to determine if a flux shift is due to wrong energy calibration or wrong effective area calculation. [^11]: Under such conditions the NSB level can be much higher than the 30 $\times \, \textit{NSB}_{\text{Dark}}$ limit until which the performance was studied here.

--- abstract: 'Search engines play an important role in our everyday lives by assisting us in finding the information we need. When we input a complex query, however, results are often far from satisfactory. In this work, we introduce a query reformulation system based on a neural network that rewrites a query to maximize the number of relevant documents returned. We train this neural network with reinforcement learning. The actions correspond to selecting terms to build a reformulated query, and the reward is the document recall. We evaluate our approach on three datasets against strong baselines and show a relative improvement of 5-20% in terms of recall. Furthermore, we present a simple method to estimate a conservative upper-bound performance of a model in a particular environment and verify that there is still large room for improvements.' author: - | Rodrigo Nogueira\ Tandon School of Engineering\ New York University\ [rodrigonogueira@nyu.edu]{} Kyunghyun Cho\ Courant Institute of Mathematical Sciences\ Center for Data Science\ New York University\ [kyunghyun.cho@nyu.edu]{} bibliography: - 'main.bib' title: 'Task-Oriented Query Reformulation with Reinforcement Learning' --- Introduction ============ Search engines help us find what we need among the vast array of available data. When we request some information using a long or inexact description of it, these systems, however, often fail to deliver relevant items. In this case, what typically follows is an iterative process in which we try to express our need differently in the hope that the system will return what we want. This is a major issue in information retrieval. For instance, @huang2009analyzing estimate that 28-52% of all the web queries are modifications of previous ones. ![A graphical illustration of the proposed framework for query reformulation. A set of documents $D_0$ is retrieved from a search engine using the initial query $q_0$. Our reformulator selects terms from $q_0$ and $D_0$ to produce a reformulated query $q'$ which is then sent to the search engine. Documents $D'$ are returned, and a reward is computed against the set of ground-truth documents. The reformulator is trained with reinforcement learning to produce a query, or a series of queries, to maximize the expected return.[]{data-label="fig:query_reformulator"}](query_reformulator){width="\columnwidth"} -4mm To a certain extent, this problem occurs because search engines rely on matching words in the query with words in relevant documents, to perform retrieval. If there is a mismatch between them, a relevant document may be missed. One way to address this problem is to automatically rewrite a query so that it becomes more likely to retrieve relevant documents. This technique is known as *automatic query reformulation*. It typically expands the original query by adding terms from, for instance, dictionaries of synonyms such as WordNet [@miller1995wordnet], or from the initial set of retrieved documents [@xu1996query]. This latter type of reformulation is known as pseudo (or blind) relevance feedback (PRF), in which the relevance of each term of the retrieved documents is automatically inferred. The proposed method is built on top of PRF but differs from previous works as we frame the query reformulation problem as a reinforcement learning (RL) problem. An initial query is the natural language expression of the desired goal, and an agent (i.e. reformulator) *learns* to reformulate an initial query to maximize the expected return (i.e. retrieval performance) through actions (i.e. selecting terms for a new query). The environment is a search engine which produces a new state (i.e. retrieved documents). Our framework is illustrated in Fig. \[fig:query\_reformulator\]. The most important implication of this framework is that a search engine is treated as a *black box* that an agent learns to use in order to retrieve more relevant items. This opens the possibility of training an agent to use a search engine for a task other than the one it was originally intended for. To support this claim, we evaluate our agent on the task of question answering (Q&A), citation recommendation, and passage/snippet retrieval. As for training data, we use two publicly available datasets (TREC-CAR and Jeopardy) and introduce a new one (MS Academic) with hundreds of thousands of *query*/*relevant document* pairs from the academic domain. Furthermore, we present a method to estimate the upper bound performance of our RL-based model. Based on the estimated upper bound, we claim that this framework has a strong potential for future improvements. Here we summarize our main contributions: - A reinforcement learning framework for automatic query reformulation. - A simple method to estimate the upper-bound performance of an RL-based model in a given environment. - A new large dataset with hundreds of thousands of *query*/*relevant document* pairs.[^1] ![An illustration of our neural network-based reformulator.[]{data-label="fig:reformulator"}](reformulator){width="\columnwidth"} -4mm A Reinforcement Learning Approach ================================= Model Description {#sec:rl} ----------------- In this section we describe the proposed method, illustrated in Fig. \[fig:reformulator\]. The inputs are a query $q_0$ consisting of a sequence of words $(w_1, ..., w_n)$ and a candidate term $t_i$ with some context words $(t_{i-k},\allowbreak ...,\allowbreak t_{i+k})$, where $k \geq 0$ is the context window size. Candidate terms are from $q_0 \cup D_0$, the union of the terms in the original query and those from the documents $D_0$ retrieved using $q_0$. We use a dictionary of pretrained word embeddings [@mikolov2013efficient] to convert the symbolic terms ${w_j}$ and ${t_i}$ to their vector representations $v_j$ and $e_i \in {\mathbb{R}}^d$, respectively. We map out-of-vocabulary terms to an additional vector that is learned during training. We convert the sequence $\{v_j\}$ to a fixed-size vector $\phi_a(v)$ by using either a Convolutional Neural Network (CNN) followed by a max pooling operation over the entire sequence [@kim2014convolutional] or by using the last hidden state of a Recurrent Neural Network (RNN).[^2] Similarly, we fed the candidate term vectors ${e_i}$ to a CNN or RNN to obtain a vector representation ${\phi_b(e_i)}$ for each term $t_i$. The convolutional/recurrent layers serve an important role of capturing context information, especially for out-of-vocabulary and rare terms. CNNs can process candidate terms in parallel, and, therefore, are faster for our application than RNNs. RNNs, on the other hand, can encode longer contexts. Finally, we compute the probability of selecting $t_i$ as: $$\label{eq:1} P(t_i|q_0) = \sigma( U^\mathsf{T} \tanh( W(\phi_a(v) \Vert \phi_b(e_i)) + b )),$$ where $\sigma$ is the sigmoid function, $\Vert$ is the vector concatenation operation, $W \in {\mathbb{R}}^{d \times 2d}$ and $U \in {\mathbb{R}}^{d}$ are weights, and $b \in {\mathbb{R}}$ is a bias. At test time, we define the set of terms used in the reformulated query as $T=\{t_i\ |\ P(t_i|q_0)>\epsilon\}$, where $\epsilon$ is a hyper-parameter. At training time, we sample the terms according to their probability distribution, $T=\{t_i\ |\ \alpha=1 \wedge \alpha \sim P(t_i|q_0)\}$. We concatenate the terms in $T$ to form a reformulated query $q'$, which will then be used to retrieve a new set of documents $D'$. Sequence Generation {#sec:seqgen} ------------------- One problem with the method previously described is that terms are selected independently. This may result in a reformulated query that contains duplicated terms since the same term can appear multiple times in the feedback documents. Another problem is that the reformulated query can be very long, resulting in a slow retrieval. To solve these problems, we extend the model to sequentially generate a reformulated query, as proposed by @buck2017ask. We use a Recurrent Neural Network (RNN) that selects one term at a time from the pool of candidate terms and stops when a special token is selected. The advantage of this approach is that the model can remember the terms previously selected through its hidden state. It can, therefore, produce more concise queries. We define the probability of selecting $t_i$ as the k-th term of a reformulated query as: $$P(t_i^k|q_0) \propto \exp(\phi_b(e_i)^\mathsf{T} h_k),$$ where $h_k$ is the hidden state vector at the k-th step, computed as: $$h_k = \tanh(W_a \phi_a(v) + W_b \phi_b(t^{k-1}) + W_h h_{k-1}),$$ where $t^{k-1}$ is the term selected in the previous step and $W_a \in {\mathbb{R}}^{d \times d}$, $W_b \in {\mathbb{R}}^{d \times d}$, and $W_h \in {\mathbb{R}}^{d \times d}$ are weight matrices. In practice, we use an LSTM [@hochreiter1997long] to encode the hidden state as this variant is known to perform better than a vanilla RNN. We avoid normalizing over a large vocabulary by using only terms from the retrieved documents. This makes inference faster and training practical since learning to select words from the whole vocabulary might be too slow with reinforcement learning, although we leave this experiment for a future work. Training -------- We train the proposed model using REINFORCE [@williams1992simple] algorithm. The per-example stochastic objective is defined as $$\label{eq:2} C_a = (R - \bar{R}) \sum_{t \in T} -\log P(t | q_0),$$ where $R$ is the reward and $\bar{R}$ is the baseline, computed by the value network as: $$\label{eq:3} \bar{R} = \sigma( S^\mathsf{T} \tanh(V ( \phi_a(v) \Vert \bar{e} ) + b)),$$ where $\bar{e} = \frac{1}{N} \sum_{i=1}^{N} \phi_b(e_i)$, $N=|q_0 \cup D_0|$, $V \in {\mathbb{R}}^{d \times 2d}$ and $S \in {\mathbb{R}}^{d}$ are weights and $b \in {\mathbb{R}}$ is a bias. We train the value network to minimize $$\label{eq:4} C_b = \alpha||R-\bar{R}||^2,$$ where $\alpha$ is a small constant (e.g. 0.1) multiplied to the loss in order to stabilize learning. We conjecture that the stability is due to the slowly evolving value network which directly affects the learning of the policy. This effectively prevents the value network to fit extreme cases (unexpectedly high or low reward.) We minimize $C_a$ and $C_b$ using stochastic gradient descent (SGD) with the gradient computed by backpropagation [@rumelhart1988learning]. This allows the entire model to be trained end-to-end directly to optimize the retrieval performance. #### Entropy Regularization We observed that the probability distribution in Eq. became highly peaked in preliminary experiments. This phenomenon led to the trained model not being able to explore new terms that could lead to a better-reformulated query. We address this issue by regularizing the negative entropy of the probability distribution. We add the following regularization term to the original cost function in Eq. : $$C_H = -\lambda \sum_{t \in q_0 \cup D_0} P(t|q_0)\log P(t|q_0),$$ where $\lambda$ is a regularization coefficient. Related Work ============ Query reformulation techniques are either based on a global method, which ignores a set of documents returned by the original query, or a local method, which adjusts a query relative to the documents that initially appear to match the query. In this work, we focus on local methods. A popular instantiation of a local method is the *relevance model*, which incorporates pseudo-relevance feedback into a language model form [@lavrenko2001relevance]. The probability of adding a term to an expanded query is proportional to its probability of being generated by the language models obtained from the original query and the document the term occurs in. This framework has the advantage of not requiring *query/relevant documents* pairs as training data since inference is based on word co-occurrence statistics. Unlike the relevance model, algorithms can be trained with supervised learning, as proposed by @cao2008selecting. A training dataset is automatically created by labeling each candidate term as relevant or not based on their individual contribution to the retrieval performance. Then a binary classifier is trained to select expansion terms. In Section \[sec:experiments\], we present a neural network-based implementation of this supervised approach. A generalization of this supervised framework is to *iteratively* reformulate the query by selecting one candidate term at each retrieval step. This can be viewed as navigating a graph where the nodes represent queries and associated retrieved results and edges exist between nodes whose queries are simple reformulations of each other [@diaz2016pseudo]. However, it can be slow to reformulate a query this way as the search engine must be queried for each newly added term. In our method, on the contrary, the search engine is queried with multiple new terms at once. An alternative technique based on supervised learning is to learn a common latent representation of queries and relevant documents terms by using a *click-through* dataset [@sordoni2014learning]. Neighboring document terms of a query in the latent space are selected to form an expanded query. Instead of using a *click-through* dataset, which is often proprietary, it is possible to use an alternative dataset consisting of anchor text/title pairs. In contrast, our approach does not require a dataset of paired queries as it learns term selection strategies via reinforcement learning. Perhaps the closest work to ours is that by @narasimhan2016improving, in which a reinforcement learning based approach is used to reformulate queries iteratively. A key difference is that in their work the reformulation component uses domain-specific template queries. Our method, on the other hand, assumes open-domain queries. Experiments {#sec:experiments} =========== In this section we describe our experimental setup, including baselines against which we compare the proposed method, metrics, reward for RL-based models, datasets and implementation details. Baseline Methods ---------------- #### Raw: The original query is given to a search engine without any modification. We evaluate two search engines in their default configuration: Lucene[^3] (Raw-Lucene) and Google Search[^4] (Raw-Google). #### Pseudo Relevance Feedback (PRF-TFIDF): A query is expanded with terms from the documents retrieved by a search engine using the original query. In this work, the top-$N$ TF-IDF terms from each of the top-$K$ retrieved documents are added to the original query, where $N$ and $K$ are selected by a grid search on the validation data. #### PRF-Relevance Model (PRF-RM): This is a popular relevance model for query expansion by @lavrenko2001relevance. The probability of using a term $t$ in an expanded query is given by: $$\begin{gathered} P(t|q_0) = (1-\lambda) P'(t|q_0)\\ + \lambda \sum_{d \in D_0} P(d) P(t|d) P(q_0|d),\end{gathered}$$ where $P(d)$ is the probability of retrieving the document $d$, assumed uniform over the set, $P(t|d)$ and $P(q_0|d)$ are the probabilities assigned by the language model obtained from $d$ to $t$ and $q_0$, respectively. $P'(t|q_0)= \frac{\text{tf}(t \in q)}{|q|}$, where $\text{tf}(t,d)$ is the term frequency of $t$ in $d$. We set the interpolation parameter $\lambda$ to 0.5, following @zhai2001study. We use a Dirichlet smoothed language model [@zhai2001study] to compute a language model from a document $d \in D_0$: $$P(t|d)=\frac{\text{tf}(t,d)+u P(t|C)}{|d| + u},$$ where $u$ is a scalar constant ($u=1500$ in our experiments), and $P(t|C)$ is the probability of $t$ occurring in the entire corpus $C$. We use the $N$ terms with the highest $P(t|q_0)$ in an expanded query, where $N$ is a hyper-parameter. #### Embeddings Similarity: Inspired by the methods proposed by @roy2016using and , the top-$N$ terms are selected based on the cosine similarity of their embeddings against the original query embedding. Candidate terms come from documents retrieved using the original query (PRF-Emb), or from a fixed vocabulary (Vocab-Emb). We use pretrained embeddings from @mikolov2013efficient, and it contains 374,000 words. Proposed Methods {#sec:proposed_methods} ---------------- #### Supervised Learning (SL): Here we detail a deep learning-based variant of the method proposed by @cao2008selecting. It assumes that query terms contribute independently to the retrieval performance. We thus train a binary classifier to select a term if the retrieval performance increases beyond a preset threshold when that term is added to the original query. More specifically, we mark a term as relevant if $(R' - R) / R > 0.005$, where $R$ and $R'$ are the retrieval performances of the original query and the query expanded with the term, respectively. We experiment with two variants of this method: one in which we use a convolutional network for both original query and candidate terms (SL-CNN), and the other in which we replace the convolutional network with a single hidden layer feed-forward neural network (SL-FF). In this variant, we average the output vectors of the neural network to obtain a fixed size representation of $q_0$. #### Reinforcement Learning (RL): We use multiple variants of the proposed RL method. RL-CNN and RL-RNN are the models described in Section \[sec:rl\], in which the former uses CNNs to encode query and term features and the latter uses RNNs (more specifically, bidirectional LSTMs). RL-FF is the model in which term and query vectors are encoded by single hidden layer feed-forward neural networks. In the RL-RNN-SEQ model, we add the sequential generator described in Section \[sec:seqgen\] to the RL-RNN variant. ---------- ---------------------- ------ ------- ------- ------ ------ ------ ------ ------ Dataset Corpus Docs Train Valid Test Avg. Std. Avg. Std. TREC-CAR Wikipedia Paragraphs 3.5M 585k 195k 195k 3.6 5.7 84 68 Jeopardy Wikipedia Articles 5.9M 118K 10k 10k 1.0 0.0 462 990 MSA Academic Papers 480k 270k 20k 20k 17.9 21.5 165 158 ---------- ---------------------- ------ ------- ------- ------ ------ ------ ------ ------ -1mm Datasets -------- We summarize in Table \[tab:datasets\] the datasets. #### TREC - Complex Answer Retrieval (TREC-CAR) This is a publicly available dataset automatically created from Wikipedia whose goal is to encourage the development of methods that respond to more complex queries with longer answers [@dietz2017trec]. A query is the concatenation of an article title and one of its section titles. The ground-truth documents are the paragraphs within that section. For example, a query is “*Sea Turtle, Diet*” and the ground truth documents are the paragraphs in the section “*Diet*” of the “*Sea Turtle*” article. The corpus consists of all the English Wikipedia paragraphs, except the abstracts. The released dataset has five predefined folds, and we use the first three as the training set and the remaining two as validation and test sets, respectively. #### Jeopardy This is a publicly available Q&A dataset introduced by @nogueira2016end. A query is a question from the *Jeopardy!* TV Show and the corresponding document is a Wikipedia article whose title is the answer. For example, a query is *“For the last eight years of his life, Galileo was under house arrest for espousing this man’s theory”* and the answer is the Wikipedia article titled *“Nicolaus Copernicus”*. The corpus consists of all the articles in the English Wikipedia. #### Microsoft Academic (MSA) This dataset consists of academic papers crawled from Microsoft Academic API.[^5] The crawler started at the paper @silver2016mastering and traversed the graph of references until 500,000 papers were crawled. We then removed papers that had no reference within or whose abstract had less than 100 characters. We ended up with 480,000 papers. A query is the title of a paper, and the ground-truth answer consists of the papers cited within. Each document in the corpus consists of its title and abstract.[^6] ------------ ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- Method R@40 P@10 MAP@40 R@40 P@10 MAP@40 R@40 P@10 MAP@40 Raw-Lucene 43.6 7.24 19.6 23.4 1.47 7.40 12.9 7.24 3.36 Raw-Google - - - 30.1 1.92 7.71 - - - PRF-TFIDF 44.3 7.31 19.9 29.9 1.91 7.65 13.2 7.27 3.50 PRF-RM 45.1 7.35 19.5 30.5 1.96 7.64 12.3 7.22 3.38 PRF-Emb 44.5 7.32 19.0 30.1 1.92 7.74 12.2 7.22 3.20 Vocab-Emb 44.2 7.30 19.1 29.4 1.87 7.80 12.0 7.21 3.21 SL-FF 44.1 7.29 19.7 30.8 1.95 7.70 13.2 7.28 3.88 SL-CNN 45.3 7.35 19.8 31.1 1.98 7.79 14.0 7.42 3.99 SL-Oracle 50.8 8.25 21.0 38.8 2.50 9.92 17.3 10.12 4.89 RL-FF 44.1 7.29 20.0 31.0 1.98 7.81 13.9 7.33 3.81 RL-CNN 47.3 7.45 20.3 33.4 **2.14** 8.02 14.9 7.63 4.30 RL-RNN **47.9** **7.52** **20.6** **33.7** 2.12 **8.07** **15.1** **7.68** **4.35** RL-RNN-SEQ 47.4 7.48 20.3 33.4 2.13 8.01 14.8 7.63 4.27 RL-Oracle 55.9 9.06 23.0 42.4 2.74 10.3 24.6 12.83 6.33 ------------ ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- -2mm Metrics and Reward ------------------ Three metrics are used to evaluate performance: #### Recall@K: Recall of the top-K retrieved documents: $$\text{R}@K = \frac{|D_K \cap D^*|}{|D^*|},$$ where $D_K$ are the top-$K$ retrieved documents and $D^*$ are the relevant documents. Since one of the goals of query reformulation is to increase the proportion of relevant documents returned, recall is our main metric. #### Precision@K: Precision of the top-K retrieved documents: $$\text{P}@K = \frac{|D_K \cap D^*|}{|D_K|}$$ Precision captures the proportion of relevant documents among the returned ones. Despite not being the main goal of a reformulation method, improvements in precision are also expected with a good query reformulation method. Therefore, we include this metric. #### Mean Average Precision: The average precision of the top-K retrieved documents is defined as: $$\text{AP}@K = \frac{\sum_{k=1}^K \text{P}@k \times \text{rel}(k)}{|D^*|},$$ where $$\text{rel}(k) = \begin{cases} 1, & \text{if the k-th document is relevant;}\\ 0, & \text{otherwise.} \end{cases}$$ The mean average precision of a set of queries $Q$ is then: $$\text{MAP}@K = \frac{1}{|Q|}\sum_{q \in Q} \text{AP}@K_q,$$ where $\text{AP}@K_q$ is the average precision at $K$ for a query $q$. This metric values the position of a relevant document in a returned list and is, therefore, complementary to precision and recall. #### Reward We use $\text{R}@K$ as a reward when training the proposed RL-based models as this metric has shown to be effective in improving the other metrics as well. #### SL-Oracle In addition to the baseline methods and proposed reinforcement learning approach, we report two oracle performance bounds. The first oracle is a supervised learning oracle (SL-Oracle). It is a classifier that perfectly selects terms that will increase performance according to the procedure described in Section \[sec:proposed\_methods\]. This measure serves as an upper-bound for the supervised methods. Notice that this heuristic assumes that each term contributes independently from all the other terms to the retrieval performance. There may be, however, other ways to explore the dependency of terms that would lead to a higher performance. #### RL-Oracle Second, we introduce a reinforcement learning oracle (RL-Oracle) which estimates a conservative upper-bound performance for the RL models. Unlike the SL-Oracle, it does not assume that each term contributes independently to the retrieval performance. It works as follows: first, the *validation* or *test* set is divided into $N$ small subsets $\{A_i\}_{i=1}^N$ (each with 100 examples, for instance). An RL model is trained on each subset $A_i$ until it overfits, that is, until the reward $R_i^*$ stops increasing or an early stop mechanism ends training.[^7] Finally, we compute the oracle performance $R^*$ as the average reward over all the subsets: $R^*= \frac{1}{N}\sum_{i=1}^{N} R_i^*$. This upper bound by the RL-Oracle is, however, conservative since there might exist better reformulation strategies that the RL model was not able to discover. ----------- ---------- ---------- ----- TREC-CAR Jeopardy MSA SL-Oracle 13% 5% 11% RL-Oracle 29% 27% 31% ----------- ---------- ---------- ----- : Percentage of relevant terms over all the candidate terms according to SL- and RL-Oracle.[]{data-label="tab:oracleterms"} -2mm Implementation Details ---------------------- #### Search engine We use Lucene and BM25 as the search engine and the ranking function, respectively, for all PRF, SL and RL methods. For Raw-Google, we restrict the search to the *wikipedia.org* domain when evaluating its performance on the Jeopardy dataset. We could not apply the same restriction to the two other datasets as Google does not index Wikipedia paragraphs, and as it is not trivial to match papers from MS Academic to the ones returned by Google Search. ![Our RL-based model continues to improve recall as more candidate terms are added, whereas a classical PRF method saturates. []{data-label="fig:feedback_terms"}](candidate_terms){width="\columnwidth"} -4mm #### Candidate terms We use Wikipedia articles as a source for candidate terms since it is a well curated, clean corpus, with diverse topics. At training and test times of SL methods, and at test time of RL methods, the candidate terms are from the first $M$ words of the top-$K$ Wikipedia articles retrieved. We select $M$ and $K$ using grid search on the validation set over $\{50,100,200,300\}$ and $\{1,3,5,7\}$, respectively. The best values are $M=300$ and $K=7$. These correspond to the maximum number of terms we could fit in a single GPU. At training time of an RL model, we use only *one* document uniformly sampled from the top-$K$ retrieved ones as a source for candidate terms, as this leads to a faster learning. For the PRF methods, the top-$M$ terms according to a relevance metric (i.e., TF-IDF for PRF-TFIDF, cosine similarity for PRF-Emb, and conditional probability for PRF-RM) from each of the top-$K$ retrieved documents are added to the original query. We select $M$ and $K$ using grid search over $\{10, 50, 100, 200, 300, 500\}$ and $\{1, 3, 5, 9, 11\}$, respectively. The best values are $M=300$ and $K=9$. #### Multiple Reformulation Rounds Although our framework supports multiple rounds of search and reformulation, we did not find any significant improvement in reformulating a query more than once. Therefore, the numbers reported in the results section were all obtained from models running two rounds of search and reformulation. #### Neural Network Setup For SL-CNN and RL-CNN variants, we use a 2-layer convolutional network for the original query. Each layer has a window size of 3 and 256 filters. We use a 2-layer convolutional network for candidate terms with window sizes of 9 and 3, respectively, and 256 filters in each layer. We set the dimension $d$ of the weight matrices $W,S,U$, and $V$ to $256$. For the optimizer, we use ADAM [@kingma2014adam] with $\alpha=10^{-4}$, $\beta_1=0.9$, $\beta_2=0.999$, and $\epsilon=10^{-8}$. We set the entropy regularization coefficient $\lambda$ to $10^{-3}$. For RL-RNN and RL-RNN-SEQ, we use a 2-layer bidirectional LSTM with 256 hidden units in each layer. We clip the gradients to unit norm. For RL-RNN-SEQ, we set the maximum possible number of generated terms to 50 and we use beam search of size four at test time. We fix the dictionary of pre-trained word embeddings during training, except the vector for out-of-vocabulary words. We found that this led to faster convergence and observed no difference in the overall performance when compared to learning embeddings during training. Results and Discussion ====================== Table \[tab:results\] shows the main result. As expected, reformulation based methods work better than using the original query alone. Supervised methods (SL-FF and SL-CNN) have in general a better performance than unsupervised ones (PRF-TFIDF, PRF-RM, PRF-Emb, and Emb-Vocab), but perform worse than RL-based models (RL-FF, RL-CNN, RL-RNN, and RL-RNN-SEQ). RL-RNN-SEQ performs slightly worse than RL-RNN but produces queries that are three times shorter, on average (15 vs 47 words). Thus, RL-RNN-SEQ is faster in retrieving documents and therefore might be a better candidate for a production implementation. The performance gap between the oracle and best performing method (Table \[tab:results\], RL-Oracle vs. RL-RNN) suggests that there is a large room for improvement. The cause for this gap is unknown but we suspect, for instance, an inherent difficulty in learning a good selection strategy and the partial observability from using a black box search engine. Relevant Terms per Document --------------------------- The proportion of relevant terms selected by the SL- and RL-Oracles over the total number of candidate terms (Table \[tab:oracleterms\]) indicates that only a small subset of terms are useful for the reformulation. Thus, we may conclude that the proposed method was able to learn an efficient term selection strategy in an environment where relevant terms are infrequent. Scalability: Number of Terms vs Recall -------------------------------------- Fig. \[fig:feedback\_terms\] shows the improvement in recall as more candidate terms are provided to a reformulation method. The RL-based model benefits from more candidate terms, whereas the classical PRF method quickly saturates. In our experiments, the best performing RL-based model uses the maximum number of candidate terms that we could fit on a single GPU. We, therefore, expect further improvements with more computational resources. {width="0.68\paperwidth"} 0.1in {width="0.73\paperwidth"} -0.1in -4mm ------------------------- ------------------------------------------------------------ Query Top-3 Retrieved Documents (Original) *The Cross* *-The Cross Entropy Method* *Entropy Method for* *for Network Reliability Estim.* *Fast Policy Search* ***-Robot Weightlifting by*** ***Direct Policy Search*** *-Off-policy Policy Search* (Reformulated) *Cross* ***-Near Optimal Reinforcement*** *Entropy Fast Policy* ***Learning in Polynom. Time*** *Reinforcement* *-The Cross Entropy Method* *Learning policies* *for Network Reliability Estim.* *global search* ***-Robot Weightlifting by*** *optimization biased* ***Direct Policy Search*** (Original) *Daikon* “*...many types of pickles are*\ *Cultivation* & *made with daikon, includ...*” “***Certain varieties of daikon***\ & ***can be grown as a winter...***” “*In Chinese cuisine, turnip*\ & *cake and chai tow kway...*” (Reformulated) *Daikon* “*...many types of pickles are*\ *Cultivation root seed* & “*made with daikon, includ...*”\ *grow fast-growing* & “***Certain varieties of daikon***\ *Chinese leaves* & ***can be grown as a winter...***”\ & “***The Chinese and Indian***\ & ***varieties tolerate higher....***” ------------------------- ------------------------------------------------------------ : Top-3 retrieved documents using the original query and a query reformulated by our RL-CNN model. In the first example, we only show the titles of the retrieved MSA papers. In the second example, we only show some words of the retrieved TREC-CAR paragraphs. **Bold** corresponds to ground-truth documents.[]{data-label="tab:return_example"} ------------ ---------------------------------------------- Trained on Selected Terms TREC-CAR *serves american national Winsted* *accreditation* Jeopardy *Tunxis Quinebaug Winsted NCCC* MSA *hospital library arts center cancer center* *summer programs* ------------ ---------------------------------------------- : Given the query *“Northwestern Connecticut Community College”*, models trained on different tasks choose different terms.[]{data-label="tab:samequery"} Qualitative Analysis -------------------- We show two examples of queries and the probabilities of each candidate term of being selected by the RL-CNN model in Fig. \[fig:probs\]. Notice that terms that are more related to the query have higher probabilities, although common words such as “*the*” are also selected. This is a consequence of our choice of a reward that does not penalize the selection of neutral terms. In Table \[tab:return\_example\] we show an original and reformulated query examples extracted from the MS Academic and TREC-CAR datasets, and their top-3 retrieved documents. Notice that the reformulated query retrieves more relevant documents than the original one. As we conjectured earlier, we see that a search engine tends to return a document simply with the largest overlap in the text, necessitating the reformulation of a query to retrieve semantically relevant documents. #### Same query, different tasks We compare in Table \[tab:samequery\] the reformulation of a sample query made by models trained on different datasets. The model trained on TREC-CAR selects terms that are similar to the ones in the original query, such as “*serves*” and “*accreditation*”. These selections are expected for this task since similar terms can be effective in retrieving similar paragraphs. On the other hand, the model trained on Jeopardy prefers to select proper nouns, such as “*Tunxis*”, as these have a higher chance of being an answer to the question. The model trained on MSA selects terms that cover different aspects of the entity being queried, such as “*arts center*” and “*library*”, since retrieving a diverse set of documents is necessary for the task the of citation recommendation. Training and Inference Times ---------------------------- Our best model, RL-RNN, takes 8-10 days to train on a single K80 GPU. At inference time, it takes approximately one second to reformulate a batch of 64 queries. Approximately 40% of this time is to retrieve documents from the search engine. Conclusion ========== We introduced a reinforcement learning framework for task-oriented automatic query reformulation. An appealing aspect of this framework is that an agent can be trained to use a search engine for a specific task. The empirical evaluation has confirmed that the proposed approach outperforms strong baselines in the three separate tasks. The analysis based on two oracle approaches has revealed that there is a meaningful room for further development. In the future, more research is necessary in the directions of (1) iterative reformulation under the proposed framework, (2) using information from modalities other than text, and (3) better reinforcement learning algorithms for a partially-observable environment. Acknowledgements {#acknowledgements .unnumbered} ================ RN is funded by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES). KC thanks support by Facebook, Google and NVIDIA. This work was partly funded by the Defense Advanced Research Projects Agency (DARPA) D3M program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of DARPA. [^1]: The dataset and code to run the experiments are available at <https://github.com/nyu-dl/QueryReformulator>. [^2]: To deal with variable-length inputs in a mini-batch, we pad smaller ones with zeros on both ends so they end up as long as the largest sample in the mini-batch. [^3]: https://lucene.apache.org/ [^4]: https://cse.google.com/cse/ [^5]: https://www.microsoft.com/cognitive-services/en-us/academic-knowledge-api [^6]: This was done to avoid a large computational overhead for indexing full papers. [^7]: The subset should be small enough, or the model should be large enough so it can overfit.

--- abstract: 'Is [*Cosmic Censorship*]{} special to General Relativity, or can it survive a violation of equivalence principle? Recent studies have shown that singularities in Lorentz violating Einstein-Aether (or Horava-Lifhsitz) theories can lie behind a [*universal horizon*]{} in simple black hole spacetimes. Even infinitely fast signals cannot escape these universal horizons. We extend this result, for an incompressible aether, to 3+1d dynamical or spinning spacetimes which possess inner killing horizons, and show that a universal horizon always forms in between the outer and (would-be) inner horizons. This finding suggests a notion of [*Cosmic Censorship*]{}, given that geometry in these theories never evolves beyond the universal horizon (avoiding potentially singular inner killing horizons). A surprising result is that there are 3 distinct possible stationary universal horizons for a spinning black hole, only one of which matches the dynamical spherical solution. This motivates dynamical studies of collapse in Einstein-Aether theories beyond spherical symmetry, which may reveal instabilities around the spherical solution.' author: - Michael Meiers - Mehdi Saravani - Niayesh Afshordi bibliography: - 'Universal.bib' title: Cosmic Censorship in Lorentz Violating Theories of Gravity --- Introduction ============ Theory of general relativity (GR) has been successful in describing a wide range of phenomena, from solar system to cosmological scales. In addition to being consistent with various experiments, the mathematical elegance of the theory is very appealing. Diffeomorphism invariance, at the core of GR, gives a straightforward constructive way of building the theory. In fact, GR is the simplest diffeomorphism invariant theory for metric. From observational point of view, there is no reason to abandon this theory. GR is compatible with a wide variety of experimental constraints[^1]. On the other hand, many attempts have shown so far that modifying GR is a tricky task, and one often faces physically unacceptable results, e.g. the appearance of Boulware-Deser ghost in massive gravity [@Boulware:1973my] and ghost degrees of freedom in quadratic gravity [@Stelle:1976gc]. However, studying non-GR theories of gravity is still valuable, and the main reason stems from quantizing gravity. GR, while being a very successful classical theory, has failed to cope with quantum mechanics. Therefore, one approach to quantum gravity has been to abandon diffeomorphism invariance (at high enough energies), as e.g., done in the celebrated Horava-Liftshitz gravity [@Horava:2009uw]. In different examples of theories with broken Lorentz invariance, superluminal degrees of freedom appear (see [@Blas:2010hb; @Jacobson:2000xp]). The existence of superluminal excitations (SLE) points out that a different causal structure exists in these theories compared to GR, even when the back-reaction of these excitations on the geometry is negligible. This property is especially of significance in the black hole (BH) solutions. While potentially SLE can escape the traditional killing horizon of a BH and make the classical theory unpredictable, it has been shown in many examples [@Sotiriou:2014gna; @Ding:2015kba; @Barausse:2012qh; @Bhattacharyya:2015gwa; @Barausse:2013nwa; @Babichev:2006vx; @Barausse:2011pu; @Blas:2011ni] that a notion of horizon (called universal horizon) still exists in these theories. Moreover, universal horizon (UH) thermally radiate and satisfies the first law of horizon thermodynamics[^2] [@Mohd:2013zca; @Berglund:2012bu; @Berglund:2012fk; @Bhattacharyya:2014kta]. Studying the notion of universal horizon and its temperature and entropy is important since it guides us to better understanding the structure of UV theory. In this paper, we study the universal horizon formation in dynamical or stationary spacetimes with an inner killing horizon, in the limit of infinite sound speed for excitations (i.e. [*incompressible*]{} limit). In order to do so, we make use of the fact that surfaces of global time (defined by the background field), in the incompressible limit, coincide with constant mean curvature (CMC) surfaces of the spacetime. Furthermore, the backreaction of the incompressible field on the spacetime geometry is negligible as long as the event horizon is much smaller than the cosmological horizon [@Saravani:2013]. In the next section, we show how the universal horizon forms in a dynamical setting, in the collapse of a charged shell, and we derive a formula for the radius of the universal horizon in terms of the charge. In Section \[geoUH\], we propose a geometric definition for universal horizon. This allows us to study the universal horizon for spinning black holes. In \[KerrUH\] we show that there are three axisymmetric surfaces which satisfy the conditions of a universal horizon. As we show, this means that two families (with infinite numbers) of axi-symmetric universal horizons in Schwarzschild case exist. Section \[conclude\] concludes the paper. Universal Horizon in Dynamic Reissner–Nordstrom Geometry {#RN} ======================================================== We start this section by finding CMC slicing of dynamic Reissner–Nordstrom (RN) geometry. As we mentioned earlier, CMC surfaces of this spacetime are the constant global time surfaces of the background incompressible field, and they define the new causal structure imposed by this field (see the analysis in [@Saravani:2013]). Once we derive the CMC slicing, we focus on the (universal) horizon formation in this geometry. CMC Surfaces in a Dynamic Reissner–Nordstrom Geometry {#CMC-RN} ----------------------------------------------------- In order to examine the formation of the universal horizon in a dynamic Reissner–Nordstrom geometry, one must first describe surfaces of constant mean curvature for a collapsing charged massive spherical shell. An examination of CMC surfaces has been similarly looked at in the restricted case of maximal surfaces ($K=0$) [@Maeda:1980]. The dynamics of the collapse itself is well known and described by Israel [@Israel:1967]. Describing the metric in the standard way has the geometry inside the shell as flat and the RN outside. We write this geometry as: $$\begin{aligned} &{\mathrm{d}}s^2 =f_-(r) {\mathrm{d}}t_-^2 -f_-(r)^{-1} {\mathrm{d}}r^2 -r^2{\mathrm{d}}\Omega^2 &(r<R)\\ &{\mathrm{d}}s^2 = f_+(r){\mathrm{d}}t_+^2 - f_+(r)^{-1}{\mathrm{d}}r^2 -r^2{\mathrm{d}}\Omega^2 &(r>R)\end{aligned}$$ where $f_-(r)=1$ and $f_+(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}$ in $G=c=1$ units. The parameters are the gravitational mass $M$ and shell’s charge $Q$. For simplicity we will often use relative charge $\mathit{q}=Q/M$. While the spherical coordinates are shared between the inner and outer regions, the time coordinates $t_-$ and $t_+$ correspond to the Minkowski and RN time respectively. Let the family of spacelike CMC surfaces be denoted by $\Sigma_K(t_g)$ where $t_g$ is a global time coordinate that is constant for each surface. The timelike normal vector to this surface is labelled $v^\mu$. The CMC condition implies $\nabla_\mu v^\mu=K$, resulting in: \[CMC\_eq\] v\^[t\_]{}+r\^2v\^r =K If we denote $B \equiv -r^2v^r$ and use the normalization condition $v_\mu v^\mu =1$ then: r\^2v\^[t\_]{}=f\_(r)\^[-1]{}. For now we use the ’$+$’ case so that for $v_t>0$ for $r\gg M$. Additional explanation and the cases where ’$-$’ is relevant will be seen in Section \[foliation\_struc\]. Combining this result with we get $$\label{eq:pde} \frac{B}{f_\pm(r)\sqrt{h(r,B)}}\frac{\partial}{\partial t_\pm}B-\frac{\partial}{\partial r}B=Kr^2$$ with $h(r,B)=B^2+f_\pm r^4$. The characteristic equations of (\[eq:pde\]) are simply: $$\label{eq:chareq} \frac{{\mathrm{d}}t_\pm}{{\mathrm{d}}s}= \frac{B}{f_\pm(r)\sqrt{h(r,B)}},~\: \frac{{\mathrm{d}}r}{{\mathrm{d}}s}=-1, \: ~\text{and}~ \: \frac{{\mathrm{d}}B}{{\mathrm{d}}s}= K r^2,$$ for some parameter $s$. Using the second equation of (\[eq:chareq\]) to integrate the first and third equations results in: $$\label{eq:coordeq} t_\pm=(t_\pm)_0-\int_{r_0}^r \frac{B{\mathrm{d}}r}{f_\pm(r)\sqrt{h(r,B)}},~~ \text{and}~~ B= \frac{K}{3} (r^3-r_0^3)+B_0,$$ where $(t_\pm)_0$, $r_0$, and $B_0$ are integration constants. In order to fix these constants, we examine the internal and external cases separately. #### 1) Inside the shell: If $r_0=0$ then $B(r=0)=B_0$. If $B_0 \neq 0$ this would lead to a contradiction, as $v^r=\frac{-B}{r^2}$ should be finite in the flat geometry. Therefore with $r_0=0$, the equation reduces to: $$\label{eq:coordeqinside} t_-=(t_-)_0+\int_{0}^r \frac{Kr^3{\mathrm{d}}r}{3\sqrt{(\frac{Kr^3}{3})^2+r^4}} \: \text{and}\: B= \frac{K}{3} r^3$$ #### 2) Ouside the shell: Let $r_0=R((t_+)_0)$, we can determine $B_0$ by looking at the boundary between the flat and RN spaces. Projecting the vector $v^\mu$ along the shell should give us continuous observable values. The shell timelike path comes from $S=R(t_\pm)-r=0$ which creates the unit normal vector and tangent vector labelled as $n^\mu$ and $u^\mu$ respectively. If we choose the sign of the normalization factors such that $u^r<0$ , the vectors take the form of: n\_-\^&&=(\_-)\_S=(,1,0,0),\ u\_-\^&&=(1,,0,0),\ N\_-\^2&&=1-()\^2, inside the shell, while outside takes the from of: n\_+\^&&=(\_+)\_S=(f\_+\^[-1]{},f\_+,0,0),\ u\_+\^&&=(1,,0,0),\ N\_+\^2&&=. We wish to find functions $C(R)$ and $D(R)$, such that: v\_-\^=C n\_-\^+D u\_-\^. From inside the shell $v^\mu(R)=(1,0,0,0)$ which means $C=\frac{-1}{N_-}\frac{{\mathrm{d}}R}{{\mathrm{d}}t_-}$ and $D=\frac{1}{N_-}$. Requiring projections ($C$ and $D$) to be the same from outside, we get: B\_0=-R\^2(C (n\_+)\^r+D (u\_+)\^r)=(-f\_+ ). So, if we specify the dynamics of the shell $\frac{dR}{dt_{\pm}}$, all the parameters are fixed. The description of the radial velocity comes from Israel and De La Cruz [@Israel:1967]: ( )\^2&&= 1-\[drdt+\],\ ( )\^2&&=f\_+\^2-,\[drdt-\] where $\epsilon=\frac{M}{\mathcal{M}}$ and $b=\frac{M(\epsilon^2\mathit{q}^2-1)}{2\epsilon}$ with $\mathcal{M}$ denoting the total rest mass. We can use (\[drdt+\]) and (\[drdt-\]) to reduce $N_+=\frac{Rf_+}{\epsilon R-b-\frac{M}{\epsilon}}$ and $N_-=\frac{R}{\epsilon R-b}$. Note that $N_+$ changes signs to enforce $u^r<0$, becoming negative only when $\frac{{\mathrm{d}}R}{{\mathrm{d}}t_\pm}$ flips signs. These choices simplifies $B_0$ to: B\_0=. ![$h(r,B)$ with sub-critical, post-critical and critical $B$.[]{data-label="fig:hsols"}](hplot){width="50.00000%"} Horizon Formation ----------------- Following the analysis of [@Saravani:2013], we examine the properties of $t_+$. The behaviour of $t_+$ heavily depends on $h(r,B)$. While $B$ is large, which corresponds to large $R$, $h(r,B)$ is never vanishing. However when a critical value $B_c$ is reached, $ h(r,B_c)$ has double root at a particular value of $r$ labelled $r_h$ (see Figure \[fig:hsols\]). Something interesting will occur when $r_h$ is larger than the radius of the shell for which $B_c$ occurs named $R_{lc}$ or radius of [*last contact*]{} ($B(R_{lc})=B_c$). A signal sent out from the shell at $R_{lc}$ will proceed out to $r_h$ but takes infinitely long time to ever reach this radius. In fact, signals sent just outside $R_{lc}$ will form an envelope around $r_h$ staying at this radius longer and longer, as $R_{lc}$ is approached, before escaping to infinity (see Figure \[fig:plot0log\]). The values of $B_c$ and $r_h$ can be found by finding the solutions to $h(r,B)=\frac{\partial h(r,B)}{\partial \mathrm{r}}=0$. We examine this equation in two different cases. ### Case 1: $K=0$ {#case-1-k0 .unnumbered} Equations for the double root reduce to: &&r\_h\^4-2Mr\_h\^3+Q\^2r\_h\^2+B\_c\^2=0\[eq1\_rh\]\ &&2r\_h\^3-3Mr\_h\^2+ Q\^2r\_h=0\[eq2\_rh\] to which the solutions with non-negative real $B_c$ are the trivial $r_h=B_c=0$ and r\_h&&=+ \[r\_h\_ns\]\ B\_c&&=r\_h=r\_h\^2. It is of interest to not that the UH is always between the inner and outer killing horizons of the metric (see Figure \[fig:uni-hor\]). ![The outer, inner, and universal horizons for $K=0$ and varying $Q$ []{data-label="fig:uni-hor"}](unihor){width="50.00000%"} ### Case 2: $K\not=0$ {#case-2-knot0 .unnumbered} Equations for the double root are written as: &&r\_h\^6+r\_h\^4+(-2M)r\_h\^3+Q\^2r\_h\^2+B\_0\^2=0,\ &&r\_h\^5+ 2r\_h\^3+(k B\_0-3M)r\_h\^2+ Q\^2r\_h=0. ![The universal horizon formation for $Q=0$ in Schwarzschild coordinates. The blue lines represent CMC surfaces, the lowest brown line is the shell’s surface, the red line is the universal horizon (UH) and the dotted black is the radius that UH asymptotes to. Here, and in all the subsequent diagrams, the red region lies behind the UH. []{data-label="fig:plot0log"}](plot0logprime){width="50.00000%"} The non-trivial solution for $B_c$ is: B\_c=r\_h-=r\_h\^2-, however the solution for $r_h$ can be at best expressed perturbatively in $K$. To linear order the solution is: r\_h=r\_h\^0-K+(K\^2), where $r_h^0=\frac{3M}{4}+\frac{M}{4}\sqrt{9-8\mathit{q}^2}$. Assuming that the expansion of the background field is negligible (for example fixed by cosmology, as $K=3 \times$Hubble constant), in the region of interest $0<r<2M$ the effect of terms containing $K$ are insignificant. From here we set $K=0$ as its effect will only come into play when looking at the causal structure when $R$ is very large. The last unknown of the universal horizon is where it begins before it asymptotes to $r_h$. To solve for $R_{lc}$ we use that $B(R_{lc})=B_c$ and it results in either: R\_[lc]{}=-, for $\epsilon=1$ or R\_[lc]{}=, for $\epsilon >1$. [0.45]{} {width="\hsize"} [0.5]{} {width="\hsize"} We take the larger of the solutions to the quadratic, as the first instance of $B_c$ will create the behaviour desired. Inside the Universal Horizon ---------------------------- The foliation can be extended for $B<B_c$, however some subtleties arise. Denote the unit tangent vector of the CMC surfaces at the point the surface intersects the shell as $s^\mu$ which in components can be written as: s\^&&=(T\_[cmc]{}’(R),1,0,0),\ N\_s\^2&&==, where the last equality comes from using the derivative of . In general $h(r,B(R))$ is a quartic that can not easily be factored, however when restricted to the surface of the shell it can be factored to: h(R,B(R))=(R\^2-MR+)\^2. Thus one can write the normalization factor as: N\_s=. As a result, between the zeroes of $1/N_s$ at $M\frac{1\pm\sqrt{1-4b/\epsilon}}{2}$ we get $s^r<0$. In particular this means that rather than increasing in $r$ the CMC surfaces that intersect between these two roots will have a strictly decreasing $r$ coordinate. Moreover it is precisely at these points where $s^t$ switches signs corresponding to the second solution for $T'_{cmc}$ which comes from the ’-’ solution to $r^2v^{t_+}$. The second complication occurs when $R$ does not lie between the roots of $N_s$ while still being less than $R_{lc}$. Here the CMC surface increases its radial coordinate initially only to encounter a zero of $h(r,B(R))$ at $r_{turn}$. The integral for $t_+$ can carried out since $T'_{cmc}$ only depends on the square root of $h(r,B(R))$ and, by construction, $r_{turn}$ is only a first order zero of $h(r,B(R))$. After this point $T'_{cmc}$ switches signs and the $r$ coordinate begins decreasing, flipping the direction of the integration taking the surface from $r_{turn}$ to $0$. Now that these subtleties are understood, we are ready to examine the structure of the complete foliation. Foliation Structure {#foliation_struc} ------------------- We will break up the discussion into several sections. For all our analysis we consider the shell to be dropped from infinity thus $\epsilon\geq1$ [@Israel:1967], in particular the inward velocity of the shell at infinity is exactly $\sqrt{\epsilon^2-1}$. There are 4 cases of interest: ### Case 1: $Q=0$ When restricted to Schwarzschild, $b=\frac{-M}{2\epsilon}$ making it strictly negative. In particular the value of $\epsilon$ is only relevant to the radius of last contact, and so without losing any depth of examination we set $\epsilon=1$. Figures \[fig:plot0log\] and \[fig:plot0kr\] illustrate the foliations created by the CMC surfaces for $R>R_{lc}$ and the final well defined CMC surface that creates the universal horizon in Schwarzchild and Kruskal-Szekeres coordinates. Once a conformal compactification has been performed the casual structure is clear in Figure \[fig:plot0pn\] with the additional sub-UH CMC surfaces (which end in singularity, rather than the space-like infinity $i^0$). ### Case 2: $Q\neq0$ & $b\leq0$ For $\epsilon$ small enough such that the numerator of $b$ remains negative, the shell is unable to rebound before collapsing to a singularity. In Schwarzchild and the charged generalization of Kruskal-Szekeres remain almost identical in their analogous charts. The causal structure in Figure \[fig:plotqpn\] reveals the distinction from case 1. The collapse ends in the coordinates, colloquially called the [*parallel universe*]{}. ![The Penrose diagram for $Q=0.99 M$ and $\epsilon=1$ collapsing shell depicting the UH. Coloured lines/region have the same meaning as Figure \[fig:plot0log\]](UHRN1prime){width="50.00000%"} with the inclusion of sub-UH CMC surfaces in blue. \[fig:plotqpn\] ### Case 3: $Q\neq 0$ & $0<b<b/\epsilon+1/\epsilon^2$ For the range of $\epsilon$ such that $0<b<\frac{M}{\epsilon^2}$, the shell rebounds at the radius of $\frac{b}{\epsilon-1}$ but in the parallel coordinates which distinguishes it from the next case. In particular this means that $t(R)$ has a stationary point between $r_+$ and $r_-$. Figure \[fig:UHRN3\] shows the collapse and the corresponding causal structure for this case. ![The Penrose diagram for $Q=0.99 M$ and $\epsilon=1.1$ values. These parameters make $b/(\epsilon-1)<b/\epsilon+1/\epsilon^2$. Coloured lines/region have the same meaning as Figure \[fig:plot0log\] with the inclusion of sub-UH CMC surfaces in blue. []{data-label="fig:UHRN3"}](UHRN3prime){width="50.00000%"} ![The Penrose diagram for $Q=0.99M$ and $\epsilon$ set to make $b/(\epsilon-1)=b/\epsilon+1/\epsilon^2$. Coloured lines/region have the same meaning as Figure \[fig:plot0log\] with the inclusion of sub-UH CMC surfaces in blue. []{data-label="fig:UHRN4"}](UHRN4prime){width="50.00000%"} ![The Penrose diagram for $Q=0.99 M$ and $\epsilon=3/2$ values. These parameters make $b/(\epsilon-1)>b/\epsilon+1/\epsilon^2$. Coloured lines/region have the same meaning as Figure \[fig:plot0log\] with the inclusion of sub-UH CMC surfaces in blue. []{data-label="fig:UHRN2"}](UHRN2prime){width="50.00000%"} ### Case 4: $Q\neq0$ & $b\geq b/\epsilon+1/\epsilon^2$ Subsequently for $b>\frac{M}{ep}$ shell rebounds at the radius of $\frac{b}{\epsilon-1}$ in the original coordinate charts or exactly where the original and parallel coordinates meet . In particular this means that $t(R)$ has a stationary point at or inside $r_-$. The Schwarzchild and Kruskal-Szekeres coordinates are again nearly indistinguishable from case 1 except when the placement of $R_{lc}$ requiring that the UH being in a second charts however this does not reveal any new structure. Figure \[fig:UHRN4\] and \[fig:UHRN2\] represents paths within this case, $b=b/\epsilon+1/\epsilon^2$ and $b>b/\epsilon+1/\epsilon^2$ respectively. In this final case $R_{lc}<r_-$ resulting in the the UH piercing the $r_-$. Nevertheless, the singularity and the parallel interior horizon, which is considered to be unstable [@Poisson:1989zz] is still hidden within the UH. Censorship in Reissner-Nordstrom -------------------------------- Ultimately, in all the above cases, the UH shields any singularity from being probed even using superluminal signals and preserves a sense of cosmic censorship in Lorentz violating theories. It is apparent from the structure that every CMC is terminated at $i^0$, $i^+$, ${i^{-}}'$ ,${i^{0}}'$ or the singularity. We would posit an analogous, although informally made, statement to the original [*weak*]{} cosmic censorship conjecture: the set of points which can be connected to $i^0$ with CMC surfaces (an analogous property of being in the causal past of $\mathcal{I}^+$) is distinct from the set of points which can be connected to the singularity. Moreover, the boundary between these two sets will exactly be the universal horizon. Even though we have plotted the maximal foliation of spacetime beyond the universal horizon, one can argue that the self-consistent evolution of the Lorentz-violating theory (including, e.g., backreaction or quantum effects, which we have ignored) stops at the universal horizon, which can be viewed as the boundary of classical spacetime (see Sec. VI in [@Saravani:2013] for more discussions). The fact that both curvature singularity and the (potentially unstable) inner killing horizon [@Poisson:1989zz] lie beyond this region, further suggests a notion of [*strong*]{} cosmic censorship. Note that in the cases where a parallel universe exists, a second UH acts as a white hole horizon for superluminal signals. Spinning Black Holes: A Tale of Three Horizons {#spinningBH} ============================================== Geometrical Definition of Universal Horizon {#geoUH} ------------------------------------------- In the previous section, we discussed the formation of universal horizon in a dynamic RN geometry. Before moving on to the spinning black hole case, it would be illuminating to acquire more intuition about the geometric nature of the universal horizon. We start by asking the following question: is there a way of finding the universal horizon in the final geometry (after collapse completed) without knowing the details of collapse? Let’s consider the Schwarzschild case ($Q=0$ collapse). CMC surfaces in the thin shell collapse geometry describe the surfaces of constant global time. As we discussed earlier, as long as we are interested in the behaviour of these surfaces near the black hole (small radii) we can treat them as maximal surfaces ($K=0$). Maximal surfaces in this geometry inside the Schwarzschild radius asymptote to $r=\frac{3}{2}M$ before escaping to infinity. This suggests that $r=\frac{3}{2}M$ itself should be a maximal surface. In fact, one can simply verify that $r=r_*$ is a maximal (space-like) surface in Schwarzschild spacetime, only if $r_*=\frac{3}{2}M$. This observation suggests a geometrical definition for (asymptotic) universal horizon in stationary spacetimes; it is a maximal space-like hypersurface which is invariant under the flow of time-like killing vector. Let’s discuss each element of this definition. First of all, UH has to be a maximal surface as we described earlier. It also has to be space-like, since it describes a constant global time surface. Secondly, it is invariant under time translation, as it is the asymptotic surface of the maximal slicing. We will show later explicitly that this definition does not pick a unique hypersuface. However, this should not be surprising, since the position of universal horizon depends on the behaviour of the background incompressible fluid (which defines the global time), unlike the killing horizon where its position is independent of the behaviour of the background fields [@Saravani:2013; @Babichev:2007dw]. Again, let’s consider Schwarzschild spacetime. If we use the given definition of UH [*with the additional assumption of spherical symmetry*]{}, there is a unique solution of $r=\frac{3}{2}M$. However, there are many non-spherical UHs in the same geometry. Before moving to the spinning case, let’s find the spherical UH of RN geometry using the definition given above. Assume $r=r_h$ to be the universal horizon and $v_{\mu}$ the unit normal vector to this surface. Solving \_ u\^=0, we get \[RNUH\] f’(r\_h)=-4f(r\_h), with $f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}$. Eq. (\[RNUH\]) has a unique solution, which coincides with our previous result for universal horizon (\[r\_h\_ns\]), and is plotted in Figure (\[fig:uni-hor\]). One can also directly check that is equivalent to system of equations and . Universal Horizon in Kerr geometry {#KerrUH} ---------------------------------- In this section, we find the asymptotic universal horizon of Kerr metric. Given our definition above, it is a static axisymmetric[^3] (space-like) maximal surface. We express the Kerr metric in the following coordinates: ds\^2=&&(1-)dt\^2-dr\^2-\^2d\^2-dt d\ &&-(r\^2+a\^2+)\^2d\^2 where \^2&=&r\^2+a\^2\^2,\ &=&r\^2-2mr+a\^2. The inner ($r_-$) and outer ($r_+$) killing horizons are the solution to $\Delta=0$. UH is the surface r=r\_h() which satisfies \[maximal\_Kerr\] \_v\^=0 where $v^\mu$ is the (time-like) normal vector to the universal horizon. In other words, v\_=(0,1,-r\_h’,0) where $'$ is the derivative w.r.t $\theta$ and $N$ is the normalization factor \[Normalization\] N\^2=-(+r\_h’\^2). Equation leads to the following conclusion: demanding UH to be a space-like surface ($v^\mu$ to be time-like) requires the UH to be positioned between the inner and the outer killing horizons N\^2&gt;0&lt;0r\_-&lt;r\_h()&lt;r\_+. Now on to finding $r_h(\theta)$: Equation takes the form \[master\_eq\] &2(r\_h-m)+-\ &=+r\_h”-r\_h’. One way to find the solution of this differential equation is to expand $r_h(\theta)$ in powers of $a$ \[perturb\_sol\] r\_h()=m\_[n=0]{}r\^[(n)]{}() and solve the differential equation order by order. At zero order ($a=0$), we expect $r^{(0)}=\frac{3}{2}$. At any higher order, we find Legendre differential equation. Requiring finite solution at $\theta=0$ and $\theta=\pi$, this gives us a unique solution at any order. Here is the solution up to the order $a^{4}$: r\^[(2n-1)]{}&=&0,   n{1,2,}\ r\^[(0)]{}&=&\ r\^[(2)]{}&=&- \^2-\ r\^[(4)]{}&=&\^4+\^2-.Surprisingly though, upon solving numerically, we have found two more solutions that are different from and do not approach to $r_h=\frac{3}{2}m$ as $a \rightarrow 0$ (see Figure \[axisymmetric\_UH\]). [0.49]{} {width="\hsize"} [0.49]{} {width="\hsize"} The case of $a=0$ is interesting, since the background geometry is spherically symmetric, and yet we have found two axisymmetric UHs. Moreover, we can always perform a rotation and get two other axisymmetric UHs. This means that there are two families (with infinite number in each family) of axisymmetric UHs in Schwarzschild spacetime. Conclusion {#conclude} ========== In the incompressible (or infinitely fast propagation speed) limit of many Lorentz-violating theories of gravity, surfaces of constant mean curvature define the preferred foliation [@Saravani:2013][^4]. For such theories, one may worry that superluminal signal propagation may lead to naked singularities. In this paper, we have shown that a [*universal*]{} horizon always forms when a charged spherical shell collapses to form a Reissner–Nordstrom black hole. Evidence that causal horizon formation will take place in Lorentz-violating theories supports a conjecture similar to cosmic censorship in General Relativity. We see that the universal horizon acts almost like an extension of $i^+$, since any observer approaching the UH will pass through all future CMC surfaces outside the UH. Consequently, the analysis conducted here is likely only valid in the classical regime (ignoring quantum effects like the evaporation of BH). As a result the region close to the UH is likely where non-classical effects will begin to become relevant. Making claims past this region may require the full UV theory. We have also presented a geometric definition for the UH which provides a tool for finding generic solutions in non-spherically symmetric geometries. This tool is additionally valuable as the full evolution of the system up to the point of UH formation is not needed to be explored. In particular, we show how the definition can be applied to the Kerr geometry, revealing a family of solutions in a non-spherical geometry. Additional horizon solutions may be real solutions and the result of different (possibly more generic) collapse histories; or just an artifact of our definition which does not single out the correct solution. This further motivates numerical dynamical studies of [*non-spherical*]{} collapse in Lorentz-violating gravitational theories, as the spherical solution (and universal horizon [@Blas:2011ni]) may be unstable. Gravitational dynamics within real black holes may yet have more surprises in store for us! [^1]: Although there have been various attempts to solve the problems of dark matter and dark energy with GR modifications, simple solutions to these problem in the context of GR exist. In other words, there is no apparent observational contradiction with GR which necessitates GR modifications. [^2]: so far only for spherically symmetric solutions [^3]: we assume that the background incompressible field obeys the axial symmetry of Kerr geometry. [^4]: For a careful study of theories with infinite sound speed see [@Bhattacharyya:2015gwa].

--- abstract: | Let $K_0(\m V/X)$ be the relative Grothendieck group of varieties over $X\in Obj(\m V)$, with $\m V=\m V^{(qp)}_k$ (resp. $\m V=\m V^{an}_c$) the category of (quasi-projective) algebraic (resp. compact complex analytic) varieties over a base field $k$. Then we constructed the motivic Hirzebruch class transformation ${T_y}_*: K_0(\m V /X) \to H_*(X) \otimes \bQ[y]$ in the algebraic context for $k$ of characteristic zero, with $H_*(X)=CH_*(X)$ (resp. in the complex algebraic or analytic context, with $H_*(X)=H^{BM}_{2*}(X)$). It “unifies" the well-known three characteristic class transformations of singular varieties: MacPherson’s Chern class, Baum–Fulton–MacPherson’s Todd class and the $L$-class of Goresky–MacPherson and Cappell–Shaneson. In this paper we construct a bivariant relative Grothendieck group $\bK_0(\m V/X \to Y)$ for $\m V=\m V^{(qp)}_k$ (resp., $\m V^{an}_c$) so that $\bK_0(\m V/X \to pt)=K_0(\m V/X)$ in the algebraic context with $k$ of characteristic zero (resp., complex analytic context). We also construct in the algebraic context (in any characteristic) two Grothendieck transformations $mC_y=\La_y^{mot}: \bK_0(\m V^{qp}/X \to Y) \to \bK_{alg}(X \to Y)\otimes \bZ[y]$ and $T_y: \bK_0(\m V^{qp}/X \to Y) \to \bH(X \to Y) \otimes \bQ[y]$ with $\bK_{alg}(f)$ the bivariant algebraic $K$-theory of $f$-perfect complexes and $\bH$ the bivariant operational Chow groups (or the even degree bivariant homology in case $k=\bC$). Evaluating at $y=0$, we get a “motivic" lift $T_0$ of Fulton–MacPherson’s bivariant Riemann–Roch transformation $\tau :\bK_{alg} \to \bH \otimes \bQ$. The covariant transformations $mC_y: \bK_0(\m V^{qp}/X \to pt) \to G_0(X)\otimes \bZ[y]$ and $T_{y*}: \bK_0(\m V^{qp}/X \to pt) \to H_*(X) \otimes \bQ[y]$ agree for $k$ of characteristic zero with our motivic Chern- and Hirzebruch class transformations defined on $K_0(\m V^{qp}/X)$. Finally, evaluating at $y=-1$, for $k$ of characteristic zero we get a “motivic" lift $T_{-1}$ of Ernström-Yokura’s bivariant Chern class transformation $\gamma: \tilde{\bF}\to CH$. address: - 'Jörg Schürmann: Westf. Wilhelms-Universität, Mathematisches Institut, Einsteinstrasse 62, 48149 Münster, Germany ' - 'Department of Mathematics and Computer Science, Faculty of Science, Kagoshima University, 21-35 Korimoto 1-chome, Kagoshima 890-0065, Japan' author: - Jörg Schürmann and Shoji Yokura title: Motivic bivariant characteristic classes --- \[section\] \[thm\][Proposition]{} \[thm\][Problem]{} \[thm\][Corollary]{} \[thm\][Conjecture]{} \[thm\][Lemma]{} \[thm\][Example]{} \[thm\][Definition]{} \[thm\][Remark]{} Introduction {#intro} ============ The classical theory of characteristic classes of vector bundles is a natural transformation from the contravariant monoid functor $(\m Vect,\oplus)$ of isomorphism classes of complex or algebraic vector bundles, or the associated Grothendieck group $K^0$, to a contravariant cohomology theory $H^*$. When it comes to characteristic classes of singular spaces, they have been so far formulated as natural transformations from certain covariant theories to a covariant homology theory $H_*$. Topologically or geometrically, the following characteristic classes of singular spaces are most important and have been well-investigated by many people. Here we work either in the category $\m V=\m V^{(qp)}_k$ of (quasi-projective) algebraic varieties (i.e. reduced separated schemes of finite type) over a base field $k$, with $H_*(X)=CH_*(X)$ the Chow homology groups, or in the category $\m V=\m V^{an}_c$ of compact reduced complex analytic spaces, with $H_*(X)=H^{BM}_{2*}(X)$ the even degree Borel-Moore homology in the complex algebraic or analytic context: - MacPherson’s Chern class transformation [@BSY1; @Ken; @MacPherson]: $$c_*: F(X) \to H_*(X),$$ defined on the group $F(X)$ of constructible functions in the algebraic context for $k$ of characteritic zero or in the compact complex analytic context. - Baum–Fulton–MacPherson’s Todd class or Riemann–Roch transformation [@BFM; @Fulton-book]: $$td_*: G_0(X) \to H_*(X)\otimes \bQ,$$ defined on the Grothendieck group $G_0(X)$ of coherent sheaves in the algebraic context in any characteristic. In the compact complex analytic context such a transformation can be deduced (compare with [@BSY1]) from Levy’s $K$-theoretical Riemann-Roch transformation [@Levy]. - Goresky– MacPherson’s homology $L$-class [@GM], which is extended as a natural transformation by Cappell-Shaneson [@CS] (see also [@BSY1; @Yokura-TAMS; @Woolf]): $$L_*: \Omega_{sd}(X) \to H_*(X)\otimes \bQ$$ defined on the cobordism group $\Omega(X)$ of selfdual constructible sheaf complexes. This transformation is only defined for compact spaces in the complex algebraic or analytic context, with $H_*$ the usual homology, since its definition is based on a corresponding signature invariant together with the Thom-Pontrjagin construction. In 1973 R. MacPherson gave a survey talk about characteristic classes of singular varieties, and his survey article [@MacPherson2] ends with the following remark:\ *“It remains to be seen whether there is a unified theory of characteristic classes of singular varieties like the classical one outlined above."*[^1]\ In our previous paper [@BSY1] (see also [@BSY2], [@SY], [@Sch-MSRI] and [@Yokura-MSRI]) we introduced in the algebraic context for $k$ of characteristic zero, as well as in the compact complex analytic context, the motivic Hirzebruch class transformation $${T_y}_*: K_0(\m V/X) \to H_*(X)\otimes \bQ[y],$$ defined on the relative Grothendieck group $K_0(\m V/X)$ of varieties over $X\in Obj(\m V)$, with $\m V=\m V^{(qp)}_k$ resp. $\m V=\m V^{an}_c$. This Hirzebruch class transformation “unifies" the above three characteristic classes $c_*, td_*, L_*$ (see also §3) in the sense that we have the following commutative diagrams of transformations: $$\xymatrix{ & K_0(\Cal V/X) \ar [dl]_{\epsilon} \ar [dr]^{{T_{-1}}_*} \\ {F(X) } \ar [rr] _{c_*}& & H_*(X)\otimes \bQ.}$$ $$\xymatrix{ & K_0(\Cal V/X) \ar [dl]_{mC_0} \ar [dr]^{{T_{0}}_*} \\ {G_0(X) } \ar [rr] _{td_*}& & H_*(X)\otimes \bQ.}$$ $$\xymatrix{ & K_0(\Cal V/X) \ar [dl]_{sd} \ar [dr]^{{T_{1}}_*} \\ {\Omega_{sd}(X) } \ar [rr] _{L_*}& & H_*(X)\otimes \bQ.}$$ This “unification" could be considered as a positive answer to the above MacPherson’s remark. The commutativity of the diagrams above follows (by the functoriality for proper morphisms) already from the normalization condition $$T_{y*}(X):=T_{y*}([id_{X}])= T^{*}_{y}(TX) \cap [X],$$ for $X$ a smooth manifold, since by “resolution of singularities" the group $ K_0(\Cal V/X)$ is generated by isomorphism classes $[V \xrightarrow {h} X]$ of proper morphisms $h:V \to X$ with $V$ smooth. Here the Hirzebruch class $T^{*}_{y}(E)$ of the complex or algebraic vector bundle $E$ over $X$ is defined to be (see [@Hirzebruch; @HBJ]): $$T^{*}_{y}(E) := \prod _{i=1}^{\op {rank} E} Q_{y}(\alpha_i)\in H^*(X) \otimes \bQ[y],$$ with $$Q_{y}(\alpha):= \frac{\alpha(1+y)}{1-e^{-\alpha(1+y)}} -\alpha y \quad \in \bQ[y][[\alpha]] \:.$$ Here $\alp _i$’s are the Chern roots of $E$, i.e., $\displaystyle c(E) = \prod_{i=1}^{\op{rank(E)}} (1 + \alp_i).$ Note that $Q_{y}$ is a normalized power series, i.e. $Q_{y}(0)=1$, with: - $T^{*}_{-1}(E) =c(E)$ the Chern class, since $Q_{-1}(\alpha)=1+\alpha$. - $T^{*}_{0}(E) =td(E)$ the Todd class, since $\displaystyle Q_{0}(\alpha)=\frac{\alpha}{1-e^{-\alpha}}$. - $T^{*}_{1}(E) =L(E)$ the Thom–Hirzebruch $L$-class, since $\displaystyle Q_{1}(\alpha)=\frac{\alpha}{\tanh \alpha}$. Moreover, we also constructed in [@BSY1] in the algebraic context for $k$ of characteristic zero, and in the compact complex analytic context, the motivic Chern class transformation $$mC_y: K_0(\m V/X) \to G_0(X)\otimes \bZ[y].$$ This satisfies the normalization condition $$mC_{y}(X):= mC_{y}([id_{X}])= \sum_{i=0}^{dim(X)} \; [\Lambda^{i} T^{*}X]\cdot y^{i} = \lambda_{y}([T^{*}X])\cap [\m O_{X}]$$ for $X$ a smooth manifold, with $\lambda_{y}$ the total $\lambda$-class. In the compact complex analytic (or complex algebraic) context, the transformation $mC_y$ could also be composed with the $K$-theoretical [*Riemann-Roch transformation*]{} $$\alpha: G_{0}(X)\to K^{top}_{0}(X)$$ to the (periodic) topological $K$-homology (in even degrees) constructed by Levy [@Levy] (generalizing the corresponding transformation of Baum-Fulton-MacPherson [@BFM2] for the quasi-projective complex algebraic context). Then the Hirzebruch class transformation ${T_y}_*$ could also be defined as the composition $td_*\circ mC_y$, renormalized by the multiplication $\times (1+y)^{-i}$ on $H_i(X)\otimes \bQ[y]$ to fit with the normalization condition above. So $mC_y$ could be considered as a $K$-theoretical refinement of ${T_y}_*$.\ Note that all the source and target functors appearing above are not only functorial for proper morphisms, but also have compatible [*cross products*]{} $\times$ and [*pullback Gysin homomorphisms*]{} $f^!$ for a smooth morphism $f$. Moreover, all the characteristic class transformations $c\ell_*$ above (like $c_*, td_*, L_*, mC_y, T_{y*}$) commute with the cross products $\times$. Similarly, they commute for a smooth morphism $f$ with the pullback Gysin homomorphisms $f^!$ only up to a correction factor $c\ell^*(T_f)$ given by the corresponding cohomological characteristic class of the tangent bundle $T_f$ to the fibers of $f$, i.e. one gets a [*Verdier-Riemann-Roch formula*]{} (see [@BSY1]): $$c\ell_*\circ f^! = c\ell^*(T_f)\cap (f^! \circ c\ell_*)\:.$$ This generalizes a corresponding normalization condition for $X$ a smooth manifold (so that the constant map $X\to pt$ is smooth). All these properties can be stated in a very efficient way by just saying that $cl_*$ is a natural transformation of [*Borel-Moore functors*]{} (with product) in the sense of [@LP; @Yokura-obt], if the Gysin maps $f^!$ of the target functors are “redefined or twisted" by the characteristic class $c\ell^*(T_f)$ of the tangent bundle $T_f$ to the fibers of $f$ (see [@Quillen] and [@LM §4.1.9]). Here it is only important that the target functors of our transformations $cl_*$ have a suitable theory of characteristic classes of (complex or algebraic) vector bundles (like first Chern classes of line bundles). So only the target functors should be an [*oriented Borel-Moore (weak) homology theory*]{} in the sense of Levine-Morel [@LM] (like $CH_*, G_0$), generalizing, in the algebraic context, the notion of a “complex oriented (co)homology theory" (like $H^{BM}_*, K^{top}_{0}$) introduced by Quillen [@Quillen] in the context of differentiable manifolds. In fact, Quillen [@Quillen] introduced in geometric terms [*complex cobordism*]{} $\Omega_*^U$ as a universal “complex oriented (co)homology theory". More recently, Levine-Morel [@LM] introduced [*algebraic cobordism*]{} $\Omega_*^{alg}$ as a universal “oriented Borel-Moore (weak) homology theory" in the algebraic context over a base field of characteristic zero (see also Levine–Pandharipande [@LP] for a more geometric approach).\ In early 1980’s William Fulton and Robert MacPherson have introduced the notion of bivariant theory as a [*categorical framework for the study of singular spaces*]{}, which is the title of their AMS Memoir book [@Fulton-MacPherson] (see also Fulton’s book [@Fulton-book]). As reviewed very quickly in §2, a bivariant theory is definded on morphisms, instead of objects, and unifies both a covariant functor and a contravariant functor. Important objects to be investigated in Bivariant Theories are what they call *Grothendieck transformations* between given two bivariant theories. A Grothendieck transformation is a bivariant version of a natural transformation. A bit more precisely, the main objective of [@Fulton-MacPherson] are bivariant-theoretic Riemann–Roch transformations or bivariant analogues of various theorems of Grothendieck–Riemann–Roch type and Verdier–Riemann–Roch type (as mentioned before). A key example of is the bivariant Riemann–Roch transformation $\tau :\bK_{alg} \to \bH \otimes \bQ$ on the category $\m V=\m V^{qp}_{\bC}$ of complex quasi-projective varieties, with $\bK_{alg}(f)$ the bivariant algebraic $K$-theory of $f$-perfect complexes and $\bH$ the even degree bivariant homology. It unifies the covariant Todd class transformation $td_*$ and the contravariant Chern character $ch$. An algebraic version on the category $\m V=\m V^{qp}_{k}$ of quasi-projective varieties over a base field $k$ of any characteristic was constructed later on in [@Fulton-book Example 18.3.19], with $\bH=CH$ the bivariant operational Chow groups. As another example, Fulton-MacPherson constructed in [@Fulton-MacPherson Part II] in the complex quasi-projective context also a Grothendieck transformation $\alpha: \bK_{alg} \to \bK_{top}$ between their bivariant algebraic and topological $K$-theory, as well as in a bivariant Whitney class transformation. And they asked in the complex algebraic context for a corresponding bivariant Chern class transformation $\gamma: \bF\to \bH$ on their bivariant theory $\bF$ of constructible functions satisfying a suitable local Euler condition, which generalizes the covariant MacPherson Chern class transformation $c_*$. For $\bH$ the even degree bivariant homology, this problem was solved by Brasselet [@Brasselet] in a suitable context (even for compact analytic spaces), whereas Ernström-Yokura [@EY1] solved it for $\bH=A^{PI} (\supset CH)$ another bivariant operational Chow group theory (for the notation $A^{PI}$ see [@EY1]). Finally, relaxing the local Euler condition, they introduced in [@EY2] a bivariant Chern class transformation $\gamma: \tilde{\bF}\to CH$ from another bivariant theory $\tilde{\bF}$ of constructible functions. This last approach is based on the usual calculus of constructible functions and the surjectivity of $c_*: F(X)\to CH_*(X)$, so it works in the algebraic context over any base field $k$ of characteristic zero (even though it was stated in [@EY2] only in the complex algebraic context). Here $\tilde{\bF}(X\to pt)=F(X)$ follows from the multiplicativity of $c_*$ with respect to cross products $\times$.\ One of the main objects of the present paper is to obtain two bivariant analogues $$mC_y=\La_y^{mot}: \bK_0(\m V^{qp}/X \to Y) \to \bK_{alg}(X \to Y)\otimes \bZ[y]$$ and $$T_y: \bK_0(\m V^{qp}/X \to Y) \to \bH(X \to Y) \otimes \bQ[y]$$ of the motivic Chern and Hirzebruch class transformations $mC_y$ and ${T_y}_*$, with $T_y$ defined as the composition $\tau \circ mC_y$, renormalized by the multiplication $\times (1+y)^i$ on $\bH^i(-)\otimes \bQ[y]$. Moreover, $T_y$ unifies the bivariant Riemann–Roch transformation $\tau :\bK_{alg} \to \bH \otimes \bQ$ (for $y=0$) and the bivariant Chern class transformation $\gamma: \tilde{\bF}\to CH$ (for $y=-1$). Note that a bivariant $L$-class transformation (corresponding to $y=1$) is still missing. In [@BSY3; @BSY4] we considered a kind of general construction of a bivariant analogue of a given natural transformation between two covariant functors, but our approach presented in this paper is quite different from it. The former is more “operational", but the latter is more “direct" and very “motivic", as outlined below.\ Let $\m V=\m V^{(qp)}_k$ be the category of (quasi-projective) algebraic varieties (i.e. reduced separated schemes of finite type) over a base field $k$ of any characteristic, or let $\m V=\m V^{an}_c$ be the category of compact reduced complex analytic spaces. On the category $\m V$ we define $$\bM(\m V/X \xrightarrow {f} Y)$$ to be the free abelian group on the set of isomorphism classes $[V \xrightarrow {h} X]$ of proper morphisms $h:V \to X$ such that the composite $f \circ h: V \to Y$ is a smooth morphism, in other words, $h: V \to X$ is “a left quotient" of a smooth morphism $s: V \to Y$ devided by the given morphism $f$: $$f \circ h = s \quad \text {or} \quad h = \frac {s}{f},$$ $$\xymatrix{ & V\ar [dl]_{h} \ar [dr]^{s} \\ X\ar [rr] _{f}& & Y.}$$ Here two morphisms $h: V \to X$ and $h': V' \to X$ are called isomorphic to each other if there exists an isomorphism $\phi: V \xrightarrow {\cong} V'$ such that the following diagram commutes $$\xymatrix{ V\ar[dr]_ {h}\ar[rr]^ {\phi} && V' \ar[dl]^{h'}\\ & X.}$$ The association $\bM(\m V/-)$ becomes a bivariant theory with natural bivariant-theoretic operations. The associated “cohomology theory" ${\bM}^*(\m V/X) = \bM(\m V/X \xrightarrow{\op {id}_X} X)$ is the free abelian group generated by the isomorphism classes of proper and smooth morphism $[V \xrightarrow{h} X]$. So it is a geometric approach to cohomology classes in the algebraic or compact complex analytic context, based on proper submersions having a tangent bundle to the fibers (as a substitute for a bundle theoretic approach to cohomology classes in topology). Moreover, the bivariant theory $\bM(\m V/-)$ based on (isomorphism classes of) “left quotients" $h = \frac {s}{f}$ with $h$ proper and $s$ smooth fits nicely with the recent approach of Emerson-Meyer [@EM2; @EM3] to “(bivariant) $KK$-theory via correspondences" (here “bivariant" has a meaning different from the notion of Fulton-MacPherson used in this paper). In fact, one can see the “left quotient" $h = \frac {s}{f}$ also as a correspondence between $X$ and $Y$ fitting with the given morphism $f: X\to Y$. Forgetting $f$, one can define the free abelian group $\bM(\m V/X,Y)$ generated by the isomorphism classes of such correspondences (with $h$ proper and $s$ smooth), with the “usual" composition $\circ$ of correspondences. Then our definition of the bivariant product $\bullet$ fits under the tautological map (forgetting $f$): $$forget: (\bM(\m V/X \xrightarrow {f} Y),\bullet) \to (\bM(\m V/X,Y),\circ)$$ with the composition product of these correspondences (and it is also functorial in $X$ with respect to the corresponding pushforwards under proper morphisms). As will be explained elsewhere (see [@BaSY]), in the context of complex varieties there is also a similar transformation $$(\bM(\m V/X,Y),\circ) \to (KK(X,Y),\circ)$$ to the “$KK$-theory via correspondences" of Emerson-Meyer [@EM2; @EM3] (and more generally to their counterpart based on a complex oriented cohomology theory). Let $\bB$ be a bivariant theory on $\m V$ such that a smooth morphism $f: X\to Y$ has a [*stable orientation*]{} $\theta(f)\in \bB(f)$, like $\bM(\m V/-)$, with $\theta(f):=[X \xrightarrow {\op {id}_X} X]$ (these notions will be explained in §2). In the algebraic context, examples for $\bB$ are given by the bivariant algebraic $K$-theory $\bK_{alg}$ of relative perfect complexes and the bivariant operational Chow groups $CH$. Examples in the complex algebraic or analytic context are given by the (even degree) bivariant topological $K$-theory $\bK^{top}$ or homology theory $\bH\otimes R$ of Fulton–MacPherson [@Fulton-MacPherson], with $R=\bZ, \bQ, \bQ[y]$. Another example is Fulton-MacPherson’s bivariant theory $\bF$ of constructible functions in the complex algebraic or analytic context, or Ernström-Yokura’s bivariant theory $\tilde{\bF}$ of constructible functions in the algebraic context over a base field of characteristic zero, with $\theta(f)=\jeden_f:=1_X$ for a smooth morphism $f: X\to Y$. \[thm:univ\] Let $\bB$ be a bivariant theory on $\m V$ such that a smooth morphism $f: X\to Y$ has a stable orientation $\theta(f)\in \bB(f)$. Then there exists a unique Grothendieck transformation $$\ga:=\ga_{\theta}: \bM(\m V/-) \to \bB(-)$$ satisfying the normalization condition that for a smooth morphism $f:X \to Y$ the following identity holds in $\bB(X \xrightarrow {f} Y)$: $$\ga([X \xrightarrow {\op {id}_X} X]) = \theta(f).$$ Let $c\ell: Vect(-) \to \bB^*(-)$ be a contravariant functorial characteristic class of algebraic (or analytic) vector bundles with values in the associated cohomology theory, which is multiplicative in the sense that $c\ell(V) = c\ell(V') c\ell(V'')$ for any short exact sequence of vector bundles $0\to V'\to V \to V'' \to 0$, with $c\ell(T_{pt})=1_{pt}\in \bB^*(\{pt\})$. Assume that $c\ell$ commutes with the stable orientation $\theta$, i.e. $$\theta(f)\bullet cl(V)=f^*cl(V)\bullet \theta(f)$$ for all smooth morphism $f: X\to Y$ and $V\in Vect(Y)$. Then there exists a unique Grothendieck transformation $$\ga_{c\ell}: \bM(\m V/-) \to \bB(-)$$ satisfying the normalization condition that for a smooth morphism $f:X \to Y$ the following identity holds in $\bB(X \xrightarrow {f} Y)$: $$\ga_{c\ell}([X \xrightarrow {\op {id}_X} X]) = c\ell(T_f) \bullet \theta(f).$$ Here $T_f$ is the relative tangent bundle of the smooth morphism $f$. This follows from Theorem \[thm:univ\] by using the new “twisted" stable orientation $\theta'(f):=c\ell(T_f) \bullet \theta(f)$ for a smooth morphism $f: X\to Y$. Similar twisting constructions are due to Quillen [@Quillen] (resp., Levine-Morel [@LM §4.1.9]) in the context of complex oriented (co)homology theories (resp., oriented Borel-Moore (weak) homology theories).\ This $\ga_{c\ell}: \bM(\m V/ -) \to \bB( -)$ should be considered as *a “pre-motivic" bivariant theory of characteristic classes*. In particular, if we consider the case of a mapping $X \to pt$ to a point, ${\bM}_ *(\m V/X) := \bM(\m V/X \to pt)$ behaves covariantly for proper morphisms and we have ${\ga_{c\ell}}_*: {\bM}_ *(\m V/-)\to \bB_*(-)$ is a unique natural transformation satisfying the “normalization condition" that for a smooth variety $X$ $${\ga_{c\ell}}_*([X \xrightarrow {\op {id}_X} X]) = c\ell(TX) \cap [X],$$ with $[X]:= \theta(p)\in \bB_*(X)$ (resp., $[X]:= p^!(1_{pt})\in \bB_*(X)$) the “fundamental class" of $X$ given by the canonical orientation (resp., the Gysin homomorphism) of the smooth morphism $p: X\to pt$. \[univ-BM\] We note that in fact here $\bB_*(-)$ does not need to be associated to a bivariant theory, e.g. it would be enough that $\bB_*(-)$ is an oriented Borel-Moore (weak) homology theory like $\Omega_*^{alg}$ (or a complex oriented (co)homology theory like $\Omega^{U}_*$). In fact ${\bM}_ *(\m V/-)$ is a [*universal Borel Moore functor*]{} with product (,but without an additivity property), see [@Yokura-obt]. Also the characteristic class $c\ell$ does not need to be multiplicative for the definition of the natural transformation ${\ga_{c\ell}}_* :{\bM}_ *(\m V/-)\to \bB_*(-)$, although we do need the multiplicativity of $c\ell$ for the multiplicativity of ${\ga_{c\ell}}_*$ with respect to cross products $\times$. Similarly, for a corresponding Verdier-Riemann-Roch formula, we need the compability $$f^!(c\ell(V)\cap -) = f^*c\ell(V)\cap f^!(-)$$ of $c\ell$ with the Gysin homomorphism $f^!$ for a smooth morphism $f: X\to Y$ and $V\in Vect(Y)$. ${\ga_{c\ell}}_*: {\bM}_ *(\m V/-)\to \bB_*(-)$ should be considered as a *“pre-motivic" characteristic class transformation of possibly singular varieties*, e.g. like $\ga_*: {\bM}_ *(\m V/-)\to \Omega_*^{alg}$ resp. $\ga_*: {\bM}_ *(\m V/-)\to \Omega^{U}_*$ associated to $c\ell(V):=1_Y$ for $V\in Vect(Y)$. These fit, in the complex algebraic context, into the following commutative diagram of transformations: $$\label{diagram} \begin{CD} {\bM}_ *(\m V/X) @> \ga_* >> \Omega_*^{alg}(X) @>>> \Omega^{U}_*(X) \\ @VVV @VVV @VVV \\ K_0(\m V/X) @> mC_y >> G_0(X)[y] @> \alpha >> K_0^{top}(X)[y] \\ @| @VVV @VVV \\ K_0(\m V/X) @> T_{y*} >> CH_*(X)\otimes \bQ[y] @>>> H^{BM}_{2*}(X)\otimes \bQ[y]\\ @V \epsilon VV @V y=-1 VV @V y=-1 VV \\ F(X) @> c_* >> CH_*(X)\otimes \bQ @>>> H^{BM}_{2*}(X)\otimes \bQ \:. \end{CD}$$ The left (resp. outer) part of this diagram is also available in the algebraic context over a base field of characteristic zero (resp. in the compact complex analytic context). The horizontal transformations in the upper line are the canonical ones associated to different universal theories, with ${\bM}_ *(\m V/-)$ the universal Borel Moore functor with product (but without an additivity property), $\Omega_*^{alg}$ the universal oriented Borel-Moore (weak) homology theory and $\Omega^{U}_*$ the universal complex oriented (co)homology theory. Similarly, the theories $H_*$ in the last two vertical lines represent different such homology theories (with a $\cap$-product action of characteristic classes of vector bundles) in the algebraic resp. topological context, like the universal theories $\Omega_*^{alg}, \Omega^{U}_*$, the $K$-theoretical theories $G_0, K_0^{top}$ or the classical theories $CH_*, H^{BM}_{2*}$. Also these six homology theories are associated to suitable bivariant theories, which are due to Fulton-MacPherson [@Fulton-MacPherson], except for algebraic cobordism $\Omega_*^{alg}$, where a corresponding “operational" bivariant version has been recently constructed by González and Karu [@GK]. In the topological context one also has [*Mayer-Vietoris and long exact homology sequences*]{}, whereas in the algebraic context one has [*short exact sequences*]{} $$\label{short} \begin{CD} H_*(Z) @> i_* >> H_*(X) @> j^* >> H_*(U)@>>> 0 \end{CD}$$ for $i: Z\to X$ the inclusion of a closed algebraic subset, with open complement $j: U:=X\backslash Z\to X$. For our unification, it is important to work with more general theories like ${\bM}_ *(\m V/-)$ and $K_0(\m V/-)$, which are [*not*]{} oriented Borel-Moore (weak) homology (or complex oriented (co)homology) theories, like the group $F(X)$ of constructible functions in relation to MacPherson’s Chern class transformation. Here we do not have such a short exact sequence (\[short\]) for ${\bM}_ *(\m V/-)$, but in the case of $K_0(\m V/-)$ (and also for $F(-)$) we even have short exact sequences $$\begin{CD} 0@>>> K_0(\m V/Z) @> i_* >> K_0(\m V/X) @> j^* >> K_0(\m V/U)@>>> 0 \:. \end{CD}$$ But another important property, which fails for them, is “homotopy invariance", e.g. $$p^*: K_0(\m V/X) \to K_0(\m V/X\times \bA^1)$$ is injective but not surjective for the projection $p: X\times \bA^1\to X$ (and similarly for $F(-)$). A true *“motivic" characteristic class transformation of possibly singular varieties* should factorize as in (\[diagram\]) over the canonical group homomorphism $$q: {\bM}_*(\m V/X) \to K_0(\m V/X),$$ like the transformations ${\ga_{c\ell}}_*$ associated to the multiplicative characteristic classes $c\ell$ given by $c, td, L, T^*_y$, or the total lambda-class $\lambda_y((-)^*)$ of the dual vector bundle, as mentioned before (in the complex analytic or algebraic context over a base field of characteristic zero). Only then we can also speak of the corresponding characteristic class $$cl_*(X) := {\ga_{c\ell}}_*([\op{id}_X])$$ of a *singular* space $X$, where ${\ga_{c\ell}}_*$ is the bottom homomorphism in the following diagram: $$\label{diagram2} \xymatrix{ & {\bM}_*(\m V/-) \ar [dl]_{q} \ar [dr]^{{\ga_{c\ell}}_*} \\ {K_0(\m V/-)} \ar [rr] _{{\ga_{c\ell}}_*}& & \bB_*(-)\,.}$$ Note that for a singular space $X$ one has the distinguished element $[X \xrightarrow {\op {id}_X} X] \in K_0(\m V/X)$, but $[X \xrightarrow {\op {id}_X} X]$ cannot be defined in ${\bM}_*(\m V/X)$. In fact in [@BSY1] we proved more in the complex analytic or algebraic context over a base field of characteristic zero, with $\bB=CH\otimes R$ or $\bB=\bH\otimes R$: The induced genus ${\ga_{c\ell}}_*: \bM(\m V/ pt) \to H_*(pt)\otimes R=R$ of a corresponding multiplicative characteristic class $c\ell$ has to be a specialization of the Hirzebruch $\chi_y$-genus characterized by $$\chi_y(\mathbb {P}^n)=1 - y + y^2 + \cdots +(-y)^n.$$ Moreover, the Hirzebruch class $T^*_y$ is for $R=\bQ[y]$ the only multiplicative characteristic class $c\ell$ with this property, which is defined by a normalized power series in $\bQ[y][[\alpha]]$. So it is the only such characteristic class $c\ell$, for which ${\ga_{c\ell}}_*: \bM(\m V/X \to pt) \to H_*(X)\otimes \bQ[y]$ can be factorized over the motivic group $K_0(\m V/X)$: $$\label{diagram2b} \xymatrix{ & {\bM}_*(\m V/X) \ar [dl]_{q} \ar [dr]^{{\ga_{c\ell}}_*} \\ {K_0(\m V/X)} \ar [rr] _{{T_y}_*}& & H_*(X)\otimes \bQ[y]\,.}$$ By “resolution of singularities", the canonical group homomorphism $q: {\bM}_*(\m V/X) \to K_0(\m V/X)$ is surjective in the complex analytic or algebraic context over a base field of characteristic zero. Moreover, using the “weak factorization theorem" of [@AKMW; @W], its kernel was described by Bittner [@Bittner] in terms of a “blow-up relation". In some sense (as mentioned by a referee), this can be seen as a counterpart of the “Conner-Floyd theorem" [@CF] in topology (or [@LM] in algebraic geometry), about recovering $K$-theory from cobordism. Here we introduce the following bivariant analogue of the “blow-up relation": \[BL\] For a morphism $f: X \to Y$ in the category $\m V=\m V^{(qp)}_k$ or $\m V=\m V^{an}_c$, we consider a blow-up diagram $$\begin{CD} E @> i'>> Bl_{S}X' \\ @VV q' V @VV q V \\ S @> i >> X' @> h >> X @> f >> Y\:, \end{CD}$$ with $h$ proper and $i$ a closed embedding such that $f \circ h$ as well as $f \circ h \circ i$ are smooth. Here $q: Bl_{S}X' \to X'$ is the blow-up of $X'$ along $S$, with $q':E \to S$ the exceptional divisor map. Then also $f \circ h \circ q$ and $f \circ h \circ i\circ q'$ are smooth (with $Bl_{S}X'$ and $E$ quasi-projective in the case $\m V=\m V^{qp}_k$). Let $\mathbb {BL}(\m V/ X \xrightarrow{f} Y)$ be the free abelian subgroup of $\bM(\m V/X \xrightarrow{f} Y)$ generated by $$[Bl_{S}X' \xrightarrow{h q} X] - [E \xrightarrow{hiq'} X] - [X'\xrightarrow{h} X] + [S\xrightarrow{hi} X]$$ for any such diagram, and define $$\bK_0(\m V/X \xrightarrow{f} Y) := \frac{\bM(\m V/X \xrightarrow{f} Y)} {\mathbb {BL}(\m V/X \xrightarrow{f} Y)}.$$ The corresponding equivalence class of $[V \xrightarrow{p} X]$ shall be denoted by $\Bigl[[V \xrightarrow{p} X] \Bigr].$ Note that for $Y=pt$ a point, the smoothness of $f \circ h$ and $f \circ h \circ i$ above is equivalent to $X'$ and $S$ are smooth manifolds. So in this case $\mathbb {BL}(\m V/ X \to pt)$ reduces to the “blow-up relation" considered by Bittner. In particular, we get a canonical group homomorphism $\bK_0(\m V/X\to pt) \to K_0(\m V/X)$ to the relative motivic Grothendieck group of varieties over $X$, which by Bittner’s theorem is an isomorphism in the complex analytic or algebraic context over a base field of characteristic zero. \[thm:main\] Let $\m V=\m V^{(qp)}_k$ be the category of (quasi-projective) algebraic varieties (i.e. reduced separated schemes of finite type) over a base field $k$ of any characteristic, or let $\m V=\m V^{an}_c$ be the category of compact reduced complex analytic spaces. 1. $\bK_0(\m V / - )$ can be given uniquely the structure of a bivariant theory so that the canonical projection $\bB q: \bM(\m V/-) \to \bK_0(\m V / - )$ is a Grothendieck transformation. 2. There exists a unique Grothendieck transformation $$mC_y=\La_y^{mot}: \bK_0(\m V_k^{qp}/ - ) \to \bK_{alg}( - )\otimes \bZ[y]$$ satisfying the normalization condition that for a smooth morphism $f: X \to Y$ the following equality holds in $\bK_{alg}(X \xrightarrow {f} Y) \otimes \bZ[y]$: $$\La_y^{mot}\Bigl(\Bigl[[X \xrightarrow{\op {id}_X} X]\Bigr]\Bigr) = \La_y(T^*_f) \bullet \theta(f).$$ 3. Let $T_y : \bK_0(\m V_k^{qp} / - ) \to \bH(-) \otimes \bQ[y]$ be defined as the composition $\tau \circ \La_y^{mot}$, renormalized by $\cdot (1+y)^i$ on $\bH^i(-)\otimes \bQ[y]$. Here $\bH$ is either the operational bivariant Chow group, or the even degree bivariant homology theory for $k=\bC$, with $\tau$ the corresponding Riemann-Roch transformation.\ Then $T_y$ is the unique Grothendieck transformation satisfying the normalization condition that for a smooth morphism $f: X \to Y$ the following equality holds in $\bH(X \xrightarrow {f} Y) \otimes \bQ[y]$: $$T_y\Bigl(\Bigl[[X \xrightarrow{\op {id}_X} X]\Bigr]\Bigr) = T^*_y(T_f) \bullet \theta(f).$$ \[cor-Grothendieck\] We have the following commutative diagrams of Grothendieck transformations: 1. $$\xymatrix{ & \bK_0(\m V_k^{qp} / - ) \ar [dl]_{mC_0} \ar [dr]^{{T_{0}}} \\ {\bK_{alg}( - ) } \ar [rr] _{\tau}& & \bH(-) \otimes \bQ.}$$ 2. $$\xymatrix{ & \bK_0(\m V_k^{qp} / - ) \ar [dl]_{\epsilon} \ar [dr]^{{T_{-1}}} \\ {\tilde{\bF}( - ) } \ar [rr] _{\gamma}& & CH(-) \otimes \bQ,}$$ if $k$ is of characteristic zero. Here $\epsilon$ is the unique Grothendieck transformation satisfying the normalization condition $\epsilon\Bigl(\Bigl[[X \xrightarrow{\op {id}_X} X]\Bigr]\Bigr)=\jeden_f$ for a smooth morphism $f: X \to Y$. And similarly for the bivariant Chern class transformation $\gamma: \bF( - ) \to A^{PI}( - )\otimes \bQ \supset CH(-) \otimes \bQ$ in case $k=\bC$. 3. Assume $k$ is of characteristic zero. Then the associated covariant transformations in Theorem \[thm:main\] (ii) and (iii) agree under the identification $$\bK_0(\m V_k^{qp}/X\to pt) \simeq K_0(\m V_k^{qp}/X)$$ with the motivic Chern and Hirzebruch class transformations $mC_y$ and ${T_y}_*$. Let us finish this introduction with some problems left open: 1. Our construction of the Grothendieck transformation $$mC_y=\La_y^{mot}: \bK_0(\m V_k^{qp}/ - ) \to \bK_{alg}( - )\otimes \bZ[y]$$ based on [@Gros Chapter IV, Theorem 1.2.1 and (1.2.6)] also works in the algebraic context without considering only quasi-projective varieties, if one uses the more sophisticated definition of $\bK_{alg}(X \xrightarrow {f} Y)=K_0(D^b_{f-perf}(X))$ as the Grothendieck goup of the triangulated category of $f$-perfect complexes. And a similar definition can also be used in the context of compact complex analytic varieties (cf. [@Fulton-MacPherson Part I, §10.10] and [@Levy2]). Then it seems reasonable, that one can also construct in a similar way in this compact complex analytic context the Grothendieck transformation $mC_y=\La_y^{mot}$. Here it would be enough to prove the analogues of in the complex analytic context. 2. Similarly one would like to further construct in this compact complex analytic context also the Grothendieck transformation $T_y$ based on Levy’s K-theoretical Riemann-Roch transformation $\alpha : \bK_{alg}(-)\to \bK^{top}_0(-)$ from algebraic to topological bivariant K-theory (see [@Levy2]). A key result missing so far is the counterpart $$\alpha(\m O_f)=\theta(f)$$ of [@Fulton-MacPherson Part II, Theorem 1.4 (3)], that $\alpha$ identifies for a smooth morphism $f: X\to Y$ the orientation $\m O_f:=[\m O_X]\in \bK_{alg}(X \xrightarrow {f} Y)$ with the orientation $\theta(f)\in \bK^{top}_0(X \xrightarrow {f} Y).$ 3. We do not know if Brasselet’s bivariant Chern class transformation $\ga: \bF(-) \to \bH(-)$ (see [@Brasselet]) satisfies for a smooth morphism $f:X \to Y$ the “strong normalization condition” $$\ga(\jeden_f) = c(T_f) \bullet \theta(f) \in \bH(X \xrightarrow{f} Y).$$ Then Corollary \[cor-Grothendieck\] (ii) would also be true for Brasselet’s bivariant Chern class transformation $\ga: \bF(-) \to \bH(-)$. 4. In a future work we will construct in the compact complex algebraic or analytic context a bivariant analogue $\bB\Omega(-)$ of the cobordism group $\Omega(-)$ of selfdual constructible sheaf complexes, together with a Grothendieck transformation $sd: \bK_0(\m V/ - ) \to \bB\Omega(-)$. This will be based on suitable Witt-groups of constructible sheaves and some other related topics different from the theme of the present paper. But what is still missing to get the counterpart of Corollary \[cor-Grothendieck\] (i) and (ii) for $y=1$ is a bivariant $L$-class transformation $\bB L: \bB\Omega(-) \to \bH(-)\otimes \bQ$. Our debt to the works of Quillen and Fulton-MacPherson should be clear after reading this introduction. In this paper we focus in the last sections on the unification of different bivariant theories of characteristic classes of singular spaces, by dividing out our universal bivariant theory by a “bivariant blow-up relation". But similar ideas (with other bivariant relations) should also work for other applications, e.g. in the algebraic geometric context for the construction of a “geometric" bivariant-theoretic version of Levine–Morel’s algebraic cobordism (different from the “operational" vivariant theory of [@GK], e.g. see [@Yokura-obt; @SY2]). Similarly, in [@BaSY] we will construct in the context of reduced differentiable spaces a “geometric" bivariant-theoretic version of Quillen’s complex cobordism (different from the abstract definition given by the general theory of Fulton-MacPherson [@Fulton-MacPherson]), and closely related to the approach of Emerson-Meyer [@EM2; @EM3] to “(bivariant) $KK$-theory via correspondences". The corresponding cohomology theory for smooth manifolds will be different and a refinement of Quillen’s geometric theory of complex cobordism [@Quillen]. Fulton–MacPherson’s bivariant theory {#FM-BT} ==================================== For the sake of the reader we quickly recall some basic ingredients of Fulton–MacPher- son’s bivariant theory [@Fulton-MacPherson].\ Let $\Cal V$ be a category which has a final object $pt$ and on which the fiber product or fiber square is well-defined, e.g. the category $\m V^{(qp)}_k$ of (quasi-projective) algebraic varieties (i.e. reduced separated schemes of finite type) over a base field $k$, or $\m V^{an}_{(c)}$ the category of (compact) reduced complex analytic spaces. We also consider a class of maps, called “confined maps" (e.g., proper maps in this algebraic or analytic geometric context), which are closed under composition and base change and contain all the identity maps. Finally, one fixes a class of fiber squares, called “independent squares" (or “confined squares", e.g., “Tor-independent" in algebraic geometry, a fiber square with some extra conditions required on morphisms of the square), which satisfy the following properties: \(i) if the two inside squares in $$\CD X''@> {h'} >> X' @> {g'} >> X \\ @VV {f''}V @VV {f'}V @VV {f}V\\ Y''@>> {h} > Y' @>> {g} > Y \endCD$$ or $$\CD X' @>> {h''} > X \\ @V {f'}VV @VV {f}V\\ Y' @>> {h'} > Y \\ @V {g'}VV @VV {g}V \\ Z' @>> {h} > Z \endCD$$ are independent, then the outside square is also independent. \(ii) any square of the following forms are independent: $$\xymatrix{X \ar[d]_{f} \ar[r]^{\op {id}_X}& X \ar[d]^f & & X \ar[d]_{\op {id}_X} \ar[r]^f & Y \ar[d]^{\op {id}_Y} \\ Y \ar[r]_{\op {id}_X} & Y && X \ar[r]_f & Y}$$ where $f:X \to Y$ is any morphism.\ A bivariant theory $\bB$ on a category $\Cal V$ with values in the category of (graded) abelian groups is an assignment to each morphism $$X \xrightarrow{f} Y$$ in the category $\Cal V$ a (graded) abelian group (in most cases we can ignore a possible grading) $$\bB(X \xrightarrow{f} Y)$$ which is equipped with the following three basic operations. The $i$-th component of $\bB(X \xrightarrow{f} Y)$, $i \in \bZ$, is denoted by $\bB^i(X \xrightarrow{f} Y)$ (with $\bB(X \xrightarrow{f} Y)=:\bB^0(X \xrightarrow{f} Y)$ in the ungraded context).\ [**Product operations**]{}: For morphisms $f: X \to Y$ and $g: Y \to Z$, the ($\bZ$-bilinear) product operation $$\bullet: \bB^i( X \xrightarrow{f} Y) \otimes \bB^j( Y \xrightarrow{g} Z) \to \bB^{i+j}( X \xrightarrow{gf} Z)$$ is defined. [**Pushforward operations**]{}: For morphisms $f: X \to Y$ and $g: Y \to Z$ with $f$ *confined*, the ($\bZ$-linear) pushforward operation $$f_*: \bB^i( X \xrightarrow{gf} Z) \to \bB^i( Y \xrightarrow{g} Z)$$ is defined. [**Pullback operations**]{}: For an *independent* square $$\CD X' @> g' >> X \\ @V f' VV @VV f V\\ Y' @>> g > Y, \endCD$$ the ($\bZ$-linear) pullback operation $$g^* : \bB^i( X \xrightarrow{f} Y) \to \bB^i( X' \xrightarrow{f'} Y')$$ is defined.\ And these three operations are required to satisfy the seven compatibility axioms (see [@Fulton-MacPherson Part I, §2.2] for details): 1. product is associative, 2. pushforward is functorial, 3. pullback is functorial, 4. product and pushforward commute, 5. product and pullback commute, 6. pushforward and pullback commute, and 7. projection formula.\ We also assume that $\bB$ has *units*, i.e., there is an element $1_X \in \bB^0( X \xrightarrow{\op {id}_X} X)$ such that $\alp \bullet 1_X = \alp$ for all morphisms $W \to X$ and $\alp \in \bB(W \to X)$; such that $1_X \bullet \beta = \beta $ for all morphisms $X \to Y$ and $\beta \in \bB(X \to Y)$; and such that $g^*1_X = 1_{X'}$ for all $g: X' \to X$.\ Let $\bB, \bB'$ be two bivariant theories on the category $\Cal V$. Then a [*Grothendieck transformation*]{} from $\bB$ to $\bB'$ $$\ga : \bB \to \bB'$$ is a collection of group homomorphisms $$\bB(X \to Y) \to \bB'(X \to Y)$$ for all morphisms $X \to Y$ in the category $\Cal V$, which preserves the above three basic operations (as well as the units, but not necessarily possible gradings): 1. $\ga (\alp \bullet_{\bB} \be) = \ga (\alp) \bullet _{\bB'} \ga (\be)$, 2. $\ga(f_{*}\alp) = f_*\ga (\alp)$, and 3. $\ga (g^* \alp) = g^* \ga (\alp)$.\ Most of our bivariant theories in this paper are *commutative* (see [@Fulton-MacPherson §2.2]), i.e., if whenever both $$\xymatrix{W \ar[d]_{f'} \ar[r]^{g'}& X \ar[d]^f && W \ar[d]_{g'} \ar[r]^{f'}& Y \ar[d]^{g} \\ Y \ar[r]_{g} & Z && X \ar[r]_f & Z}$$ are independent squares, then for $\alp \in \bB(X \xrightarrow {f} Z)$ and $\be \in \bB(Y \xrightarrow {g} Z)$ $$g^*(\alp) \bullet \be = f^*(\be) \bullet \alp.$$ This is for example the case for all bivariant theories mentioned in the introduction in the algebraic or analytic geometric context, except for the bivariant operational Chow group $CH$, with bivariant algebraic K-theory $\bK_{alg}$ and bivariant constructible functions $\bF, \tilde{\bF}$ examples of ungraded theories. Here $CH$ is at least commutative in the context of a base field $k$ of characteristic zero, by (using resolution of singularities). Similarly the bivariant homology $\bH$ is commutative, if we restrict ourselves to the even degree part only (otherwise $\bH$ would be *skew-commutative*, i.e. $g^*(\alp) \bullet \be = (-1)^{\op {deg}(\alp) \op{deg}(\be)} f^*(\be) \bullet \alp$ in the situation above).\ $\bB_*(X):= \bB(X \to pt)$ becomes a covariant functor for [*confined*]{} morphisms and $\bB^*(X) := \bB(X \xrightarrow{id} X)$ becomes a contravariant ring valued functor for [*any*]{} morphisms, with $\bB_*(X)$ a left $\bB^*(X)$-module under the product $\cap:=\bullet: \bB^*(X)\otimes \bB_*(X)\to \bB_*(X)$. As to a possible grading, one sets $\bB_i(X):= \bB^{-i}(X \to {pt})$ and $\bB^j(X):= \bB^j(X \xrightarrow{id} X)$ so that $\bB^*(X)$ becomes a graded ring with $\cap: \bB^j(X)\otimes \bB_i(X)\to \bB_{i-j}(X)$.\ The following notion of an *orientation* makes $\bB_*$ a contravariant functor and $\bB^*$ a covariant functor with the corresponding Gysin (or transfer) homomorphisms: \[canonical\]([@Fulton-MacPherson Part I, Definition 2.6.2]) Let $\Cal S$ be a class of maps in $\Cal V$, which is closed under compositions and contains all identity maps. Suppose that to each $f: X \to Y$ in $\Cal S$ there is assigned an element $\theta(f) \in \bB(X \xrightarrow {f} Y)$ satisfying that 1. $\theta (g \circ f) = \theta(f) \bullet \theta(g)$ for all $f:X \to Y$, $g: Y \to Z \in \Cal S$ and 2. $\theta(\op {id}_X) = 1_X $ for all $X$ with $1_X \in \bB^*(X):= \bB(X \xrightarrow{\op {id}_X} X)$ the unit element. Then $\theta(f)$ is called an [*orientation*]{} of $f$. If we need to refer to which bivariant theory we consider, we denote $\theta_{\bB}(f)$ instead of the simple notation $\theta (f)$. Since there can be different choices of such orientations (e.g., compare with our “twisting” construction later on), we prefer to call the above $\theta$ simply an *orientation* for what is called a “canonical orientation” in [@Fulton-MacPherson]. If we want to emphasize the class $\Cal S$, it is called an *$\Cal S$-orientation*, and if we want to emphasize the bivariant theory $\bB$ as well, it is called a *$\bB$-valued $\Cal S$-orientation*. For example the class $\Cal S$ of [*smooth*]{} morphisms in the algebraic or analytic geometric context has orientations for all the bivariant theories mentioned in the introduction, with all cartesian squares independent. For the composite $X \xrightarrow{f} Y \xrightarrow{g} Z$, if $f \in \m S$ has an orientation $\theta_{\bB}(f)$, then we have the Gysin homomorphism (or transfer) defined by $f^!(\alp) :=\theta(f) \bullet \alp$: $$f^!: \bB(Y \xrightarrow{g} Z) \to \bB(X \xrightarrow{gf} Z),$$ which is functorial, i.e., $(gf)^! =f^! g^!$ and $id^!=id$. In particular, when $Z = pt$, we have the Gysin homomorphism: $$f^!: \bB_*(Y) \to \bB_*(X).$$ For an independent square $$\CD X' @> g' >> X \\ @V f' VV @VV {f}V\\ Y'@>> g > Y, \endCD$$ if $g \in \m C \cap \m S$ and $g$ has an orientation $\theta_{\bB}(g)$, then we have the Gysin homomorphism defined by $g_!(\alp) :=g'_*(\alp \bullet \theta(g))$: $$g_!: \bB(X' \xrightarrow{f'} Y') \to \bB(X \xrightarrow{f} Y),$$ which is functorial, i.e., $(gf)_! = g_! f_!$ and $id_!=id$. In particular, for an independent square $$\CD X @> f >> Y \\ @V {\op {id}_X} VV @VV {\op {id}_Y}V\\ X @>> f > Y, \endCD$$ with $f \in \m C \cap \m S$, we have the Gysin homomorphism: $$f_!: \bB^*(X) \to \bB^*(Y).$$ The symbols $f^!$ and $g_!$ should carry the information of $\Cal S$ and the orientation $\theta$, but it will be usually omitted if it is not necessary to be mentioned.\ Suppose that we have a Grothendieck transformation $\ga: \bB \to \bB'$ of two bivariant theories $\bB, \bB'$. This induces natural transformations $\ga_*: \bB_* \to \bB_*'$ and $\ga^*: \bB^* \to {\bB'}^*$, i.e., we have the following commutative diagrams:\ For any morphism $f: X \to Y$ we have the commutative diagram $$\CD \bB^*(X) @> \ga^* >> \bB'^*(X) \\ @V {f^*} VV @VV {f^*}V\\ \bB^*(Y) @>> \ga^*> \bB'^*(Y) . \endCD$$ For a confined morphism $f:X \to Y$ we have the commutative diagram $$\CD \bB_*(X) @> \ga_* >> \bB'_*(X) \\ @V {f_*} VV @VV {f_*}V\\ \bB_*(Y) @>> \ga_*> \bB'_*(Y) . \endCD$$ And these are related by the *module property* $$\ga_*(\beta\cap \alpha) = \ga^*(\beta) \cap \ga_*(\alpha) \quad \text{for all} \quad \beta\in \bB^*(X), \alpha \in \bB_*(X).$$ Assume now that $f: X\to Y$ has an orientation for both bivariant theories. Then a bivariant element $u_f \in \bB'^*(X) = \bB'(X \xrightarrow {\op {id}_X} X)$ with $$\ga (\theta_{\bB}(f)) = u_f \bullet \theta_{\bB'}(f)$$ is called a *Riemann–Roch formula* (see [@Fulton-MacPherson]) comparing these orientations with respect to the bivariant theories $\bB, \bB'$. Such a Riemann–Roch formula gives rise to the following (wrong-way) commutative diagrams with respect to the above two Gysin homomorphisms $f_!, f^!$ : $$\begin{CD} \bB^*(X) @> \ga^* >> \bB'^*(X) \\ @V {f_!} VV @VV {f_!( \;-\; \bullet u_f)}V\\ \bB^*(Y) @>> \ga^*> \bB'^*(Y) . \end{CD} \hspace{2cm} \begin{CD} \bB_*(Y) @> \ga_* >> \bB'_*(Y) \\ @V {f^!} VV @VV {u_f \bullet f^!}V\\ \bB_*(X) @>> \ga_*> \bB'_*(X) . \end{CD}$$ The most important and motivating example of such a Grothendieck transformation is Baum–Fulton–MacPherson’s bivariant *Riemann–Roch transformation* ([@Fulton-MacPherson Part II]): $$\tau: \bK_{alg} \to \bH\otimes \bQ\:,$$ or its algebraic counterpart of [@Fulton-book Example 18.3.19]. Here $\m V=\m V^{qp}_{k}$ is the category of quasi-projective varieties over a base field $k$ of any characteristic, with $\bH=CH$ the bivariant operational Chow groups, or $\bH$ the even degree bivariant homology in case $k=\bC$. The independent squares in this context are the *Tor-independent* fiber squares. $\bK_{alg}$ is the bivariant algebraic K-theory of relative perfect complexes, so that ${\bK_{alg}}_*(X) = K_0(X)$ is the Grothendieck group of coherent sheaves and ${\bK_{alg}}^*(X) = K^0(X)$ is the Grothendieck group of algebraic vector bundles. The associated contravariant transformation is the *Chern character* $$\tau^*=ch: K^0(X) \to H^*(X)\otimes \bQ,$$ and the associated covariant transformation is the *Todd class transformation* $$\tau_*=td_*: G_0(X)\to H_*(X)\otimes \bQ,$$ which is functorial for proper morphisms $f:X \to Y$. Moreover, they are related by the *module property* $$\label{module} td_*(\beta\cap \alpha) = ch^*(\beta) \cap td_*(\alpha) \quad \text{for all} \quad \beta\in K^0(X), \alpha \in G_0(X).$$ This generalizes the original Grothendieck–Riemann–Roch Theorem and Hirzebruch–Riemann–Roch Theorem. Both bivariant theories $\bK_{alg}$ and $H_*(-)\otimes \bQ$ are oriented for the class $\Cal S$ of smooth (or more generally of local complete intersection) morphism, with $\theta_{\bK}(f)= \m O_f:=[\m O_X]\in \bK_{alg}(X \xrightarrow {f} Y)$ the class of the structure sheaf, and $\theta_{\bH}(f)=[f]\in \bH(X \xrightarrow {f} Y)$ the corresponding “relative fundamental class”. And these are related by the *Riemann–Roch formula* $$\label{RR-formula} \tau(\m O_f) = td(T_f)\bullet [f] \:,$$ with $u_f:=td(T_f)\in H^*(X)\otimes \bQ$ (compare with [@Fulton-MacPherson (\*) on p.124] for $\bH$ the bivariant homology in case $k=\bC$. For $\bH=CH$ the bivariant Chow group and $k$ of any characteristic, this follows from [@Fulton-book Theorem 18.2] as we explain in the last section of our paper). Here $T_f$ is the (virtual) tangent bundle of $f$. This implies the following two results:\ \ : The following diagram commutes for a proper smooth morphism $f: X \to Y$: $$\CD K(X) @> ch >> H^*(X)\otimes \bQ \\ @V {f_!} VV @VV {f_!(td(T_f) \cup \;-\;) }V\\ K(Y) @>> ch> H^*(Y)\otimes \bQ . \endCD$$ : The following diagram commutes for a smooth morphism $f: X \to Y$: $$\CD G_0(Y) @> td_* >> H_*(Y)\otimes \bQ \\ @V {f^!} VV @VV {td(T_f)\cap f^!}V\\ G_0(X) @>> td_* > H_*(X)\otimes \bQ . \endCD$$ Of course both formulae are more generally true for $f$ a local complete intersection morphism, which is special to the Grothendieck transformation $\tau$. In this paper only the case of a smooth morphism will be used, and then similar results are also true for the other considered Grothendieck transformations. It should also be remarked that *one motivation of Fulton–MacPherson’s bivariant theory was to unify the above three Riemann–Roch theorems ...* (see [@Fulton-MacPherson Part II, §0.1.4]). \[stable\] (i) Let $\Cal S$ be another class of maps in $\Cal V$ , called “specialized maps" (e.g., smooth maps in algebraic geometry), which is closed under composition and under base change and containing all identity maps. Let $\bB$ be a bivariant theory. If $\Cal S$ has orientations in $\bB$, then we say that $\Cal S$ is [*$\bB$-oriented* ]{}and an element of $\Cal S$ is called a [*$\bB$-oriented*]{} morphism. \(ii) Assume furthermore, that the orientation $\theta$ on $\Cal S$ satisfies for any independent square with $f \in \Cal S$ $$\CD X' @> g' >> X\\ @Vf'VV @VV f V \\ Y' @>> g > Y \endCD$$ the condition $$\label{nice} \theta (f') = g^* \theta (f)$$ (which means that the orientation $\theta$ is preserved under the pullback operation). Then we call $\theta$ a [*stable orientation*]{} and say that $\Cal S$ is [*stably $\bB$-oriented*]{}. Similarly an element of $\Cal S$ is called a [*stably $\bB$-oriented*]{} morphism. Consider for example the class $\Cal S$ of all [*smooth*]{} morphisms for $\m V=\m V^{(qp)}_{k}$ the category of (quasi-projective) varieties over a base field $k$ of any characteristic, with all fiber squares as the independent squares. Then this class has a stable orientation $\theta$ with respect to $\bK_{alg}$ or $CH$ in any characteristic (with $\theta(f)=\m O_f$ or $[f]$), to $\tilde{\bF}$ in characteristic zero (with $\theta(f)=\jeden_f$) and to $\bF$ or bivariant homology $\bH$ for $k=\bC$ (with $\theta(f)=\jeden_f$ or $[f]$). A universal bivariant theory on the category of varieties ========================================================= Let $\Cal V$ be the category $\m V=\m V^{(qp)}_{k}$ of (quasi-projective) varieties over a base field $k$ of any characteristic, or the category $\m V=\m V^{an}_{c}$ of compact reduced analytic spaces, with all fiber squares as the independent squares. As the “confined" resp. “specialized" maps we take the class $\Cal Prop $ of [*proper*]{} resp. $\Cal Sm$ of [*smooth*]{} morphisms. \[UBT\] We define $$\bM(\m V/X \xrightarrow{f} Y)$$ to be the free abelian group generated by the set of isomorphism classes of proper morphisms $h: W \to X$ such that the composite of $h$ and $f$ is a smooth morphism: $$h \in \Cal Prop \quad \text {and} \quad f \circ h: W \to Y \in \Cal Sm.$$ Then the association $\bM$ is a *bivariant theory*, if the three operations are defined as follows: [**Product operation**]{}: For morphisms $f: X \to Y$ and $g: Y \to Z$, the product operation $$\bullet: \bM (\m V/X \xrightarrow{f} Y) \otimes \bM (\m V/ Y \xrightarrow{g} Z) \to \bM ( \m V/X \xrightarrow{gf} Z)$$ is defined for $[V \xrightarrow{p} X] \in \bM (\m V/X \xrightarrow{f} Y)$ and $[W \xrightarrow{k} Y] \in \bM (\m V/ Y \xrightarrow{g} Z)$ by $$[V \xrightarrow{p} X] \bullet [W \xrightarrow{k} Y] := [V' \xrightarrow{p \circ {k}''} X],$$ and bilinearly extended. Here we consider the following fiber squares $$\label{cd:product}\CD V' @> {p'} >> X' @> {f'} >> W \\ @V {k''}VV @V {k'}VV @V {k}VV\\ V@>> {p} > X @>> {f} > Y @>> {g} > Z .\endCD$$ [**Pushforward operation**]{}: For morphisms $f: X \to Y$ and $g: Y \to Z$ with $f \in \m Prop$, the pushforward operation $$f_*: \bM (\m V/ X \xrightarrow{gf} Z) \to \bM (\m V/ Y \xrightarrow{g} Z)$$ is defined by $$f_*([V \xrightarrow{p} X]) := [V \xrightarrow{f \circ p} Y]$$ and linearly extended. [**Pullback operation**]{}: For an independent square $$\CD X' @> g' >> X \\ @V f' VV @VV f V\\ Y' @>> g > Y, \endCD$$ the pullback operation $$g^* : \bM (\m V/ X \xrightarrow{f} Y) \to \bM(\m V/ X' \xrightarrow{f'} Y')$$ is defined by $$g^*([V \xrightarrow{p} X] ):= [V' \xrightarrow{p'} X']$$ and linearly extended. Here we consider the following fiber squares: $$\label{cd:pullback}\CD V' @> g'' >> V \\ @V {p'} VV @VV {p}V\\ X' @> g' >> X \\ @V f' VV @VV f V\\ Y' @>> g > Y. \endCD$$ The proof is left for the reader. Note that $\theta(f):=[X \xrightarrow {\op {id}_X} X]$ for the smooth morphism $ f: X\to Y$defines a stable orientation on $\bM(\m V/-)$. We call the bivariant theory $\bM(\m V/-)$ a *pre-motivic bivariant relative Grothendieck group* on the category $\m V$ of varieties. \(1) ${\bM}_*(\m V/X) = \bM(\m V/X \to pt)$ is the free abelian group generated by the isomorphism classes $[V \xrightarrow{h} X]$, where $h$ is proper and $V$ is smooth. ${\bM}_*(\m V/-)$ is a covariant functor for proper morphisms, i.e., if $f: X \to Y$ is proper, we have the covariant pushforward $$f_*: {\bM}_*(\m V/X) \to {\bM}_*(\m V/Y)\:.$$ ${\bM}_*(\m V/-)$ is also a contravariant functor for smooth morphisms, i.e., if $f:X \to Y$ is a smooth morphism, we have the contravaraint Gysin homomorphism $$f^!: {\bM}_*(\m V/Y) \to {\bM}_*(\m V/X) \:.$$ (2) ${\bM}^*(\m V/X) = \bM(\m V/X \xrightarrow{\op {id}_X} X)$ is the free abelian group generated by the isomorphism classes $[V \xrightarrow{h} X]$, where $h$ is . It gets a ring structure $\cup$ by fiber products, with unit $1_X=[X \xrightarrow {\op {id}_X} X]$. ${\bM}^*(\m V/-)$ is a contravariant functor for any morphism, i.e., for any morphism $f:X \to Y$ we have the contravariant pullback (preserving $\cup$ and the units) $$f^*: {\bM}^*(\m V/Y) \to {\bM}^*(\m V/X)\:.$$ ${\bM}^*(\m V/-)$ is also a covariant functor for morphisms which are smooth and proper, i.e., if $f:X \to Y$ is a smooth proper morphism, we have the covariant Gysin homomorphism $$f_!: {\bM}^*(\m V/X) \to {\bM}^*(\m V/Y)\:.$$ (3) The bivariant product induces the following “cap product": $$\cap :{\bM}^*(\m V/X) \times {\bM}_*(\m V/X) \to {\bM}_*(\m V/X).$$ In particular, when $X$ itself is a *smooth* variety, with $[X]:=\cap[X \xrightarrow{\op {id}_X} X]\in {\bM}_*(\m V/X)$, we have the “Poincaré duality" homomorphism $$\cap [X]: {\bM}^*(\m V/X) \to {\bM}_*(\m V/X)\:,$$ which is nothing but $[W \xrightarrow{k} X ]\cap[X] = [W \xrightarrow{k} X ].$ More generally, the isomorphism class $[V \xrightarrow{h} X]\in {\bM}_*(\m V/X)$ of any proper morphism $h:V \to X$ from a *smooth* variety $V$ to $X$ gives rise to the homomorphism $$\cap{[V \xrightarrow{h} X]}: {\bM}^*(\m V/X) \to {\bM}_*(\m V/X)$$ defined by $[W \xrightarrow{k} X ]\cap[V \xrightarrow{h} X]= [W \times _X V \to X].$ The bivariant theory $\bM(\m V/-)$ has the following universal property (see [@Yokura-obt Theorem 3.1] for the proof of a more general result): \[univ\] Let $\bB$ be a bivariant theory on $\m V$ such that a smooth morphism $f: X\to Y$ has a stable orientation $\theta(f)\in \bB(f)$. Then there exists a unique Grothendieck transformation $$\ga:=\ga_{\theta}: \bM(\m V/-) \to \bB(-)$$ satisfying the normalization condition that for a smooth morphism $f:X \to Y$ the following identity holds in $\bB(X \xrightarrow {f} Y)$: $$\ga([X \xrightarrow {\op {id}_X} X]) = \theta(f).$$ Note that in [@Yokura-obt] only [*commutative*]{} bivariant theories are considered, but the result and proof of [@Yokura-obt Theorem 3.1] works without this assumption. \[twisting\] Let $c\ell: Vect(-) \to \bB^*(-)$ be a contravariant functorial characteristic class of algebraic (or analytic) vector bundles with values in the associated cohomology theory, which is multiplicative in the sense that $c\ell(V) = c\ell(V') c\ell(V'')$ for any short exact sequence of vector bundles $0\to V'\to V \to V'' \to 0$, with $c\ell(T_{pt})=1_{pt}\in \bB^*(\{pt\})$. Assume that $c\ell$ commutes with the stable orientation $\theta$, i.e. $$\theta(f)\bullet cl(V)=f^*cl(V)\bullet \theta(f)$$ for all smooth morphism $f: X\to Y$ and $V\in Vect(Y)$. Then there exists a unique Grothendieck transformation $$\ga_{c\ell}: \bM(\m V/-) \to \bB(-)$$ satisfying the normalization condition that for a smooth morphism $f:X \to Y$ the following identity holds in $\bB(X \xrightarrow {f} Y)$: $$\ga_{c\ell}([X \xrightarrow {\op {id}_X} X]) = c\ell(T_f) \bullet \theta(f).$$ Here $T_f$ is the relative tangent bundle of the smooth morphism $f$. This follows from Theorem \[univ\] by using the next result (similar twisting constructions are due to Quillen [@Quillen] (resp., Levine-Morel [@LM §4.1.9]) in the context of complex oriented (co)homology theories (resp., oriented Borel-Moore (weak) homology theories): The definition $\theta'(f):=c\ell(T_f) \bullet \theta(f)$ for a smooth morphism $f: X\to Y$ defines a new “twisted" stable orientation. First note that $T_{id_X}$ is the zero vector bundle $p^*T_{pt}$ for $p: X\to pt$ the constant map so that by functoriality $c\ell(T_{id})= p^*c\ell(T_{pt}) = p^*(1_{pt})=1_X \in \bB^*(X)$. This implies (ii): $\theta'(id_X)=1_X\in \bB^*(X)$ for all $X$. Let us now proof the multiplicativity (i): $$\theta'(g\circ f) = \theta'(g) \bullet \theta'(f)$$ for all smooth morphism $f: X\to Y$ and $g: Y\to Z$. Here we have $c\ell (T_{gf})= c\ell (T_{f}) \bullet f^* c\ell(T_{g})$ by the functoriality and multiplicativity of $c\ell$, due to the short exact sequence of vector bundles $$0\to T_{f } \to T_{gf} \to f^*T_{g}\to 0\:.$$ Similarly $\theta(gf)=\theta(f) \bullet \theta(g) $, since $\theta$ is a canonical orientation. Moreover, $c\ell$ commutes by assumption with the orientation $\theta$ so that $$\theta(f) \bullet c\ell(T_{g}) = f^* c\ell(T_{g}) \bullet \theta(f).$$ So we get $$\begin{aligned} \theta'(g\circ f) &:= c\ell(T_{g\circ f}) \bullet \theta(g\circ f)\\ & = \left( c\ell (T_{f}) \bullet f^* c\ell(T_{g})\right) \bullet \left( \theta(f) \bullet \theta(g) \right) \\ & = c\ell (T_{f}) \bullet \left( f^* c\ell(T_{g}) \bullet \theta(f) \right) \bullet \theta(g) \\ & = \left( c\ell (T_{f}) \bullet \theta(f) \right) \bullet \left( c\ell(T_{g}) \bullet \theta(g) \right) \\ & = \theta'(g) \bullet \theta'(f) \:.\end{aligned}$$ Finally we show that $\theta'$ is a stable orientation, i.e. (iii): $$\theta'(f')=g^* \theta'(f)$$ in the context of Definition \[stable\](ii). This follow from $T_{f'}\simeq g^*(T_f)$ by the functoriality $ c\ell(T_{f'}) = g^*c\ell(T_{f})$ of $ c\ell$ and the stability $\theta(f')=g^* \theta(f)$ of $\theta$: $$\begin{aligned} \theta'(f')&:= c\ell(T_f') \bullet \theta(f')\\ &= g^*c\ell(T_{f}) \bullet g^* \theta(f)\\ &= g^*\left( c\ell(T_{f}) \bullet \theta(f) \right)\\ &= g^* \theta'(f)\:.\end{aligned}$$ Note that the assumption, that the characterisic class $c\ell$ commutes with the orientation $\theta$, is true for $\bB$ commutative, or $\bB$ graded-commutative with $c\ell$ taking values in even degree cohomology classes. Similarly it is true for the trivial class $c\ell(V)=1$ the unit in $\bB^*(-)$, as well as for $\bB=CH$ the bivariant Chow homology , with $c\ell$ a “usual" multiplicative characteristic class given in terms of Chern class operators as in [@Fulton-book §3.2]. This covers all cases we need in this paper. Finally, the Grothendieck transformation $$\ga_{c\ell}: \bM(\m V/-) \to \bB(-)$$ from Corollary \[twisting\] satisfies by the normalization condition $$\ga_{c\ell}([X \xrightarrow {\op {id}_X} X]) = c\ell(T_f) \bullet \theta(f)$$ the *Riemann-Roch formula* with $u_f=c\ell(T_f)$ for a smooth morphism $f:X \to Y$. So by the general theory we get the\ : The following diagram commutes for a proper smooth morphism $f: X \to Y$: $$\CD {\bM}^*(\m V/X) @> {\ga_{c\ell}}^* >> \bB^*(X)\\ @V {f_!} VV @VV {f_!(c\ell(T_f) \cup \;-\;) }V\\ {\bM}^*(\m V/Y) @>> {\ga_{c\ell}}^* > \bB^*(Y) . \endCD$$ : The following diagram commutes for a smooth morphism $f: X \to Y$: $$\CD {\bM}_*(\m V/X) @> {\ga_{c\ell}}_* >> \bB_*(X)\\ @V {f^!} VV @VV {c\ell(T_f)\cap f^!}V\\ {\bM}_*(\m V/Y) @>> {\ga_{c\ell}}_* > \bB_*(Y) . \endCD$$\ 1. $\ga_{c\ell}:\bM(\m V/X \xrightarrow{f} Y) \to \bB(X \xrightarrow{f} Y)$ can be called a *bivariant pre-motivic characteristic class transformation*. When $Y$ is a point $pt$, $${\ga_{c\ell}}_*: \bM (\m V/ X \to pt) \to \bH(X \to pt) = \bB_*(X)$$ is the *unique natural transformation* satisfying the *normalization condition* that for a smooth variety $${\ga_{c\ell}}_*([X \xrightarrow {id_X} X]) = c\ell(TX) \cap [X].$$ In other words, this gives rise to a *pre-motivic characteristic class transformation for singular varieties*. In a sense, this could be also a very general answer to the forementioned MacPherson’s question about the existence of a unified theory of characteristic classes for singular varieties. As mentioned in Remark \[univ-BM\], ${\bM}_ *(\m V/-)$ is in fact a [*universal Borel Moore functor*]{} with product (but without an additivity property), see [@Yokura-obt].\ 2. In particular, we have the following commutative diagrams: $$\xymatrix{ &{\bM}_* (\m V/ X) \ar [dl]_{\epsilon} \ar [dr]^{{\ga_{c}}_*} \\ {F(X) } \ar [rr] _{c_*}& & H_*(X)},$$ with $H_*(X)=CH_*(X)$ in the algebraic context over a base field of characteristic zero, or $H_*(X)=H^{BM}_{2*}(X)$ in the complex algebraic or compact complex analytic context. Here $\epsilon ([V \xrightarrow{h} X]) := h_* \jeden_V$. $$\xymatrix{ & {\bM}_* (\m V/ X) \ar [dl]_{mC_0} \ar [dr]^{{\ga_{td}}_*} \\ {G_0(X) } \ar [rr] _{td_*}& & H_*(X)\otimes \bQ},$$ with $H_*(X)=CH_*(X)$ in the algebraic context over a base field of any characteristic, or $H_*(X)=H^{BM}_{2*}(X)$ in the complex algebraic or compact complex analytic context. Here $mC_0([V \xrightarrow{h} X]) := [h_* \mathcal O_V]$. $$\xymatrix{ & {\bM}_* (\m V/ X) \ar [dl]_{sd} \ar [dr]^{{\ga_L}_*} \\ {\Omega_{sd}(X) } \ar [rr] _{L_*}& & H_{2*}^{BM}(X)\otimes \bQ.}$$ Here $X$ has to be a compact complex algebraic or analytic variety, with $$sd([V \xrightarrow{h} X]) := [h_* \bQ_V[dim(X)]]\:.$$ Note that all these covariant theories come from a suitable bivariant theory, except $\Omega_{sd}(X)$. So the right slant arrows follow e.g. from Corollary \[twisting\] applied to the canonical orientation $\theta(f)=[f]\in \bH(f)$ given by the relative fundamental class of the smooth morphism $f$. As mentioned already before, the characteristic classes $c\ell= c^*, td^*$ and $L^*$ are multiplicative and commute with $\theta$ by general reasons. The first two left slant arrows follow from Theorem \[univ\] applied to the following canonical orientation of a smooth morphism $f$: $\theta(f)=\jeden_f\in \bF$ resp. $\jeden_f\in \tilde{\bF}$, and $\theta(f)= \m O_f \in \bK_{alg}(f)$. The third left slant arrows $sd$ follows e.g. from the universal property of ${\bM}_* (\m V/ -)$ as a universal Borel Moore functor (or by direct construction). 3. It follows from Hironaka’s resolution of singularities ([@Hironaka]) that there exists a surjection $${\bM}_* (\m V/ X) \to K_0(\m V /X)$$ in the algebraic context over a base field of characteristic zero, or in the compact complex analytic context. As already explained in the introduction, it then turns out that if (under a certain requirement) the natural transformation ${\ga_{c\ell}}_*: {\bM}_* (\m V/ X)\to H_*(X)\otimes \bQ[y]$ can be pushed down to the relative Grothendieck group $K_0(\m V /X)$, then it has to be the Hirzebruch class transformation, i.e., the following diagram commutes: $$\xymatrix{ & {\bM}_* (\m V/ X) \ar [dl]_{q} \ar [dr]^{{\ga_{c\ell}}_*} \\ {K_0(\Cal V/X) } \ar [rr] _{{T_y}_*}& & H_*(X)\otimes \bQ[y].}$$ And one of the main results of our previous paper [@BSY1] claims that in this context the above three diagrams also commute with ${\bM}_* (\m V/ X)$ being replaced by the smaller group $K_0(\m V/X)$. Thus we are led to the following natural problem: \[problem\] Formulate a reasonable bivariant analogue $\bK_0(\m V/ X\xrightarrow{f} Y)$ of the relative Grothendieck group $K_0(\m V/ X)$ so that the following hold: 1. There is a natural group homomorphism $q: \bK_0(\m V/ X\xrightarrow{} pt) \to K_0(\m V/X)$, which is an isomorphism in the algebraic context over a base field of characteristic zero, or in the compact complex analytic context. 2. $\bB q: \bM(\m V/ X\xrightarrow{f} Y) \to \bK_0(\m V/ X\xrightarrow{f} Y)$ is a certain quotient map, which specializes for $Y$ a point to the quotient map $q: {\bM}_*(\m V/ X) \to K_0(\m V/ X)$. 3. $T_y:\bK_0(\m V/ X\xrightarrow{f} Y) \to \bH(X\xrightarrow{f} Y)\otimes \bQ[y]$ is a Grothendieck transformation, which specializes for $Y$ a point (in the algebraic context over a base field of characteristic zero, or in the compact complex analytic context) to the motivic Hirzebruch class transformation ${T_y}_*:K_0(\Cal V/X) \to H_*(X)\otimes \bQ[y]$. 4. The following diagram commutes: $$\xymatrix{ & \bM(\m V/ X\xrightarrow{f} Y) \ar [dl]_{\bB q} \ar [dr]^{\ga_{T_{y}^*}} \\ {\bK_0(\m V/ X\xrightarrow{f} Y)} \ar [rr] _{T_y}& & \bH(X\xrightarrow{f} Y)\otimes \bQ[y].}$$ If such a bivariant theory $\bK_0(\m V/ X\xrightarrow{f} Y)$ is obtained, then its associated contravariant functor $K^0(\m V/ X): = \bK_0(\m V/ X\xrightarrow{\op {id}_X} X)$ can be considered as a contravariant counterpart of the relative Grothendieck group $K_0(\m V/X)$ (at least in the algebraic context over a base field of characteristic zero, or in the compact complex analytic context). Similarly, the natural transformation $T_y^*: K^0(\m V/ -) \to H^*(-)\otimes \bQ[y]$ is a contravariant counterpart of the Hirzebruch class transformations $T_{y*}$ satisfying the *module property*. A bivariant relative Grothendieck group $\bK_0(\m V/ X\xrightarrow{f} Y)$ ========================================================================= First we recall the following result of Franziska Bittner [@Bittner]: \[bittner\] Let $K_0(\m V/X)$ be the relative Grothendieck group of varieties over $X\in obj(\m V)$, with $\m V=\m V^{(qp)}_k$ (resp. $\m V=\m V^{an}_c$) the category of (quasi-projective) algebraic (resp. compact complex analytic) varieties over a base field $k$ of characteristic zero. Then $K_{0}(\m V/X)$ is isomorphic to ${\bM}_*(X)$ modulo the “blow-up” relation $$\label{eq:bl} [\emptyset \to X]=0 \quad \text{and} \quad [Bl_{Y}X'\to X] - [E\to X]= [X'\to X] - [Y\to X] \:,\tag{bl}$$ for any cartesian diagram (which shall be called the “blow-up diagram" from here on) $$\begin{CD} E @> i'>> Bl_{Y}X' \\ @VV q' V @VV q V \\ Y @> i >> X' @> f >> X \:, \end{CD}$$ with $i$ a closed embedding of smooth spaces and $f:X'\to X$ proper. Here $Bl_{Y}X'\to X'$ is the blow-up of $X'$ along $Y$ with exceptional divisor $E$. Note that all these spaces other than $X$ are also smooth (and quasi-projective in case $X', Y\in ob(\m V^{qp}_k)$). The proof of this theorem requires Abramovich et al’s “Weak Factorisation Theorem" [@AKMW; @W]. The kernel of the canonical quotient map $q:{\bM}_* (\m V/ X) \to K_0(\m V/X)$ is the subgroup $BL(\m V/X)$ of ${\bM}_* (\m V/ X)$ generated by $$[Bl_{Y}X'\to X] - [E\to X] -[X'\to X] + [Y\to X]$$ for any blow-up diagram as above. Thus what we want is a bivariant analogue of the subgroup $BL(\m V/X)$. For that purpose we first observe the following result, working in the category $\m V=\m V^{(qp)}_k$ (resp. $\m V=\m V^{an}_c$) of (quasi-projective) algebraic (resp. compact complex analytic) varieties over a base field $k$ of any characteristic. \[key-lemma\] Let $h:X' \to X$ be a *smooth* morphism, with $i:S \to X'$ a closed embedding such that the composite $h \circ i : S \to X$ is also *smooth* morphism. Consider the cartesian diagram $$\label{relative bl} \begin{CD} E @> i'>> Bl_{S}X' \\ @V q' VV @VV q V \\ S @>> i > X' @>>h > X \:, \end{CD}$$ with $q: Bl_{S}X' \to X'$ the blow-up of $X'$ along $S$ and $q':E \to S$ the exceptional divisor map. Then: 1. $h \circ q: Bl_{S}X' \to X$ and $h\circ q\circ i': E \to X$ are also *smooth* morphisms, with $ Bl_{S}X', E$ quasi-projective in case $X', S\in ob(\m V^{qp}_k)$. 2. This blow-up diagram *commutes with any base change* in $X$, i.e. the corresponding fiber-square induced by pullback along a morphism $\tilde{X}\to X$ is isomorphic to the corresponding blow-up diagram of $\tilde{S}\to \tilde{X}'$. 3. The closed embeddings $i,i'$ are *regular* embeddings, and the projection map $q$ as well as $i,i'$ are of *finite Tor-dimension*. Note that all results are (étale) local in $X'$. Since both morphisms $h: X' \to X$ and $S \to X' \to X$ are smooth, we can assume that $h$ is the projection $h=pr_2: X'= \bA^n \times X \to X$, with $i: S= \bA^m \times X \to \bA^n \times X $ induced from a standard inclusion $\bA^m \hookrightarrow \bA^n$ of affine spaces ($m\leq n$), and the blow-up diagram (\[relative bl\]) isomorphic to $$\begin{CD} E \times X @> i'>> Bl_{\bA^m}\bA^n \times X \\ @V q' VV @VV q V \\ \bA^m \times X @>> i > \bA^n \times X @>>h=pr_2 > X \:. \end{CD}$$ Here we use the fact that $$Bl_{ \bA^m \times X}(\bA^n \times X)\simeq Bl_{\bA^m}\bA^n \times X \:,$$ since blowing up commutes with flat base change for the flat projection map $h=pr_2: X'=\bA^n \times X \to X$. Then (1) and (3) are well known, whereas (2) follows again from the fact that blowing up commutes with flat base change for the flat projection maps $h=pr_2: X'=\bA^n \times X \to X$ and $\tilde{h}=pr_2: \tilde{X}'=\bA^n \times \tilde{X} \to \tilde{X}$. Now we are ready to define a bivariant analogue ${\mathbb {BL}(\m V/X \xrightarrow{f} Y)}$ of the subgroup $BL(\m V/X)$ and thus a bivariant analogue $\bK_0(\m V/X \xrightarrow{f} Y)$ of $K_0(\m V/X)$. For a morphism $f: X \to Y$ in the category $\m V=\m V^{(qp)}_k$ or $\m V=\m V^{an}_c$, we consider a blow-up diagram $$\begin{CD} E @> i'>> Bl_{S}X' \\ @VV q' V @VV q V \\ S @> i >> X' @> h >> X @> f >> Y\:, \end{CD}$$ with $h$ proper and $i$ a closed embedding such that $f \circ h$ as well as $f \circ h \circ i$ are smooth. Let $\mathbb {BL}(\m V/ X \xrightarrow{f} Y)$ be the free abelian subgroup of $\bM(\m V/X \xrightarrow{f} Y)$ generated by $$\label{eq:rbl} [Bl_{S}X' \xrightarrow{h q} X] - [E \xrightarrow{hiq'} X] - [X'\xrightarrow{h} X] + [S\xrightarrow{hi} X] \tag{rbl}$$ for any such diagram, and define $$\bK_0(\m V/X \xrightarrow{f} Y) := \frac{\bM(\m V/X \xrightarrow{f} Y)} {\mathbb {BL}(\m V/X \xrightarrow{f} Y)}.$$ The corresponding equivalence class of $[V \xrightarrow{p} X]$ shall be denoted by $\Bigl[[V \xrightarrow{p} X] \Bigr].$ Note that by Lemma \[key-lemma\] (1) $f \circ h \circ q$ and $f \circ h \circ i\circ q'$ are smooth (with $Bl_{S}X'$ and $E$ quasi-projective in the case $\m V=\m V^{qp}_k$), so that the “relative blow-up relation” (\[eq:rbl\]) makes sense in $\bM(\m V/X \xrightarrow{f} Y)$. \[theorem\] Let $\m V=\m V^{(qp)}_k$ be the category of (quasi-projective) algebraic varieties (i.e. reduced separated schemes of finite type) over a base field $k$ of any characteristic, or let $\m V=\m V^{an}_c$ be the category of compact reduced complex analytic spaces. $\bK_0(\m V/X \xrightarrow{f} Y)$ becomes a bivariant theory with the following three operations, so that the canonical projection $\bB q: \bM(\m V/-) \to \bK_0(\m V / - )$ is a Grothendieck transformation. [**Product operation**]{}: For morphisms $f: X \to Y$ and $g: Y \to Z$, the product operation $$\bigstar: \bK_0 (\m V/X \xrightarrow{f} Y) \otimes \bK_0 (\m V/ Y \xrightarrow{g} Z) \to \bK_0 ( \m V/X \xrightarrow{gf} Z)$$ is defined by $$\Bigl[[V \xrightarrow{h} X]\Bigr] \bigstar \Bigl[[W \xrightarrow{k} Y]\Bigr]:= \Bigl[[V \xrightarrow{h} X] \bullet [W \xrightarrow{k} Y]\Bigr]$$ and bilinearly extended. [**Pushforward operation**]{}: For morphisms $f: X \to Y$ and $g: Y \to Z$ with $f \in \m Prop$, the pushforward operation $$f_*: \bK_0 (\m V/ X \xrightarrow{gf} Z) \to \bK_0 (\m V/ Y \xrightarrow{g} Z)$$ is defined by $$f_*\left (\, \, \Bigl[[V \xrightarrow{p} X]\Bigr]\, \, \right ) := \Bigl[ f_*([V \xrightarrow{p} X])\Bigr]$$ and linearly extended. [**Pullback operation**]{}: For an independent square $$\CD X' @> g' >> X \\ @V f' VV @VV f V\\ Y' @>> g > Y, \endCD$$ the pullback operation $$g^* : \bK_0 (\m V/ X \xrightarrow{f} Y) \to \bK_0 (\m V/ X' \xrightarrow{f'} Y')$$ is defined by $$g^*\left (\, \,\Bigl[[V \xrightarrow{p} X]\Bigr] \, \, \right ) := \Bigl[g^*([V \xrightarrow{p} X])\Bigr]$$ and linearly extended. It suffices to show the well-definedness of these three operations.\ (i) $ \Bigl[V \xrightarrow {h} X]\Bigr]\bigstar \Bigl[W \xrightarrow {k} Y]\Bigr] := \Bigl[V \xrightarrow{h} X] \bullet [W \xrightarrow{k} Y]\Bigr]$ is well-defined: Let $$\alpha = [Bl_{S_1}X'\to X] - [E_1\to X] - [X'\to X] + [S_1\to X] \in \mathbb {BL}(\m V/ X \xrightarrow{f} Y)$$ and $$\beta = [Bl_{S_2}Y'\to Y] - [E_2\to Y] - [Y'\to Y] + [S_2\to Y] \in \mathbb {BL}(\m V/ Y \xrightarrow {g} Z)$$ be given. Then we have $$\begin{aligned} & \left ([V \xrightarrow {h} X] + \alpha \right ) \bullet \left ([W \xrightarrow {k} Y] + \beta \right) \\ & = [V \xrightarrow {h} X] \bullet [W \xrightarrow {k} Y] + [V \xrightarrow {h} X] \bullet \beta + \alp \bullet \left ([W \xrightarrow {k} Y] + \beta \right),\end{aligned}$$ and we show that $$[V \xrightarrow{h} X] \bullet \beta + \alp \bullet \left ([W \xrightarrow{k} Y] + \beta \right) \in \mathbb {BL}(\m V/ X \xrightarrow{g \circ f} Z).$$ For this end it suffices to show that $$[V \xrightarrow{h} X] \bullet \beta \in \mathbb {BL}(\m V/ X \xrightarrow{g \circ f} Z)$$ and $$\alp \bullet [H \xrightarrow{j} Y] \in \mathbb {BL}(\m V/ X \xrightarrow{g \circ f} Z)$$ for any $[H \xrightarrow{j} Y] \in \bM(\m V/ Y \xrightarrow{g} Z)$. For the proof of $ \alp \bullet [H \xrightarrow{j} Y] \in \mathbb {BL}(\m V/ X \xrightarrow{g \circ f} Z)$, consider the following diagram: $$\xymatrix{ & \widetilde {E_1} \ar[ld] \ar'[d][dd]_{\widetilde{q'}} \ar [rr]^{\widetilde {i'}} && Bl_{\widetilde{S_1}}\widetilde{X'} \ar[ld] \ar[dd]^{\widetilde q} \\ E_1 \ar[dd]_{q'} \ar[rr]^(.65){i'} && Bl_{S_1}X' \ar[dd]^(.65){q} \\ & \widetilde {S_1} \ar[ld] \ar'[r][rr]^{\widetilde i} && \widetilde {X'} \ar[ld] \ar[r]^{\widetilde {h}} & \,\,\widetilde {X} \ar[ld]^{k} \ar[r] & \, \, H \ar[ld]^{j}\\ S_1 \ar[rr]_{i} && X' \ar[r]_{h} & X \ar[r]_{f} & Y \ar[r]_{g} & Z,}$$ which by Lemma \[key-lemma\] (2) is the pullback by the proper morphism $j: H \to Y$ of the following blow-up diagram: $$\label{bd} \begin{CD} E_1 @> i'>> Bl_{S_1}X' \\ @V q' VV @VV q V \\ S_1 @> i >> X' @> h >> X @> f >> Y. \end{CD}$$ Then we have that $$\begin{aligned} & \alp \bullet [H \xrightarrow{j} Y] \\ & = [Bl_{\widetilde{S_1}}\widetilde{X'} \xrightarrow{k\widetilde{h}\widetilde{q}} X] - [\widetilde {E_1} \xrightarrow{k\widetilde{h} \widetilde{q} \widetilde {i'}} X] - [\widetilde {X'} \xrightarrow {k\widetilde{h}} X] + [\widetilde {S_1} \xrightarrow{k\widetilde{h}\widetilde {i}} X], \end{aligned}$$ which is in $\bM(\m V/ X \xrightarrow{g \circ f} Z).$ In the same way one gets $$[V \xrightarrow{h} X] \bullet \beta \in \mathbb {BL}(\m V/ X \xrightarrow{g \circ f} Z).$$ Here we are using the fact that the pullback of the corresponding blow-up diagram for $\beta$ under the morphism $fh$ is again a similar blow-up diagram, since $fh$ is smooth and therefore flat. \(ii) The well-definedness of $f_*\left[[V \xrightarrow{p} X]\right ] := \left[[V \xrightarrow{f \circ p} Y] \right]$ is obvious. \(iii) $g^*\left[[V \xrightarrow{p} X]\right ] := \left [g^*[V \xrightarrow{p} X] \right ]$ is well-defined. The proof based on Lemma \[key-lemma\] (2) is similar to that of (i) above, so omitted. Note that the proof of the well-definedness of the product- and pullback operations above used Lemma \[key-lemma\] (2), as well as the fact that the smooth and therefore flat pullback of a blow-up diagram is again a blow-up diagram. Here we note (cf. [@EH]) that in general the pullback of a blow-up is *not* the blow-up of the pullback, i.e., consider the following pullback diagram, which is obtained by pulling back a blow-up diagram by the morphism $\widetilde X \to X$: $$\xymatrix{ & \widetilde {E} \ar[ld] \ar'[d][dd]_{\widetilde{q'}} \ar [rr]^{\widetilde {i'}} &&\widetilde {Bl_{S}X} \ar[ld] \ar[dd]^{\widetilde q} \\ E \ar[dd]_{q'} \ar[rr]^(.65){i'} && Bl_{S}X \ar[dd]^(.65){q} \\ & \widetilde {S} \ar[ld] \ar'[r][rr]^{\widetilde i} && \widetilde {X} \ar[ld] \\ S \ar[rr]_{i} && X}$$ Then the diagram $$\begin{CD} \widetilde {E} @> \widetilde {i'}>> \widetilde {Bl_{S}X} \\ @VV \widetilde {q'} V @VV \widetilde {q} V \\ \widetilde {S} @> \widetilde {i} >> \widetilde {X} \end{CD}$$ is in general *not* a blow-up diagram, i.e., $\widetilde {Bl_{S}X}$ is not the blow-up of $\widetilde {X}$ along $\widetilde {S}$. A typical example is the situation that $S$ is a point of the $2$-dimensional projective space $X = \bP^2$, $\widetilde X$ is a smooth curve going through the point $S$ and $h:\widetilde X \to X$ is the inclusion map. Let us finish this section with the following In the case when $Y$ is a point, the blow-up diagram defining ${\mathbb {BL}(\m V/X \xrightarrow{f} pt)}$ is nothing but the following: $$\begin{CD} E @> i'>> Bl_{S}X' \\ @VV q' V @VV q V \\ S @> i >> X' @> h >> X \:, \end{CD}$$ such that $h: X' \to X$ is proper, $X'$ and $S$ are nonsingular, and $q: Bl_{S}X' \to X'$ is the blow-up of $X'$ along $S$ with $q':E \to S$ the exceptional divisor map. Hence ${\mathbb {BL}(\m V/X \xrightarrow{f} pt)}$ is nothing but $BL(\m V/X)$, i.e., we have by Bittner’s theorem $$\bK_0(\m V /X \to pt) \simeq K_0(\m V/X)$$ in the compact complex analytic context, as well as in the algebraic context over a base field of characteristic zero. Finally note that we always have a group homomorphism $$\bK_0(\m V /X \to pt) \to K_0(\m V/X)\:,$$ since $Bl_{S}X'\backslash E\simeq X'\backslash S$ in the diagram above so that $$[Bl_{S}X'\to X] - [E\to X]= [X'\to X] - [S\to X] \in K_0(\m V/X)\:.$$ Motivic bivariant Chern and Hirzebruch class transformations ============================================================ Now we are ready to prove the following main theorem, which is about the *motivic bivariant Chern and Hirzebruch class transformations*. \[thm:main2\] Let $\m V=\m V^{qp}_k$ be the category of quasi-projective algebraic varieties over a base field $k$ of any characteristic. 1. There exists a unique Grothendieck transformation $$mC_y=\La_y^{mot}: \bK_0(\m V_k^{qp}/ - ) \to \bK_{alg}( - )\otimes \bZ[y]$$ satisfying the normalization condition that for a smooth morphism $f: X \to Y$ the following equality holds in $\bK_{alg}(X \xrightarrow {f} Y) \otimes \bZ[y]$: $$\La_y^{mot}\Bigl(\Bigl[[X \xrightarrow{\op {id}_X} X]\Bigr]\Bigr) = \La_y(T^*_f) \bullet \m O_{f}.$$ 2. Let $T_y : \bK_0(\m V_k^{qp} / - ) \to \bH(-) \otimes \bQ[y]$ be defined as the composition $\tau \circ \La_y^{mot}$, renormalized by $\cdot (1+y)^i$ on $\bH^i(-)\otimes \bQ[y]$. Here $\bH$ is either the operational bivariant Chow group, or the even degree bivariant homology theory for $k=\bC$, with $\tau$ the corresponding Riemann-Roch transformation.\ Then $T_y$ is the unique Grothendieck transformation satisfying the normalization condition that for a smooth morphism $f: X \to Y$ the following equality holds in $\bH(X \xrightarrow {f} Y) \otimes \bQ[y]$: $$T_y\Bigl(\Bigl[[X \xrightarrow{\op {id}_X} X]\Bigr]\Bigr) = T^*_y(T_f) \bullet [f].$$ Uniqueness follows from $$\Bigl[[V \xrightarrow{h} X]\Bigr]= h_*\Bigl(\Bigl[[V \xrightarrow {\op {id}_V} V]\Bigr]\Bigr)\in \bK(\m V/X \xrightarrow{f} Y)$$ for $h: V\to X$ a proper morphism with $f \circ h$ smooth. So we simply define in this case $$\ga_{c\ell}\Bigl(\Bigr[[V \xrightarrow{h} X]\Bigr]\Bigr) := h_* (c\ell (T_{fh}) \bullet \theta(fh)).$$ Here the Grothendieck transformation $\ga_{c\ell}$ is the following: - The motivic bivariant Chern class transformation in (i) $$mC_y=\La_y^{mot}: \bK_0(\m V_k^{qp}/ - ) \to \bK_{alg}( - )\otimes \bZ[y]$$ corresponds to the multiplicative characteristic class $$c\ell(W):= \La_y(W^*)\in K^0(-)[y] \subset K^0(-)[y,(1+y)^{-1}]$$ given by the total $\lambda$-class of the dual vector bundle, with $\theta(fh)=\m O_{fh}=[\m O_{V}]$. - The bivariant Hirzebruch class transformation in (ii) $$T_y : \bK_0(\m V_k^{qp} / - ) \to \bH(-) \otimes \bQ[y]$$ corresponds to the multiplicative characteristic class $$c\ell(W):= T^*_y(W)\in \bH^*(-)\otimes \bQ[y]$$ given by the Hirzebruch class, with $\theta(fh)=[fh]$ the relative fundamental class. Moreover, these characteristic classes commute with the corresponding orientations $\theta$ of a smooth morphism (as already explained before). So we only have to show that - the corresponding Grothendieck transformation $$\ga_{c\ell}=:\La_y^{mot}: \bM(\m V/X \xrightarrow{f} Y) \to \bK(X \xrightarrow{f} Y)$$ from Corollary \[twisting\] vanishes on the subgroup $\mathbb {BL}(\m V/ X \xrightarrow{f} Y)$, and - the relation $\ga_{T^*_y}=\tau\circ \La_y^{mot}$ up to the renormalization by the multiplication with $(1+y)^i$ on $\bH^i(-)\otimes \bQ[y]$. \(i) : Let us identify the vector bundle $T^*_{fh}$ for the smooth morphism $fh: V\to Y$ with the corresponding locally free sheaf $\Omega^1_{fh}$ of sections given by the relative one-forms, so that $$\Lambda_y^{mot}([V \xrightarrow{h} X]) := \sum_{p \geq 0} h_*([\Omega^p_{fh}] \bullet \m O_{fh}) \cdot y^p\:.$$ Note that by the definition of relative perfectness, $D^b_{id-perf}(V)=D^b_{fh-perf}(V)$ for the smooth morphism $fh$, so that $$\bullet \m O_{fh}: \bK(V \xrightarrow{id_V} V)=K_0(D^b_{id-perf}(V)) \xrightarrow{\sim} K_0(D^b_{fh-perf}(V))=\bK(V \xrightarrow{fh} X)\:,$$ with $h_*(\;-\; \bullet \m O_{fh})$ induced by the total direct image $$Rh_*: D^b_{id-perf}(V)= D^b_{fh-perf}(V)\to D^b_{f-perf}(X)\:.$$ Consider now a blow-up diagram $$\begin{CD} E @> i'>> Bl_{S}X' \\ @V q' VV @VV q V \\ S @>> i > X' @>>h > X @>> f > Y\:, \end{CD}$$ with $h$ proper and $i$ a closed embedding such that $fh$ and $fhi$ are smooth. Then we have by [@Gros Chapter IV, Theorem 1.2.1 and (1.2.6) on p.74] that the following natural morphisms are quasi-isomorphisms for all $p\geq 0$ (and note that Gros is working in [@Gros Chapter IV, §1.2] with the corresponding *relative De Rham complexes*): 1. $\Omega^p_{fh} \xrightarrow{\sim} R^0q_*\Omega^p_{fhq}$. 2. $R^kq_*\Omega^p_{fhq} \xrightarrow{\sim} i_*R^kq'_*\Omega^p_{fhiq'}$ for all $k\geq 1$. 3. $\Omega^p_{fhi} \xrightarrow{\sim} R^0q'_*\Omega^p_{fhiq'}$. Here (c) can be checked (étale) locally, so that it follows from [@Gros (1.2.6) on p.74 and (4.2.12) on p.23]. Moreover all coherent sheaves $\Omega^p_{fh}, \Omega^p_{fhi}$ and $\Omega^p_{fhiq'}$ for $p\geq 0$ are locally free, since the corresponding morphisms are smooth. Similarly all direct image sheaves $R^kq'_*\Omega^p_{fhiq'}$ for $k,p\geq 0$ are locally free, since $q': E\to S$ is a projective bundle (e.g. compare [@Gros (1.2.6) on p.74 and (4.2.12) on p.23]). Finally the morphisms $i$ and $q$ are of finite Tor-dimension by Lemma \[key-lemma\] (3), with $i$ exact, so that (a) and (b) resp.(c) can be considered as quasi-isomorphisms in $ D^b_{fh-perf}(X')$ resp. $ D^b_{fhi-perf}(S)$. So one gets for all $p\geq 0$ the following equalities in $\bK_{alg}(X' \xrightarrow{fh} Y)$: $$\begin{aligned} q_* [\Omega^p_{fhq}] - i_*q'_* [\Omega^p_{fhiq'}] &= \sum_{k\geq 0}\: (-1)^k\Bigl( [R^kq_* \Omega^p_{fhq}] - [i_*R^kq'_* \Omega^p_{fhiq'}]\Bigr)\\ &=[R^0q_* \Omega^p_{fhq}] -[i_*R^0q'_* \Omega^p_{fhiq'}] \\ &=[\Omega^p_{fh}] -i_* [\Omega^p_{fhi}] \:.\end{aligned}$$ And this implies the needed vanishing result: $$\begin{aligned} & \Lambda_y^{mot} \left ([Bl_{S}X' \xrightarrow{hq} X] - [E \xrightarrow{hiq'} X] - [X' \xrightarrow{h} X] + [S \xrightarrow{hi} X] \right ) \\ & = \sum_{p \geq 0} \Bigl( \;h_*q_*([\Omega^p_{fhq}])y^p - h_*i_*q'_*([\Omega^p_{fhiq'}])y^p - h_*([\Omega^p_{fh}])y^p + h_*i_*([\Omega^p_{fhi}])y^p \;\Bigr) \\ &= \sum_{p \geq 0} h_* \left ( q_*[\Omega^p_{fhq}] - i_*q'_* [\Omega^p_{fhiq'}] - [\Omega^p_{fh}] + i_*[\Omega^p_{fhi}] \right) y^p = 0 \:.\end{aligned}$$ \(ii) : By composition with the bivariant Riemann–Roch transformation $\tau: \bK_{alg}(X \xrightarrow{f} Y) \to \bH(X \xrightarrow{f} Y)$, and extending linearly with respect to the coefficients $\bZ[y]$, we get a Grothendieck transformation $$\tau\circ \Lambda_y^{mot}: \bK_0(\m V_k^{qp} / - ) \to \bH(-) \otimes \bQ[y]\:.$$ Similarly, the renormalization $\Psi_{(1+y)}: \bH(-) \otimes \bQ[y]\to \bH(-) \otimes \bQ[y,(1+y)^{-1}]$ given by $$\cdot (1+y)^i: \bH^i(-) \otimes \bQ[y]\to \bH^i(-) \otimes \bQ[y,(1+y)^{-1}]$$ is a Grothendieck transformation, since $\bH(-)$ is a graded bivariant theory. Now we show that our looking-for transformation $T_y=\ga_{T_y^*}$ can be defined as $$T_y := \Psi_{(1+y)}\circ \tau \circ \Lambda_y^{mot} :\bK_0(\m V / -) \to \bH(-) \otimes \bQ[y] \subset \bH(-) \otimes \bQ[y,(1+y)^{-1}]\:.$$ It suffices to check that for a smooth morphism $f:X \to Y$ $$T_y ([X \xrightarrow{id} X]) = T^*_y(T_f) \bullet [f] \in \bH(X \xrightarrow{f} Y) \otimes \bQ[y] \:.$$ And this can be seen as follows: $$\begin{aligned} \tau \circ \Lambda_y^{mot} ([X \xrightarrow {\op {id}} X]) &= \tau(\lambda_y(T^*_f)\bullet \m O_{f} )\\ & =ch(\lambda_y(T^*_f)) \bullet \tau(\Cal O_f)\\ & = ch(\lambda_y(T^*_f)) \bullet td(T_f) \bullet [f]\end{aligned}$$ by the *Riemann–Roch formula* $$\tau(\Cal O_f)= td(T_f) \bullet [f]\:.$$ Compare with [@Fulton-MacPherson (\*) on p.124] for $\bH$ the bivariant homology in case $k=\bC$. For $\bH=CH$ the bivariant Chow group and $k$ of any characteristic, this follows from [@Fulton-book Theorem 18.2], as we explain later on in Remark \[strong-todd\]. So we get $$\tau \circ \Lambda_y^{mot} ([X \xrightarrow {\op {id}} X])= \left( \prod_{j=1}^{\op {rank} T_f} (1+ye^{-\alpha _j}) \prod_{j=1}^{\op {rank} T_f} \frac {\alpha_j}{1 - e^{-\alpha_j}} \right) \bullet [f] \:,$$ with $\alpha_j$ the Chern roots of $T_f$. Here it should be noted that $[f] \in \bH^{-\op {rank} T_f}(X \xrightarrow{f} Y)$ by [@Fulton-MacPherson Part II, §1.3]) resp. [@Fulton-book (1) on p.326]. Moreover, the substitution $\alpha_j\mapsto \alpha_j(1+y)$ corresponds to the renormalization $$\Psi_{(1+y)}: \bH^*(-)\otimes \bQ[y]\to \bH^*(-)\otimes \bQ[y,(1+y)^{-1}]\:,$$ since $\alpha_j\in \bH^1(-)$. So we get $$\begin{aligned} & T_y ([X \xrightarrow {\op {id}} X]) = \Psi_{(1+y)}\circ \tau \circ \Lambda_y^{mot} ([X \xrightarrow {\op {id}} X])\\ & = \left (\prod_{j=1}^{\op {rank} T_f} \left(1+ye^{-\alpha _j(1+y)} \right) \frac {\alpha_j (1+y)}{1 - e^{-\alpha_j(1+y)}} \right) \bullet [f]\cdot(1+y)^{-\op {rank} T_f}\\ & = \left (\prod_{j=1}^{\op {rank} T_f} \left(1+ye^{-\alpha _j (1+y)}\right) \frac {\alpha_j }{1 - e^{-\alpha_j(1+y)}} \right) \bullet [f]\\ & = \left (\prod_{j=1}^{\op {rank} T_f} \left(\frac{\alpha_j(1 +y)}{1 - e^{-\alpha_j(1 +y)}} - \alpha_jy \right)\right) \bullet [f]\\ & = T^*_y(T_f) \bullet [f] \in \bH(X \xrightarrow{f} Y) \otimes \bQ[y]\:.\end{aligned}$$ \(1) Our construction of the Grothendieck transformation $mC_y=\La_y^{mot}: \bK_0(\m V_k^{qp}/ - )$ $\to \bK_{alg}( - )\otimes \bZ[y]$ based on [@Gros Chapter IV, Theorem 1.2.1 and (1.2.6)], i.e. on the properties (a),(b) and (c) in the proof above, also works in the algebraic context without considering only quasi-projective varieties, if one uses the more sophisticated definition of $\bK_{alg}(X \xrightarrow {f} Y)=K_0(D^b_{f-perf}(X))$ as the Grothendieck goup of the triangulated category of $f$-perfect complexes. And a similar definition can also be used in the context of compact complex analytic varieties (compare [@Fulton-MacPherson Part I, §10.10] and [@Levy2]). Then it seems reasonable that one can also construct in a similar way in this compact complex analytic context the Grothendieck transformation $mC_y=\La_y^{mot}$. Here it would be enough to prove the analogues of the properties (a), (b) and (c) in the complex analytic context.\ (2) Similarly one would like to further construct in this compact complex analytic context also the Grothendieck transformation $T_y$ based on Levy’s K-theoretical Riemann-Roch transformation $$\alpha : \bK_{alg}(-)\to \bK^{top}_0(-)$$ from algebraic to topological bivariant K-theory (see [@Levy2]), together with the topological bivariant Riemann-Roch transformation $$\bK^{top}_0(-)\to \bH(-)\otimes \bQ$$ from [@Fulton-MacPherson Part I, Example 3.2.2]. A key result missing so far is the counterpart $\alpha(\m O_f)=\theta(f)$ of [@Fulton-MacPherson Part II, Theorem 1.4 (3)], that $\alpha$ identifies for a smooth morphism $f: X\to Y$ the orientation $\m O_f:=[\m O_X]\in \bK_{alg}(X \xrightarrow {f} Y)$ with the orientation $\theta(f)\in \bK^{top}_0(X \xrightarrow {f} Y).$ Comparing the different normalization conditions for a smooth morphism $f:X\to Y$, from Theorem \[thm:main2\] one gets the following corollary: \[cor-Grothendieck2\] Let $\m V=\m V^{qp}_k$ be the category of quasi-projective algebraic varieties over a base field $k$ of any characteristic. Then we have the following commutative diagrams of Grothendieck transformations: 1. $$\xymatrix{ & \bK_0(\m V_k^{qp} / - ) \ar [dl]_{mC_0} \ar [dr]^{{T_{0}}} \\ {\bK_{alg}( - ) } \ar [rr] _{\tau}& & \bH(-) \otimes \bQ.}$$ 2. $$\xymatrix{ & \bK_0(\m V_k^{qp} / - ) \ar [dl]_{\epsilon} \ar [dr]^{{T_{-1}}} \\ {\tilde{\bF}( - ) } \ar [rr] _{\gamma}& & CH(-) \otimes \bQ,}$$ if $k$ is of characteristic zero. Here $\epsilon$ is the unique Grothendieck transformation satisfying the normalization condition $\epsilon\Bigl(\Bigl[[X \xrightarrow{\op {id}_X} X]\Bigr]\Bigr)=\jeden_f$ for a smooth morphism $f: X \to Y$. And similarly for the bivariant Chern class transformation $\gamma: \bF( - ) \to A^{PI}( - )\otimes \bQ \supset CH(-) \otimes \bQ$ in case $k=\bC$. 3. Assume $k$ is of characteristic zero. Then the associated covariant transformations in Theorem \[thm:main2\] (i) and (ii) agree under the identification $$\bK_0(\m V_k^{qp}/X\to pt) \simeq K_0(\m V_k^{qp}/X)$$ with the motivic Chern and Hirzebruch class transformations $mC_y$ and ${T_y}_*$. Everything follows from the different normalization conditions for a smooth morphism $f:X\to Y$, except (ii). First we explain the existence of the Grothendieck transformation $$\epsilon: \bK_0(\m V_k^{qp} / - )\to \tilde{\bF}( - )$$ to Ernström–Yokura’s bivariant theory of constructible functions, resp. in case $k=\bC$ to Fulton–MacPherson’s bivariant theory $\bF( - )$ of constructible functions satisfying the local Euler condition. \(a) Let us first consider the last case. Since $f: X \to Y$ is a smooth morphism, it satisfies trivially the local Euler condition so that $ \jeden_f:=\jeden_X \in \bF(X \xrightarrow{f } Y)$. Moreover, $\theta(f):=\jeden_f$ is a stable orientation for the smooth morphism $f$, which commutes with the trivial multiplicative characteristic class $c\ell(V):=1_X \in \bF(X \xrightarrow{id_X} X)$ of a vector bundle $V$ on $X$. So by Theorem \[univ\], we get a unique Grothendieck transformation $$\epsilon: \bM(\m V/-) \to \bF(-)$$ satisfying for the smooth morphism $f: X \to Y$ the normalization condition $$\epsilon([X \xrightarrow {\op {id}_X} X]) = \jeden_f \:.$$ Finally, $\epsilon$ vanishes on the subgroup $\mathbb {BL}(\m V/ X \xrightarrow{f} Y)$: Consider a blow-up diagram $$\begin{CD} E @> i'>> Bl_{S}X' \\ @V q' VV @VV q V \\ S @>> i > X' @>>h > X @>> f > Y\:, \end{CD}$$ with $h$ proper and $i$ a closed embedding such that $fh$ and $fhi$ are smooth. Then $q: U':=Bl_{S}X'\backslash E \xrightarrow{\sim} X'\backslash S=:U$ so that $$(fhq)_*\jeden_{fhq}-(fhiq')_*\jeden_{fhiq'}=(fhq)_*1_{U'}=(fh)_*1_{U} =(fh)_*\jeden_{fh}-(fhi)_*\jeden_{fhi}\:.$$ (b) The same argument works for Ernström–Yokura’s bivariant theory $\tilde{\bF}( - )$, once we know $ \jeden_f:=\jeden_X \in \tilde{\bF}(X \xrightarrow{f } Y)$ for a smooth morphism $f: X\to Y$. Consider a fiber square $$\begin{CD} X''@> h'>> X' @> g'>> X \\ @V f'' VV @V f' VV @VV f V \\ Y'' @>> h > Y' @>> g > Y \:, \end{CD}$$ with $h$ and therefore also $h'$ flat. Then the following diagram is commutative by the *Verdier Riemann-Roch theorem* for the smooth morphism $f'$ (see [@Yokura-VRR], as well as [@Fulton-MacPherson §10.4, p.111] and the proof of [@BSY1 Corollary 2.1 (4)]): $$\label{VRR-c} \begin{CD} F(Y') @> (g^*\jeden_f)\bullet = f'^* >> F(X') \\ @V c_* VV @VV c_* V \\ CH_*(Y') @>> c(T_{f'})\cap f'^* > CH_*(X') \:. \end{CD}$$ So $\alpha:=\jeden_f$ satisfies the condition ($\tilde{\bF}-1$) of [@EY1] with $c_g(\jeden_f)=c(T_{f'})\cap f'^*$. But it also satisfies the condition ($\tilde{\bF}-2$) of [@EY1], since $c(T_{f'})\cap$ commutes with flat pullback (by [@Fulton-book Theorem 3.2(d)]) so that $$h'^*\circ c_g(\jeden_f)=h'^*(c(T_{f'})\cap f'^*)=c(T_{f''})\cap (f''^*\circ h^*) =c_{g\circ h}(\jeden_f)\circ h^*\:.$$ And this implies $ \jeden_f \in \tilde{\bF}(X \xrightarrow{f } Y)$, together with commutativity of the diagram in (ii) by the following “strong normalization condition” for the smooth morphism $f: X\to Y$, which by the definition of the right hand side given in [@Fulton-book p.325–326] is equivalent to $c_g(\jeden_f)=c(T_{f'})\cap f'^*$ for all base changes $g$: $$\label{nor-gamma} \gamma( \jeden_f )= c(T_{f})\bullet [f] \in CH(X \xrightarrow{f } Y)\:.$$ \[strong-todd\] In the same way as above one can get the *Riemann–Roch formula* $$\tau(\Cal O_f)= td(T_f) \bullet [f]$$ for a smooth (or local complete intersection) morphism $f: X\to Y$ and the bivariant Riemann-Roch transformation $\tau: \bK_{alg}(-)\to CH(-)\otimes \bQ$ from [@Fulton-book Example 18.3.16]. By the definition of $\tau$, the associated covariant transformation $\tau_*$ agrees with the Todd class transformation $$\tau_*=td_*: G_0(X')\to CH^{-*}(X'\to pt)\otimes \bQ\simeq CH_{*}(X')\otimes \bQ \:,$$ with the last isomorphism given by [@Fulton-book Proposition 17.3.1]. Since $\tau$ commutes with the bivariant product $\bullet$, one gets for a base change $g$ as above by the *Verdier Riemann-Roch theorem* a commutative diagram $$\label{VRR-td} \begin{CD} G_0(Y') @> (g^*\Cal O_{f'} )\bullet = f'^* >> G_0(X') \\ @V td_* VV @VV td_* V \\ CH_*(Y')\otimes \bQ @>> td(T_{f'})\cap f'^* > CH_*(X')\otimes \bQ \:. \end{CD}$$ But $(td_*)\otimes \bQ$ is surjective (in fact even an isomorphism) by [@Fulton-book Corollary 18.3.2], which implies $\tau_g(\Cal O_{f'})=td(T_{f'})\cap f'^*$ for any such base change $g$. And this is equivalent to the Riemann-Roch formula $\tau(\Cal O_f)= td(T_f) \bullet [f]$ by the definition of the right hand side given in [@Fulton-book p.325–326]. 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--- abstract: 'In this era of “big” data, not only the large amount of data keeps motivating distributed computing, but concerns on data privacy also put forward the emphasis on distributed learning. To conduct feature selection and to control the false discovery rate in a distributed pattern with multi-[*machines*]{} or multi-[*institutions*]{}, an efficient aggregation method is necessary. In this paper, we propose an adaptive aggregation method called ADAGES which can be flexibly applied to any machine-wise feature selection method. We will show that our method is capable of controlling the overall FDR with a theoretical foundation while maintaining power as good as the Union aggregation rule in practice.' author: - 'Yu Gui[^1]' bibliography: - 'ygbib.bib' title: 'ADAGES: adaptive aggregation with stability for distributed feature selection' --- =1 Introduction ============ In recent decades, the idea of distributed learning and data decentralization has been frequently discussed. On one hand, the notion of distributed learning is motivated by the advanced techniques of data collection and storage which leads to a large amount of accessible data. Distributed storage and parallel computing are put forward to address the concerns, which further requires statistical learning methods in this distributed scenario. On the other hand, statisticians focus on distributed learning since privacy protection is of main interest nowadays. A representative example is the collaborative clinical research among different hospitals on certain diseases, where hospitals will not share patients’ data for privacy protection. Therefore, statisticians have to deal with certain “encoded’’ statistics collected from distributed institutions. Many recent works focusing on different statistical perspectives have contributed to this field. Estimation is the most fundamental topic in statistics, some works adopt the divide and conquer algorithm for distributed estimation and also study the accuracy of estimation under various contexts, among which are [@battey2015distributed], [@JMLR:v16:zhang15d], [@zhao2014general] and [@cai2020distributed]. Distributed hypothesis testing is discussed in works such as [@ramdas], [@sreekumar2018distributed], [@gilani2019distributed] and is also covered in [@battey2015distributed] and [@zhao2014general]. Specifically, [@su2015communicationefficient], [@Emery2019ControllingTF] and [@nguyen2020aggregation] have studied the aggregated feature selection based on multiple knockoffs. Originated from applications, communication constraints and privacy constraints ought to be taken into consideration, [@zhangandberger], [@10.1145/2897518.2897582], [@cai2020distributed] study the tradeoff between communication constraints and estimation accuracy. In addition, many other works contribute to distributed learning theories such as [@garg2014communication], [@dobriban2018distributed], [@doi:10.1080/01621459.2018.1429274] and [@kipnis2019mean]. [1.5]{} **Controlled feature selection.** In addition to feature selection methods such as regularized regression (e.g. [@10.2307/2346178],[@doi:10.1198/016214501753382273]), controlled feature selection aims to select important features and reduce false selections under some criteria. In this paper, we focus on a fundamental criterion in feature selection: false discovery rate ($\FDR$). The notion of $\FDR$ is introduced in [@benjamini1995controlling]. With the definition of the subset $\cS \subset \{1,\dots,d\}$ of relevant features, feature selection is equivalent to recovering $\cS$ based on observations. When the estimated set $\hat{\cS}$ is produced, the false discoveries can be denoted as $\hat{\cS} \cap \cS^{c}$ and [*false discovery proportion*]{} ($\FDP$) is defined in the form $$\begin{aligned} {\rm FDP} = \frac{|\hat{\cS} \cap \cS^{c}|}{|\hat{\cS}|}.\end{aligned}$$ The expectation of $\FDP$ is called the [*false discovery rate*]{} ($\FDR$), i.e. $ {\rm FDR} = \mathbb{E}\left[{\rm FDP}\right]$. In addition, power of feature selection illustrates the ability to recover true features and thus is defined as $$\begin{aligned} {\rm Power} = \EE[\frac{|\hat{\cS} \cap \cS|}{|\cS|}],\end{aligned}$$ which is the expected number of true discoveries over the total number of true features $|\cS |$. A series of $\FDR$-based methods originate from the invention of $\FDR$ in [@benjamini1995controlling] which utilizes the rank of z-scores for selecting important features. Based on this, [@benjamini2001control] relaxes the independence assumption as an extension. Knockoff filter is introduced in [@barber2015controlling] with exact control of $\FDR$ and can be extended in a model-free way in [@Cands2016PanningFG]. Recently, methods based on mirror statistics are put forward under this topic: [@xing2019gm] creates Gaussian mirror variables for all features that get rid of the conditional correlation within each mirrored pair; [@dai2020false] utilizes the data splitting and multiple splitting techniques to ensure the recovery of feature importance with stability. {width="\textwidth"} [1.5]{} **Stability selection.** As an improvement to general feature selection methods, the notion of stability selection is introduced by [@Meinshausen2008StabilityS] which conducts subsampling of size $\left[n/2\right]$ and identifies the most frequently selected features. The idea is close to a “voting process” where each sub-sample votes for each feature once and it is in line with our belief that important features will stably become outstanding with more votes. The spirit of stability selection later motivates works such as [@Shah2011VariableSW], [@Hofner2015ControllingFD] and also stimulates our idea of adaptive aggregation in distributed feature selection. [1.5]{} **Our contribution.** With the belief in the future of data decentralization, in this paper, we consider the topic of distributed feature selection with a controlled error rate. We present a general aggregation method for distributed feature selection called ADAGES (**AD**aptive **AG**gr**E**gation with **S**tability) that can apply for any controlled feature selection method. Without looking into the original datasets, we operate on Boolean variables in $\left\{0,1\right\}^{d}$ that is equivalent to subset of features of dimension $d$. Therefore, there is no complex communication or privacy concern in this context. Unlike [@su2015communicationefficient], [@Emery2019ControllingTF] and [@nguyen2020aggregation] that transfer knockoff statistics for aggregation, ADAGES does not depend on any specific feature selection method and is thus more flexible in application. Besides, in this paper, we assume the feature selection procedures of all the machines are independent of each other, i.e. as random Boolean vectors, $\hat{\cS}_i \perp \hat{\cS}_j$ for all $i,j \in [k]$. It is noticeable that in practice, the dependence exists due to the overlap of samples for different machines, e.g. the common patients for different hospitals. The generalized case to study the dependence is a promising topic for future work. [1.5]{} **Outline.** We begin with the problem formulation in section \[back\] and then in section \[method\], we introduce the detail of ADAGES as an adaptive improvement on empirical rules. In section \[result\], the main theorem will be established to guarantee the exact control of overall , theoretical proofs of which are in section \[proof\]. The results of numerical experiments are shown in section \[simu\]. [1.5]{} **Notations.** Suppose the dimension of the $n$ observed features is $d$, i.e. $\bx \in \RR^{d}$. Define $\cS \subset \left\{1,\dots,d\right\}$ is the subset of true features of interest. There are $k$ different machines or institutions contributing to the problem and we denote them as $M_1, \dots, M_k$. For each $i \in \left\{1,\dots,k\right\}$, machine $M_i$ produces an estimated subset $\hat{\cS}_i$ before aggregation and our goal is to obtain $\hat{\cS}$ based on $\left\{\hat{\cS}_1,\dots,\hat{\cS}_k\right\}$. Notation $\hat{\cS}_{(c)}$ refers to the subset produced by the aggregation method with threshold $c$, which will be introduced in section \[method\]. Also, $\hat{\cS}_{I}$ and $\hat{\cS}_{U}$ are aggregated subsets of the Intersection rule and the Union rule respectively. Background {#back} ========== In the context of distributed learning, imagine there is a central machine (the yellow one in Figure \[fig:plot0\]) and $k$ machines $\left\{M_i:i=1,\dots,k\right\}$ which can be $k$ hospitals or servers. In the current task, the dataset of interest is distributed over all $k$ machines due to concerns of privacy or distance and assume the $i$th machine deals with a sub-dataset $D_i$ with $n_i$ observations. All the machines share the same set of features in the same task, i.e. $\left\{X_j:j=1,\dots,d\right\}$ and they focus on $\FDR$ control with the universal pre-defined level of $q \in \left(0,1\right)$. Suppose the selection result for the $i$th machine is $\hat{\cS}_i$. We should note that the feature selection method adopted for each machine can be arbitrary and the only requirement is that the method should be capable of exact $\FDR$ control. With our adaptive aggregation with stability, we produce the final selection result $\hat{\cS}$ based on controlled selections $\left\{\hat{\cS}_i:i=1,\dots,k\right\}$. For each machine $M_i,i=1,\dots,k$, we define ${\rm FDR}_i = \EE\left[\frac{|\hat{\cS}_i \cap \cS^{c}|}{|\hat{\cS}_i|}\right]$. Empirical aggregation methods for distributed feature selection --------------------------------------------------------------- First, three empirical aggregation methods are introduced and we will later cover them as special cases in a generalized family. Define $z^{\left(i\right)}_j = \bfm{1}_{\left\{j \in \hat{\cS}_i\right\}}$ for each feature, then $\hat{\cS}_i$ is equivalent to an indicator vector $\bz^{\left(i\right)} = \left(z^{\left(i\right)}_1,\dots,z^{\left(i\right)}_d\right)^\top$ and aggregation algorithms can be viewed as operation rules for Boolean variables. Also, in the sense of privacy protection, the selected subset $\hat{\cS}_i$ as the statistics with less sensitive information can be publicly transferred to the “center machine” for aggregation. Among aggregation methods, union and intersection of sets are usually adopted empirically. As the simplest rule similar to the OR rule in Boolean operation, we obtain the Union rule $$\begin{aligned} \hat{\cS}_{U} = \bigcup_{i=1}^{k} \hat{\cS}_i.\end{aligned}$$ Also, the intersection of all selected subsets produces the Intersection rule: $$\begin{aligned} \hat{\cS}_{I} = \bigcap_{i=1}^{k} \hat{\cS}_i.\end{aligned}$$ The Union rule is not strict, thus requires the stricter $\FDR$ control for each machine. It indicates that if each machine has $\FDR$ control at $q$, then the overall $\FDR$ may far exceeds the pre-defined level. On the other hand, the Intersection rule is far more stricter and will result in the loss of power in aggregation. The phenomenon is illustrated in the left plot of Figure \[fig:plot1\]. We will show that these two rules will have a more general representation and are thus included in a family of threshold-based aggregation rules. Generalized threshold-based aggregation --------------------------------------- As an extension to the operation of Boolean variables, we first define $$\begin{aligned} m_j = \sum_{i=1}^{k} \bfm{1}_{\left\{j \in \hat{\cS}_{i}\right\}},\;\;j=1,\dots,d.\end{aligned}$$ Then the [*threshold-based rule*]{} is conducted as $$\begin{aligned} \hat{\cS}_{\left(c\right)} = \left\{j \in \left[d\right]: m_j \geq c\right\}\end{aligned}$$ for an integer $c$. We should notice that [*the Union rule*]{} is a special case of [*the threshold-based rule*]{} with $\hat{\cS}_{U} = \hat{\cS}_{\left(c=1\right)}$. And for [*the Intersection rule*]{}, $\hat{\cS}_{I} = \bigcap_{i=1}^{k} \hat{\cS}_i = \hat{\cS}_{\left(c=k\right)}$. Lying between the Intersection and the Union rules, the threshold $c = \left[\left(k+1\right)/2\right]$ can be adopted as a mild rule and we call it “median-aggregation”. However, we rarely have prior information to determine a universal threshold $c$ and the suitable threshold may also vary in different cases. Therefore, we introduce ADAGES, the adaptive aggregation method in the following section. Adaptive aggregation for distributed feature selection {#method} ====================================================== Based on the definition of $\hat{\cS}_{\left(c\right)}$, $\hat{\cS}_{\left(c_1\right)} \subseteq \hat{\cS}_{\left(c_2\right)}$ for any $c_1 \geq c_2$, thus $|\hat{\cS}_{\left(c\right)}|$ is a decreasing function of $c$. Further, adaptive information aggregation from $k$ machines utilizes the data-driven threshold which is determined conditionally on $\left\{\hat{\cS}_i,i=1,\dots,k\right\}$, thus it is meaningful to look into the behavior of $\hat{\cS}_{\left(c\right)}|\left(\hat{\cS}_1,\dots,\hat{\cS}_k\right)$. Denote $\bar{s} = \frac{1}{k} \sum_{i=1}^{k} |\hat{\cS}_i|$ and $M = \max_{i=1}^{k} |\hat{\cS}_i|$. Candidate region for threshold ------------------------------ Restrictions on the size of $\hat{\cS}_{(c)}$ is one traditional way to regularize model complexity, and in the first step, we determine the candidate region for threshold $c$ by restricting the model complexity measure $|\hat{\cS}_{(c)}|$. In the contrast to the usual upper bounds for model complexity, we use the mean $\bar{s} = \frac{1}{k} \sum_{i=1}^{k} |\hat{\cS}_i|$ as a lower bound for $|\hat{\cS}_{\left(c\right)}|$, which is in line with the purpose of power maintenance in multiple testing. We define $c_0$ as an upper bound as $$\begin{aligned} \label{c0} c_0 = \max \left\{c: |\hat{\cS}_{\left(c\right)}| \geq \bar{s}\right\}%\wedge \left[\frac{k+1}{2}\right],\end{aligned}$$ and it is trivial that $c_0 \geq 1$ since $$\begin{aligned} |\hat{\cS}_{U}| = |\hat{\cS}_{\left(c=1\right)}| \geq \max \left\{|\hat{\cS}_i|:i=1,\dots,k\right\} \geq \bar{s}.\end{aligned}$$ Therefore, we can choose any integer $c \leq c_0$ as a mild threshold for aggregation, but in the meanwhile, a threshold ought to be chosen to balance the tradeoff between false discovery rate and power. {width="\textwidth"} {width="\textwidth"} Choice of threshold for recovery accuracy ----------------------------------------- Besides, to improve the tradeoff between $\FDR$ and Power, we adopt the following rule emphasizing stable recovery. With $c_0$ as an upper bound, smaller threshold leads to higher selection power as well as more false discoveries. [1.5]{} **Complexity ratio.** First, we consider the complexity ratio $$\eta_c = \begin{cases} \frac{|\hat{\cS}_{\left(c\right)}|}{|\hat{\cS}_{\left(c+1\right)}|},& |\hat{\cS}_{\left(c+1\right)}| > 0,\\ \infty,& |\hat{\cS}_{\left(c+1\right)}| = 0, \end{cases}$$ for thresholds decreasing from $c_0$ and the minimum of complexity ratio is a sign of stable and accurate recovery. Then, the adaptive threshold $c^*$ for aggregation can be chosen by $$\begin{aligned} c^* = {\rm argmin} \left\{\eta_c: 1 \leq c \leq c_0,\right\}.\end{aligned}$$ In practice, to avoid infinite values, we can also use a surrogate $(1+|\hat{\cS}_{\left(c\right)})|/(1+|\hat{\cS}_{\left(c+1\right)}|)$. As is shown in a simple example in the right plot of Figure \[fig:plot1\] with true $|\cS | = 20$, threshold with the minimum ratio $\eta_c$ produces a more stable recovery of the true $\cS$, and in this figure we adopt a modified form $20 \times {\rm log}\left(\eta_c\right)$ to represent the magnitude of ratio. To illustrate the complexity ratio, the idea is similar to the eigenvalue ratio in PCA for determining the number of meaningful eigen-components. We can also consider a toy example where $|\cS \cap \hat{\cS}| \sim B(k,p)$ with $p = \PP(j \in \cS: j \in \hat{\cS}_i)$. In this case, minimizing the ratio $\eta_c$ approximately produces the mode of Bernoulli distribution that recovers the threshold in line with the most likely frequency for important features. It is noticeable that another rule with theoretical intuition for choosing the threshold is given by $$\begin{aligned} \widetilde{c} = {\rm argmin}_{1 \leq c \leq c_0} c |\hat{\cS}_{(c)}|,\end{aligned}$$ which explicitly focuses on the tradeoff between the magnitude of threshold and the size of selected subset. As we will show in Lemma \[lem:shrink\], the power shrinkage term $\left(\frac{c|\hat{\cS}_{(c)}|}{k|\cS|} \cdot {\rm FDP}\right)$ plays the leading role in the lower bound for the true positive proportion. Then, for ${\rm FDP}$ at a certain level, minimizing the product $c |\hat{\cS}_{(c)}|$ is equivalent to maximizing the true positive proportion. Details of numerical simulations will be discussed in section \[simu\] and the implementation of adaptive aggregation based on the complexity ratio is shown in the Algorithm \[alg1\]. Aggregated feature selection is an initial case dealing with binary variables. It is more exciting to extend this threshold-based aggregation method to estimation and inference based on communication of more informative statistics, and we leave this for future work. **Input** $\left\{\hat{\cS}_i:i=1,\dots,k\right\}$: $\hat{\cS}_i \subset \left[d\right]$ is the selected subset for the $i$th machine **Output** $\hat{\cS} = \hat{\cS}_{\left(c^*\right)}$ as an estimation for $\cS$ Calculate $m_j = \sum_{i=1}^{k} \bfm{1}_{\left\{j \in \hat{\cS}_{i}\right\}}$, $\forall j \in \left\{1,\dots,d\right\}$ Calculate $\bar{s} = \frac{1}{k} \sum_{i=1}^{k} |\hat{\cS}_i|$ $\hat{\cS}_{\left(c\right)} = \left\{j \in \left[d\right]: m_j \geq c\right\}$ Calculate the complexity ratio $\eta_c = \frac{|\hat{\cS}_{\left(c\right)}|+1}{|\hat{\cS}_{\left(c+1\right)}|+1}$, $c \leq k-1$;   $\eta_k = \infty$ Determine $c_0 = \max \left\{c: |\hat{\cS}_{\left(c\right)}| \geq \bar{s}\right\}$ Produce adaptive threshold $c^* = {\rm argmin} \left\{\eta_c: 1 \leq c \leq c_0\right\}$\ $\hat{\cS} = \hat{\cS}_{(c^*)} = \left\{j \in \left[d\right]: m_j \geq c^*\right\}$ Main result {#result} =========== In this section, we will show the theoretical properties of ADAGES for adaptive aggregation in the scenario of distributed feature selection. First, we obtain the control of overall false discovery rate in theorem \[thm1\]; besides, we establish the connection of overall power and machine-wise power: theorem \[thm2\] shows the simultaneous control of $\FDR$ and a power shrinkage term and theorem \[thm3\] compares the power of ADAGES with the “optimal” power produced by the Union rule. Distributed FDR control ----------------------- Based on the adaptive threshold for aggregation, the ADAGES produces exact control of the false discovery rate. \[thm1\] For a pre-defined level $q \in \left(0,1\right)$, suppose machine-wise ${\rm FDR}_i \leq q$ for $i=1,\dots,k$ and $\lambda \geq \max_{1\leq i \leq k}\frac{|\hat{\cS}_i|}{c^*} \sum_{j=1}^{k} \frac{1}{|\hat{\cS}_j|}$. Then, ADAGES with $c^* \in \left[1,c_0\right] \cap \ZZ$ produces $$\begin{aligned} {\rm FDR}_{\left(c^*\right)} = \EE\left[\frac{|\hat{\cS}_{\left(c^*\right)} \cap \cS^{c}|}{|\hat{\cS}_{\left(c^*\right)}|}\right] \leq \lambda q.\end{aligned}$$ Then, we discuss two special cases with fixed thresholds $c=1$ and $c=k$ respectively, which may reveal their shortcomings to some extend. \[union\] For a pre-defined level $q \in \left(0,1\right)$, if machine-wise ${\rm FDR}_i \leq q$ for all $i=1,\dots,k$, the Union rule produces $$\begin{aligned} {\rm FDR}_{U} = \EE\left[\frac{|\hat{\cS}_{U} \cap \cS^{c}|}{|\hat{\cS}_{U}|}\right] \leq kq.\end{aligned}$$ More generally, as is pointed out in [@xie2019aggregated], if there is a sequence of pre-defined FDR levels $\left(q_1,\dots,q_k\right)$ such that ${\rm FDR}_i \leq q_i$ for all $i \in \left[k\right]$, then the overall $\FDR$ can be exactly controlled at level $q=\sum_{i=1}^{k} q_i$. If we would like to have overall FDR controlled at level $q$, it requires that $\sum_{i=1}^{k} q_i = q$ and a simple case is $q_i = q/k$ for all $k$ machines. Besides, in the case with $c = k$, based on $k|\hat{\cS}_{I} \cap \cS^{c}| \leq \sum_{i=1}^{k} |\hat{\cS}_{i} \cap \cS^{c}|$, we have the following proposition: \[intersec\] For a pre-defined level $q \in \left(0,1\right)$, if machine-wise ${\rm FDR}_i \leq q$ for $i=1,\dots,k$ and there is a constant $\kappa \geq 1$such that $\max_{i \in \left[k\right]} \frac{|\hat{\cS}_i|}{|\hat{\cS}_{I}|} \leq \kappa$, then the Intersection rule produces $$\begin{aligned} {\rm FDR}_{I} = \EE\left[\frac{|\hat{\cS}_{I} \cap \cS^{c}|}{|\hat{\cS}_{I}|}\right] \leq \kappa q.\end{aligned}$$ Comparing the overall $\FDR$ bounds, the Union rule as a less strict aggregation rule produces $\FDR$ at an expected level as high as $kq$. Instead, the Intersection rule is the most conservative and has theoretical $\FDR$ control at $q$ multiplied by a factor $\kappa$. However, with an adaptive threshold, ADAGES summarizes machine-wise information more efficiently and has the control of overall $\FDR$ at level $\lambda q$. Here, as an illustration, we compare the magnitude of $k,\lambda,\kappa$ to show the abilities of $\FDR$ control of the three methods. First, if $c^*/k$ has a positive lower bound such that $c^* \geq b\cdot k$ and $\max_{i \in [k]} \frac{|\hat{\cS}_i|}{|\hat{\cS}_j|} = O(1)$ for all $j$, then we obtain $\lambda = o(k)$. Comparison between $\lambda$ and $\kappa$ is of more interest, which is summarized in the following proposition. Denote the tight bound $\bar{\lambda} = \max_{i \in [k]}\frac{|\hat{\cS}_i|}{c^*} \sum_{j=1}^{k} \frac{1}{|\hat{\cS}_j|}$ and $\bar{\kappa} = \max_{i \in \left[k\right]} \frac{|\hat{\cS}_i|}{|\hat{\cS}_{I}|}$. Then, we have $$\begin{aligned} \frac{\bar{\lambda}}{\bar{\kappa}} = \frac{1}{c^*} \sum_{j=1}^{k} \frac{|\hat{\cS}_{I}|}{|\hat{\cS}_j|}.\end{aligned}$$ Further, if $(1-\epsilon)c^* < \sum_{j=1}^{k} \frac{|\hat{\cS}_{I}|}{|\hat{\cS}_j|} < (1+\epsilon)c^*$ for any $\epsilon \in (0,1)$, then $|\frac{\bar{\lambda}}{\bar{\kappa}} - 1| < \epsilon$. Power analysis -------------- We also establish a lower bound for the Power based on $\left\{ {\rm Power}_i\right\}$, $i=1,\dots,k$ as well as the power produced by the Union bound, before which we introduce the basic lemma to establish the connection between overall true positive proportion (TPP) with machine-wise $\TPP_i$, $i=1,\dots,k$. \[lem:shrink\] Based on the ADAGES algorithm, we obtain $$\begin{aligned} \TPP \geq \frac{1}{k} \sum_{i=1}^{k} \TPP_i - \frac{c^*}{k} \frac{| \cS^c \cap \hat{\cS}_{(c^*)} |}{| \cS |}.\end{aligned}$$ The second term $\frac{c^*}{k} \frac{| \cS^c \cap \hat{\cS}_{(c^*)} |}{| \cS |}$ acts as the term of “power shrinkage” and can be connected with ${\rm FDP}$ in the form: $$\begin{aligned} {\rm Power~shrinkage} = \frac{c^*|\hat{\cS}_{(c^*)} |}{k| \cS |}{\rm FDP},\end{aligned}$$ which involves a tradeoff between $c^*$ and $|\hat{\cS}_{(c^*)} |$. Therefore, with proper restriction on $|\hat{\cS}_{(c^*)} |$, i.e. a proper choice of $c^*$, we can simultaneously control $\FDR$ and the power shrinkage term, which is shown in theorem \[thm2\]. \[thm2\] Denote ${\rm Power}_i$ as the selection of for the $i$th machine. Suppose there exists constant $\gamma \in (0,1/2)$ such that $|\hat{\cS}_{(c^*)} | \leq (1+\gamma)|\cS|$ and $c^* \leq k/2$. If the overall $\FDR$ is controlled at level $q \in (0,1)$, then for a constant $\alpha \leq 3/4$, we have $$\begin{aligned} {\rm Power} \geq \frac{1}{k} \sum_{i=1}^{k} {\rm Power}_i - \alpha q.\end{aligned}$$ It is noticeable that power produced by the Union bound is the maximum power one aggregation method can achieve. Denote ${\rm diff} = |(\hat{\cS}_{U}\cap \cS)\backslash(\hat{\cS}_{U}\cap \cS)| = |(\hat{\cS}_{U}\cap \cS)| - |(\hat{\cS}\cap \cS)|$, with which we obtain the following theorem. \[thm3\] Suppose we have a uniform lower bound for ${\rm Power}_i,i\in [k]$ that $\PP(j \in \cS, j \in \hat{\cS}_i) \geq \eta_{n,d}$ for $i\in [k],j \in[d]$. If we further have $c^* \leq k/2$, then $\exists \xi \leq 2$ such that $$\begin{aligned} \EE[{\rm diff}] \leq \xi (1-\eta_{n,d})|\cS|.\end{aligned}$$ Further, if the selection method has the property that $\eta_{n,d} \rightarrow 1$ as $n,d \rightarrow \infty$, we have $|{\rm Power} - {\rm Power}_{U}| \rightarrow 0$ as $n,d \rightarrow \infty$. Numerical simulation {#simu} ==================== In this section, we study the performance of our adaptive aggregation method by comparisons with the empirical Union, Intersection and median-aggregation rules in simulations. We also compare with the performance of the aggregation method in [@xie2019aggregated], which is a modified version of the Union rule. In numerical simulations, we use model-X knockoffs with second-order construction for each machine which produces exact $\FDR$ control, so the method can be named as “model-X knockoffs + ADAGES” to illustrate the procedure. In this case, we are also interested in the comparison between our algorithm-free ADAGES and the knockoff-based aggregation method AKO in [@nguyen2020aggregation]. We consider the AKO with BY step-up with theoretical guarantee and use $\gamma = 0.3$ that is adopted in [@nguyen2020aggregation]. In experiments, ADAGES refers to our adaptive method with $c^* = {\rm argmin} \left\{\eta_c: 1 \leq c \leq c_0\right\}$ while ${\rm ADAGES}_m$ is the modified method with threshold $\widetilde{c} = {\rm argmin}_{1 \leq c \leq c_0} c |\hat{\cS}_{(c)}|$. A simple linear model is adopted for feature selection: $$\begin{aligned} \mathbf{y} = \mathbf{X} \bfm{\beta} + \mathbf{\epsilon},\end{aligned}$$ where $\mathbf{X} \in \RR^{n \times d} \sim \cN(\mathbf{0},\bSigma)$ is the design matrix, where $\bSigma \in \RR^{d \times d}$ and $\bSigma_{ls} = \rho^{|l-s|}$ for all $l,s \in [d]$. $\mathbf{y} \in \RR^{n}$ is the vector of $n$ responses and elements in the noise vector $\mathbf{\epsilon}$ are drawn [*i.i.d.*]{} from standard Gaussian distribution. Feature importance is revealed in $\mathbf{\beta}$ and $\cS = \left\{j \in \left[d\right]: \beta_j \neq 0\right\}$. Comparisons are conducted in the following two aspects, in which the repetition number is $r=100$ and $\rho = 0.25$. We use the criteria of averaged FDP and averaged power as the sample-versions of FDR and power respectively. {width="\textwidth"} Varying the number of machines $k$ ---------------------------------- Since the number of machines is a vital factor in the context of distributed learning, in the first experiment, we vary $k$ among $\left\{1,2,5,8,10,20\right\}$ with $n=1000$, $d=50$ and $s=|\cS | = 20$ fixed. Here nonzero elements in true $\mathbf{\beta}$ is drawn [*i.i.d.*]{} and uniformly from $\left\{\pm 2\right\}$. From Figure \[fig:num\_k\], we can see that ADAGES obtains a desirable tradeoff between the averaged FDP and power. As an adaptive aggregation method, ADAGES controls FDP exactly under $q=0.2$ while achieves power nearly as good as that of the Union rule, which meets the goal of power maintenance for controlled feature selection. For the three empirical methods, although the Union rule maintains power at the highest level, it produces FDP exceeding the pre-defined level $q=0.2$; the Intersection rule has conservative control of FDP but results in a serious loss of power in feature selection while the power loss of median-aggregation occurs earlier than ADAGES. As an improvement for the Union rule on $\FDR$ control, the method in [@xie2019aggregated] obtains comparable FDP with the Intersection rule; but since the pre-defined level for each machine becomes $q_i = q/k$, this method will sacrifice power as shown in Figure \[fig:num\_k\] and is thus limited in application. On the other hand, in this case without ultra-high dimension or strict sparsity, the AKO that transforms more informative “p-values” in aggregation is capable of controlling the averaged FDP around the level $\kappa q$ where $\kappa \leq 3.24$ is given in [@nguyen2020aggregation]; power of AKO is lower than other algorithm-free methods when $k < 10$, but remains stable as $k$ increases. However, the modified ADAGES with $\widetilde{c} = {\rm argmin}_{1 \leq c \leq c_0} c |\hat{\cS}_{(c)}|$ does not produce higher power in experiments since the power shrinkage term indicates the tradeoff between FDP and $c |\hat{\cS}_{(c)}|$. Here, FDP is also a function of $c$ which ought not to be ignored in the choice of $\widetilde{c}$. {width="\textwidth"} Varying dimension $d$ --------------------- In the second experiment, we vary dimension $d$ in the set\ $\left\{15,30,45,60,75,90\right\}$ while fix model parameters as $n=1000$, $k=10$ and $|\cS | = 10$. True signal $\beta_j$ is generated in the same way mentioned above. In Figure \[fig:dim\], both ADAGES, median-aggregation and the Intersection rule have exact FDP control under $q=0.2$, but the Union rule suffers from “uncensored” aggregation and cannot control the overall FDP. Partially dependent on the property of the feature selection method adopted for each machine, the power goes down as $d$ increases. But it is noticeable that the Union rule can always achieve the highest power after aggregation and ADAGES shows comparable performance due to the use of an adaptive threshold based on $\eta_c$ on an interval with an upper bound. In addition, the aggregation method in [@xie2019aggregated] tends to make null discovery that is $|\hat{\cS}| = 0$ which naturally control $\FDR$ at 0 but also have no power. Similar to our findings with varying $k$, the AKO performs better than empirical aggregation methods as $d$ increases, especially in power; but ADAGES shows better performance in both averaged FDP and empirical power. Discussion ========== In this paper, we present an adaptive aggregation method called ADAGES for distributed false discovery rate control. Our method utilizes selected subsets from all machines to determine the aggregation threshold and shows better performance in the tradeoff of $\FDR$ control and power maintenance compared with empirical aggregation methods. The ADAGES is algorithm-free, which means it can be applied to any machine-wise feature selection method, and is thus more flexible than aggregation rules based on specific statistics produced by each machine-wise method. It is motivating to further study the modified method based on the power shrinkage term, which has theoretical intuition for power maintenance and requires a good estimation of overall FDP. Besides, as potential extensions, we can adopt this adaptive method with stability in other statistical aspects in distributed learning. Selected subsets are binary vectors consisting of limited but private information and we can further take communication constraints and privacy into consideration, which are left for our future work. More importantly, there is a tradeoff between information communication and selection power, thus it is meaningful to study aggregation methods with machines transferring encoded but more informative statistics. As the distributed pattern becomes more common in the statistical community, to promote inter-institutional collaboration, efficient aggregation methods are necessary for distributed computing as well as privacy protection. With the idea of adaptive aggregation, collaboration can adapt to specific scenarios while each institution simply needs to focus on its specific statistical problem, which greatly contributes to the new collaboration mode in data science. However, another direction for future research is to relax the independence assumption among institutions in the learning procedure and to study the influence of inter-institutional dependence in the statistical context. Implementation of ADAGES with R is available and raw codes can be accessed on <https://github.com/yugjerry/ADAGES/blob/master/code_ADAGES.R>. Technical proofs are presented in the following sections. Technical proofs {#proof} ================ In this section, we present the proofs for the main theorems and propositions in this paper. **Proof for Theorem \[thm1\].** With $\FDR_i = \EE\left[\frac{|\hat{\cS}_i \cap \cS^{c}|}{|\hat{\cS}_i|}\right] \leq q$, observe the overall $\FDR$: $$\begin{aligned} \FDR_{\left(c^*\right)} &= \EE\left[\frac{|\hat{\cS}_{(c^*)} \cap \cS^{c}|}{|\hat{\cS}_{(c^*)}|}\right] \nonumber\\&= \EE\left\{\EE\left[\frac{|\hat{\cS}_{(c^*)} \cap \cS^{c}|}{|\hat{\cS}_{(c^*)}|}\Big|\left(\hat{\cS}_1,\dots,\hat{\cS}_k\right)\right]\right\}.\end{aligned}$$ First, we have $c^* |\hat{\cS}_{(c^*)}| \leq \sum_{j:m_j \geq c^*} m_j \leq \sum_{j=1}^{d} m_j = \sum_{i=1}^{k} |\hat{\cS}_i|$, and similarly for features $j \in \cS^{c}$, $$\begin{aligned} c^* |\hat{\cS}_{(c^*)} \cap \cS^{c}| \leq \sum_{j \in \cS^{c}: m_j \geq c^*} m_j \leq \sum_{j \in \cS^{c}} m_j = \sum_{i=1}^{k} |\hat{\cS}_i \cap \cS^{c}|.\end{aligned}$$ Therefore, the overall $\FDR$ can be linked to the machine-wise $\FDR$’s as $$\begin{aligned} \FDR_{\left(c^*\right)} &= \EE\left\{\EE\left[\frac{|\hat{\cS}_{(c^*)} \cap \cS^{c}|}{|\hat{\cS}_{(c^*)}|}\Big|\left(\hat{\cS}_1,\dots,\hat{\cS}_k\right)\right]\right\} \nonumber\\&\leq %\EE\left\{\frac{1}{c^*}\EE\left[\frac{\sum_{i=1}^{k} |\hat{\cS}_i \cap \cS^{c}|}{|\hat{\cS}_{(c^*)}|}|\left(\hat{\cS}_1,\dots,\hat{\cS}_k\right)\right]\right\}\nonumber\\ %& = \EE\left\{\frac{1}{c^*}\sum_{i=1}^{k} \EE\left[\frac{|\hat{\cS}_i \cap \cS^{c}|}{|\hat{\cS}_{(c^*)}|}\Big|\left(\hat{\cS}_1,\dots,\hat{\cS}_k\right)\right]\right\}.\end{aligned}$$ In addition, by definition of $c^*$ in the theorem: $|\hat{\cS}_{(c^*)}| \geq \frac{1}{k} \sum_{i=1}^{k} |\hat{\cS}_i| \geq \frac{k}{\sum_{i=1}^{k} 1/|\hat{\cS}_i|}$, we then obtain $$\begin{aligned} \FDR_{\left(c^*\right)} & \leq \EE\left\{\frac{1}{c^*}\sum_{i=1}^{k} \EE\left[\frac{|\hat{\cS}_i \cap \cS^{c}|}{|\hat{\cS}_{(c^*)}|}\Big|\left(\hat{\cS}_1,\dots,\hat{\cS}_k\right)\right]\right\} \nonumber\\ &\leq \EE\left\{\frac{1}{k c^*}\sum_{j=1}^{k} \sum_{i=1}^{k} \EE\left[\frac{|\hat{\cS}_i \cap \cS^{c}|}{|\hat{\cS}_j|}\Big|\left(\hat{\cS}_1,\dots,\hat{\cS}_k\right)\right]\right\}\nonumber\\ & = \EE\left\{\frac{1}{k c^*}\sum_{1 \leq i,j \leq k} \EE\left[\frac{|\hat{\cS}_i \cap \cS^{c}|}{|\hat{\cS}_i|} \cdot \frac{|\hat{\cS}_i|}{|\hat{\cS}_j|}\Big|\{\hat{\cS}_l\}_{l=1}^{k}\right]\right\}\nonumber\\ & = \EE\left\{\frac{1}{k c^*}\sum_{i=1}^{k} \sum_{j=1}^{k}\EE\left[{\rm FDP}_i\frac{|\hat{\cS}_i|}{|\hat{\cS}_j|} \Big|\left(\hat{\cS}_1,\dots,\hat{\cS}_k\right)\right]\right\} \nonumber\\ & \leq \sum_{i = 1}^{k} \frac{1}{kc^*} {\rm FDP}_i \cdot \left(\max_{1\leq i \leq k}|\hat{\cS}_i| \sum_{j=1}^{k} \frac{1}{|\hat{\cS}_j|}\right)\nonumber\\ & \leq \sum_{i = 1}^{k} \frac{1}{kc^*} {\rm FDP}_i \cdot \lambda c^* \leq \lambda q.\end{aligned}$$ Here, $\lambda$ is a bound that $$\begin{aligned} \lambda \geq \max_{1\leq i \leq k}\frac{|\hat{\cS}_i|}{c^*} \sum_{j=1}^{k} \frac{1}{|\hat{\cS}_j|}.\end{aligned}$$ $\ep$\ \ **Proof for theorem \[thm2\].** We consider the expected number of true discoveries $\EE| \hat{\cS} \cap \cS |$ and denote ${\rm TPP}_i = \frac{|\hat{\cS}_i \cap \cS|}{| \cS |},~i \in \left[k\right]$ and $\TPP = \frac{|\hat{\cS}_{(c^*)} \cap \cS|}{| \cS |}$. Then we have $$\begin{aligned} |\cS| \sum_{i=1}^{k} \TPP_i &= \sum_{i=1}^{k} | \hat{\cS}_i \cap \cS| \nonumber\\&= \sum_{i=1}^{k} \sum_{j \in \cS} \bfm{1}_{\left\{j \in \hat{\cS}_i\right\}} = \sum_{j \in \cS} m_j \nonumber\\&= \sum_{j \in \cS \cap \hat{\cS}_{(c^*)}} m_j + \sum_{j \in \cS^c \cap \hat{\cS}_{(c^*)}} m_j \nonumber\\&\leq k| \cS \cap \hat{\cS}_{(c^*)} | + c^* | \cS^c \cap \hat{\cS}_{(c^*)} |,\end{aligned}$$ which is equivalent to $$\begin{aligned} \TPP \geq \frac{1}{k} \left(\sum_{i=1}^{k} \TPP_i - c^* \frac{| \cS^c \cap \hat{\cS}_{(c^*)} |}{| \cS |} \right).\end{aligned}$$ Based on the assumption with an upper bound on $\hat{\cS}_{(c^*)}$ with $\gamma \in (0,1/2)$, $$\begin{aligned} |\hat{\cS}_{(c^*)}|\leq (1+\gamma) |\cS|,\end{aligned}$$ we take expectation for the inequality and then obtain $$\begin{aligned} {\rm Power} \geq \frac{1}{k} \sum_{i=1}^{k} {\rm Power}_i - \alpha q,\end{aligned}$$ where $\alpha = \frac{c^*}{k}(1+\gamma) < \frac{3}{4}$. $\ep$\ \ **Proof for theorem \[thm3\].** We can write explicitly that $$\begin{aligned} {\rm diff} = \sum_{j \in \cS} \bfm{1}_{\left\{0 < m_j < c^*\right\}}.\end{aligned}$$ Then, for positive $m_j$, $\EE[{\rm diff}] = \sum_{j \in \cS} \PP(m_j < c^*)$ with $$\begin{aligned} \PP(m_j < c^*, j \in \cS) &\leq \frac{k - \EE[m_j|j \in \cS]}{k - c^*} \nonumber\\&= \frac{k - \EE[\sum_{i=1}^{k} \bfm{1}_{\{j \in \hat{\cS}_i\}}|j \in \cS]}{k - c^*} \nonumber\\&\leq \frac{k(1-\eta_{n,d})}{k - c^*} \nonumber\\ &\leq \xi (1-\eta_{n,d}),\end{aligned}$$ where $c^* \leq k/2$ by definition and thus $\xi \leq 2$. $\ep$\ \ **Proof for proposition \[union\].** With the Union rule, $\hat{\cS}_{U} = \bigcup_{i=1}^{k} \hat{\cS}_i$ and thus $\hat{\cS}_i \subset \hat{\cS}_{U}$ for all $i=1,\dots,k$. Since $|\bigcup_{i=1}^{k} A_i| \leq \sum_{i=1}^{k} |A_i|$, we apply this fact to $\hat{\cS}_{U} \cap \cS^c = \bigcup_{i=1}^{k} (\hat{\cS}_i \cap \cS^c)$ and consider the overall FDR: $$\begin{aligned} \FDR &= \EE\left[\frac{|\hat{\cS}_{U} \cap \cS^c|}{|\hat{\cS}_{U}|}\right] \leq \EE\left[\sum_{i=1}^{k} \frac{|\hat{\cS}_{i} \cap \cS^c|}{|\hat{\cS}_{U}|}\right] \nonumber\\&\leq \sum_{i=1}^{k} \EE\left[\frac{|\hat{\cS}_{i} \cap \cS^c|}{|\hat{\cS}_{i}|} \right] \leq \sum_{i=1}^{k} \FDR_i \nonumber\\&\leq \sum_{i=1}^{k} q_i.\end{aligned}$$ $\ep$\ \ **Proof for proposition \[intersec\].** With $\hat{\cS}_{I} = \bigcap_{i=1}^{k} \hat{\cS}_i$, we have $m_j = \sum_{i=1}^{k} \bfm{1}_{\left\{j \in \hat{\cS}_{i}\right\}} = k$ for $j \in \hat{\cS}$. Therefore, we have $$\begin{aligned} k|\hat{\cS}_{I} \cap \cS^c| &= \sum_{j \in \hat{\cS}_{I} \cap \cS^c} m_j \leq \sum_{j \in \cS^c} m_j \nonumber\\&= \sum_{j \in \cS^c} \sum_{i=1}^{k} \bfm{1}_{\left\{j \in \hat{\cS}_{i}\right\}} = \sum_{i=1}^{k} |\hat{\cS}_{i} \cap \cS^c|\end{aligned}$$ We then consider the overall FDR, $$\begin{aligned} \FDR &= \EE\left[\frac{|\hat{\cS}_{I} \cap \cS^c|}{|\hat{\cS}_{I}|}\right] \leq \EE\left[\frac{1}{k}\frac{|\hat{\cS}_{i} \cap \cS^c|}{|\hat{\cS}_{I}|}\right]\nonumber\\ &\leq \EE\left[\frac{1}{k}\frac{|\hat{\cS}_{i} \cap \cS^c|}{|\hat{\cS}_{i}|} \cdot \frac{|\hat{\cS}_{i}}{|\hat{\cS}_{I}|}\right] \nonumber\\&\leq \frac{\kappa}{k} \sum_{i=1}^{k} \EE\left[\frac{|\hat{\cS}_{i} \cap \cS^c|}{|\hat{\cS}_{i}|}\right]\leq \kappa q.\end{aligned}$$ Here $\kappa \geq 1$ is a constant such that $\max_{i \in \left[k\right]} \frac{|\hat{\cS}_i|}{|\hat{\cS}_{I}|} \leq \kappa$. $\ep$ Illustration of the aggregation process ======================================= In this part, results in four cases are provided to illustrate the connection between overall FDR/power and the machine-wise ones. The $k$ grey bars in each plot are the $ \FDR_i$s or ${\rm Power }_i$s for $k$ machines. From the four cases together with the simulation results in our paper, we can see that ADAGES has a better tradeoff than other methods (the Union rule, the Intersection rule, median aggregation and method in [@xie2019aggregated]). For FDR, all methods except the Union rule produce the exact control whenever machine-wise FDR is controlled at the pre-defined level. The Union rule, however, as is shown in proposition \[union\], is only able to control FDR at a higher level. When $k$ or the dimension $d$ is large, strict aggregation methods will cause the power loss, such as the results of the Intersection rule and method in [@xie2019aggregated]. We should note that “strict” refers to strict pre-defined levels for each machine as well as strict aggregation rules. As is shown in the results, ADAGES produces power very close to that of the Union rule, which is the highest power an aggregation method can achieve. ![Representation of the aggregation process: barplot of machine-wise FDR(left)/power(right) and aggregation results under different rules ($q=0.2,k=5,d=20,n=1000,n_i=200$).[]{data-label="fig:p1"}](Bar5_20.pdf){width="80.00000%"} ![Representation of the aggregation process: barplot of machine-wise FDR(left)/power(right) and aggregation results under different rules ($q=0.2,k=5,d=80,n=1000,n_i=200$).[]{data-label="fig:p2"}](Bar5_80.pdf){width="80.00000%"} ![Representation of the aggregation process: barplot of machine-wise FDR(left)/power(right) and aggregation results under different rules ($q=0.2,k=10,d=20,n=1000,n_i=100$).[]{data-label="fig:p3"}](Bar10_20.pdf){width="80.00000%"} ![Representation of the aggregation process: barplot of machine-wise FDR(left)/power(right) and aggregation results under different rules ($q=0.2,k=10,d=80,n=1000,n_i=100$).[]{data-label="fig:p4"}](Bar10_80.pdf){width="80.00000%"} [^1]: The author finished this paper when he was an undergraduate at the University of Science and Technology of China

--- abstract: 'In sub-Riemannian geometry the coefficients of the Jacobi equation define curvature-like invariants. We show that these coefficients can be interpreted as the curvature of a canonical Ehresmann connection associated to the metric, first introduced in [@lizel]. We show why this connection is naturally nonlinear, and we discuss some of its properties.' address: - '$^\flat$Institut de Mathématiques de Jussieu-Paris Rive Gauche UMR CNRS 7586, Université Paris-Diderot, Batiment Sophie Germain, Case 7012, 75205 Paris Cedex 13, France' - '$^\sharp$CMAP École Polytechnique, Palaiseau and Équipe INRIA GECO Saclay Île-de-France, Paris, France' author: - 'Davide Barilari$^\flat$' - 'Luca Rizzi$^\sharp$' bibliography: - 'Connection-Biblio.bib' title: 'On Jacobi fields and a canonical connection in sub-Riemannian geometry' --- Introduction ============ A key tool for comparison theorems in Riemannian geometry is the Jacobi equation, i.e. the differential equation satisfied by Jacobi fields. Assume $\gamma_{{\varepsilon}}$ is a one-parameter family of geodesics on a Riemannian manifold $(M,g)$ satisfying $$\label{eq:geo0} \ddot{\gamma}_{{\varepsilon}}^{k}+\Gamma_{ij}^{k}(\gamma_{{\varepsilon}})\dot{\gamma}^{i}_{{\varepsilon}}\dot{\gamma}^{j}_{{\varepsilon}}=0.$$ The corresponding Jacobi field $J=\left.\frac{\partial}{\partial {\varepsilon}}\right|_{{\varepsilon}=0} \gamma_{{\varepsilon}}$ is a vector field defined along $\gamma=\gamma_{0}$, and satisfies the equation $$\label{eq:jacobo0} \ddot{J}^{k}+2\Gamma_{ij}^{k}\dot J^{i}\dot{\gamma}^{j}+\frac{\partial \Gamma_{ij}^{k}}{\partial x^{\ell}}J^{\ell} \dot{\gamma}^{i}\dot{\gamma}^{j}=0.$$ The Riemannian curvature is hidden in the coefficients of this equation. To make it appear explicitly, however, one has to write in terms of a parallel transported frame $X_{1}(t),\ldots,X_{n}(t)$ along ${\gamma}(t)$. Letting $J(t)=\sum_{i=1}^{n} J_{i}(t)X_{i}(t)$ one gets the following normal form: $$\label{eq:jacobo} \ddot{J}_{i}+R_{ij}(t) J_{j}=0.$$ Indeed the coefficients $R_{ij}$ are related with the curvature $R^{\nabla}$ of the unique linear, torsion free and metric connection $\nabla$ (Levi-Civita) as follows $$R_{ij}=g(R^{\nabla}(X_{i},\dot {\gamma})\dot {\gamma},X_{j}).$$ Eq.  is the starting point to prove many results in Riemannian geometry. In particular, bounds on the curvature (i.e. on the coefficients $R$, or its trace) have deep consequences on the analysis and the geometry of the underlying manifold. In the sub-Riemannian setting this construction cannot be directly generalized. Indeed, the analogous of the Jacobi equation is a first-order system on the cotangent bundle that cannot be written as a second-order equation on the manifold. Still one can put it in a normal form, analogous to , and study its coefficients [@lizel]. These appear to be the correct objects to bound in order to control the behavior of the geodesic flow and get comparison-like results (see for instance [@lizel2; @BR-comparison]). Nevertheless one can wonder if these coefficients can arise, as in the Riemannian case, as the curvature of a suitable connection. We answer to this question, by showing that these coefficients are part of the curvature of a nonlinear canonical Ehresmann connection associated with the sub-Riemannian structure. In the Riemannian case this reduces to the classical, linear, Levi-Civita connection. The general setting ------------------- A sub-Riemannian structure is a triple $(M,{\mathcal{D}},g)$ where $M$ is smooth $n$-dimensional manifold, ${\mathcal{D}}$ is a smooth, completely non-integrable vector sub-bundle of $TM$ and $g$ is a smooth scalar product on ${\mathcal{D}}$. Riemannian structures are included in this definition, taking ${\mathcal{D}}=TM$. The sub-Riemannian distance is the infimum of the length of absolutely continuous admissible curves joining two points. Here admissible means that the curve is almost everywhere tangent to the distribution ${\mathcal{D}}$, in order to compute its length via the scalar product $g$. The totally non-holonomic assumption on ${\mathcal{D}}$ implies, by the Rashevskii-Chow theorem, that the distance is finite on every connected component of $M$, and the metric topology coincides with the one of $M$. A more detailed introduction on sub-Riemannian geometry can be found in [@montgomerybook; @nostrolibro; @noterifford; @notejean]. In Riemannian geometry, it is well-known that the geodesic flow can be seen as a Hamiltonian flow on the cotangent bundle $T^{*}M$, associated with the Hamiltonian $$H(p,x)=\frac{1}{2}\sum_{i=1}^{n} {\langle}p, X_{i}(x){\rangle}^{2},\qquad (p,x)\in T^{*}M,$$ where $X_{1},\ldots,X_{n}$ is any local orthonormal frame for the Riemannian structure[, and the notation $\langle p, v\rangle$ denotes the action of a covector $p \in T_x^*M$ on a vector $v \in T_x M$.]{} In the sub-Riemannian case, the Hamiltonian is defined by the same formula, where the sum is taken over a local orthonormal frame $X_{1},\ldots,X_{k}$ for ${\mathcal{D}}$, with $k=\operatorname{\mathrm{rank}}{\mathcal{D}}$. The restriction of $H$ to each fiber is a degenerate quadratic form, but Hamilton’s equations are still defined. These can be written as a flow on $T^*M$ $$\dot\lambda = \vec{H}({\lambda}),\qquad {\lambda}\in T^{*}M,$$ where $\vec{H}$ is the Hamiltonian vector field associated with $H$. This system cannot be written as a second order equation on $M$ as in . The projection $\pi : T^*M \to M$ of its integral curves are geodesics, i.e. locally minimizing curves. In the general case, some geodesics may not be recovered in this way. These are the so-called strictly abnormal geodesics [@montgomeryabnormal], and they are related with hard open problems in sub-Riemannian geometry [@agrachevopen]. In what follows, with a slight abuse of notation, the term “geodesic” refers to the not strictly abnormal ones. An integral line of the Hamiltonian vector field $\lambda(t)=e^{t\vec{H}}({\lambda}) \in T^{*}M$, with initial covector ${\lambda}$ is called *extremal*. Notice that the same geodesic may be the projection of two different extremals. For these reasons, it is convenient to see the Jacobi equation as a first order equation for vector fields on $T^*M$, associated with an extremal, rather then a second order system on $M$, associated with a geodesic. Jacobi equation revisited ========================= For any vector field $V(t)$ along an extremal $\lambda(t)$ of the sub-Riemannian Hamiltonian flow, a dot denotes the Lie derivative in the direction of $\vec{H}$: $$\dot{V}(t) := \left.\frac{d}{d{\varepsilon}}\right|_{{\varepsilon}=0} e^{-{\varepsilon}\vec{H}}_* V(t+{\varepsilon}).$$ A vector field ${\mathcal{J}}(t)$ along ${\lambda}(t)$ is called a *sub-Riemannian Jacobi field* if it satisfies $$\label{eq:defJF} \dot{{\mathcal{J}}} = 0.$$ The space of solutions of is a $2n$-dimensional vector space. The projections $J=\pi_{*}{\mathcal{J}}$ are vector fields on $M$ corresponding to one-parameter variations of ${\gamma}(t)=\pi({\lambda}(t))$ through geodesics; in the Riemannian case, they coincide with the classical Jacobi fields. We intend to write using the natural symplectic structure $\sigma$ of $T^{*}M$. First, observe that on $T^*M$ there is a natural smooth sub-bundle of Lagrangian[^1] spaces: $${\mathcal{V}}_{\lambda} := \ker \pi_*|_{\lambda} = T_\lambda(T^*_{\pi(\lambda)} M).$$ We call this the *vertical subspace*. Then, pick a Darboux frame $\{E_i(t),F_i(t)\}_{i=1}^{n}$ along $\lambda(t)$. It is natural to assume that $E_1,\ldots,E_n$ belong to the vertical subspace. To fix the ideas, one can think at the canonical basis $\{\partial_{p_i}|_{\lambda(t)},\partial_{x_i}|_{\lambda(t)}\}$ induced by a choice of coordinates $(x_1,\ldots,x_n)$ on $M$. In terms of this frame, ${\mathcal{J}}(t)$ has components $(p(t),x(t)) \in {\mathbb{R}}^{2n}$: $${\mathcal{J}}(t) = \sum_{i=1}^n p_{i}(t) E_{i}(t) + x_{i}(t) F_{i}(t).$$ The elements of the frame satisfy $$\label{eq:Jacobiframe} \begin{pmatrix} \dot{E} \\ \dot{F} \end{pmatrix} = \begin{pmatrix} C_1(t)^{*} & -C_2(t) \\ R(t) & -C_1(t) \end{pmatrix} \begin{pmatrix} E\\ F \end{pmatrix},$$ for some smooth families of $n\times n$ matrices $C_1(t),C_2(t),R(t)$, where $C_2(t) = C_2(t)^*$ and $R(t)= R(t)^*$. We stress that the particular structure of the equations is implied solely by the fact that the frame is Darboux, that is $$\sigma(E_i,E_j) = \sigma(F_i,F_j) = \sigma(E_i,F_j) -\delta_{ij} = 0, \qquad i,j=1,\ldots,n.$$ Moreover, $C_2(t) \geq 0$ as a consequence of the non-negativity of the sub-Riemannian Hamiltonian. To see this, for a bilinear form $B: V\times V \to {\mathbb{R}}$ and $n$-tuples $v,w \in V$ let $B(v,w)$ denote the matrix $B(v_i,w_j)$. With this notation $$C_2(t) = \sigma(\dot{E},E)|_{\lambda(t)} = 2H(E,E)|_{\lambda(t)} \geq 0,$$ where we identified ${\mathcal{V}}_{\lambda(t)} \simeq T_{\gamma(t)}^*M$ and we see the Hamiltonian as a symmetric bilinear form on fibers. In the Riemannian case, $C_2(t) > 0$. In turn, the Jacobi equation, written in terms of the components $(p(t),x(t))$, becomes $$\begin{pmatrix}\label{eq:Jacobicoord} \dot{p} \\ \dot{x} \end{pmatrix} = \begin{pmatrix} - C_1(t) & -R(t) \\ C_2(t) & C_1(t)^{*} \end{pmatrix} \begin{pmatrix} p \\ x \end{pmatrix}.$$ The Riemannian case =================== In the Riemannian case one can choose a suitable frame to simplify  as much as possible. Let $X_1,\ldots,X_n$ be a parallel transported frame along the geodesic $\gamma(t)$. Let $h_i:T^*M \to {\mathbb{R}}$ be the fiber-wise linear functions, defined by $h_i(\lambda):= \langle \lambda, X_i\rangle$. Indeed $h_{1},\ldots,h_{n}$ define coordinates on each fiber, and the vectors $\partial_{h_{i}}$. We define a moving frame along the extremal $\lambda(t)$ as follows $$E_i:=\partial_{h_i}, \qquad F_i := -\dot{E}_i.$$ One can recover the original parallel transported frame by projection, namely $\pi_* F_i|_{\lambda(t)} = X_i|_{{\gamma}(t)}$. We state here the properties of the moving frame. \[p:riemcan\] The smooth moving frame $\{E_i,F_i\}_{i=1}^n$ satisfies: - $\pi_{*}E_i|_{\lambda(t)}=0$. - It is a Darboux basis, namely $$\sigma(E_i,E_j) = \sigma(F_i,F_j) = \sigma(E_i,F_j) - \delta_{ij} = 0, \qquad i,j=1,\ldots,n.$$ - The frame satisfies the structural equations $$\dot{E}_i = - F_i, \qquad \dot{F}_i = \sum_{j=1}^n R_{ij}(t) E_j,$$ for some smooth family of $n\times n$ symmetric matrices $R(t)$. If $\{{\widetilde{E}}_i,{\widetilde{F}}_j\}_{i=1}^n$ is another smooth moving frame along $\lambda(t)$ satisfying (i)-(iii), for some matrix ${\widetilde{R}}(t)$ then there exist a constant, orthogonal matrix $O$ such that $$\label{eq:orthonormal} {\widetilde{E}}_i|_{\lambda(t)} = \sum_{j=1}^n O_{ij}E_j|_{\lambda(t)}, \qquad {\widetilde{F}}_i|_{\lambda(t)} = \sum_{j=1}^nO_{ij}F_j|_{\lambda(t)}, \qquad {\widetilde{R}}(t) = O R(t) O^*.$$ Thanks to this proposition, the symmetric matrix $R(t)$ induces a well defined quadratic form $\mathfrak{R}_{{\lambda}(t)}:T_{\gamma(t)}M \times T_{\gamma(t)}M\to {\mathbb{R}}$ $${\mathfrak{R}}_{{\lambda}(t)}(v,v) := \sum_{i,j=1}^n R_{ij}(t) v_{i}v_j , \qquad v = \sum_{i=1}^n v_i X_i|_{\gamma(t)}.$$ Indeed one can prove that $$\label{eq:trc} {\mathfrak{R}}_{{\lambda}(t)}(v,v) = g(R^\nabla(v,\dot{\gamma})\dot{\gamma},v), \qquad v \in T_{\gamma(t)}M.$$ The proof is a standard computation that can be found, for instance, in [@BR-comparison Appendix C]. Then, in the Jacobi equation , one has $C_1(t) =0$, $C_2(t) = \mathbb{I}$ (in particular, they are constant matrices), and the only non-trivial block $R(t)$ is the curvature operator along the geodesic: $$\dot x=p, \qquad \dot{p} = -R(t) x,$$ The sub-Riemannian case ======================= The problem of finding a the set of Darboux frames normalizing the Jacobi equation has been first studied by Agrachev-Zelenko in [@agzel1; @agzel2] and subsequently completed by Zelenko-Li in [@lizel] in the general setting of curves in the Lagrange Grassmannian. A dramatic simplification, analogous to the Riemannian one, cannot be achieved in the general sub-Riemannian setting. Nevertheless, it is possible to find a normal form of where the matrices $C_{1}$ and $C_{2}$ are constant. Moreover, the very block structure of these matrices depends on the geodesic and already contains important geometric invariants, that we now introduce. Geodesic flag and Young diagram {#s:gfyd} ------------------------------- Let $\gamma(t)$ be a geodesic. Recall that $\dot{\gamma}(t) \in {\mathcal{D}}_{{\gamma}(t)}$ for every $t$. Consider a smooth admissible extension of the tangent vector, namely a vector field ${\mathsf{T}}\in \Gamma({\mathcal{D}})$ such that ${\mathsf{T}}|_{\gamma(t)} = \dot{\gamma}(t)$. \[d:flag\] The *flag of the geodesic* $\gamma(t)$ is the sequence of subspaces $${\mathcal{F}}_{\gamma(t)}^i := \operatorname{\mathrm{span}}\{{\mathcal{L}}_{\mathsf{T}}^j (X)|_{\gamma(t)} \mid X \in \Gamma({\mathcal{D}}),\, j \leq i-1\} \subseteq T_{\gamma(t)} M, \qquad \forall\, i \geq 1,$$ where ${\mathcal{L}}_{{\mathsf{T}}}$ denotes the Lie derivative in the direction of ${\mathsf{T}}$. By definition, this is a filtration of $T_{\gamma(t)}M$, i.e. ${\mathcal{F}}_{\gamma(t)}^i \subseteq {\mathcal{F}}_{\gamma(t)}^{i+1}$, for all $i \geq 1$. Moreover, ${\mathcal{F}}_{\gamma(t)}^1 = {\mathcal{D}}_{\gamma(t)}$. Definition \[d:flag\] is well posed, namely does not depend on the choice of the admissible extension ${\mathsf{T}}$ (see [@curvature Sec. 3.4]). The *growth vector* of the geodesic $\gamma(t)$ is the sequence of integer numbers $$\mathcal{G}_{\gamma(t)} := \{\dim {\mathcal{F}}_{\gamma(t)}^1,\dim {\mathcal{F}}_{\gamma(t)}^2,\ldots\}.$$ A geodesic $\gamma(t)$, with growth vector $\mathcal{G}_{\gamma(t)}$, is said - *equiregular* if $\dim {\mathcal{F}}_{\gamma(t)}^i$ does not depend on $t$ for all $i \geq 1$, - *ample* if for all $t$ there exists $m \geq 1$ such that $\dim {\mathcal{F}}_{\gamma(t)}^{m} = \dim T_{\gamma(t)}M$. Equiregular (resp. ample) geodesics are the microlocal counterpart of equiregular (resp. bracket-generating) distributions. Let $d_i:= \dim {\mathcal{F}}_\gamma^i - \dim {\mathcal{F}}_\gamma^{i-1}$, for $i\geq 1$, be the increment of dimension of the flag of the geodesic at each step (with the convention $\dim\mathcal{F}^0=0$). \[l:decreasing\] For an equiregular, ample geodesic, $d_1 \geq d_2 \geq \ldots \geq d_m$. The generic geodesic is ample and equiregular. More precisely, the set of points $x \in M$ such that there a exists non-empty Zariski open set $A_{x} \subseteq T_{x}^*M$ of initial covectors for which the associated geodesic is ample and equiregular with the same (maximal) growth vector, is open and dense in $M$. See [@curvature; @lizel] for more details. For an ample, equiregular geodesic we can build a tableau ${D}$ with $m$ columns of length $d_{i}$, for $i=1,\ldots,m$, as follows: Indeed $\sum_{i=1}^m d_i = n=\dim M$ is the total number of boxes in ${D}$. Consider an ample, equiregular geodesic, with Young diagram ${D}$, with $k$ rows, of length $n_1,\ldots,n_k$. Indeed $n_1+\ldots+n_k = n$. The moving frame we are going to introduce is indexed by the boxes of the Young diagram. The notation $ai \in {D}$ denotes the generic box of the diagram, where $a=1,\ldots,k$ is the row index, and $i=1,\ldots,n_a$ is the progressive box number, starting from the left, in the specified row. We employ letters $a,b,c,\dots$ for rows, and $i,j,h,\dots$ for the position of the box in the row. We collect the rows with the same length in ${D}$, and we call them *levels* of the Young diagram. In particular, a level is the union of $r$ rows ${D}_1,\ldots,{D}_r$, and $r$ is called the *size* of the level. The set of all the boxes $ai \in{D}$ that belong to the same column and the same level of ${D}$ is called *superbox*. We use Greek letters $\alpha,\beta,\dots$ to denote superboxes. Notice that that two boxes $ai$, $bj$ are in the same superbox if and only if $ai$ and $bj$ are in the same column of ${D}$ and in possibly distinct row but with same length, i.e. if and only if $i=j$ and $n_a = n_b$ (see Fig. \[f:Yd2\]). The following theorem is proved in [@lizel]. \[p:can\] Assume ${\lambda}(t)$ is the lift of an ample and equiregular geodesic ${\gamma}(t)$ with Young diagram ${D}$. Then there exists a smooth moving frame $\{E_{ai},F_{ai}\}_{ai \in {D}}$ along $\lambda(t)$ such that - $\pi_{*}E_{ai}|_{\lambda(t)}=0$. - It is a Darboux basis, namely $$\sigma(E_{ai},E_{bj}) = \sigma(F_{ai},F_{bj}) = \sigma(E_{ai},F_{bj}) = \delta_{ab}\delta_{ij}, \qquad ai,bj \in {D}.$$ - The frame satisfies structural equations $$\label{zelframe} \displaystyle\begin{cases} \dot{E}_{ai} = E_{a(i-1)} & a = 1,\dots,k,\quad i = 2,\dots, n_a,\\[0.1cm] \dot{E}_{a1} = -F_{a1} & a= 1,\dots,k, \\[0.1cm] \dot{F}_{ai} = \sum_{bj \in {D}} R_{ai,bj}(t) E_{bj} - F_{a(i+1)} & a=1,\dots,k,\quad i = 1,\dots,n_a-1,\\[0.1cm] \dot{F}_{an_a} = \sum_{bj \in {D}} R_{an_a,bj}(t) E_{bj} & a = 1, \dots,k, \end{cases}$$ for some smooth family of $n\times n$ symmetric matrices $R(t)$, with components $R_{ai,bj}(t) = R_{bj,ai}(t)$, indexed by the boxes of the Young diagram ${D}$. The matrix $R(t)$ is *normal* in the sense of [@lizel] (see Appendix \[s:appendixnormal\]). If $\{{\widetilde{E}}_{ai},{\widetilde{F}}_{ai}\}_{ai \in {D}}$ is another smooth moving frame along $\lambda(t)$ satisfying (i)-(iii), with some normal matrix ${\widetilde{R}}(t)$, then for any superbox $\alpha$ of size $r$ there exists an orthogonal constant $r\times r$ matrix $O^\alpha$ such that $${\widetilde{E}}_{ai} = \sum_{bj \in \alpha} O^\alpha_{ai,bj} E_{bj}, \qquad {\widetilde{F}}_{ai} = \sum_{bj \in \alpha} O^\alpha_{ai,bj} F_{bj}, \qquad ai \in \alpha.$$ \[rmk:notation\] For $a=1,\dots,k$, the symbol $E_a$ denotes the $n_a$-dimensional column vector $ E_a = (E_{a1},E_{a2},\dots,E_{an_a})^*, $ with analogous notation for $F_a$. Similarly, $E$ denotes the $n$-dimensional column vector $ E = (E_1,\dots,E_k)^*, $ and similarly for $F$. Then, we rewrite the system as follows (compare with ) $$\label{eq:Jacobiframe2} \begin{pmatrix} \dot{E} \\ \dot{F} \end{pmatrix} = \begin{pmatrix} C^*_1 & -C_2 \\ R(t) & -C_1 \end{pmatrix} \begin{pmatrix} E\\ F \end{pmatrix},$$ where $C_1 = C_1({D})$, $C_2=C_2({D})$ are $n\times n$ matrices, depending on the Young diagram ${D}$, defined as follows: for $a,b = 1,\dots,k$, $i=1,\dots,n_a$, $j=1,\dots,n_b$: $$[C_1]_{ai,bj} := \delta_{ab}\delta_{i,j-1}, \label{eq:G1},\qquad [C_2]_{ai,bj} := \delta_{ab}\delta_{i1}\delta_{j1}. $$ It is convenient to see $C_1$ and $C_2$ as block diagonal matrices: $$C_i({D}) := \begin{pmatrix} C_i({D}_1) & & \\ & \ddots & \\ & & C_i({D}_k) \end{pmatrix}, \qquad i =1,2,$$ the $a$-th block being the $n_a\times n_a$ matrices $$\label{eq:Gamma} C_1({D}_a) := \begin{pmatrix} 0 & \mathbb{I}_{n_a-1} \\ 0 & 0 \end{pmatrix} , \qquad C_2({D}_a) := \begin{pmatrix} 1 & 0 \\ 0 & 0_{n_a-1} \end{pmatrix},$$ where $\mathbb{I}_{m}$ is the $m \times m$ identity matrix and $0_{m}$ is the $m \times m$ zero matrix. Notice that the matrices $C_{1},C_{2}$ satisfy the Kalman rank condition $$\label{eq:Kalman} \operatorname{\mathrm{rank}}\{C_{2},C_{1}C_{2},\ldots,C_{1}^{n-1}C_{2}\}=n.$$ Analogously, the matrices $C_{i}(D_{a})$ satisfy with $n=n_{a}$. Let $\{X_{ai}\}_{ai \in {D}}$ be the moving frame along $\gamma(t)$ defined by $X_{ai}|_{\gamma(t)}=\pi_{*}F_{ai}|_{{\lambda}(t)}$, for some choice of a canonical Darboux frame. Theorem \[p:can\] implies that the following definitions are well posed. The *canonical splitting* of $T_{\gamma(t)} M$ is $$T_{\gamma(t)}M = \bigoplus_{\alpha}S_{\gamma(t)}^{\alpha}, \qquad S_{\gamma(t)}^{\alpha}:=\operatorname{\mathrm{span}}\{ X_{ai}|_{\gamma(t)}\mid \, ai \in \alpha\},$$ where the sum is over the superboxes $\alpha$ of ${D}$. Notice that the dimension of $S_{\gamma(t)}^{\alpha}$ is equal to the size $r$ of the level to which the superbox $\alpha$ belongs. \[d:curv\] The *canonical curvature* (along $\lambda(t)$), is the quadratic form ${\mathfrak{R}}_{{\lambda}(t)}: T_{\gamma(t)} M \times T_{\gamma(t)} M\to {\mathbb{R}}$ whose representative matrix, in terms of the basis $\{X_{ai}\}_{ai \in {D}}$, is $R_{ai,bj}(t)$. In other words $$\label{eq:SR-dircurv} {\mathfrak{R}}_{{\lambda}(t)}(v,v) := \sum_{ai,bj\in{D}} R_{ai,bj}(t) v_{ai}v_{bj}, \qquad v = \sum_{ai\in{D}} v_{ai} X_{ai}|_{\gamma(t)} \in T_{\gamma(t)}M.$$ We denote the restrictions of $\mathfrak{R}_{\lambda(t)}$ on the appropriate subspaces by: $$\mathfrak{R}^{\alpha\beta}_{\lambda(t)} : S^{\alpha}_{\gamma(t)} \times S^{\beta}_{\gamma(t)} \to {\mathbb{R}}.$$ For any superbox $\alpha$ of $D$, the *canonical Ricci curvature* is the partial trace: $$\mathfrak{Ric}_{\lambda(t)}^\alpha:= \sum_{ai \in \alpha} {\mathfrak{R}}_{\lambda(t)}^{\alpha\alpha}(X_{ai},X_{ai}).$$ The Jacobi equation, written in terms of the components $(p(t),x(t))$ with respect to a canonical Darboux frame $\{E_{ai},F_{ai}\}_{ai \in {D}}$, becomes $$\begin{pmatrix}\label{eq:Jacobicoord2} \dot{p} \\ \dot{x} \end{pmatrix} = \begin{pmatrix} - C_1 & -R(t) \\ C_2 & C_1^* \end{pmatrix} \begin{pmatrix} p \\ x \end{pmatrix}.$$ This is the sub-Riemannian generalization of the classical Jacobi equation seen as first-order equation for fields on the cotangent bundle. Its structure depends on the Young diagram of the geodesic through the matrices $C_{i}({D})$, while the remaining invariants are contained in the curvature matrix $R(t)$. Notice that this includes the Riemannian case, where ${D}$ is the same for every geodesic, with $C_{1}=0$ and $C_{2}=\mathbb{I}$. Homogeneity properties ---------------------- For all $c>0$, let $H_c := H^{-1}(c/2)$ be the Hamiltonian level set. In particular $H_1$ is the unit cotangent bundle: the set of initial covectors associated with unit-speed geodesics. Since the Hamiltonian function is fiber-wise quadratic, we have the following property for any $c>0$ $$\label{eq:commutation} e^{t \vec{H}}(c \lambda) = c e^{c t\vec{H}}(\lambda),$$ where, for $\lambda \in T^*M$, the notation $c \lambda$ denotes the fiber-wise multiplication by $c$. Let $P_c : T^*M \to T^*M$ be the map $P_c(\lambda) = c\lambda$. Indeed $\alpha \mapsto P_{e^\alpha}$ is a one-parameter group of diffeomorphisms. Its generator is the *Euler vector field* $\mathfrak{e} \in \Gamma({\mathcal{V}})$, and is characterized by $P_c = e^{(\ln c)\mathfrak{e}}$. We can rewrite  as the following commutation rule for the flows of $\vec{H}$ and $\mathfrak{e}$: $$e^{t\vec{H}} \circ P_c = P_c \circ e^{ c t \vec{H}}.$$ Observe that $P_c$ maps $H_1$ diffeomorphically on $H_{c}$. Let $\lambda \in H_1$ be associated with an ample, equiregular geodesic with Young diagram ${D}$. Clearly also the geodesic associated with $\lambda^c:=c \lambda \in H_{c}$ is ample and equiregular, with the same Young diagram. This corresponds to a reparametrization of the same curve: in fact $\lambda^c(t) =e^{t\vec{H}}(c{\lambda})= c(\lambda(ct))$, hence $\gamma^c(t) = \pi(\lambda^c(t))= \gamma(ct)$. \[t:homogR\] For any superbox $\alpha \in {D}$, let $|\alpha|$ denote the column index of $\alpha$. Denoting ${\lambda}^{c}(t)=e^{t\vec{H}}(c{\lambda})$ we have, for any $c>0$ $$\mathfrak{R}^{\alpha\beta}_{\lambda^{c}(t)} = c^{|\alpha|+ |\beta|} \mathfrak{R}^{\alpha\beta}_{\lambda(ct)},$$ In the Riemannian setting, ${D}$ has only one superbox with $|{\alpha}|=1$ (see Fig. \[f:Yd2\]). Then $\mathfrak{R}_{\lambda}:=\mathfrak{R}^{\alpha\alpha}_{\lambda(0)}$ is homogeneous of degree $2$ as a function of ${\lambda}$. Theorem \[t:homogR\] follows directly from the next result and Definition \[d:curv\]. In the next proposition, for any $\eta \in T^*M$ and $c >0$, we denote with $d_{\eta} P_c : T_{\eta}(T^*M) \to T_{c\eta}(T^*M)$ the differential of the map $P_c$, computed at $\eta$. \[p:framescaling\] Let $\lambda \in H_1$ and $\{E_{ai},F_{ai}\}_{ai \in {D}}$ be the associated canonical frame along the extremal $\lambda(t)$. Let $c>0$ and define, for $ai \in {D}$ $$E^c_{ai}(t):=\frac{1}{c^i}(d_{\lambda(ct)} P_{c}) E_{ai}(ct), \qquad F_{ai}^c(t):=c^{i-1}(d_{\lambda(ct)} P_c) F_{ai}(ct).$$ The moving frame $\{E^c_{ai}(t),F^c_{ai}(t)\}_{ai \in {D}} \in T_{\lambda^c(t)}(T^*M)$ is a canonical frame associated with the initial covector $\lambda^c =c\lambda\in H_{c}$, with curvature matrix $$\label{eq:homog} R^{\lambda^{c}}_{ai,bj}(t) = c^{i+j} R^\lambda_{ai,bj}(ct).$$ We check all the relations of Theorem \[p:can\]. Indeed $P_\alpha$ sends fibers to fibers, hence (i) is trivially satisfied. For what concerns (ii), let $\theta$ be the Liouville one-form, and $\sigma =d\theta$. Indeed $P_c^* \theta = c\theta$. Hence $P_c^*\sigma = c\sigma$. It follows that $\{E^c_{ai}(t),F^c_{ai}(t)\}_{ai \in {D}}$ is a Darboux frame at $\lambda^c(t)$: $$\sigma_{\lambda^c(t)}(E^c_{ai}(t),F^c_{bj}(t)) = \tfrac{1}{c} (P_c^* \sigma)_{\lambda(t)}(E_{ai}(t),F_{bj}(t)) = \delta_{ab}\delta_{ij},$$ and similarly for the others Darboux relations. For what concerns (iii) (the structural equations), let $\xi(t)$ be any vector field along $\lambda(t)$, and $(d_{\lambda(t)} P_c) \xi(ct)$ be the corresponding vector field along $\lambda^c(t)$. Then $$\begin{aligned} \left.\frac{d}{d{\varepsilon}}\right|_{{\varepsilon}=0} e^{-{\varepsilon}\vec{H}}_* \circ (d_{\lambda(t)} P_c) \xi (c(t+{\varepsilon})) & =\left.\frac{d}{d{\varepsilon}}\right|_{{\varepsilon}=0} (e^{-{\varepsilon}\vec{H}} \circ P_c)_* \xi(c(t+{\varepsilon})) \\ & =\left.\frac{d}{d{\varepsilon}}\right|_{{\varepsilon}=0} (P_c \circ e^{-c{\varepsilon}\vec{H}})_* \xi(c(t+{\varepsilon})) \\ & =\left.c\frac{d}{d\tau}\right|_{\tau =0} (P_c \circ e^{-\tau \vec{H}})_* \xi(ct+\tau) \\ & = c (d_{\lambda(ct)}P_c) \dot\xi(ct).\end{aligned}$$ Applying the above identity to compute the derivatives of the new frame, and using , one finds that $\{E^c_{ai}(t),F^c_{ai}(t)\}_{ai \in {D}}$ satisfies the structural equations, with curvature matrix given by . For example $$\begin{aligned} \dot{F}_{ai}^c(t) & = c^{i-1} c (d_{\lambda(ct)} P_c) \dot{F}_{ai}(ct) \\ & = c^i (d_{\lambda(ct)}P_c) [R^{{\lambda}}_{ai,bj}(ct) E_{bj}(ct) - F_{a(i+1)}(ct)] \\ & = c^i [c^j R^{{\lambda}}_{ai,bj}(ct) E_{bj}^c(t) - c^{-i} F^c_{a(i+1)}(t)] \\ & = c^{i+j}R^{{\lambda}}_{ai,bj}(ct) E_{bj}^c(t) - F^c_{a(i+1)}(t),\end{aligned}$$ where we suppressed a summation over $bj \in {D}$. Proposition \[p:can\] defines not only a curvature, but also a (non-linear) connection, in the sense of Ehresmann, that we now introduce. Ehresmann curvature and curvature operator ========================================== For any smooth vector bundle $N$ over $M$, let $\Gamma(N)$ denote the smooth sections of $N$. Recall that ${\mathcal{V}}:=\ker \pi_* \subset T(T^*M)$ is the *vertical distribution*. An *Ehresmann connection* on $T^{*}M$ is a smooth distribution ${\mathcal{H}}\subset T (T^*M)$ such that $$T(T^*M) = {\mathcal{H}}\oplus {\mathcal{V}}.$$ We call ${\mathcal{H}}$ the *horizontal distribution*[^2]. An Ehresmann connection ${\mathcal{H}}$ is *linear* if ${\mathcal{H}}_{c{\lambda}}=(d_{{\lambda}}P_{c} ) {\mathcal{H}}_{{\lambda}}$ for every ${\lambda}\in T^{*}M$ and $c>0$. For any $X \in \Gamma(TM)$ there exists a unique *horizontal lift* $\nabla_X $ in $ \Gamma({\mathcal{H}})$ such that $\pi_* \nabla_X = X$. A function $h \in C^\infty(T^*M)$ is fiber-wise linear if it can be written as $h({\lambda})={\langle}{\lambda},Y{\rangle}$, for some $Y\in \Gamma(TM)$. Such an $Y$ is clearly unique, and for this reason we denote $h_Y := {\lambda}\mapsto {\langle}{\lambda},Y{\rangle}$ the fiber-wise linear function associated with $Y \in \Gamma(TM)$. A connection $\nabla$ is linear if, for every $X\in \Gamma(TM)$, the derivation $\nabla_{X}$ maps fiber-wise linear functions to fiber-wise linear functions. In this case, we recover the classical notion of covariant derivative by defining $\nabla_{X}Y=Z$ if $\nabla_{X}h_{Y}=h_{Z}$, where $Y,Z\in \Gamma(TM)$. We recall the definition of curvature of an Ehresmann connection [@KNFDG]. The *Ehresmann curvature* of the connection $\nabla$ is the $C^\infty(M)$-linear map $R^{\nabla}: \Gamma(TM) \times \Gamma(TM) \to \Gamma({\mathcal{V}})$ defined by $$R^{\nabla}(X,Y) = [\nabla_X,\nabla_Y] - \nabla_{[X,Y]}, \qquad X,Y \in \Gamma(TM).$$ $R^{\nabla}$ is skew-symmetric, namely $R^{\nabla}(X,Y) = -R^{\nabla}(Y,X)$. Notice that $R^{\nabla}=0$ if and only if ${\mathcal{H}}$ is involutive. Canonical connection -------------------- Let ${\gamma}(t)$ be a fixed ample and equiregular geodesic with Young diagram ${D}$, projection of the extremal ${\lambda}(t)$, with initial covector ${\lambda}$. Let $\{E_{ai}(t),F_{ai}(t)\}$ be a canonical frame along $\lambda(t)$. For $t=0$, this defines a subspace at ${\lambda}\in T^*M$, namely $$\label{eq:cancon} {\mathcal{H}}_\lambda := \operatorname{\mathrm{span}}\{F_{ai}|_{\lambda}\}_{ai \in {D}}, \qquad \lambda \in T^*M .$$ Indeed this definition makes sense on the subset of covectors $N\subset T^{*}M$ associated with ample and equiregular geodesics. In the Riemannian case, every non-trivial geodesic is ample and equiregular, with the same Young diagram. Hence $N=T^{*}M\setminus H^{-1}(0)$. A posteriori one can show that this connection is linear and can be extended smoothly on the whole $T^*M$. In the sub-Riemannian case, $N\subset T^{*}M\setminus H^{-1}(0)$. In general, using the results of [@curvature Section 5.2] and [@lizel Section 5], one can prove that $N$ is open and dense in $T^{*}M$. Moreover, the elements of the frame depend rationally (in charts) on the point $\lambda$, hence ${\mathcal{H}}$ is smooth on $N$. For simplicity, we assume that it is possible to extend ${\mathcal{H}}$ to a smooth distribution on the whole $T^*M$. This is indeed possible in some cases of interest: on corank $1$ structures with symmetries [@lizel2] and on contact sub-Riemannian structures [@nostrocontact] (see also [@RS-3-Sasakian] for fat structures). In the general case, we replace $T^*M$ with $N$. The *canonical Ehresmann connection* associated with the sub-Riemannian structure is the horizontal distribution ${\mathcal{H}}\subset T (T^{*}M)$ defined by . As a consequence of Proposition \[p:framescaling\], ${\mathcal{H}}$ is non-linear, in general. However, if the structure is Riemannian, one has ${\mathcal{H}}_{c{\lambda}}=(d_{{\lambda}}P_{c} ) {\mathcal{H}}_{{\lambda}}$ and the connection is linear. Let $H$ be the sub-Riemannian Hamitonian and ${\mathcal{H}}$ the canonical connection. Then $\nabla_{X} H=0$ for every $X\in\Gamma(TM)$. Equivalently, $\vec{H}\in {\mathcal{H}}$. The above condition is the compatibility of the canonical connection with the sub-Riemannian metric. In the Riemannian setting, ${\mathcal{H}}$ is linear and this condition can be rewritten, in the sense of covariant derivative, as $\nabla g=0$. The equivalence of the two statements follows from the definition of Hamiltonian vector field and the fact that ${\mathcal{H}}$ is Lagrangian, by construction. Indeed $$\nabla_{X}H=dH(\nabla_{X})=\sigma(\vec{H},\nabla_{X}).$$ Then we prove that $\vec H\in {\mathcal{H}}$. \[l:dere\] Let $\mathfrak{e}$ be the Euler vector field. Then $\dot{\mathfrak{e}} = -\vec{H}$. Let $P_s = e^{(\ln s) \mathfrak{e}}$ be the dilation along the fibers. We have the following commutation rule for the flows of $\vec{H}$ and $\mathfrak{e}$ $$P_{-s} \circ e^{-t\vec{H}} \circ P_s = e^{ -ts\vec{H}}.$$ Computing the derivative w.r.t $t$ and $s$ at $(t,s) = (0,1)$ we obtain $[\vec{H},\mathfrak{e}] = - \mathfrak{e}$, that implies the statement. \[l:mform0\] Since $\mathfrak{e}$ is vertical, then $\mathfrak{e} = v(t)^* E(t)$ for some smooth $v(t) \in {\mathbb{R}}^n$. Accordingly with the decomposition of Remark \[rmk:notation\], we set $$v(t) = (v_1(t),\ldots,v_k(t))^{*}, \quad \text{with} \quad v_a(t) = (v_{a1}(t),\ldots,v_{an_a}(t))^{*}.$$ Then $v(t)$ is constant and we have $$\mathfrak{e}=\sum_{\substack{ai\in {D}\\ n_{a}=1}} v_{ai} E_{ai}.$$ As a consequence of Lemma \[l:dere\], $\ddot{\mathfrak{e}} = 0$. Using the structural equations , we obtain $$\begin{aligned} C_1^* C_2v - C_2 C_1 v - 2 C_2 \dot{v} & = 0, \label{eq:Fpart}\\ \ddot{v} + 2 C_1 \dot{v} + C_1^2 v - RC_2 v &= 0. \label{eq:Epart}\end{aligned}$$ We show that for any row index of the Young diagram $a=1,\ldots,k$ $$v_a = \begin{cases} (0,\ldots,0)^{*} & n_a > 1, \\ \text{constant} & n_a = 1. \end{cases}$$ Let us focus on . For each $a=1,\ldots,k,$ we take its $a$-th block. By the block structure of $C_1$ and $C_2$, this is $$\label{eq:Fpart2} C_1^* C_2 v_a - C_2 C_1 v_a - 2 C_2 \dot{v}_a = 0, \qquad \forall\, a=1,\ldots,k,$$ where here $C_1 = C_1({D}_a)$ and $C_2 = C_2({D}_a)$. If $n_a = 1$, then $C_1 = 0$ and $C_2 = 1$. In this case  implies $v_a(t) = v_a$ is constant. Now let $n_a > 1$. In this case, the particular form of $C_1,C_2$ for  yields $$C_1^* C_2 v_a = 0, \qquad \text{and} \qquad C_2 C_1 v_a + 2C_2 \dot{v}_a = 0, \qquad (n_a > 1).$$ Indeed the kernel of $C_1^*$ is orthogonal to the image of $C_2$. Hence $C_1^* C_2 v_a=0$ implies $C_2 v_a = 0$. In particular  is equivalent to $$\label{eq:Fpart3} C_2 v_a = 0, \qquad C_2 C_1 v_a = 0, \qquad (n_a > 1).$$ More explicitly, $v_a = (0,0,v_{a3},\ldots,v_{an_a})$. For the case $n_a =2$ this is sufficient to completely determine $v_a$. In all the other cases, let us turn to . The latter does not split immediately, as the curvature matrix $R$ is not block-diagonal. However, let us consider a copy of  multiplied by $C_2 C_1^i$. For each $a$ such that $n_{a}>2$ we consider its $a$-th block, obtaining the following: $$\label{eq:Epart2} C_2 C_1^i \ddot{v}_a + 2 C_2 C^{i+1}_1 \dot{v}_a + C_2 C_1^{i+2} v_a - [C_2 C_1^i RC_2 v]_a = 0, \qquad (n_a > 2).$$ We claim that $[C_2 C_1^i RC_2 v]_a = 0$ if $n_a > 2$ and $i < n_a -2$. By setting the matrix $[R_{ab}]_{ij} := R_{ai,bj}$, with $ai,bj \in {D}$ (this is a block of $R$, corresponding to the rows $a,b$ of the Young diagram ${D}$), we compute $$\begin{aligned} [C_2 C_1^i R C_2 v]_a & = \sum_{b,c,d=1}^k [C_2 C_1^i]_{ab} R_{bc} [C_2]_{cd} v_d = \sum_{b=1}^k (C_2 C_1^i) R_{ab} (C_2 v_b) \\ & = \sum_{n_b =1} (C_2 C_1^i) R_{ab} (C_2 v_b) = \sum_{n_b =1} R_{a(i+1),b1} v_{b1},\end{aligned}$$ where we used the block structure of the $C_i$’s and . The last sum involves only $R_{a(i+1),b1}$ with $n_b =1$ and $n_a > 2$. If $i< n_a -2$, then $R_{a(i+1),b1}$ is *not* in the last $2n_b = 2$ elements of Table \[t:normaltable\], and vanishes by the normal conditions (see Appendix \[s:appendixnormal\]). Thus we have: $$\label{eq:Epart3} C_2 C_1^i \ddot{v}_a + 2 C_2 C^{i+1}_1 \dot{v}_a + C_2 C_1^{i+2} v_a=0 , \qquad (n_a > 2, \quad i < n_a -2).$$ In particular using , and taking $i=0,\ldots,n_a-3$ we see that  is equivalent to $C_2 C_1^{i+2} v_a = 0$ for all $i=0,\ldots,n_{a}-3$. Combining all the cases $$v_a \in \ker\{ C_2, C_2 C_1, C_2 C_1^2,\ldots,C_2 C_1^{n_a -1}\}, \qquad (n_a > 1).$$ This yields $v_a = 0,$ by Kalman rank condition . Lemma \[l:mform0\] implies our statement since $$\vec{H}=-\dot{\mathfrak{e}}=-\sum_{\substack{ai\in {D}\\ n_{a}=1}} v_{ai} \dot E_{ai}=\sum_{\substack{ai\in {D}\\ n_{a}=1}} v_{ai} F_{ai}\in {\mathcal{H}},$$ where we used the structural equations for the $E_{ai}$’s with $n_{a}=1$. Relation with the canonical curvature ------------------------------------- We now discuss the relation between the curvature of the canonical Ehresmann connection and the sub-Riemannian curvature operator. In what follows we denote by ${\mathfrak{R}}_{\lambda}:={\mathfrak{R}}_{{\lambda}(0)}$, where ${\lambda}(t)$ is the extremal with initial datum ${\lambda}$. Then ${\mathfrak{R}}$ extends to a well defined map $$\label{eq:cancurva} \begin{gathered} {\mathfrak{R}}: \Gamma(T^*M) \times \Gamma(TM) \times \Gamma(TM) \to C^\infty(M), \\ (\lambda, X, Y) \mapsto {\mathfrak{R}}_\lambda(X,Y). \end{gathered}$$ We stress that here the first argument is a section ${\lambda}\in\Gamma(T^*M)$. Although ${\mathfrak{R}}$ is $C^\infty(M)$-linear in the last two arguments by construction, it is in general non-linear in the first argument, so it does not define a $(1,2)$ tensor. Nevertheless, for any fixed section $\lambda \in \Gamma(T^*M)$, the restriction ${\mathfrak{R}}_\lambda : \Gamma(TM) \times \Gamma(TM) \to C^\infty(M)$ is a $(0,2)$ symmetric tensor. \[t:canehr\] Let $R^{\nabla} : \Gamma(TM) \times \Gamma(TM) \to \Gamma({\mathcal{V}})$ be the curvature of the canonical Ehresmann connection, and let ${\mathfrak{R}}: \Gamma(T^*M) \times \Gamma(TM) \times \Gamma(TM) \to C^\infty(M)$ be the canonical curvature map . Then $$\label{eq:relation} {\mathfrak{R}}_\lambda(X,Y) = \sigma_\lambda(R^{\nabla}({\mathsf{T}},X),\nabla_Y), \qquad \forall\,\lambda \in \Gamma(T^*M), \quad X,Y \in \Gamma(TM),$$ where ${\mathsf{T}}= \pi_*\vec{H}|_\lambda \in \Gamma(TM)$. We evaluate the right hand side of at the point $x$, for any fixed section $\lambda={\lambda}(x) \in \Gamma(T^*M)$. By linearity, it is sufficient to take $X = X_{ai}$ and $Y=Y_{bj}$, projections of a canonical frame $F_{ai}|_{{\lambda}},F_{bj}|_{{\lambda}}$ at $t=0$. Indeed, by definition, $ \nabla_{X_{ai}}|_{{\lambda}} = F_{ai}|_{{\lambda}}$. Then $$\begin{aligned} \sigma_\lambda(R^\nabla({\mathsf{T}},X_{ai}),\nabla_{X_{bj}}) & = \sigma_\lambda([\nabla_{{\mathsf{T}}},F_{ai}],F_{bj}) = \sigma_\lambda([\vec{H},F_{ai}],F_{bj}) \\ & = \sigma_\lambda(\dot{F}_{ai}, F_{bj}) = R^\lambda_{ai,bj}(0).\end{aligned}$$ Here we used the structural equations and that $\vec{H} \in {\mathcal{H}}$, thus $\nabla_{{\mathsf{T}}} = \vec{H}$. By definition of canonical curvature map, we obtain the statement. For $\lambda \in \Gamma(T^*M)$, the corresponding tangent field ${\mathsf{T}}\in \Gamma({\mathcal{D}}) \subsetneq \Gamma(TM)$. Therefore, ${\mathfrak{R}}$ recovers only part of the whole Ehresmann connection. As we proved in , we have $${\mathfrak{R}}_\lambda(X,Y) =R^\nabla({\mathsf{T}},X,Y,{\mathsf{T}}),$$ where ${\mathsf{T}}= \pi_*\vec{H}|_\lambda$ is the tangent vector associated with the covector $\lambda$. For completeness, let us recover the same formula by the r.h.s. of . Indeed, for any vertical vector $V \in {\mathcal{V}}_\lambda$ and $W \in T_{\lambda}(T^*M)$, we have $\sigma_{\lambda}(V,W) = V(h_{\pi_*W})|_{\lambda}$ as one can check from a direct computation. Thus the r.h.s. of is $$\begin{aligned} \sigma_\lambda([\nabla_{\mathsf{T}},\nabla_X]-\nabla_{[{\mathsf{T}},X]},\nabla_Y) & = \left(\nabla_{\mathsf{T}}\nabla_X (h_Y) - \nabla_X \nabla_{\mathsf{T}}(h_Y) - \nabla_{[{\mathsf{T}},X]} (h_Y)\right)|_{\lambda} \\ & = h_{\nabla_{{\mathsf{T}}}\nabla_X Y - \nabla_X \nabla_{\mathsf{T}}Y - \nabla_{[{\mathsf{T}},X]} Y}(\lambda) \\ & = \langle \lambda, \nabla_{{\mathsf{T}}}\nabla_X Y - \nabla_X \nabla_{\mathsf{T}}Y - \nabla_{[{\mathsf{T}},X]} Y \rangle \\ & = g(\nabla_{{\mathsf{T}}}\nabla_X Y - \nabla_X \nabla_{\mathsf{T}}Y - \nabla_{[{\mathsf{T}},X]} Y, {\mathsf{T}}) \\ & = R^\nabla({\mathsf{T}},X,Y,{\mathsf{T}}).\end{aligned}$$ Normal condition for the canonical frame {#s:appendixnormal} ======================================== Here we rewrite the *normal* condition for the matrix $R_{ai,bj}$ mentioned in Theorem \[p:can\] (and defined in [@lizel]) according to our notation. The matrix $R_{ai,bj}$ is *normal* if it satisfies: - global symmetry: for all $ai,bj\in {D}$ $$R_{ai,bj}=R_{bj,ai}.$$ - partial skew-symmetry: for all $ai,bi\in {D}$ with $n_{a}=n_{b}$ and $i<n_{a}$ $$R_{ai,b(i+1)}=- R_{bi,a(i+1)}.$$ - vanishing conditions: the only possibly non vanishing entries $R_{ai,bj}$ satisfy - $n_{a}=n_{b}$ and $|i-j|\leq 1$, - $n_{a}>n_{b}$ and $(i,j)$ belong to the last $2n_{b}$ elements of Table \[t:normaltable\]. $i$ $1$ $1$ $2$ $\cdots$ $\ell$ $\ell$ $\ell+1$ $\cdots$ $n_{b}$ $n_{b}+1$ $\cdots$ $n_a-1$ $n_a$ ----- ----- ----- ----- ---------- -------- ---------- ---------- ---------- --------- ----------- ---------- --------- ------- $j$ $1$ $2$ $2$ $\cdots$ $\ell$ $\ell+1$ $\ell+1$ $\cdots$ $n_b$ $n_b$ $\cdots$ $n_b$ $n_b$ : Vanishing conditions. \[t:normaltable\] The sequence is obtained as follows: starting from $(i,j)=(1,1)$ (the first boxes of the rows $a$ and $b$), each next even pair is obtained from the previous one by increasing $j$ by one (keeping $i$ fixed). Each next odd pair is obtained from the previous one by increasing $i$ by one (keeping $j$ fixed). This stops when $j$ reaches its maximum, that is $(i,j) = (n_b,n_b)$. Then, each next pair is obtained from the previous one by increasing $i$ by one (keeping $j$ fixed), up to $(i,j) = (n_a,n_b)$. The total number of pairs appearing in the table is $n_b+n_a-1$. Acknowledgments {#acknowledgments .unnumbered} =============== This research has been supported by the European Research Council, ERC StG 2009 “GeCoMethods”, contract number 239748, by the iCODE institute (research project of the Idex Paris-Saclay), and by the Grant ANR-15-CE40-0018 of the ANR. This research benefited from the support of the “FMJH Program Gaspard Monge in optimization and operation research” and from the support to this program from EDF. [^1]: A Lagrangian subspace $L \subset \Sigma$ of a symplectic vector space $(\Sigma,\sigma)$ is a subspace with $\dim L = \dim\Sigma/2$ and $\sigma|_{L} = 0$. [^2]: Note that this is a distribution on $T^{*}M$, i.e. a sub-bundle of $T(T^*M)$ and should not be confused with the sub-Riemannian distribution ${\mathcal{D}}$, that is a subbundle of $TM$.

--- abstract: 'Energy harvesting (EH) in wireless communications has become the focus of recent transmission technology studies. Herein, energy storage modeling is one of the crucial design benchmarks that must be treated carefully. Understanding the energy storage dynamics and the throughput levels is essential especially for communication systems in which the performance depends solely on harvested energy. While energy outages should be avoided, energy overflows should also be prevented in order to utilize all harvested energy. Hence, a simple, yet comprehensive, analytical model that can represent the characteristics of a general class of EH wireless communication systems needs to be established. In this paper, invoking tools from large deviation theory along with Markov processes, a firm connection between the energy state of the battery and the data transmission process over a wireless channel is established for an EH transmitter. In particular, a simple exponential approximation for the energy overflow probability is formulated, with which the energy decay rate in the battery as a measure of energy usage is characterized. Then, projecting the energy outages and supplies on a Markov process, a discrete state model is established and an expression for the energy outage probability for given energy arrival and demand processes is provided. Finally, under energy overflow and outage constraints, the *average data service (transmission) rate* over the wireless channel is obtained and the *effective capacity* of the system, which characterizes the maximum data arrival rate at the transmitter buffer under quality-of-service (QoS) constraints imposed on the data buffer overflow probability, is derived.' author: - | \ [^1] [^2] [^3] bibliography: - 'IEEEabrv.bib' - 'references.bib' title: On the Energy and Data Storage Management in Energy Harvesting Wireless Communications --- Energy harvesting, energy overflow, energy outage, effective capacity, queueing theory, large deviation theory, Markov process. Introduction ============ Due to the unprecedented growth in the number of wireless devices and systems, requiring an ever-increasing amount of energy, research efforts in recent years have focused on more sophisticated energy management techniques. In particular, energy harvesting (EH) technology has attracted significant interest from the research community as a means both to reduce the carbon footprint of communication networks and to provide increased autonomy to wireless devices [@energy_harvesting_ulukus_2015; @ozel2015fundamental; @gunduz2014designing; @tadayon2013power]. The challenges in these studies arise from the complexity in modeling practical energy storage technologies, variable energy consumption patterns, and the stochastic aspects of natural energy sources, such as wind and solar. Researchers have employed various mathematical models to study energy storage mechanisms for EH systems. Among these models, queueing-theoretic energy quantization models [@tadayon2013power] are arguably the most common and well-established ones. These models can reflect the practical characteristics of energy storage systems relatively accurately, and are generally amenable to mathematical analysis when invoked in communication systems. Concurrently, understanding the energy consumption profile of wireless devices is also essential, as the energy retrieval rate from a storage unit may affect the lifetime of a device. However, while it was relatively easy to model and estimate the energy consumption behavior of a communication device in the past because mostly the circuit, baseband, radio frequency and power amplifier components consumed the energy, nowadays it is much more difficult to accurately understand and model the energy needs of communication devices due to the ever-increasing complexity of modern devices and applications [@rice2010measuring]. Therefore, along with the introduction of EH wireless communication technology, which relies on environment-friendly techniques to generate energy from renewable resources, the effective use of the generated energy to guarantee energy availability when required, led to a paradigm shift in research on radio resource allocation [@ahmed2015survey]. Particularly, in addition to spectral efficiency and quality-of-service (QoS) constraints, economic use of energy has emerged as another requirement. The concern lies in estimating the periodicity and magnitude of the exploited energy source, deciding which parameters to tune, and simultaneously avoiding premature energy depletion before the next recharge cycle [@sudevalayam2011energy]. Hence, the goal is to characterize a tradeoff between the performance levels and lifespan of built-in energy units. This new perspective compels the need to understand energy storage technologies and the associated performance levels. Related Work ------------ Since the natural energy sources are stochastic, the power and data management policies in EH wireless communication systems are different from their counterparts that depend on the grid energy or non-rechargeable batteries. Moreover, considering a system that has a data buffer as well as a battery, the optimal control of an EH wireless communication system requires managing the transmission rate by monitoring both the traffic load and the stored energy, and respecting the causality constraints in both the data and energy arrivals. The authors in [@ozel2011transmission] and [@6202352] controlled the transmit power levels subject to energy storage capacity and causality constraints, and introduced a directional offline water-filling algorithm that optimizes the delay-constrained throughput. Separately, the authors in [@ozel2011transmission] considered a transmitter model with a battery and a data buffer. Assuming that both data and harvested energy packets randomly arrive at the data buffer and the battery, respectively, the authors in [@yang2012optimal] developed optimal offline scheduling policies to adaptively change transmission rates under a deterministic system setting. The authors in [@6144764] and [@6253061] extended this analysis to a transmitter with finite-capacity battery, where the latter also considered energy leakage at the transmitter. The authors in [@5992840] considered the offline minimization of transmission completion time in broadcast links under EH constraints. A hybrid energy storage system model with one unlimited battery and a limited super-capacitor is considered in [@6620534], where the authors maximize the throughput under a data transmission deadline constraint. Moreover, having a transmitter utilizing both the harvested energy and the grid energy, the authors in [@liu2015delay] provided an analysis on the average data queueing delay and the average power consumption from the grid by formulating the data queueing and energy storage as a two-dimensional Markov chain. The authors in [@arafa2015optimal] and the authors in [@arafa2017energy] investigated the effects of decoding and processing costs in one-way and two-way channels, respectively, where the transmitter needs to adjust its transmission power policies regarding not only its own energy values but also the receiver’s energy values. One more fundamental consideration in using certain EH sources, which is different from those in using non-rechargeable batteries, is the maximum rate at which we can utilize the harvested energy [@Kansal:2007:PME:1274858.1274870]. Therefore, we have to capture the uncertainty in not only the energy source but also the consumption. With this motivation in mind, the authors in [@993375] modeled a battery similarly to a server with a finite service capacity, and the data packets similarly to customers to be served, and analyzed the performance using a queueing-theoretic approach. This work is one of the earlier attempts to model the battery using queueing theory. Regarding a sensor node as a paired queueing system with two buffers, one for the accumulated energy and the other for the arriving data, Cuypere *et al.* analyzed its performance numerically, and investigated the energy-information tradeoff [@3229100]. A more comprehensive and practical, yet simple enough, model, which also takes the variations in harvested energy into consideration, appeared in [@Kansal:2007:PME:1274858.1274870] and [@Kansal:2004:PAT:1012888.1005714]. We can easily incorporate the aforementioned queueing model to a general class of stochastic energy sources. Bounding the energy arrival from an energy source with lower and upper bounds, and bounding the energy consumption with an upper bound, they showed that a device can operate forever as long as it has a storage capacity greater than the total burstiness defined by the upper and lower bounds [@Kansal:2004:PAT:1012888.1005714]. They provided the stability condition such that the average energy arrival rate is less than or equal to the average energy demanded for consumption. Although these deterministic bounds provide a framework to study the steady-state behavior of energy production and consumption processes, they indeed capture the worst-case scenario. Also, these bounds do not take advantage of the statistical nature of the energy arrival and consumption processes. Separately, considering statistical bounds, Srivastava *et al.* invoked large deviation theory and showed that the energy underflow probability, i.e., the probability that the energy level in the battery is below a defined threshold, scales exponentially as a function of the battery size and a constant in the asymptotic regime of large battery size [@srivastava2013basic]. Finally, the authors in [@jia2017data] characterized the average backlog for both constant and random data arrivals at a finite-size data buffer considering a Bernoulli energy arrival process. Last but not least, considering a network of energy harvesters, namely energy packet networks with energy harvesting, the authors in [@gelenbe2016energy] and the ones in [@kadioglu2018product] invoked a branch of queuing theory called G-networks and queueing networks with product-form solution, respectively, in order to compute relevant performance metrics of such networks operating with intermittent energy. Another research direction in energy harvesting wireless communication headed the simultaneous wireless information and power transfer. We refer interested readers to [@tran2018resource; @8629017; @hu2018swipt; @8620264]. Finally, we refer to studies where energy harvesting is investigated in multiple-input multiple-output systems [@7790901; @le2018joint; @8629017; @Le2019]. Contributions ------------- In EH communications, storage modeling is one of the crucial design benchmarks that we have to treat diligently. Especially in communication systems whose operations depend solely on the harvested energy, understanding the storage dynamics becomes fundamental from a design viewpoint. While a tenable energy storage is of paramount importance for us to avoid energy outages, preventing energy overflows due to the limited battery size is also necessary in order to fully utilize the harvested energy. Accordingly, we need a simple, yet comprehensive analytical model that can encapsulate the characteristics of a generic EH wireless communication system. In this paper, we introduce an analytical framework that system designers can use in order to understand the performance levels in a general class of EH communication systems under energy overflow and outage probability constraints, and QoS requirements. Similarly to [@srivastava2013basic] and [@zhang2015joint], we take advantage of large deviation theory and Markov processes; however, different from these studies, we consider simultaneously the energy overflow and outage probabilities in an EH communication system, and perform the throughput analysis. Specifically, while [@srivastava2013basic] provides the energy underflow probability in the battery using large deviation theory , the authors in [@zhang2015joint] maximize the effective capacity under the energy underflow probability constraint. On the other hand, we formulate the energy overflow probability in the battery as an exponential function of the battery size by invoking large deviation theory, and the energy outage probability by employing the Markov process analysis. Particularly, we consider that the transmitter initially sets a transmission power policy taking into account the energy overflow and outage probability constraints, and the energy arrival statistics. It sets the transmission rate based on the power allocation policy. We assume that the transmitter knows the channel statistics, but is unaware of the instantaneous channel state. Specifically, we have the following contributions: 1. Using large deviation theory and queueing theory, we formulate a simple exponential approximation for the energy overflow probability, where we characterize the energy decay rate (i.e., the decay rate of the tail distribution of the stored energy) in the battery as a measure of energy utilization. 2. Mapping the system evolution to a Markov chain, in which we have one state representing the energy outage, and other states representing the number of time frames since the last energy outage event, we provide an expression for the energy outage probability. 3. Under energy overflow and outage constraints, we obtain the average data service rate in the wireless channel. Subsequently, we identify the effective capacity of the system, which characterizes the maximum data arrival rate at the transmitter buffer when there are QoS requirements in the form of constraints on the buffer overflow probability. Apart from this paper, we invoke large deviation theory and queueing theory in [@akin2017energy; @akin2018energy], where we characterize the energy underflow probability. Specifically in this paper, we have a focus primarily on controlling energy waste and then energy outages under QoS constraints, whereas in [@akin2017energy; @akin2018energy], we principally control energy underflows, and hence outages, in communication settings under stricter QoS constraints, where transmission interruptions are not tolerated at any time. The rest of the paper is organized as follows. We introduce the transmission system model consisting of an EH device, a storage unit, and a data buffer in Section \[sec:system\_model\]. We characterize the energy storage performance measures such as energy overflow and energy outage in Section \[sec:Energy\_Storage\_Characterization\]. We provide the throughput analysis in Section \[sec:transmission\_throughput\], i.e., formulate the average data service rate in the channel in Subsection \[sec:reliable\_throughput\] and the effective capacity in Subsection \[sec:effective\_capacity\]. We present the numerical results in Section \[sec:numerical\_results\]. Our conclusions are provided in Section \[sec:conclusion\]. We relegate the proofs to the Appendix. System Model {#sec:system_model} ============ We consider a discrete-time system model consisting of two separate queues at the transmitter corresponding to the energy and data buffers, respectively. Please, see Fig. \[res:fig\_2\] for an illustration of the system model. We describe each of these components separately. ![EH transmitter model consisting of a data buffer and a battery acting as an energy buffer.[]{data-label="res:fig_2"}](resim_4_new.png){width="55.00000%"} Energy Harvesting and Storage {#sec:Energy_Harvesting_Storage} ----------------------------- We denote the amount of the harvested energy in the $i^{\text{th}}$ time frame by $u(i)$ and the amount of energy demanded for data transmission by $p(i)$ for $i=1,2,\cdots$. We assume that $u(i)$ is stochastic and varies from one time frame to another, whereas the amount of energy demanded, $p(i)$, is a stochastic process[^4], and its parameters[^5] are pre-determined by the transmitter. When setting $p(i)$, the transmitter regards certain constraints and performance requirements, e.g., energy overflow and outage probability constraints, which will be defined below. The energy level in the battery at the end of the $i^{\text{th}}$ time frame, denoted by $e(i)$, $i\in\mathbb{Z}^{+}$, is governed by the following update rule: $$\label{actual_capacity_2} e(i)=\min\{[e(i-1)+u(i)-p(i)]^{+},e_{\max}\},$$ where $[\cdot]^{+}\triangleq\max\{0,\cdot\}$, $e_{\max}$ denotes the battery capacity, and $e(0)$ is the initial energy level of the battery. As seen from (\[actual\_capacity\_2\]), the energy harvested in the $i^{\text{th}}$ time frame, $u(i)$, is ready for consumption in the same time frame. If $e(i-1)+u(i)-p(i)>e_{\max}$, we will have an *energy overflow* in the $i^{\text{th}}$ time frame, wasting some of the harvested energy. On the other hand, when $e(i-1)+u(i)<p(i)$, we will have an *energy outage*. While $p(i)$ is the rate of the energy demanded by the transmitter, the battery may not be always able to provide energy at this level. When there is enough energy in the battery, i.e., $e(i-1)+u(i)\geq p(i)$, the battery satisfies the energy demand completely. Otherwise, the battery provides what is left in it for the transmission of data. We denote the actual consumed energy in the $i^{\text{th}}$ time frame by $p_{\text{c}}(i)$, where $$\label{eq:power_level} p_{\text{c}}(i) = \begin{cases} p(i), &\text{if $e(i-1)+u(i)\geq p(i)$},\\ e(i-1)+u(i), &\text{otherwise}. \end{cases}$$ Moreover, because the transmitter cannot spend more energy than what has been generated (i.e., energy causality), we have the following causality constraints: $$\label{energy_causality} e(0)+U(t)\geq P_{c}(t),\quad\forall t,$$ where $U(t)\triangleq\sum_{i=1}^{t}u(i)$ and $P_{c}(t)\triangleq\sum_{i=1}^{t}p_{\text{c}}(i)$ are the total energy harvested and consumed in the first $t$ time frames, respectively. Here, we assume that the transmitter knows the battery state perfectly. Data Buffer ----------- In the $i^{\text{th}}$ frame, $a(i)$ bits of data arrive at the transmitter, and is stored in the transmitter buffer. The buffer has a capacity of $d_{\max}$ bits, and the number of bits in the buffer in the $i^{\text{th}}$ time frame is $d(i)$. When a packet of data is transmitted and decoded by the receiver correctly, the data packet is removed from the transmitter buffer, and it is considered to be served. Let $s(i)$ denote the amount of data served in the $i^{\text{th}}$ frame. Assuming that the transmitter sends $r(i)$ bits over the channel in the $i^{\text{th}}$ time frame, we can clearly see that $s(i)=r(i)$ if the receiver is able to decode the transmitted data, and $s(i)=0$ otherwise. In order to ensure the delivery of the data, we assume that a simple automatic-repeat-request (ARQ) mechanism exists to acknowledge a successful reception, or to trigger the retransmission of erroneously decoded data. We assume that the transmitter does not have the channel state information, but knows the channel statistics. If the number of transmitted bits is smaller than the maximum rate that can be supported by the channel capacity in the $i^{\text{th}}$ time frame, transmission is assumed to be successful; otherwise, a decoding error occurs. Energy Storage Characterization {#sec:Energy_Storage_Characterization} =============================== We explore the battery dynamics taking into consideration energy overflows and outages, and establish limits on the energy overflow and outage probabilities as the statistical battery (or energy management) constraints. Therefore, observing the structural similarity between a single server queuing system and the aforementioned storage model, we invoke queueing theory, large deviations theory [@chang_book] and network calculus [@jiang_book] to understand the energy overflows in the battery. Then, projecting subsequent energy demands that have been satisfied, and energy outages on a Markov process, we characterize the energy outage probability. Energy Overflow {#sec:energy_overflow} --------------- We know that given a single service provider, the steady-state queue length tail distribution in a queueing system with a first in-first out policy, assuming it exists, has a characteristic decay rate, i.e., the decay rate of the tail probability of the queue [@chang1994effective]. When the queueing capacity is infinite, we obtain a simple exponential expression for the queue overflow probability (i.e., the probability that the queue is greater than a threshold), which is a function of the desired threshold and the characteristic decay rate. In practical systems, when the queueing capacity is large, this exponential expression approximates the queue overflow probability very closely. Likewise, for the EH transmitter model introduced in Section \[sec:system\_model\], we assume that the battery size is infinite, i.e., $e_{\max}=\infty$, and define the decay rate of the tail distribution of the energy in the battery as follows. \[def:definition\_decay\_rate\] Given a stationary and ergodic energy arrival process, $u(i)$, and a stationary and ergodic energy demand process, $p(i)$, under the stability condition[^6], i.e., $\mathbb{E}_{u}\left[u(i)\right]<\mathbb{E}_{p}\left[p(i)\right]$, where $\mathbb{E}\left[\cdot\right]$ is the expected value operator, the *energy decay rate* of the battery is defined as $$\label{decay_rate} \mu\triangleq-\lim_{e_{\text{th}}\to\infty}\frac{\ln\Pr\{e\geq e_{\text{th}}\}}{e_{\text{th}}},$$ where $e_{\text{th}}$ is the desired energy level, and random variable $e$ corresponds to the steady-state distribution of the energy level in the battery. The definition of $\mu$ in (\[decay\_rate\]) suggests an approximation for the energy overflow probability in the steady-state given a large battery size[^7], i.e., $\Pr\{e\geq e_{\text{th}}\}\approx \exp\big(-\mu e_{\text{th}}\big)$. In particular, the energy decay rate characterizes the exponential decay rate of the tail distribution of the energy level in the battery, and the exponential approximation holds when we have stationary and ergodic energy arrival and demand processes [@chang1995effective_conf]. With a large battery size and a target overflow probability, we can use the energy decay rate, $\mu$, as a tool to identify the probability that the energy level in the battery is above a defined threshold. The energy decay rate primarily depends on the energy arrival and demand processes. Large $\mu$ refers to an energy demand process that consumes the stored energy rapidly, while smaller $\mu$ means a moderate energy demand process. For an infinite-size battery the instantaneous energy level is given by $e(i)=[e(i-1)+u(i)-p(i)]^{+}$. Moreover, because energy in our model is used for data transmission only, we further consider a work-conserving energy demand process[^8] in the following analysis. Hence, noting that $u(i)$ and $p(i)$ are independent of each other, we have a unique $\mu^{\star}$ that satisfies [@chang_book Remark 9.1.2] $$\begin{aligned} \label{eq:arrival_demand_balance} \Lambda_{u}(\mu^{\star})+\Lambda_{p}(-\mu^{\star})=0,\end{aligned}$$ where $\Lambda_{u}(\mu)\triangleq\lim_{t\to\infty}\frac{1}{t}\ln\mathbb{E}_{u}\left[\exp\left(\mu U(t)\right)\right]$ and $\Lambda_{p}(\mu)\triangleq\lim_{t\to\infty}\frac{1}{t}\ln\mathbb{E}_{p}\left[\exp\left(\mu P(t)\right)\right]$ are the Gärtner-Ellis limits, and are differentiable for $\mu\in\mathbb{R}$, when the moment generating functions $\mathbb{E}_{u}\left[\exp\left(\mu U(t)\right)\right]$ and $\mathbb{E}_{p}\left[\exp\left(\mu P(t)\right)\right]$ exist for $t>0$, respectively. $U(t)$ is the cumulative harvested energy as defined before, and $P(t)\triangleq\sum_{i=1}^{t}p(i)$. Noting that (\[eq:arrival\_demand\_balance\]) holds for any stationary and ergodic energy arrival and demand processes with finite mean and variance, we provide the following two cases as examples in order to gain more insights. ### Constant energy demand Given an energy arrival process, let us assume that the energy demanded by the transmitter is constant over time. Then, we can easily find the following relation for a given energy decay rate, $\mu$: $p^{\star}=\lim_{t\to\infty}\frac{1}{t\mu}\ln\mathbb{E}_{u}\left[\exp\left(\mu U(t)\right)\right]$, where $p^{\star}$ is the minimum constant energy demand such that the steady-state energy overflow probability is $\Pr\{e(i)\geq e_{\text{th}}\}\approx\exp\big(-\mu e_{\text{th}}\big)$ for a given threshold value, $e_{\text{th}}$. In other words, targeting an energy decay rate, $\mu$, we should have the constant energy demand greater than or equal to $p^{\star}$ such that we can keep the steady-state energy overflow probability less than or equal to $\exp\big(-\mu e_{\text{th}}\big)$ for given $e_{\text{th}}$. We note that keeping the energy demand above $p^{\star}$ decreases the energy overflow probability while it causes an increase in the energy outage probability. \[pro:mu\_zero\_and\_infinity\] When the energy decay rate, $\mu$, goes to zero, the minimum constant energy demand goes to the average energy arrival level, $\mathbb{E}_{u}\left[u(i)\right]$. In particular,$$p^{\star}=\lim_{\mu\to0}\lim_{t\to\infty}\frac{1}{t\mu}\ln\mathbb{E}_{u}\left[\exp\left(\mu U(t)\right)\right]=\mathbb{E}_{u}\left[u(i)\right].$$Furthermore, when the energy decay rate, $\mu$, goes to infinity, the minimum constant energy demand goes to the maximum energy arrival level, $\max\{u(i)\}$. In particular, $$p^{\star}=\lim_{\mu\to\infty}\lim_{t\to\infty}\frac{1}{t\mu}\ln\mathbb{E}_{u}\left[\exp\left(\mu U(t)\right)\right]=\max\{u(i)\}.$$ *Proof*: See Appendix \[app:proposition\_mu\_zero\_and\_infinity\]. ### Constant energy arrival Given an energy demand process, let us now assume that the energy arrival at the battery is constant[^9] over time. Then, we can easily find the following relation for a given energy decay rate, $\mu$: $u^{\star}=-\lim_{t\to\infty}\frac{1}{t\mu}\ln\mathbb{E}_{p}\left[\exp\left(-\mu P(t)\right)\right]$, where $u^{\star}$ is the maximum constant energy arrival such that the steady-state energy overflow probability is $\Pr\{e\geq e_{\text{th}}\}\approx\exp\left(-\mu e_{\text{th}}\right)$ for given $e_{\text{th}}$. In other words, targeting an energy decay rate, $\mu$, we should have the constant energy arrival less than or equal to $u^{\star}$ such that we can keep the steady-state energy overflow probability less than or equal to $\exp\big(-\mu e_{\text{th}}\big)$. Moreover, it is important to note that while having the energy arrival below $u^{\star}$ decreases the energy overflow probability, it causes an increase in the energy outage probability. \[pro:mu\_zero\_and\_infinity\_2\] When the energy decay rate, $\mu$, goes to zero, the maximum constant energy arrival approaches the average energy demand, $\mathbb{E}_{p}\left[p(i)\right]$. In particular,$$u^{\star}=-\lim_{\mu\to0}\lim_{t\to\infty}\frac{1}{t\mu}\ln\mathbb{E}_{p}\left[\exp\left(-\mu P(t)\right)\right]=\mathbb{E}_{p}\left[p(i)\right].$$Furthermore, when the energy decay rate, $\mu$, goes to infinity, the maximum constant energy arrival approaches the minimum energy demand, $\min\{p(i)\}$. In particular,$$u^{\star}=-\lim_{\mu\to\infty}\lim_{t\to\infty}\frac{1}{t\mu}\ln\mathbb{E}_{p}\left[\exp\left(-\mu P(t)\right)\right]=\min\{p(i)\}.$$ *Proof*: See Appendix \[app:proposition\_mu\_zero\_and\_infinity\_2\]. Proposition \[pro:mu\_zero\_and\_infinity\] suggests that when the constant energy demand is less than or equal to the average value of the energy arrival process, energy overflows are inevitable. It also shows that, as long as the energy demand is greater than or equal to the maximum possible energy arrival in any time frame, there is no energy overflow in the battery. Likewise, according to Proposition \[pro:mu\_zero\_and\_infinity\_2\], when the value of a constant energy arrival process is greater than or equal to the average value of the energy demand process, the energy overflows are imminent, and when the constant energy arrival is less than or equal to the minimum energy demand, there is no energy overflow in the battery. Although the statements in Propositions \[pro:mu\_zero\_and\_infinity\] and \[pro:mu\_zero\_and\_infinity\_2\] are rather intuitive, they are confirmed by the characterization in (\[eq:arrival\_demand\_balance\]), where we express the probability that the energy level in the battery is above a defined threshold as an exponential function of the energy decay rate and the defined threshold. In other words, assuming that there is a constant energy demand while the energy arrival process has a stochastic nature, we can express the energy overflow probability with an exponential function as long as the constant energy demand is between the average and maximum values of the energy arrival process. Equivalently, given that there is a constant energy arrival process while the energy demand process is stochastic, we can express the energy overflow probability as an exponential function when the constant value of the energy arrival process is between the minimum and average values of the energy demand process. Energy Outage {#sec:energy_outage} ------------- Recall that when there is not enough energy in the battery, i.e., when the energy in the battery is less than what the transmitter demands, an *energy outage* occurs; the transmitter consumes all the energy available in the battery, leaving the battery empty. We say that following an energy outage event, the battery enters state 0. Herein, we define a state space $\mathcal{W}=\{0,1,\cdots\}$ consisting of non-negative integers, where the state $w(i)$ at time $i$ denotes the consecutive number of times the harvested and stored energy successfully meet the energy demand since the last energy outage event. More specifically, assume that the $i^{\text{th}}$ time frame results in an energy outage, and we have $e(i)=0$; and hence, the battery is in state $w(i)=0$. In the subsequent $(i+1)^{\text{th}}$ time frame, if the harvested energy is greater than the demanded energy, i.e., $u(i+1)\geq p(i+1)$, the battery enters state $w(i+1)=1$, and stores the excess energy. However, if $u(i+1)<p(i+1)$, another energy outage occurs, and the battery remains in state 0, i.e., $w(i+1)=0$. Similarly, given that $w(i+1)=1$, the transition from state 1 to state 2 occurs when $u(i+1)+u(i+2)-p(i+1)\geq p(i+2)$. On the other hand, when $u(i+1)+u(i+2)-p(i+1)<p(i+2)$, the battery goes from state 1 to state 0. Similarly, given that $w(i+2)=2$, the battery transitions to state 3, i.e., $w(i+3)=3$, if $u(i+1)+u(i+2)+u(i+3)-p(i+1)-p(i+2)\geq p(i+3)$, and $w(i+3)=0$ otherwise. Generalizing the above observations and considering again an infinite-size battery, we can state that given $w(i+m-1)=m-1$, we have $w(i+m)=m$ if $\sum_{j=i+1}^{i+m}u(j)\geq\sum_{j=i+1}^{i+m}p(j)$, and $w(i+m)=0$ otherwise. Notice that the battery in state $m-1$ for $m\in\{1,2,\cdots\}$ will go to either state $m$ or state 0. Given that the battery is in state $m-1$ in the $(i+m-1)^{\text{th}}$ time interval, we can express the state transition probability from state $m-1$ to state $m$ for $m\in\{1,2,\cdots\}$, denoted by $q_{m}(i+m)$, as $$\begin{aligned} &q_{m}(i+m)\triangleq\Pr\{w(i+m)=m|w(i+m-1)=m-1\}\label{eq:state_transition_probability_2_i}\\ &=\Pr\{w(i+m)=m|w(i+m-1)=m-1,w(i+m-2)=m-2,\cdots,w(i)=0\}\label{eq:state_transition_probability_2_f}\\ &=\frac{\Pr\{w(i+m)=m,\cdots,w(i)=0\}}{\Pr\{w(i+m-1)=m-1,\cdots,w(i)=0\}}\label{eq:state_transition_probability_2_e}\\ &=\frac{\Pr\left\{U(i+m,i)\geq P(i+m,i),\cdots,U(i+1,i)\geq P(i+1,i),e(i)=0\right\}}{\Pr\left\{U(i+m-1,i)\geq P(i+m-1,i),\cdots,U(i+1,i)\geq P(i+1,i),e(i)=0\right\}},\label{eq:state_transition_probability_2}\end{aligned}$$ where $U(i+j,i)=U(i+j)-U(i)$ and $P(i+j,i)=P(i+j)-P(i)$ for $j\in\mathbb{Z}^{+}$. The transition probability from state $m-1$ to state 0 is $1-q_{m}(i+m)$. Above, (\[eq:state\_transition\_probability\_2\_f\]) follows from the fact that the battery being in state $m-1$ in the $(i+m-1)^{\text{th}}$ frame has already been in states $0$ to $m-2$ in the time frames from the $i^{\text{th}}$ to the $(i+m-2)^{\text{th}}$. This also means that the battery was empty in the $i^{\text{th}}$ time frame, i.e., $e(i)=0$. In (\[eq:state\_transition\_probability\_2\_e\]), we invoke Bayes’ theorem. Moreover, since the energy arrival and demand processes are stochastic and ergodic, we can re-write (\[eq:state\_transition\_probability\_2\]) as $$\begin{aligned} q_{m}=\frac{\Pr\left\{U(m)\geq P(m),\cdots,U(1)\geq P(1)\right\}}{\Pr\left\{U(m-1)\geq P(m-1),\cdots,U(1)\geq P(1)\right\}}\label{eq:state_transition_probability_2_add}\end{aligned}$$ for $m\in\{2,3,\cdots\}$, and $q_{1}=\Pr\left\{u(1)\geq p(1)\right\}$, where the battery is initially empty. Now, modeling the energy outages and consecutive energy supply guarantees as a Markov process, we have the state transition diagram given in Fig. \[fig:res\_2\]. Correspondingly, we can write the state transition matrix as follows: $$\begin{aligned} \label{M} M=\begin{pmatrix} 1-q_1 & 1-q_2 & \cdots & 1-q_{m} & \cdots\\ q_1 & 0 & \cdots & 0 & \cdots\\ 0 & q_2 & \cdots & 0 & \cdots\\ \vdots &\vdots & \ddots&\vdots&\vdots\;\vdots\;\vdots\\ 0 & 0 & \cdots & q_{m} & \cdots\\ \vdots &\vdots & \vdots\;\vdots\;\vdots&\vdots&\vdots\;\vdots\;\vdots \end{pmatrix}.\end{aligned}$$ Now, let $\boldsymbol{\pi}$ be the vector of steady-state probabilities, i.e., $\boldsymbol{\pi}=\{\pi_{0},\pi_{1},\cdots,\pi_{m},\cdots\}^{T}$ satisfying $\boldsymbol{\pi}=M\boldsymbol{\pi}$ and $\sum_{i=0}^{\infty}\pi_{i}=1$, where $[\cdot]^{T}$ is the transpose operator. We can notice that the steady-state probability $\pi_{0}$ gives us the energy outage probability, which is $\pi_{0}=\frac{1}{1+\sum_{m=1}^{\infty}\prod_{i=1}^{m}q_{i}}$. Having $\pi_{0}$ and $M$, we can easily provide the other steady-state probabilities as follows: $\pi_{k}=\pi_{k-1}q_{k}=\pi_{0}\prod_{i=1}^{k}q_{i}=\frac{\prod_{i=1}^{k}q_{i}}{1+\sum_{m=1}^{\infty}\prod_{i=1}^{m}q_{i}}$ for $k\in\{1,2,\cdots\}$. Recalling that both $u(i)$ and $p(i)$ are stationary and ergodic, and that the stability condition given in Definition \[def:definition\_decay\_rate\] is satisfied, we can show that $\pi_{0}\geq\pi_{1}\geq\pi_{2}\geq\cdots\geq\pi_{m}\geq\cdots$. Furthermore, using (\[eq:state\_transition\_probability\_2\_add\]) and assuming that consecutive energy arrivals and consecutive energy demands are independent and identically distributed (i.i.d.), we can employ an inductive method and realize that $q_{1}\leq q_{2}\leq q_{3}\leq\cdots\leq q_{m}\leq\cdots$, and find an upper bound on the outage probability as follows: $$\begin{aligned} \pi_{0}&=\frac{1}{1+\sum_{m=1}^{\alpha-1}\prod_{i=1}^{m}q_{i}+\sum_{m=\alpha}^{\infty}\prod_{i=1}^{m}q_{i}}=\frac{1}{1+\sum_{m=1}^{\alpha-1}\prod_{i=1}^{m}q_{i}+\prod_{i=1}^{\alpha}q_{i}\left[1+\sum_{m=\alpha+1}^{\infty}\prod_{i=\alpha+1}^{m}q_{i}\right]}\nonumber\\ &\leq\frac{1}{1+\sum_{m=1}^{\alpha-1}\prod_{i=1}^{m}q_{i}+\frac{\prod_{i=1}^{\alpha}q_{i}}{1-q_{\alpha+1}}}\label{eq:upper_bound_pi_0}\end{aligned}$$ for $\alpha\in\{1,2,\cdots\}$. The upper bound in (\[eq:upper\_bound\_pi\_0\]) comes from the fact that $\sum_{m=\alpha+1}^{\infty}\prod_{i=\alpha+1}^{m}q_{i}\geq\sum_{m=\alpha+1}^{\infty}\prod_{i=\alpha+1}^{m}q_{\alpha+1}$. With increasing $\alpha$, the bound becomes tighter. Performing the sum in the denominator in (\[eq:upper\_bound\_pi\_0\]) up to a finite number provides us the energy outage probability closely. Transmission Throughput {#sec:transmission_throughput} ======================= In this section, we concentrate on the performance levels of the aforementioned EH wireless communication system. Therefore, we invoke the average data service rate in the wireless channel and the effective capacity in the data link layer as performance measures. The average data service rate is defined as the average number of bits the transmitter forwards to the receiver reliably. On the other hand, the effective capacity is the maximum constant data arrival rate at the transmitter buffer that the data service process can support under prescribed QoS constraints, i.e., data buffer overflow and buffering delay violation probability constraints. In the sequel, we provide expressions for the average data service rate and the effective capacity given a general class of energy arrival and demand processes, and then exemplify our results by constant energy demand rate and fixed data transmission rate for a clear presentation of our analysis. Average Data Service Rate {#sec:reliable_throughput} ------------------------- We assume that the transmitter sets beforehand the energy demand process, $p(i)$, considering the energy overflow and outage probability constraints. Then, it adjusts the transmission strategy over the channel, $r(i)$, to maximize the average data service rate. As long as there is no energy outage, the transmitter does not change the transmission strategy. On the other hand, when there is an energy outage, the transmitter uses all the available energy in the battery and re-adjusts the transmission rate according to the available power, again to maximize the average data service rate[^10]. Although, the transmission rate policy is pre-determined along with the energy demand policy, it is indeed a function of the available energy in the battery given in (\[eq:power\_level\]), and hence it is stochastic. Another challenge in the wireless medium is the channel fading. When the transmission is exposed to deep fading in one frame, reliable decoding at the receiver may not be possible. In other words, if the instantaneous channel capacity falls below the transmission rate, a *transmission outage* occurs; that is, the receiver fails to decode the data. Therefore, even if the desired energy rate is met, a transmission outage may occur due to channel fading. We consider a block-fading channel model, and assume that the fading coefficient, $h(i)$, stays constant during one time frame and changes independently from one frame to the next. Now, given that the transmission power is as given in (\[eq:power\_level\]), the instantaneous channel capacity in the $i^{\text{th}}$ time frame is $\mathcal{I}(i)\triangleq N\log_{2}\left(1+\frac{|h(i)|^2p_c(i)}{N\sigma_{w}^{2}}\right)$ bits/frame, assuming Gaussian codebooks[^11] are employed [@book_information_theory Ch. 9.1][@caire1999optimum]. $N$ is the number of transmitted symbols in one time frame, which is assumed to be sufficiently large, and $\sigma_{w}^{2}$ is the variance of the zero-mean noise in the wireless channel. If $r(i)\leq \mathcal{I}(i)$, we assume that reliable transmission takes place and the receiver decodes the data successfully. Otherwise, a transmission outage occurs, and the effective transmission rate becomes zero. Recalling that $s(i)$ is the data service rate from the buffer in the $i^{\text{th}}$ time frame, we have: $s(i)=0$ if $r(i)>\mathcal{I}(i)$, and $s(i)=r(i)$ otherwise. Now, we can express the average data service rate in the steady-state as follows: $$\begin{aligned} \hspace{-0.5cm}s_{\text{avg}}=&\mathbb{E}_{u(i),p(i)}\left[s(i)\right] =\mathbb{E}_{u(i),p(i)}\left[r(i)\cdot\textbf{1}\left[r(i)\leq\mathcal{I}(i)\right]\right] =\mathbb{E}_{u(i),p(i)}\left[r(i)\cdot\textbf{1}\left[\kappa(i)\leq|h(i)|^{2}\right]\right],\label{eq:avg_srvc_rate_all}\end{aligned}$$ where $\kappa(i)=\left(2^{\frac{r(i)}{N}}-1\right)\frac{N\sigma_{w}^{2}}{p_c(i)}$, and $\textbf{1}[x]$ is the indicator function, which is $1$ if $x$ is correct, and $0$ otherwise. Noting that the result in (\[eq:avg\_srvc\_rate\_all\]) is applicable in any setting with stationary and ergodic energy arrival and demand processes with finite means and variances, we consider the following specific case with constant energy demand and fixed data transmission rate in order to gain more insights. ### Constant energy demand rate and a fixed transmission rate {#subsubsec:constant_energy_rate} Given a Rayleigh fading channel model[^12], i.e., the absolute power of the channel fading, $|h(i)|^2$, is exponentially distributed with parameter $\frac{1}{\sigma_{h}^{2}}$, where $\sigma_{h}^{2}$ is the variance of the channel fading, let us assume that we have an energy demand process with constant rate[^13], i.e., $p(i)=p$, and $\frac{p}{N}$ is the average symbol power. Hence, the consumed energy given in (\[eq:power\_level\]) is $p_{\text{c}}(i)= p$ if $p\leq\xi(i)$, and $p_{\text{c}}(i)=\xi(i)$ otherwise, where $\xi(i)=e(i-1)+u(i)$. Noting that the transmitter knows only the channel statistics but not its realizations, and that the transmitter sets the transmission rate according to the available energy, we assume that the data is transmitted at a fixed rate[^14] as a function of the consumed energy, i.e., $r(i)=g(p_{c}(i))$ bits/frame, where $g(\cdot)$ is the pre-determined transmission rate function, which is monotonically increasing[^15]. Then, we can express the average data service rate in the steady-state[^16] as $$\begin{aligned} \label{constant_transmission_rate_and_power} s_{\text{avg}}=&\mathbb{E}_{\xi(i)}\left[r(i)\textbf{1}\left[r(i)\leq \mathcal{I}(i)\right]\right]=\int_{0}^{\infty}r(i)\Pr\left\{r(i)\leq \mathcal{I}(i)\right\}f_{\xi(i)}(\xi(i))d\xi(i)\\ =&\sum_{m=0}^{\infty}\int_{0}^{\infty}g(p_{c}(i))\Pr\left\{g(p_{c}(i))\leq \mathcal{I}(i)\right\}\Pr\{w(i-1)=m\}\nonumber\\ &\times f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i)\label{eq:constant_transmission_rate_3}\\ =&\sum_{m=0}^{\infty}\pi_{m}\int_{0}^{\infty}g(p_{c}(i))\Pr\left\{g(p_{c}(i))\leq \mathcal{I}(i)\right\}f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i)\\ =&\sum_{m=0}^{\infty}\pi_{m}\int_{0}^{p}g(\xi(i))\Pr\left\{g(\xi(i))\leq \mathcal{I}(i)\right\}f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i)\nonumber\\ &+\sum_{m=0}^{\infty}\pi_{m}\int_{p}^{\infty}g(p)\Pr\left\{g(p)\leq \mathcal{I}(i)\right\}f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i)\label{eq:constant_transmission_rate_5}\\ =&\sum_{m=0}^{\infty}\pi_{m}\int_{0}^{p}g(\xi(i))\Pr\left\{\kappa(\xi(i))\leq|h(i)|^{2}\right\}f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i)\nonumber\\ &+\sum_{m=0}^{\infty}\pi_{m}\int_{p}^{\infty}g(p)\Pr\left\{\kappa(p)\leq|h(i)|^2\right\}f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i)\\ =&\sum_{m=0}^{\infty}\pi_{m}\int_{0}^{p}g(\xi(i))e^{\frac{-\kappa(\xi(i))}{\sigma_{h}^{2}}}f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i)\nonumber\\ &+\sum_{m=0}^{\infty}\pi_{m}g(p)e^{\frac{-\kappa(p)}{\sigma_{h}^{2}}}\int_{p}^{\infty}f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i)\\ =&\sum_{m=0}^{\infty}\pi_{m}\int_{0}^{p}g(\xi(i))e^{\frac{-\kappa(\xi(i))}{\sigma_{h}^{2}}}f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i)\nonumber\\ &+g(p)e^{\frac{-\kappa(p)}{\sigma_{h}^{2}}}\sum_{m=0}^{\infty}\pi_{m}q_{m+1}\\ =&\sum_{m=0}^{\infty}\pi_{m}\int_{0}^{p}g(\xi(i))e^{\frac{-\kappa(\xi(i))}{\sigma_{h}^{2}}}f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i)\nonumber\\ &+g(p)e^{\frac{-\kappa(p)}{\sigma_{h}^{2}}}(1-\pi_{0}),\label{eq:avg_throughput_final}\end{aligned}$$ where $f_{\xi(i)}\left(\xi(i)\right)$ and $f_{\xi(i)|w(i-1)=m}\left(\xi(i)|w(i-1)=m\right)$ are the probability density function of $\xi(i)$, and the conditional probability density function of $\xi(i)$ in the $i^{\text{th}}$ time frame, respectively. In (\[eq:constant\_transmission\_rate\_5\]), $g(p)$ is the fixed transmission rate when the battery satisfies the constant energy demand rate, and $g(\xi(i))$ is the re-adjusted data transmission rate according to the available energy in the battery when the battery cannot sustain the constant energy demand rate. Moreover, we define $\kappa(a)\triangleq\left(2^{\frac{g(a)}{N}}-1\right)\frac{N\sigma_{w}^{2}}{a}$ for $a\in{\mathbb{R}^{+}}$. In (\[eq:constant\_transmission\_rate\_3\]), $\pi_{m}\triangleq\Pr\{w(i-1)=m\}$ is the probability that the battery is in state $m$ in the $(i-1)^{\text{th}}$ time frame in the steady-state. We obtain the average data service rate in (\[eq:avg\_throughput\_final\]) for an arbitrary transmission rate; and therefore, it can be maximized with the appropriate choice of the transmission rate. Particularly, we have the maximized average data service rate as follows: $$\begin{aligned} \label{eq:s_avg_max_ad} s_{\text{avg}}^{\max}=&\max_{g(p)}\left\{g(p)e^{\frac{-\kappa(p)}{\sigma_{h}^{2}}}\right\}(1-\pi_{0})\nonumber\\ &+\sum_{m=0}^{\infty}\pi_{m}\int_{0}^{p}\max_{g(\xi(i))}\left\{g(\xi(i))e^{\frac{-\kappa(\xi(i))}{\sigma_{h}^{2}}}\right\}f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i).\end{aligned}$$ Notice that (\[eq:avg\_throughput\_final\]) is a sum of infinite number of terms. Therefore, it may be difficult obtain a closed-form solution. However, we can lower-bound the expression in (\[eq:avg\_throughput\_final\]) for any $\alpha>0$ as follows: $$\begin{aligned} s_{\text{avg}}=&g(p)e^{\frac{-\kappa(p)}{\sigma_{h}^{2}}}(1-\pi_{0})+\sum_{m=0}^{\infty}\pi_{m}\int_{0}^{p}g(\xi(i))e^{\frac{-\kappa(\xi(i))}{\sigma_{h}^{2}}}f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i)\nonumber\\ =&g(p)e^{\frac{-\kappa(p)}{\sigma_{h}^{2}}}(1-\pi_{0})+\sum_{m=0}^{\alpha}\pi_{m}\int_{0}^{p}g(\xi(i))e^{\frac{-\kappa(\xi(i))}{\sigma_{h}^{2}}}f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i)\nonumber\\ &+\sum_{m=\alpha+1}^{\infty}\pi_{m}\int_{0}^{p}g(\xi(i))e^{\frac{-\kappa(\xi(i))}{\sigma_{h}^{2}}}f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i)\nonumber\\ \geq&g(p)e^{\frac{-\kappa(p)}{\sigma_{h}^{2}}}(1-\pi_{0})+\sum_{m=0}^{\alpha}\pi_{m}\int_{0}^{p}g(\xi(i))e^{\frac{-\kappa(\xi(i))}{\sigma_{h}^{2}}}f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i)\label{eq:lower_bound_avg_rate}\end{aligned}$$ The lower bound in (\[eq:lower\_bound\_avg\_rate\]) converges to the actual average data service rate with increasing $\alpha$ because the steady-state probability, $\pi_{i}$, decreases with the increasing battery state index. Therefore, we can approximate the average data service rate by taking the sum up to a finite value. In our simulations, we observe that after $\alpha=100$, the average data service rate values do not change. Herein, one can find the data transfer rate function, $g(\cdot)$, that maximizes the lower bound similarly as in (\[eq:s\_avg\_max\_ad\]). Effective Capacity {#sec:effective_capacity} ------------------ Recall that we store the data arriving at the transmitter in the data buffer before sending it in frames of $N$ symbols. Therefore, buffer overflow and delay bounds can be addressed by imposing statistical constraints on the queue length and delay in the buffer. Thus, we set the effective capacity as the performance measure in order to take into account the data queueing constraints of the aforementioned delay-limited EH system. We can define the effective capacity as the maximum constant data arrival rate that a given stochastic service process can support in order to satisfy the desired QoS requirements specified by the QoS exponent $\theta$. We can formulate the effective capacity as [@wu_negi] $$\begin{aligned} \label{effective_capacity} C_{E}(\theta)=-\lim_{t\to\infty}\frac{1}{t\theta}\ln\mathbb{E}\left[\exp\left(-\theta S(t)\right)\right],\end{aligned}$$ where $S(t)\triangleq\sum_{i=1}^{t}s(i)$ is the time-accumulated service process, and $s(i)$ is the discrete-time stationary and ergodic data service process. Notice that $\lim_{t\to\infty}\frac{1}{t}\ln\mathbb{E}\left[\exp\left(\theta S(t)\right)\right]$ is the asymptotic log-moment generating function of $S(t)$. Further notice that when $\theta$ goes to zero in limit, i.e., when there are no QoS constraints, the effective capacity in (\[effective\_capacity\]) converges to the average data service rate. We express the QoS exponent, which describes the decay rate of the tail distribution of the queue length, $d(i)$, as $$\label{theta_p} \theta=-\lim_{d_{\text{th}}\to\infty}\frac{\ln\Pr\{d\geq d_{\text{th}}\}}{d_{\text{th}}},$$ where $d$ denotes the steady-state queue length, and $\Pr\{d\geq d_{\text{th}}\}$ is the data buffer overflow probability for a given threshold $d_{\text{th}}$. Similarly to the approximation of the energy overflow probability, we can have an exponential approximation also for the data buffer overflow probability when we have a large data buffer size as follows: $\Pr\{d\geq d_{\text{th}}\}\approx \exp(-\theta d_{\text{th}})$. We can now easily infer that larger $\theta$ describes stricter QoS requirements because the data buffer overflow probability for a given threshold decreases with increasing $\theta$, while smaller $\theta$ indicates less strict QoS constraints due to the fact that the data buffer overflow probability increases with decreasing $\theta$. As seen in (\[effective\_capacity\]), the effective capacity depends on the data service process, $s(i)$, and hence, the data transmission rate in the channel, $r(i)$. Because the transmission rate is a function of the consumed energy, it is also a function of the energy demand and the energy level in the battery. Therefore, we can easily see that the effective capacity, which is the maximum data arrival rate that the service can support under QoS constraints, is affected by the energy arrival and demand processes. In the following, we provide the effective capacity of the aforementioned delay-limited wireless EH system. \[theo:effective\_capacity\] [The effective capacity of the point-to-point wireless communication system in which the transmitter performs EH, stores the harvested energy in a battery, and is subject to a given data QoS exponent, $\theta$, and an energy decay rate, $\mu$, is given as $$\label{effective_capacity_main} C_{E}(\theta,\mu)=-\frac{1}{N\theta}\ln\left(\chi^{\star}\right)\text{ bits/channel use,}$$ where $\chi^{\star}$ is the unique real positive root of $z(\chi)$, which is defined as $$\begin{aligned} \label{characteristic_function} z(\chi)=&\lim_{m\to\infty}\Bigg[\chi^{m}-\phi_{0}(-\theta)\sum_{n=1}^{m}\chi^{m-n}(1-q_{n})\prod_{j=1}^{n-1}q_{j}\phi_{j}(-\theta)\Bigg],\end{aligned}$$ where $$\begin{aligned} \phi_{0}(\theta)=&\frac{1}{\pi_{0}}\sum_{m=0}^{\infty}\pi_{m}\mathbb{E}_{p(i)}\Bigg[\int_{0}^{p(i)}\Big[\exp\left(\theta g(\xi(i))\right)\Pr\left\{\kappa(\xi(i))\leq|h(i)|^{2}\right\}+\Pr\left\{\kappa(\xi(i))>|h(i)|^{2}\right\}\Big]\nonumber\\ &\times f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i)\Bigg]\label{moment_generating_function_zero_final}\end{aligned}$$ and $$\begin{aligned} \phi_{j}(\theta)=&\frac{1}{q_{j}}\mathbb{E}_{p(i)}\Bigg[\int_{p(i)}^{\infty}\Big[\exp\left(\theta g(p(i))\right)\Pr\left\{\kappa(p(i))\leq|h(i)|^2\right\}+\Pr\left\{\kappa(p(i))>|h(i)|^2\right\}\Big]\nonumber\\ &\times f_{\xi(i)|w(i-1)=j-1}(\xi(i)|w(i-1)=j-1)d\xi(i)\Bigg]\label{moment_generating_function_j_final}\end{aligned}$$ for $j\in\{1,\cdots\}$, where the buffer size, $d_{\text{th}}$, is large. Recall that $p_{c}(i)=\xi(i)=e(i-1)+u(i)$ when $e(i-1)+u(i)<p(i)$ and $p_{c}(i)=p(i)$ when $e(i-1)+u(i)\geq p(i)$. Moreover, an upper bound to the effective capacity is given by $C_{E}(\theta,\mu)=-\frac{1}{N\theta}\ln\left(\chi^{\star}\right)\leq-\frac{1}{N\theta}\ln\left(\chi_{\alpha}^{\star}\right)$, where $\chi_{\alpha}^{\star}$ is the unique real positive root of $z^{\star}(\chi_{\alpha})$, defined as $$\begin{aligned} \label{characteristic_function_upper_bound} z^{\star}(\chi_{\alpha})=&(\chi_{\alpha})^{\alpha}-\phi_{0}(-\theta)\sum_{n=1}^{\alpha}(\chi_{\alpha})^{\alpha-n}(1-q_{n})\prod_{j=1}^{n-1}q_{j}\phi_{j}(-\theta).\end{aligned}$$ The aforementioned upper bound converges to the effective capacity as $\alpha$ is increased.]{} *Proof*: See Appendix \[app:theorem\_effective\_capacity\]. ### Constant energy demand rate and fixed transmission rate Let us consider the channel and transmission settings described in Section \[subsubsec:constant\_energy\_rate\]. Then, we can express the moment generating functions in (\[moment\_generating\_function\_zero\_final\]) and (\[moment\_generating\_function\_j\_final\]) as follows: $$\begin{aligned} \phi_{0}(\theta)=&\frac{1}{\pi_{0}}\sum_{m=0}^{\infty}\pi_{m}\int_{0}^{p}\bigg[\exp\left(\theta g(\xi(i))\right)\exp\left(-\frac{\kappa(\xi(i))}{\sigma_{h}^{2}}\right)+1-\exp\left(-\frac{\kappa(\xi(i))}{\sigma_{h}^{2}}\right)\bigg]\\ &\times f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i),\end{aligned}$$ and $\phi_{j}(\theta)=\exp\left(\theta g(p)\right)\exp\left(-\frac{\kappa(p)}{\sigma_{h}^{2}}\right)+1-\exp\left(-\frac{\kappa(p)}{\sigma_{h}^{2}}\right)$, respectively. Numerical Results {#sec:numerical_results} ================= We substantiate our theoretical results with numerical demonstrations. In the following, we initially show results regarding the battery constraints, i.e., the energy overflow and outage probabilities. Particularly, we compare the theoretical approximations with finite-size and infinite-size battery simulations. Then, we plot the average data service rate as a function of the energy decay rate, $\mu$. Finally, we have the effective capacity results under QoS constraints with different energy arrival processes. Energy Overflow and Outage Probabilities ---------------------------------------- We model the energy arrival process using a Weibull distribution, which is widely employed in the wind industry as the preferred approach for modeling the wind speed for energy assessment due to its high versatility, flexibility, and accuracy for describing the wind speed variations [@olaofe2012statistical]. Hence, we have the following probability density function for the energy arrival samples for $u\geq0$: $f_{u}(u)=\frac{k}{\lambda}\left(\frac{u}{\lambda}\right)^{k-1}\exp\left(-\left(\frac{u}{\lambda}\right)^{k}\right)$, where $k$ and $\lambda$ are shape and scale parameters, respectively. We also note that we employ i.i.d. energy samples. In Fig. \[fig:fig\_1\], we set the desired energy overflow probability to $\Pr\{e\geq e_{\max}\}=10^{-4}$ for a threshold value of $e_{\max}=500$ energy units. Then, using the exponential approximation, i.e., $\Pr\{e\geq e_{\text{th}}\}\approx\exp(-\mu e_{\text{th}})$, we find $\mu=0.0184$. We initially plot the energy overflow probability using the exponential approximation, and then compare it with the simulation results where we obtain the energy overflow probability with an infinite-size battery and a finite-size battery with $e_{\max}=500$ units. In the simulations, we have $k=5$ and $\lambda=2$ for the shape and scale parameters, respectively. We further run an energy demand process with constant rate that provides the equality in (\[eq:arrival\_demand\_balance\]) for desired $\mu$ and is greater than the average energy arrival rate in order to guarantee the stability of the battery. However, one can implement the simulations with varying energy demand rates as long as (\[eq:arrival\_demand\_balance\]) is provided and the stability condition, i.e., $\mathbb{E}_{u}\left[u(i)\right]<\mathbb{E}_{p}\left[p(i)\right]$, is guaranteed. The energy overflow probability approximation captures the simulation performances with the infinite-size and finite-size batteries very closely, while the energy overflow probability with the finite-size battery is less than the approximation for threshold values close to the battery size. However, we can accurately approximate the energy overflow probability for threshold values less than the battery size. Our approximation matches the finite-size battery simulations very closely for energy threshold values up to $80\%$ of the battery capacity, which is $400$ energy units in our simulations. In a real setting, one should not charge a battery completely but up to $80\%$ in order to improve the battery life-span and the energy efficiency [@battery_uni_2; @website_2]. In Fig. \[fig:fig\_2\], we plot the energy outage probability, $\pi_{0}$ in (\[eq:upper\_bound\_pi\_0\]), as a function of the energy decay rate, $\mu$, for different scale parameters, $\lambda_{\text{dB}}=10\log_{10}\frac{\lambda}{N\sigma_{w}^{2}}=5$, $4$ and $3$ dB, when the shape parameter is $k=1$. Note that the mean value of the Weibull distribution is $\lambda\Gamma(1+1/k)$, where $\Gamma(\cdot)$ is the Gamma function. So, $\lambda$ becomes the average energy arrival rate when $k=1$. If we use all the energy for data transmission, we can consider $\lambda_{\text{dB}}$ as the average signal-to-noise ratio in the channel. In addition, given the energy decay rate, we determine the constant energy demand rate. Particularly, given that the energy packets arriving at the battery are i.i.d., the constant energy demand rate is $p=\frac{1}{\mu}\ln\mathbb{E}_{u}\left[e^{\mu u}\right]=\frac{1}{\mu}\ln\left(\frac{1}{1-\lambda\mu}\right)$ for $0<\mu<\frac{1}{\lambda}$. Notice that when $\mu$ goes to zero, $p$ approaches the average energy arrival rate, $\lambda$, whereas when $\mu$ goes to $\frac{1}{\lambda}$, $p$ approaches infinity. Therefore, the outage probability becomes 1 once $\mu$ is greater than $\frac{1}{\lambda}$ for a given energy arrival process with the aforementioned distribution, which is displayed in Fig. \[fig:fig\_2\]. We can also infer that the energy outage probability goes to 1 when we consume the energy in the battery faster, i.e., as $\mu$ increases. The black dashed vertical lines in Fig. \[fig:fig\_2\] indicate the energy decay rate above which the constant energy demand rate is infinite for the defined energy arrival process, i.e., $p=\infty$. Therefore, when the energy storage conditions are stricter, energy demand processes without a constant rate should be favored. However, having a transmission system with varying transmission power capability leads to complexity in the transmitter design and a large number of calculation steps during data transmission. Furthermore, having an energy decay rate (or an energy overflow probability for a fixed battery threshold), we can infer from the results that with the increasing scale parameter in the energy arrival process, the energy outage probability increases, which is due to the increased scattering in the probability distribution of the energy arrival samples with the increasing scale parameter. Average Data Service Rate {#average-data-service-rate} ------------------------- In Fig. \[fig:fig\_3\] and Fig. \[fig:fig\_4\], we plot the average data service rate, $s_{\text{avg}}$, as a function of the energy decay rate, $\mu$, in AWGN and Rayleigh fading channels, respectively, given that we have an energy demand process with constant rate, and a constant transmission rate policy. In the case of an AWGN channel, we have a constant, unit-valued channel gain. Since the channel does not change, we set the transmission rate equal to the instantaneous mutual information, i.e., $r(i)=g(p)=N\log_{2}\Big(1+\frac{p}{N\sigma_{w}^{2}}\Big)$, when the power demand is satisfied. On the other hand, the transmission rate is set to $r(i)=g(\xi(i))=N\log_{2}\Big(1+\frac{\xi(i)}{N\sigma_{w}^{2}}\Big)$ when there is a power outage. Notice that $s(i)$ is always equal to $r(i)$, i.e., $s(i)=r(i)$, because there is no transmission outage since $r(i)=\mathcal{I}(i)$. When we have Rayleigh fading with unit variance, i.e., $\sigma_{h}^{2}=1$, we choose the transmission rate in a way to maximize the average data service rate in the corresponding time frame, i.e., $r(i)=\arg\max_{g(p)}\{g(p)\exp(\kappa(p))\}$ when the battery sustains the energy demand or $r(i)=\arg\max_{g(\xi(i))}\{g(\xi(i))\exp(\kappa(\xi(i)))\}$ when there is an energy outage, where $\kappa(a) = \left(2^{\frac{g(a)}{N}}-1\right)\frac{N\sigma_{w}^{2}}{\xi(i)}$ for $a\geq0$. Note that $\exp(\kappa(a))$ is the probability that the data transmission is successful in the corresponding time frame. The time frames are equal to 100 channel uses, i.e., $N=100$. For both channels, we again obtain results[^17] with different scale parameters. We observe that increasing battery overflow constraints above certain values causes a sharp decrease in the average data service rate. Noting that the black dashed vertical lines indicate the energy decay rate above which the constant energy demand rate is infinite for the defined energy arrival process, i.e., the transmitter consumes the energy packets in the time frame they arrive at the battery when $\mu\geq\frac{1}{\lambda}$, the average data service rate becomes constant for $\mu\geq\frac{1}{\lambda}$. Effective Capacity {#effective-capacity} ------------------ In Fig. \[fig:fig\_5\] and Fig. \[fig:fig\_7\], we plot the effective capacity, $C_{E}(\theta,\mu)$, versus the energy decay rate, $\mu$, in AWGN and Rayleigh fading channels, respectively, again for energy processes with different average values. We further set the QoS exponent to $\theta=0.1$. Similar to the results in Fig. \[fig:fig\_3\], the effective capacity of AWGN channels increases with decreasing $\mu$ as seen in Fig. \[fig:fig\_5\], whereas unlike the results in Fig. \[fig:fig\_4\], the effective capacity of Rayleigh fading channels rises with $\mu$ up to a certain value, and then it starts decreasing until $\mu$ reaches $\frac{1}{\lambda}$ as seen in Fig. \[fig:fig\_7\]. The only concern in AWGN channel is the energy outages, and the frequency of energy outages decreases with the decreasing energy demand rate, i.e., decreasing $\mu$. The effective capacity increases with decreasing $\mu$ in AWGN channel because the service process becomes more deterministic. On the other hand, there are two concerns in Rayleigh fading channel, namely, transmission outages and energy outages. While the frequency of energy outages decreases with decreasing $\mu$, the occurrence of transmission outages increases with decreasing $\mu$, and hence the effective capacity decreases. With increasing $\mu$, the energy outages become dominant; as a result, the transmitter cannot take advantage of higher channel fading gains when there is possibly very little energy in the battery. Therefore, the effective capacity first increases and then decreases with increasing $\mu$. Recall that when $\mu\geq\frac{1}{\lambda}$, the transmitter utilizes the energy packets as soon as they arrive at the battery. Specifically, when the buffer overflow concerns are of importance in data transmission with an energy demand process with constant rate, it is strategic to set the average transmission power to the average energy arrival rate in AWGN channels, while it is necessary to set to a value that is greater than the average energy arrival rate in Rayleigh fading channels. Subsequently in Fig. \[fig:fig\_6\] and Fig. \[fig:fig\_8\], keeping the average energy arrival rate fixed at $\lambda_{\text{dB}}=5$ dB, we plot the effective capacity versus the energy decay rate for different QoS exponents, e.g., $\theta=0.09$, $0.10$ and $0.11$. We can see that the QoS exponent does not impact the range of the energy decay rate, while it affects the performance levels substantially. We can remark that if the battery conditions are not very strict, we can increase the effective capacity by minimizing the energy decay rate in AWGN channels even though it will cause an increase in energy overflow probability. In Rayleigh fading channels, by setting the energy decay rate to a value between zero and $\frac{1}{\lambda}$, we can increase the effective capacity even though the average data service rate in Rayleigh fading channels is maximized when the energy decay rate is minimized as seen in Fig. \[fig:fig\_8\]. Depending on the transmission objective, a system designer can opt for an energy consumption and battery sustaining policy. One additional note is that when we compare the results in Fig. \[fig:fig\_3\] and Fig. \[fig:fig\_4\] with the results in Fig. \[fig:fig\_5\] and Fig. \[fig:fig\_7\], we see a significant difference in performance levels of the average data service rate and the effective capacity. In other words, the service rate in the wireless channel from the transmitter to the receiver is much greater than the data arrival rate at the transmitter. This is because of the scattering in the data service rates that follows due to the energy outages and the transmission outages. More specifically, if there are no random energy outages and random transmission outages, the data arrival rate at the transmitter buffer will be equal to the constant transmission rate in the channel. Moreover, if we are to operate under data buffering constraints in Rayleigh channels, we need to have a moderate energy demand process. Otherwise, we can maximize the average data service rate by decreasing $\mu$. Different from the existing effective capacity studies in wireless fading channels, for instance see [@tang; @musavian2010effective1; @akin_eura; @musavian2010effective; @akin2010effective; @elalem], that invoke average and/or peak average power constraints, we do not have an energy source that provides guaranteed energy levels when needed, and due to the stochastic nature of the energy source, we rather employ the energy overflow and outage probability constraints in the battery. We further provide the effective capacity performance analysis as a function of the energy decay rate but not the signal-to-noise ratio since the energy decay rate is a tool that captures the balance between the energy overflow and outage probabilities, i.e., smaller energy decay rate implies increased energy overflow probability and decreased energy outage probability whereas higher energy decay rate indicates decreased energy overflow probability and increased energy outage probability. Conclusion {#sec:conclusion} ========== We have considered an energy harvesting transmitter equipped with a rechargeable battery and a data buffer. We have assumed that the transmitter harvests energy in random amounts and stores it in its battery, while the data also arrives at the data buffer in a random manner. We have provided a methodology to derive the relationship between the energy arrival and energy demand processes, and its impact on the throughput performance. We have initially approximated the energy overflow probability in the battery as an exponential function using tools from large deviation theory. Subsequently, projecting the energy availability at the battery on a Markov process, we have obtained the energy outage probability in the battery. Then, under the energy overflow and outage constraints, we have characterized the average data service rate over the wireless channel and the effective capacity in the data buffer under QoS constraints. We have substantiated our analytical results by numerical simulations considering AWGN and Rayleigh fading channels. Our results show that a strategy that stores the harvested energy and utilizes it regarding energy overflow and outage constraints results in better performance levels when compared to a strategy that consumes energy as soon as it is harvested. Finally, our results reveal that a strategy that saves energy as much as possible and avoids energy outages completely does not yield the maximum throughput performance under QoS constraints in Rayleigh fading channels. However, a strategy that consumes energy neither moderately nor greedily results in the maximum effective capacity performance. Proof of Proposition \[pro:mu\_zero\_and\_infinity\] {#app:proposition_mu_zero_and_infinity} ---------------------------------------------------- When the energy decay rate, $\mu$, goes to zero, the minimum constant energy demand rate goes to the average energy arrival rate. In particular, $$\begin{aligned} p^{\star}&=\lim_{\mu\to0}\frac{1}{\mu}\Lambda_{u}(\mu)=\lim_{\mu\to0}\lim_{t\to\infty}\frac{1}{\mu t}\ln\mathbb{E}_{u}\Big[\exp\Big(\mu\sum_{i=1}^{t}u(i)\Big)\Big]\nonumber\\ &=\mathbb{E}_{u}\big[u(i)\big]+\lim_{\mu\to0}\frac{\mu}{2t_{k}}\sigma_{t_{k}}^{2}=\mathbb{E}_{u}\big[u(i)\big],\end{aligned}$$ where $t_{k}$ is a large positive integer and $\sigma_{t_{k}}^{2}$ is the variance of $\sum_{i=1}^{t_{k}}u(i)$ [@soret2010capacity Section III]. When the decay rate, $\mu$, goes to infinity, the minimum constant energy demand rate approaches the maximum energy arrival rate, i.e., $p^{\star}=\max\{u(i)\}=u_{\max}$. Noting that the limit when $\mu$ goes to infinity exists uniformly for any given $t>0$, we can again show this by exchanging the limits by invoking [@kadelburg2005interchanging Theorem 1]. Specifically, $$\begin{aligned} p^{\star}&=\lim_{\mu\to\infty}\lim_{t\to\infty}\frac{1}{\mu t}\ln\mathbb{E}_{u}\Big[\exp\Big(\mu\sum_{i=1}^{t}u(i)\Big)\Big]=\lim_{t\to\infty}\frac{1}{t}\lim_{\mu\to\infty}\frac{1}{\mu}\ln\mathbb{E}_{u}\Big[\exp\Big(\mu\sum_{i=1}^{t}u(i)\Big)\Big]\label{eq:limit_change_2}\\ &=\lim_{t\to\infty}\frac{1}{t}\lim_{\mu\to\infty}\frac{1}{\mu}\ln\Big(\Pr\{U(t)=t\cdot u_{\max}\}\exp\left(\mu t u_{\max}\right)\Big)\label{eq:limit_change_3}\\ &=\lim_{t\to\infty}\frac{1}{t}\lim_{\mu\to\infty}\frac{1}{\mu}\ln\Big(\exp\left(\mu t u_{\max}\right)\Big)=\lim_{t\to\infty}\frac{1}{t}\left(tu_{\max}\right)=u_{\max}.\label{part_1_u_max}\end{aligned}$$ In (\[eq:limit\_change\_2\]), the first limit goes to $t\cdot u_{\max}$ for given $t$, and is primarily affected by the distribution of the energy arrival process [@kelly1996notes]. Proof of Proposition \[pro:mu\_zero\_and\_infinity\_2\] {#app:proposition_mu_zero_and_infinity_2} ------------------------------------------------------- The proof of the proposition is similar to the proof in Appendix \[app:proposition\_mu\_zero\_and\_infinity\]. Proof of Theorem \[theo:effective\_capacity\] {#app:theorem_effective_capacity} --------------------------------------------- In [@chang_book Chap. 7, Example 7.2.7], it is shown for Markov modulated processes that $\frac{\Lambda(\theta)}{\theta}=\frac{1}{\theta}\ln{\text{sp}}\{M\Phi(\theta)\}=\frac{1}{\theta}\ln{\text{sp}}\{\Upsilon(\theta)\}$ where ${\text{sp}}\{\Upsilon(\theta)\}$ is the spectral radius of the matrix $\Upsilon(\theta)=M\times\Phi(\theta)$; $M$ is the transition matrix of the underlying Markov process and $\Phi(\theta)={\text{diag}}\{\phi_{0}(\theta),\cdots,\phi_{m-1}(\theta)\}$ is a diagonal matrix; components of which are the moment generating functions of the processes in $m$ states. We can see the rates (number of bits leaving the queue, $s(i)$) supported by the above channel model with the state transition model described in Section \[sec:energy\_outage\] as a Markov modulated process, and hence we can apply the setup considered in [@chang_book] immediately in our setting. Given that the battery moves to state 0 from any state, we have $s(i)=r(i)$ bits served from the data buffer in state 0 if $r(i)\leq \mathcal{I}(i)$ in the $i^{\text{th}}$ time frame. Otherwise, the effective transmission rate is zero, i.e., $s(i)=0$. Then, we have the following moment generating function in state 0: $$\begin{aligned} \phi_{0}(\theta)=&\mathbb{E}\left[\exp\left(\theta s(i)\right)|w(i)=0\right]=\mathbb{E}\left[\exp\left(\theta r(i)\textbf{1}\left[r(i)\leq\mathcal{I}(i)\right]\right)|w(i)=0\right]\\ =&\mathbb{E}\Big[\exp\left(\theta r(i)\right)\Pr\left\{r(i)\leq\mathcal{I}(i)\right\}+\Pr\left\{r(i)>\mathcal{I}(i)\right\}|w(i)=0\Big]\\ =&\frac{1}{\pi_{0}}\mathbb{E}_{p(i)}\Bigg[\int_{0}^{p(i)}\Big[\exp\left(\theta g(\xi(i))\right)\Pr\left\{g(\xi(i))\leq\mathcal{I}(i)\right\}+\Pr\left\{g(\xi(i))>\mathcal{I}(i)\right\}\Big]f_{\xi(i)}(\xi(i))d\xi(i)\Bigg]\nonumber\\ =&\frac{1}{\pi_{0}}\sum_{m=0}^{\infty}\pi_{m}\mathbb{E}_{p(i)}\Bigg[\int_{0}^{p(i)}\Big[\exp\left(\theta g(\xi(i))\right)\Pr\left\{g(\xi(i))\leq\mathcal{I}(i)\right\}+\Pr\left\{g(\xi(i))>\mathcal{I}(i)\right\}\Big]\nonumber\\ &\times f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i)\Bigg]\nonumber\\ =&\frac{1}{\pi_{0}}\sum_{m=0}^{\infty}\pi_{m}\mathbb{E}_{p(i)}\Bigg[\int_{0}^{p(i)}\Big[\exp\left(\theta g(\xi(i))\right)\Pr\left\{\kappa(\xi(i))\leq|h(i)|^{2}\right\}+\Pr\left\{\kappa(\xi(i))>|h(i)|^{2}\right\}\Big]\nonumber\\ &\times f_{\xi(i)|w(i-1)=m}(\xi(i)|w(i-1)=m)d\xi(i)\Bigg].\label{eq:mgf_0}\end{aligned}$$ Because the battery moves into state 0 when the amount of energy in the battery is less than what the transmitter demands, i.e., $e(i-1)+u(i)<p(i)$, the integral is taken from 0 to $p(i)$. The expression in (\[eq:mgf\_0\]) converges to a finite value with the increasing number of summands. Regarding the convergence, we refer to the methodology provided in Section \[subsubsec:constant\_energy\_rate\]. Hence, it is enough to obtain a sum up to a certain value of $m$. Similarly, given that the battery is in state $j-1$ in the $(i-1)^{\text{th}}$ time frame and that it enters state $j$ in the $i^{\text{th}}$ time frame, we have $r(i)$ bits served in state $j$ if $r(i)\leq\mathcal{I}(i)$ and zero otherwise. Hence, we have the moment generating function in state $j$ as follows: $$\begin{aligned} \phi_{j}(\theta)&=\mathbb{E}\left[\exp\left(\theta s(i)\right)|w(i)=j\right]=\mathbb{E}\left[\exp\left(\theta r(i)\textbf{1}\left[r(i)\leq\mathcal{I}(i)\right]\right\}|w(i)=j\right]\\ =&\mathbb{E}\Big[\exp\left(\theta r(i)\right)\Pr\left\{r(i)\leq\mathcal{I}(i)\right\}+\Pr\left\{r(i)>\mathcal{I}(i)\right\}|w(i)=j\Big]\\ =&\frac{1}{\pi_{j}}\mathbb{E}_{p(i)}\Bigg[\int_{p(i)}^{\infty}\Big[\exp\left(\theta g(p(i))\right)\Pr\left\{\kappa(p(i))\leq|h(i)|^2\right\}+\Pr\left\{\kappa(p(i))>|h(i)|^2\right\}\Big]f_{\xi(i)}(\xi(i))d\xi(i)\Bigg]\nonumber\\ =&\frac{\pi_{j-1}}{\pi_{j}}\mathbb{E}_{p(i)}\Bigg[\int_{p(i)}^{\infty}\Big[\exp\left(\theta g(p(i))\right)\Pr\left\{\kappa(p(i))\leq|h(i)|^2\right\}+\Pr\left\{\kappa(p(i))>|h(i)|^2\right\}\Big]\nonumber\\ &\times f_{\xi(i)|w(i-1)=j-1}(\xi(i)|w(i-1)=j-1)d\xi(i)\Bigg]\\ =&\frac{1}{q_{j}}\mathbb{E}_{p(i)}\Bigg[\int_{p(i)}^{\infty}\Big[\exp\left(\theta g(p(i))\right)\Pr\left\{\kappa(p(i))\leq|h(i)|^2\right\}+\Pr\left\{\kappa(p(i))>|h(i)|^2\right\}\Big]\nonumber\\ &\times f_{\xi(i)|w(i-1)=j-1}(\xi(i)|w(i-1)=j-1)d\xi(i)\Bigg],\end{aligned}$$ where the battery moves into state $j$ when the amount of energy in the battery is greater than or equal to what the transmitter demands, i.e., $e(i-1)+u(i)\geq p(i)$, the integral is taken from $p(i)$ to infinity. In addition, we observe that $\Upsilon(\theta)$ is the Leslie matrix [@Hansen]. Hence, the characteristic function of $\Upsilon(-\theta)$ is given in (\[characteristic\_function\]). In order to analyze the roots of $z(\chi)$, we invoke the following theorem: Let $z(\chi)=\chi^{n}-b_1\chi^{n-1}-\cdots-b_n$, where all the numbers $b_i$ are non-negative and at least one of them is non-zero. The polynomial $z(\chi)$ has a unique positive root $\chi^{*}$, and the absolute values of the other roots do not exceed $\chi^{*}$ [@Prasolov]. Note that $z(\chi)$ in (\[characteristic\_function\]) has coefficients that are non-negative, and at least one of them is non-zero. Therefore, there is one unique real positive root of $z(\chi)$, denoted by $\chi^{\star}$, which gives us the spectral radius of $\Upsilon(-\theta)$. We know from [@Hansen Corollary 8.1.20] that any principal sub-matrix of $\Upsilon(-\theta)$, which is denoted by $\widehat{\Upsilon}(-\theta)$, has a spectral radius less than or equal to the spectral radius of $\Upsilon(-\theta)$, i.e., ${\text{sp}}\{\widehat{\Upsilon}(-\theta)\}\leq{\text{sp}}\{\Upsilon(-\theta)\}$ because $\Upsilon(-\theta)$ is non-negative matrix. Particularly, truncating the matrix, $\Upsilon(-\theta)$, to a finite size matrix $\Upsilon_{\alpha}(-\theta)$, i.e., from row number 1 to row number $\alpha$ and from column number 1 to column number $\alpha$, we will obtain an upper bound to the effective capacity, because ${\text{sp}}\{\Upsilon_{\alpha}(-\theta)\}\leq{\text{sp}}\{\Upsilon(-\theta)\}$ and $-\frac{1}{\theta}\ln{\text{sp}}\{{\text{sp}}\{\Upsilon_{\alpha}(-\theta)\}\}\geq-\frac{1}{\theta}\ln{\text{sp}}\{\Upsilon(-\theta)\}$. Hence, we reach the expression in (\[characteristic\_function\_upper\_bound\]), which is the characteristic function of $\Upsilon_{\alpha}(-\theta)$. Moreover, by increasing the truncated matrix size, the upper bound converges to the effective capacity. We can show this by noting that any defined truncated matrix is a principal sub-matrix of a bigger truncated matrix following the aforementioned definition of the truncated matrix. [^1]: S. Ak[i]{}n is with the Institute of Communications Technology, Leibniz Universität Hannover, 30167 Hanover, Germany, (E-mail: sami.akin@ikt.uni-hannover.de). [^2]: M. C. Gursoy is with the Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244 USA (E-mail: mcgursoy@syr.edu) [^3]: This work was supported by the German Research Foundation (DFG) – FeelMaTyc (FI 1236/6-1) [^4]: One can consider adaptive power control and modulation schemes, where transmitters adjust their power and the data modulation according to the signal-to-noise ratio at the corresponding receivers. Since the received signal-to-noise ratio at a receiver is a function of channel fading, the adjusted power levels, hence the energy demands, become stochastic. [^5]: With parameters, we refer to distribution, mean, variance, and maximum and minimum values, etc. [^6]: Note that while the energy causality constraint remains between the amount of the accumulated energy and the amount of the consumed energy, it is assumed that the stability condition, in which the average demanded energy is greater than the average accumulated energy, exists so that the amount of energy in the battery in the steady-state does not go to infinity; thus, we control the energy waste. On the other hand, if our primary concern is to control transmission interruptions, we can impose stability conditions to avoid battery being depleted. In particular, we set $\mathbb{E}\{p(i)\}<\mathbb{E}\{u(i)\}$ as the stability condition, similarly to [@srivastava2013basic; @akin2017energy; @akin2018energy]. [^7]: In certain practical scenarios, the battery capacity can be regarded as infinite with respect to the energy arrival and demand processes. For example, some transceivers require output power levels on the order of $2-100$ mW to communicate within a range of $30$ meters, while some solar cells produce $15$ mW/$\text{cm}^2$ [@safak2014wireless], in which case an AAA alkaline battery [@website2] can be considered as having a very large capacity that can closely approximate infinite capacity. [^8]: We consider that the transmitter always has data to transmit. Therefore, it consumes a certain amount of energy as long as there is energy in the battery, and we regard the energy demand process as work-conserving. We have this assumption because the harvested energy is utilized for data transmission only, and the control of energy overflows and outages becomes important for an efficient use of the harvested energy when there is data in the transmitter buffer. Otherwise, when there is no data in the buffer, the transmitter harvests energy until the battery becomes full, and then stops harvesting energy. [^9]: Although energy arrivals in general vary drastically in nature, recalling that our analytical framework works when both energy arrivals and demands are stochastic, ergodic and stationary processes with finite mean and variance given that the two processes are independent of each other, we consider a constant energy demand process as an example for mathematical tractability of (\[eq:arrival\_demand\_balance\]). We refer interested readers to [@rajesh2011capacity; @siddiqui2017performance] as well, where a practical sensor node prototype that assumes solar energy to be constant for optimal event detection probability is considered. [^10]: We assume that the transmitter is aware of the instantaneous battery state. Therefore, it is able to re-adjust its transmission rate according to the available energy in the battery. [^11]: One can easily adopt practical modulation techniques, e.g., binary phase-shift keying and quadrature amplitude modulations, into the framework provided in this paper. When practical modulation techniques are employed, one should consider the instantaneous mutual information between the channel input and output rather than the instantaneous channel capacity. We note that the instantaneous channel capacity is the maximum mutual information between the channel input and output. [^12]: Although the result in (\[eq:avg\_srvc\_rate\_all\]) and the result in Theorem \[theo:effective\_capacity\] in the sequel are valid for channel models with arbitrary distributions with finite mean and variance, we focus on Rayleigh fading due to its practical relevance. [^13]: In order to provide a smooth presentation of the aforementioned framework, we have considered the special case of constant energy demand rate as an example. However, one easily implement other energy demand policies. For instance, we have implemented and simulated a water-filling power allocation policy based energy demand policy in [@akin2017energy] under energy underflow probability constraint. Energy underflow refers to the case the energy level in the battery falls below a certain level. [^14]: Although our framework is good for data transmission settings with varying data traffic rate, we consider a fixed data transmission rate for mathematical tractability. We refer interested readers to [@pasch1993comparing] for more details on data traffic types. Voice and video traffic can be modeled with a constant data service rate. [^15]: We do not define a specific function but consider a general definition for the function that sets the transmission rate. In other words, $g(\cdot)$ is a matter of system design and may depend on the statistics of the energy demand policy. However, for instance, one can set $g(\cdot)$ as a linear function of the energy demand or the available energy in the battery. [^16]: After the battery reaches the steady-state, the probability of the battery being in state $m$ in any time frame becomes $\pi_{m}$, as defined in Section \[sec:energy\_outage\], and it does not change with state transitions in the consecutive time frames. [^17]: Here, we simulate the energy arrivals and the channel fading gains with respect to their distributions assuming that Gaussian codebooks are employed. However, one can do simulations considering practical modulation techniques and their corresponding instantaneous mutual information values given the channel conditions.

--- abstract: 'Let $p$ be an odd prime, $m\in {{\mathbb N}}$ and set $q=p^m$, ${\boldsymbol{G}}=\PSL_n(q)$. Let $\theta$ be a standard graph automorphism of ${\boldsymbol{G}}$, $d$ be a diagonal automorphism and ${\operatorname{Fr}}_q$ be the Frobenius endomorphism of $\PSL_n(\overline{{{\mathbb F}}_q})$. We show that every $(d\circ \theta)$-conjugacy class of a $(d\circ \theta,p)$-regular element in ${\boldsymbol{G}}$ is represented in some ${\operatorname{Fr}}_q$-stable maximal torus of $\PSL_n(\overline{{{\mathbb F}}_q})$ and that most of them are of type D. We write out the possible exceptions and show that, in particular, if $n\geq5$ is either odd or a multiple of $4$ and $q>7$, then all such classes are of type D. We develop general arguments to deal with twisted classes in finite groups.' author: - 'Giovanna Carnovale$^1$, Agustín García Iglesias$^{2}$' title: '$\theta$-semisimple classes of type D in $\PSL_n(q)$' --- Introduction ============ This paper belongs to the series started in [@ACGa], in which we intend to determine all racks related to (twisted) conjugacy classes in simple groups of Lie type which are of type D [*cf.*]{} , as proposed in [@AFGaV2 Question 1]. This, although being mainly a group-theoretical question, is intimately related with the classification of finite-dimensional pointed Hopf algebras over non-abelian groups, see below. In this article we will focus on racks which arise as non-trivial twisted conjugacy classes in $\PSL_n(q)$ for $q=p^m$, $p$ an odd prime. Recall that a rack is a non-empty set $X$ together with a binary operation $\rhd$ satisfying faithfulness and self-distributive axioms, see \[subsec:racks\]. The prototypical example of a rack is a twisted conjugacy class ${{\mathcal O}}_x^\psi$ with respect to an automorphism $\psi\in{\operatorname{Aut}}(G)$ inside a finite group $G$, $x\in G$, with $$\label{eqn:twisted-rack} y\rhd z=y\psi(zy^{-1}), \qquad y,z\in {{\mathcal O}}_x^\psi.$$ This is in fact a quandle, as $y\rhd y=y$, $\forall\,y\in {{\mathcal O}}_x^\psi$. A rack $X$ is said to be [*of type D*]{} when there exists a decomposable subrack $Y=R\bigsqcup S\subseteq X$and elements $r\in R$, $s\in S$ such that $r\rhd(s\rhd (r\rhd s))\neq s$, see Section \[subsec:racks\]. Their study is deeply connected with the classification problem of finite-dimensional pointed Hopf algebras, as follows. Let $H$ be a finite dimensional pointed Hopf algebra over an algebraically closed field ${\Bbbk}$ and assume the coradical of $H$ is ${\Bbbk}G$, for a finite non-abelian group $G$. Following [@AG Section 6.1], there exist a rack $X$ and a 2-cocycle ${\bf q}$ with values in $\PL(n, {\Bbbk})$ such that ${\operatorname{gr}}H$, the associated graded algebra with respect to the coradical filtration, contains as a subalgebra the bosonization ${\mathfrak B }(X,{\bf q})\#{\Bbbk}G$. See [*loc. cit.*]{} for unexplained notation. Therefore, it is central for the classification of such Hopf algebras to know when $\dim{\mathfrak B }(X,{\bf q})<\infty$ for given $X$, ${\bf q}$. A rack $X$ is said to [*collapse*]{} when ${\mathfrak B }(X, {\bf q})$ is infinite dimensional for any ${\bf q}$. A remarkable result is that [*if $X$ is of type D, then it collapses*]{}. This is the content of [@AFGV Theorem 3.6], also [@HS1 Theorem 8.6], both of which follow from results in [@AHS]. Now every rack $X$ admits a rack epimorphism $\pi: X \to S$ with $S$ simple and it follows that $X$ is of type $D$ if $S$ is so. Hence, determining all simple racks of type D is a drastic reduction indeed for the classification problem, as many groups can be discarded and only a few conjugacy classes in simple groups remain. Only for such classes one needs to compute the possible cocycles that yield a finite dimensional Nichols algebras. Simple racks are classified into three classes [@AG], also [@J], namely [*affine*]{}, [*twisted homogeneous*]{} and that of [*non-trivial twisted conjugacy classes on finite simple groups*]{}, see [@AG] for definitions. Most (twisted) conjugacy classes in sporadic groups are of type D [@AFGV2], [@FV]. This is also the case for non-semisimple classes in $\PSL_n(q)$ [@ACGa], for unipotent classes in symplectic groups [@ACGa2] and for (twisted) classes in alternating groups [@AFGV]. Similar results follow for twisted homogeneous racks [@AFGaV]. Affine racks seem to be not of type D. In this article we begin the analysis of twisted classes of type D in $\PSL_n(q)$, for $q$ odd and automorphisms induced by algebraic group automorphisms of $\SL_n(\overline{{{\mathbb F}}_q})$. Recall that the automorphisms in $\PSL_n(q)$ are compositions of automorphisms induced by conjugation in $\PL_n(q)$ (diagonal and inner automorphisms), powers of a standard graph automorphism $\theta$ of the Dynkin diagram and powers of the Frobenius automorphism ${\operatorname{Fr}}_p$. Inner automorphisms may be neglected [@AFGaV §3.1]. Diagonal and graph automorphisms are induced by algebraic group automorphisms of $\SL_n(\overline{{{\mathbb F}}_q})$, whereas ${\operatorname{Fr}}_p$ is induced by an abstract group endomorphism. Their behaviour is therefore different [@steinberg-endo 10.13] and this is reflected in the structure of the twisted classes. In addition, if the $d\circ \theta^a$-class of $x$ in $\PSL_n(p)$ is of type D, $d$ a diagonal automorphism and $a=0,1$, then the ${\operatorname{Fr}}_p^m\circ\, d\circ \theta^a$-class of $x$ in $\PSL_n(q)$ is of type D for every $m$ and every $q$. Thus, we will focus on twisted classes for automorphisms $\psi=d\circ \theta^a$. The analysis of standard conjugacy classes in simple groups of Lie type (corresponding to $a=0$) has been started in [@ACGa; @ACGa2]. For these reasons the first twisted classes to look at are the $\psi$-classes in $\PSL_n(q)$, where $\psi$ is a composition of a diagonal automorphism $d$ with $\theta$. In analogy to the case of standard conjugacy classes, it is possible to reduce most of the analysis to the study of classes whose behaviour resembles that of semisimple or unipotent ones. However, in contrast to that case, the choices to be made depend on the gcd of $|\psi|$ and $p$ [ *cf.*]{} Subsection \[subsec:semis\]. Therefore, the cases of $p$ even and odd must be handled with different methods. The diagonal automorphisms always satisfy $(|\psi|,p)=1$ so we restrict to the case $(|\psi|,p)=1$ and we will require $p$ to be odd. Set ${\boldsymbol{G}}=\PSL_n(q)$, $\psi=d\circ\theta\in{\operatorname{Aut}}({\boldsymbol{G}})$, for $d$ a diagonal automorphism. The study of $(\psi,p)$-regular classes in ${\boldsymbol{G}}$, [*i. e.*]{}, of those classes replacing semisimple ones, can be reduced to the study of $(\theta,p)$-regular ${\boldsymbol{G}}$-orbits of elements in ${\operatorname{PGL}}_n(q)$. Such classes have a representative in a maximal torus ${\overline\Tt}_w^{{\operatorname{Fr}}_q}$ of ${\operatorname{PGL}}_n(q)$, for some $w\in W^\theta$, where we can take $w$ up to conjugation [*cf.*]{} Theorem \[thm:x-in-calT\]. It turns out that in most cases, the property of being of type D depends on $n$, $q$ and the conjugacy class of $w$ in $W^\theta$. Such classes are parametrized by a partition $\lambda=(\lambda_1, \dots, \lambda_r)$ of $h=\left[\frac{n}{2}\right]$, with $r\in{{\mathbb N}}$, $h_i>0$, and a certain vector $\varepsilon\in {\mathbb Z}_2^r$. Hence our result depends on the number of cycles $r$ of $\lambda$ and on the vector $\varepsilon=(\varepsilon_1,\dots,\varepsilon_r)\in {\mathbb Z}_2^r$. Let ${\boldsymbol{1}}$ stand for the partition $(1,\dots,1)$. \[thm:one\] Let $q$ be as above. Let $x\in {\overline\Tt}_w^{{\operatorname{Fr}}_q}$. Then the class ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}$ is of type D, with the possible exceptions of classes fitting into the following table: [| c | c |c| c | c| m[3cm]{} |]{} & $n$& $q$ & $x$\ & $r=2$& $\varepsilon=(0,\varepsilon_2)$ &&3,5&\ & &&4&3,7&\ &&&&&\ &&&&&\ &&&&&\ & any\*& 3,5& any\*\ &3& 7,13 & any\ & 4& $\equiv3(4)$ & any\ & 4& 9 & any\ \* Actually, some of the classes listed on the table are of type D, for instance when $n\geq 6$, $n\neq 7$ and $\varepsilon=(0,\ldots,0)$, see Lemma \[lem:weyl-j\]. See also Remark \[rem:r=2\]. We present this result in the language of Nichols algebras, as a partial answer in this cases to [@AFGaV2 Question 2], see also [@AFGV Theorem 3.6], and [*loc. cit.*]{} for unexplained notation. Consider the classes ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}$ in Theorem \[thm:one\] as racks with the rack structure . These are simple racks. Let $X={{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}$, $x\in {\overline\Tt}_w^{{\operatorname{Fr}}_q}$. Then $\dim {\mathfrak B }({{\mathcal O}}_x^{\theta,{\boldsymbol{G}}},{\bf q})=\infty$ for any cocycle ${\bf q}$ on $X$, with the possible exceptions of the classes in Table \[table:1\]. Also, an extract of Theorem \[thm:one\] can be rephrased as follows. Let $p$ be an odd prime, $m\in{{\mathbb N}}$, $q=p^m$. Set ${\boldsymbol{G}}=\PSL_n(q)$, $\psi=d\circ\theta\in{\operatorname{Aut}}({\boldsymbol{G}})$, for $d$ a diagonal automorphism. If $n\geq 5$, $q\geq 7$, then any $(\psi,p)$-regular class ${{\mathcal O}}$ is of type D with the possible exception $n=2\times$odd, ${{\mathcal O}}\simeq {{\mathcal O}}_1^{{\operatorname{Ad}}({\nu}^{-1})\circ\theta,{\boldsymbol{G}}}$, ${\nu}$ as in . When $\psi=\theta$, we obtain the following for classes with trivial $(\theta,p)$-regular part (also called $\theta$-semisimple part) which is the content of Propositions \[pro:trivial-even\] and \[pro:trivial-odd\]: Let ${{\mathcal O}}$ be a $\theta$-twisted conjugacy class with trivial $\theta$-semisimple part. Then ${{\mathcal O}}$ is of type D provided 1. $n>2$ is even, the unipotent part is nontrivial, and $q>3$. 2. $n>3$ is odd and the Jordan form of its $p$-part in ${\boldsymbol{G}}^\theta$ corresponds to the partition $(n)$. The paper is organized as follows. In Section \[sec:preliminaries\] we fix the notation and recall some generalities about racks and the group $\PSL_n(q)$. In Section \[sec:general\] we discuss some general techniques to deal with twisted conjugacy classes in a finite group. In Section \[sec:PSL\] we focus on $\PSL_n(q)$ and we begin a systematic approach to the study of its twisted classes, that includes an analysis of the Weyl group. In Section \[sec:regular\] we concentrate on $\theta$-semisimple classes and obtain the main results of the article. In Section \[sec:unipotent\] we present some results on classes with trivial $\theta$-semisimple part. Preliminaries {#sec:preliminaries} ============= Let $H$ be a group, $\psi\in {\operatorname{Aut}}(H)$. A $\psi$-twisted conjugacy class, or simply a twisted conjugacy class, is an orbit for the action of $H$ on itself by $h\cdot_\psi x=hx\psi(h)^{-1}$. We denote this class by $\mO_h^\psi$. If $K< H$ is $\psi$-stable, we will write $\mO_h^{\psi,K}$ to denote the orbit of $h$ under the restriction of the $\cdot_\psi$-action to $K$. In particular, $\mO_h=\mO_h^{\operatorname{id}}$ denotes the (standard) conjugacy class of $h\in H$. The stabilizer in $K<H$ of an element $x\in H$ for the twisted action will be denoted by $K_\psi(x)$ so that $H_{{\operatorname{id}}}(x)$ is $H_x$, the usual centralizer of $x$. For any automorphism $\psi$ of a group $H$, we write $H^\psi$ for the set of $\psi$-invariants in $H$. The inner automorphism given by conjugation by $x\in H$ will be denoted by ${\operatorname{Ad}}(x)$. If $K\triangleleft H$ is normal, then we also denote by ${\operatorname{Ad}}(x)$ the automorphism induced from the conjugation by $x\in H$. ${{\mathcal Z}}(H)$ will denote the center of $H$. Recall that the group $\mu_n({{\mathbb F}}_q)$ of roots of unity in a finite field ${{\mathbb F}}_q$ is isomorphic to ${\mathbb Z}_{{\mathbf{d}}}$, for ${\mathbf{d}}:=(n, q-1)$. We denote by ${\mathbb S_n}$, $n\in{{\mathbb N}}$, the symmetric group on $n$ letters. We also set ${{\mathbb I}}_n:=\{1,2,\,\ldots,\,n\}$ and $(b)_a=1+a+a^2+\dots+a^{b-1}$, $a,b\in {{\mathbb N}}$. Racks {#subsec:racks} ----- A *rack* $(X, \rhd)$ is a non-empty finite set $X$ together with a function $\rhd:X\times X\to X$ such that $i\rhd (\cdot):X\to X$ is a bijection for all $i\in X$ and $$i\rhd(j\rhd k)=(i\rhd j)\rhd (i\rhd k), \, \forall i,j,k\in X.$$ Recall that a rack $(X, \rhd)$ is a *quandle* when $i{\triangleright}i=i$, $\forall\,i\in X$. We shall write simply $X$ when the function $\rhd$ is clear from the context. If $H$ is a group, then the conjugacy class $\mO_h$ of any element $h\in H$ is a rack, with the function $\rhd$ given by conjugation. More generally, if $\psi\in{\operatorname{Aut}}(H)$, any twisted conjugacy class in $H$ is a rack with rack structure given by , see [@AG Theorem 3.12, (3.4)]. These are indeed examples of quandles. A subrack $Y$ of a rack $X$ is a subset $Y\subseteq X$ such that $Y\rhd Y\subseteq Y$. A rack is said to be *indecomposable* if it cannot be decomposed as the disjoint union of two subracks. A rack $X$ is said to be *simple* if ${\operatorname{card}}X > 1$ and for any surjective morphism of racks $\pi: X \to Y$, either $\pi$ is a bijection or ${\operatorname{card}}Y = 1$. ### Racks of type D A rack $X$ is of type D when there exists a decomposable subrack $Y=R\bigsqcup S$ of $X$ and elements $r\in R$, $s\in S$ such that $$\label{eqn:typeD} r\rhd(s\rhd (r\rhd s))\neq s.$$ If a rack $X$ has a subrack of type D, or if there is a rack epimorphism $X\twoheadrightarrow Z$ and $Z$ is of type D, then $X$ is again so. In particular, if $X$ is decomposable and $X$ has a component of type D, then $X$ is of type D. On the other hand, if $X$ is indecomposable, then it admits a projection $X\twoheadrightarrow Z$, with $Z$ simple. Hence, in the quest of racks of type D it is enough to focus on simple racks. The classification of simple racks is given in [@AG Theorems 3.9, 3.12], see also [@J]. A big class consists of twisted conjugacy classes in finite simple groups. \[rem:typeD\] Let ${{\mathcal O}}$ be a $\psi$-twisted conjugacy class. Then ${{\mathcal O}}$ is of type D if there are $r,s\in {{\mathcal O}}$ such that $r\notin {{\mathcal O}}_s^{\psi,L}$, for $L$ the $\psi$-stable closure of the subgroup generated by $r$ and $s$, and $$\label{eq:rs} r\psi(s)\psi^2(r)\psi^3(s)\neq s\psi(r)\psi^2(s)\psi^3(r).$$ In fact, if the above conditions hold, we set $S=\mO^{\psi,L} _s$ and $R=\mO^{\psi,L} _r$ and then $Y=R\bigsqcup S$ is a decomposable subrack of ${{\mathcal O}}$ which satisfies . If $\psi={\operatorname{id}}$ then the condition is also necessary: if ${{\mathcal O}}$ is of type D, then there are $r,s\in{{\mathcal O}}$, $r\notin {{\mathcal O}}_s^{L}$, satisfying [@ACGa Remark 2.3]. The group ${\boldsymbol{G}}=\PSL_n(q)$ {#sec:Gb} -------------------------------------- Fix $n\in{{\mathbb N}}$. Let $p\in{{\mathbb N}}$ be a prime number and let ${\Bbbk}=\overline{{{\mathbb F}}_p}$. Fix $m\in{{\mathbb N}}$, $q=p^m$. We assume throughout the paper that $n>2$ or $q\neq 2,3$. We fix once and for all the following notation: $$\begin{aligned} \label{eqn:gps} &{{\mathbb G}}=\SL_n({\Bbbk}), &&{\overline{{\mathbb G}}}=\PSL_n({\Bbbk}), && {\boldsymbol{G}}:=\PSL_n(q).\end{aligned}$$ We also fix $\pi\colon \PL_n({\Bbbk})\to {\operatorname{PGL}}_n({\Bbbk})\simeq {\overline{{\mathbb G}}}$ the usual projection. We shall keep the name $\pi:=\pi_{|{{\mathbb G}}}\colon {{\mathbb G}}\to {\overline{{\mathbb G}}}$ for the restriction of $\pi$ to ${{\mathbb G}}$. We fix the subgroups of diagonal matrices $$\begin{aligned} \label{eqn:tori} &\Tt\leq \PL_n({\Bbbk}), &&{{\mathbb{T}}}\leq{{\mathbb G}}, &&{\overline\Tt}:=\pi(\Tt)\leq {\overline{{\mathbb G}}}.\end{aligned}$$ ### General properties of ${\boldsymbol{G}}$ Consider the exact sequence: $$\begin{aligned} \label{eqn:exact} & 1\longrightarrow {{\mathcal Z}}({{\mathbb G}})\longrightarrow {{\mathbb G}}\overset{\pi}\longrightarrow {\overline{{\mathbb G}}}\longrightarrow 1\end{aligned}$$ and let $F={\operatorname{Fr}}_p^m$ be the endomorphism of $\PL_n({\Bbbk})$ raising every entry in $X\in\PL_n({\Bbbk})$ to the $q$-th power. Taking $F$-points, yields: $$\begin{aligned} &1\longrightarrow {{\mathcal Z}}(\SL_n(q))\longrightarrow \SL_n(q)\longrightarrow {\operatorname{PGL}}_n(q).\end{aligned}$$ Then ${\boldsymbol{G}}\leq {\operatorname{PGL}}_n(q)$ is the image of the last arrow: $${\boldsymbol{G}}=\PSL_n(q)\simeq \SL_n(q)/{{\mathcal Z}}(\SL_n(q))\simeq \SL_n(q)/{\mathbb Z}_{{\mathbf{d}}},$$ for ${\mathbf{d}}=(n,q-1)$. The group ${\boldsymbol{G}}$ is simple[^1]. We will denote by ${{\mathbb{B}}}, {\mathbb{U}}, {\mathbb{U}}^-\leq{{\mathbb G}}$ be the subgroups of ${{\mathbb G}}$ of upper triangular, unipotent upper-triangular, unipotent lower-triangular matrices. Set $$W:=N_{{{\mathbb G}}}({{\mathbb{T}}})/{{\mathbb{T}}}\simeq N_{{\overline{{\mathbb G}}}}({\overline\Tt})/{\overline\Tt}\simeq {\mathbb S_n}.$$ Recall that $[\SL_n(q),\SL_n(q)]=\SL_n(q)$ and $[{\operatorname{PGL}}_n(q),{\operatorname{PGL}}_n(q)]={\boldsymbol{G}}$, for $n>2$ or $q\neq 2,3$. Also, we have the identifications: $$\begin{gathered} {\overline{{\mathbb G}}}^F={\operatorname{PGL}}_n(q)={\overline\Tt}^F [{\operatorname{PGL}}_n(q),{\operatorname{PGL}}_n(q)]={\overline\Tt}^F {\boldsymbol{G}}\\ \simeq \PL_n(q)/{{\mathcal Z}}(\PL_n(q))\simeq \PL_n(q)/{{\mathbb F}}_q^\times.\end{gathered}$$ ### Automorphisms of ${\boldsymbol{G}}$ Recall that a [*diagonal automorphism*]{} of ${\boldsymbol{G}}$ is an automorphism induced by conjugation by an element in ${\overline\Tt}^F$. The [*graph automorphism*]{} $\theta\colon \PL_n({\Bbbk})\to \PL_n({\Bbbk})$ is given by $x\mapsto {{\tt{J}}}_n\,^t x^{-1}{{\tt{J}}}_n^{-1}$, for $$\begin{aligned} \label{eqn:J} {{\tt{J}}}_n=\left(\begin{smallmatrix} 0&\dots&0&1\\ 0&\dots &-1&0\\ \vdots&\Ddots&\vdots&\vdots\\ (-1)^{n-1}&\dots&0&0\end{smallmatrix}\right).\end{aligned}$$ It induces a non-trivial automorphism of ${{\mathbb G}}$ for $n\geq3$ and it is unique up to inner automorphisms[^2]. It also induces automorphisms of $\PL_n(q), \SL_n(q)$, ${\operatorname{PGL}}_n(q)$ and ${\boldsymbol{G}}$. We will drop the subscript $n$ and write ${{\tt{J}}}={{\tt{J}}}_n$ when it can be deduced from the context. By [@MT Theorem 24.24] every automorphism of ${\boldsymbol{G}}$ is the composition of an inner, a diagonal, a power of ${\operatorname{Fr}}_p$ and a power of $\theta$, so the elements in group of outer automorphisms of ${\boldsymbol{G}}$ have representatives in ${\operatorname{Out}}({\boldsymbol{G}}):=\langle {\operatorname{Fr}}_p,\theta, {\operatorname{Ad}}(t)\,:t\in {\overline\Tt}^F\rangle$. General arguments {#sec:general} ================= In this section we present some general techniques to deal with twisted conjugacy classes in finite groups. We start with a well-known lemma. \[lem:inner\] Let $H$ be a finite group, $\varphi\in {\operatorname{Aut}}(H)$. Let $K,N< H$ be $\varphi$-stable subgroups, with $N\lhd H$. Fix $x\in H$. \(1) The set ${{\mathcal O}}_x^{\varphi, K}$ is a subrack of ${{\mathcal O}}_x^{\varphi, H}$ if and only if for every $k\in K$ there is $t\in H_\varphi(x)$ such that $xkx^{-1}t\in K$. (2)[@AFGaV §3.1] Assume $\varphi={\operatorname{Ad}}(x)\circ\psi$, for some $\psi\in {\operatorname{Aut}}(H)$. Then for every $g\in H$ there are racks isomorphisms ${{\mathcal O}}_g^{\varphi,H}\simeq {{\mathcal O}}_{gx}^{\psi,H}$ and ${{\mathcal O}}_g^{\varphi,N}\simeq{{\mathcal O}}_{gx}^{\psi,N}$. \(3) Let $y\in H$ with $y\in{{\mathcal O}}_x^{\varphi,H}$. Then ${{\mathcal O}}_x^{\varphi,N}\simeq {{\mathcal O}}_y^{\varphi,N}$. \(1) is straightforward. In (2), we have the equality of sets ${{\mathcal O}}_g^{\varphi,H}={{\mathcal O}}_{gx}^{\psi,H}x^{-1}$ and right multiplication by $x$ defines the rack isomorphism. The second isomorphism follows by restriction. As for (3), let $g\in H$ be such that $g\cdot_\varphi x=y$. Then the map $T:{{\mathcal O}}_x^{\varphi,N}\to {{\mathcal O}}_y^{\varphi,N}$ given by $T(z)=g\cdot_\varphi z$ is a rack isomorphism. Observe that if $z=h\cdot_\varphi x$ then $T(z)=(ghg^{-1})\cdot_\varphi y$. \[rem:comm\] Notice that the assumption in (1) in Lemma \[lem:inner\] holds if $x\in N_H(K)$. In particular, it always holds for $K\lhd H$. Also, (2) allows us to neglect inner automorphisms of $H$. The following slight generalization of [@FV Lemma 2.5] will be very useful. \[lem:vendra\] Let $H$ be a finite group and let $K\lhd H$. Let $s\in H$ be a non-trivial involution. Then ${{\mathcal O}}_s^K$ is a rack of type D if and only if there is $r$ in ${{\mathcal O}}_s^K$ such that $|rs|$ is even and greater than $4$. By Lemma \[lem:inner\], Remark \[rem:comm\], ${{\mathcal O}}_s^K$ is a rack. Observe first that, if $r\in {{\mathcal O}}_s^K$, then the racks ${{\mathcal O}}_s^{\langle r,s\rangle}$ and ${{\mathcal O}}_r^{\langle r,s\rangle}$ are subracks of ${{\mathcal O}}_s^K$. Indeed, if $r=k{\triangleright}s=ksk^{-1}$, then a generic element of $\langle s,\, r\rangle$ has the form $y_{a,b}=s^a ksk^{-1}s\cdots ksk^{-1}s^b$ for $a,b\in\{0,1\}$. Let $sks=l\in K$. Then, if $a=1$ we have $$\begin{aligned} &y_{1,b}{\triangleright}s=y_{1,0}{\triangleright}s=lk^{-1}\cdots l k^{-1}{\triangleright}s\in {{\mathcal O}}_s^K,\\ &y_{1,b}{\triangleright}r=lk^{-1}\cdots l k^{-1} s^b k s^b{\triangleright}s\in{{\mathcal O}}_s^{K},\end{aligned}$$ whereas if $a=0$ we have $$\begin{aligned} &y_{0,b}{\triangleright}s=y_{0,1}{\triangleright}s=kl^{-1}\cdots k l^{-1}{\triangleright}s\in {{\mathcal O}}_s^K,\\ &y_{0,b}{\triangleright}r=kl^{-1}\cdots kl^{-1}s^{b-1}ks^{b-1}{\triangleright}s\in {{\mathcal O}}_s^K,\end{aligned}$$ so the racks ${{\mathcal O}}_s^{\langle r,s\rangle}, {{\mathcal O}}_r^{\langle r,s\rangle}\subset {{\mathcal O}}_s^K$. Now, if an $r$ as in the statement exists, then $r{\triangleright}(s{\triangleright}(r{\triangleright}s))\neq s$ and ${{\mathcal O}}_s^{\langle r,s\rangle}$ and ${{\mathcal O}}_r^{\langle r,s\rangle}$ are disjoint, so ${{\mathcal O}}_s^K$ is of type D by Remark \[rem:typeD\] for $\psi={\operatorname{id}}$. Conversely, if there is no such an $r$, then for every $x\in {{\mathcal O}}_s^K$ either $|xs|\leq 4$ or it is odd, so either $(xs)^2=(sx)^2$ or ${{\mathcal O}}_s^{\langle s,x\rangle}={{\mathcal O}}_x^{\langle s,x\rangle}$ and Remark \[rem:typeD\] for $\psi={\operatorname{id}}$ applies once more. \[rem:reductions\]Let $H$ be a finite group, $\phi\in{\operatorname{Aut}}(H)$, $h\in H$. 1. Assume $K=H_h$ is $\phi$-stable. If $k\in K$, then ${{\mathcal O}}_{kh}^{\phi,K}= {{\mathcal O}}_k^{\phi, K} h$ as sets and right multiplication by $h^{-1}$ gives a rack isomorphism ${{\mathcal O}}_{kh}^{\phi,K}\simeq{{\mathcal O}}_k^{\phi,K}$. 2. Let $L=H\rtimes\langle \phi\rangle$. Then, for $x=g\phi$, we have the equality of sets: ${{\mathcal O}}_g^{\phi,H}={{\mathcal O}}_x^L\,\phi^{-1}$ and $y\mapsto y\phi$ induces a rack isomorphism ${{\mathcal O}}_g^{\phi,H}\simeq{{\mathcal O}}_x^L$. \[rem:reductions2\] Let $H$ be a finite group, $\phi\in{\operatorname{Aut}}(H)$. Let $A$ be a $\phi$-stable abelian subgroup of $H$, $a\in A$. 1. \[item:abelian\] By Remark \[rem:reductions\] (1), ${{\mathcal O}}_a^{\phi, A}\simeq {{\mathcal O}}_1^{\phi,A}$ as racks. Moreover $\gamma\colon A\to A$, $b\mapsto b\phi(b^{-1})$, is a group morphism and ${{\mathcal O}}_1^{\phi,A}={\operatorname{Im}}(\gamma)\simeq A/A^\phi$ as groups. 2. \[item:abelian2\] If $\phi$ is an involution, then ${{\mathcal O}}_a^{\phi, A}$ is of type D if and only if there is $b\in A/A^\phi$ such that $|b|$ is even, $>4$ by Remark \[rem:reductions\] (2) and Lemma \[lem:vendra\]. 3. \[item:p-part\] Let $p$ be a prime number dividing $|H|$. Let $h=us=su\in H$ be the (unique) decomposition of $h$ as a product of a $p$-element $u$ and a $p$-regular element $s$. If ${{\mathcal O}}_{u}^{H_s}$ is of type D, then ${{\mathcal O}}_h$ is again so, as ${{\mathcal O}}_{u}^{H_s}$ identifies with a subrack of ${{\mathcal O}}_h^H$. \[rem:stable-orbit\] Let $H$ be a group, let $\phi,\psi\in {\operatorname{Aut}}(H)$, with $\phi\psi=\psi\phi$, and let $N\lhd H$ be $\phi$-stable and $\psi$-stable. \(1) If ${{\mathcal O}}_{h}^{\phi,N}\cap H^\psi\neq\emptyset$, then $\psi({{\mathcal O}}_{h}^{\phi,N})={{\mathcal O}}_{h}^{\phi,N}$. Indeed, let $x\in {{\mathcal O}}_t^{\phi,N}$ with $\psi(x)=x$. Now, if $y=kx\phi(k^{-1})\in {{\mathcal O}}_x^{\phi,N}={{\mathcal O}}_h^{\phi,N}$, $k\in N$, then $\psi(y)=\psi(k)x\phi(\psi(h)^{-1})\in {{\mathcal O}}_h^{\phi,N}$. \(2) Conversely, if $\psi({{\mathcal O}}_{h}^{\phi,N})={{\mathcal O}}_{h}^{\phi,N}$ and the map $N\to N$, given by $x\mapsto x^{-1}\psi(x)$, $x\in N$, is surjective, then ${{\mathcal O}}_{h}^{\phi,N}\cap H^\psi\neq\emptyset$. To see this, fix $g\in N$ such that $\psi(h)=gh\phi(g^{-1})$ and let $x\in N$ be such that $g^{-1}=x^{-1}\psi(x)$. Then it follows that $x\cdot_\phi h\in H^\psi\cap {{\mathcal O}}_{h}^{\phi,N}$. $(\psi,p)$-elements and $(\psi,p)$-regular elements {#subsec:semis} --------------------------------------------------- Let $H$ be a finite group, $p$ be a prime number dividing $|H|$ and let $\psi\in {\operatorname{Aut}}(H)$, with $\ell:=|\psi|$. Set $\widehat{H}=H\rtimes\langle\psi\rangle$. An element $h\in H$ is called $(\psi,p)$-regular if $h\psi$ is $p$-regular in $\widehat{H}$, [*i. e.*]{} if $(|h\psi|,p)=1$. An element $h\in H$ is called a $(\psi,p)$-element if $h\psi$ is a $p$-element in $\widehat{H}$, [*i. e.*]{} if $|h\psi|=p^a$ for some $a\in{{\mathbb N}}$. Let $\psi=\psi_r\psi_p$ be the decomposition of $\psi$ as a product of its usual $p$-regular part and its $p$-part in ${\operatorname{Aut}}(H)$. Then for every $h\psi$ in $\widehat{H}$ we have $h\psi=s\psi_r(u)\psi=u\psi_p(s)\psi$ where $s$ is $(\psi_r,p)$-regular and $u$ is a $(\psi_p,p)$-element in $H$. In the quest of $\psi$-classes of type D, a first analysis can be done by looking at subracks given by the orbits with respect to $H^{\psi_r}$ or $H^{\psi_p}$. For this reason, the analysis should begin with the cases in which either $\psi_p=1$, [*i. e.*]{} when $(\ell,p)=1$, or when $\psi_r=1$, [*i. e.*]{} when $\ell$ is a power of $p$. If $(\ell,p)=1$, then for every $h\in H$ there is a unique decomposition $h=us=s\psi(u)$ with $u$ a $p$-element in $H$ and $s$ a $(\psi,p)$-regular element. In this case $s$ is $(\psi,p)$-regular if and only if the norm ${\rm Norm}_\psi(s):= s\psi(s)\cdots\psi^{\ell-1}(s)$ is $p$-regular in $H$. Here, if $C=H_{\psi}(s)$ and $C'=\widehat{H}_{s\psi}$, then Remarks \[rem:reductions\] (2) and \[rem:reductions2\] give the rack inclusions $$\label{eqn:unipotent-inclusions} {{\mathcal O}}_h^{\psi,H}\simeq {{\mathcal O}}_{h\psi}^{\widehat{H}}\supset {{\mathcal O}}_{u}^{C'}\supset{{\mathcal O}}_u^{C}.$$ So if ${{\mathcal O}}_{u}^{C}$ is of type D, then ${{\mathcal O}}_h^{\psi,H}$ is again so. Hence the first classes to be attacked are either standard conjugacy classes of $p$-elements in $C$ or twisted classes of $(\psi,p)$-regular elements in $H$. The latter are dealt with in Section \[sec:regular\]. Similarly, if $\ell=p^b$ for some $b>0$, then for each $h\in H$ there is a unique decomposition $h=su=u\psi(s)$ with $s$ a usual $p$-regular element in $H$ and $u$ a $(\psi,p)$-element. In this case $u$ is a $(\psi,p)$-element if and only if ${\rm Norm}_\psi(u)$ is a $p$-element in $H$. The first reduction is to look at classes of $(\psi,p)$-elements and the standard $p$-regular classes in $H_{\psi}(u)$. We will not pursue this analysis in this paper. Notice that, when dealing with twisted classes in simple groups of Lie type, there is a privileged choice for $p$, namely, the defining characteristic. Twisted classes and $\PSL_n(q)$ {#sec:PSL} =============================== In this section we collect some results that contribute to establish a systematic approach to twisted classes in $\PSL_n(q)$. This in particular requires a detailed study of the conjugacy classes in the subgroup of $\theta$-invariant elements of the Weyl group, and of the corresponding $F$-stable maximal tori in ${\boldsymbol{G}}$, that we develop in §\[subsec:weyl\]. Recall the notation from §\[sec:Gb\], specially in , . Next proposition deals with diagonal automorphisms $d={\operatorname{Ad}}(t)$, $t\in {\overline\Tt}^F$. \[pro:x-in-PGL\] Let $x\in{\boldsymbol{G}}$, $\varphi={\operatorname{Ad}}(t)\circ \psi\in{\operatorname{Aut}}({\boldsymbol{G}})$, $t\in {\overline\Tt}^F$. Let $y=t^{-1}x\in{\overline{{\mathbb G}}}^F$. Then ${{\mathcal O}}_x^{\varphi,{\boldsymbol{G}}}\simeq {{\mathcal O}}_y^{\psi,{\boldsymbol{G}}}$. If, in addition, $\psi\in{\operatorname{Out}}({\boldsymbol{G}})$ and $z\in {{\mathcal O}}_y^{\psi,{\operatorname{PGL}}_n(q)}$, then ${{\mathcal O}}_x^{\varphi,{\boldsymbol{G}}}\simeq {{\mathcal O}}_z^{\psi,{\boldsymbol{G}}}$. In this case, $x=ty$ and ${{\mathcal O}}_x^{\varphi,{\boldsymbol{G}}}\simeq {{\mathcal O}}_y^{\psi,{\boldsymbol{G}}}$ by Lemma \[lem:inner\] (2). The last assertion is Lemma \[lem:inner\] (3). Let $\psi={\operatorname{Fr}}_p^a\circ \theta^b\in {\operatorname{Aut}}(\PL_n(q))$ and let $\ell:=|\psi|$. Then $\psi$ induces an automorphism of $\SL_n(q), \PSL_n(q)$ and ${\operatorname{PGL}}_n(q)$ of the same order. Let $H$ be either $\PL_n(q)$, $\SL_n(q), \PSL_n(q)$, or ${\operatorname{PGL}}_n(q)$, $\widehat{H}=H\rtimes\langle \psi\rangle$. If $(\ell,p)=1$, then the $(\psi,p)$-elements in $H$ are the unipotent elements in $H$. The $(\psi,p)$-regular elements are those $g\in H$ such that ${\rm Norm}_\psi(g)$ is semisimple. If, instead, $\ell=p^b$ for some $b>0$, then the $(\psi,p)$-regular elements in $H$ are the semisimple elements in $H$, while the $(\psi,p)$-elements are those $g\in H$ such that ${\rm Norm}_\psi(g)$ is a $p$-element. We will concentrate on the case $(\ell,p)=1$. We have the following equivalence. \[lem:equivalence-1\] Let $\psi\in {\operatorname{Aut}}(\PL_n(q))$ with $(|\psi|,p)=1$. Then ${{\tt{x}}}\in \PL_n(q)$ is $(\psi,p)$-regular if and only if $x=\pi({{\tt{x}}})\in {\operatorname{PGL}}_n(q)$ is $(\psi,p)$-regular. ${\rm Norm}_\psi(x)$ is semisimple if and only $\pi({\rm Norm}_\psi(x))={\rm Norm}_\psi({{\tt{x}}})$ is so. The case $\psi=\theta$, $p\neq 2$ --------------------------------- We intend to study twisted classes for automorphisms induced from algebraic group automorphisms. By Remark \[rem:comm\] and Proposition \[pro:x-in-PGL\], we may reduce to the case $\psi=\theta$. We will focus on the case of $p$ odd and we shall investigate $(\psi,p)$-regular classes. \[rem:vinberg\]It was pointed to us by Prof. Vinberg that when the group is $\PL_n({\overline{F_q}})$ and $\psi=\theta$, then the map $x\mapsto x{{\tt{J}}}$ allows to identify the $\theta$-twisted conjugacy class of $x$ with the equivalence classes of the non-degenerate bilinear form with associated matrix $x{{\tt{J}}}$. Thus, the classification of twisted classes in this case can be deduced from the classification of bilinear forms on $\overline{{{\mathbb F}}_q}^n$. The latter, in turn, goes over in odd characteristic, as the classification in characteristic zero which is to be found for instance in [@HoP]. From this, $\SL_n(q)$-orbits could be also classified. However, since the action of the center by twisted conjugation is non-trivial, the step to $\PSL_n(q)$-orbits of elements in ${\operatorname{PGL}}_n(q)$ would need slight care. The main reason for our apparently less natural approach is related to the general problem of detecting twisted classes of type D in all finite simple groups. One of the aims in this paper is to propose a general systematic approach that could be applied, at least, to all finite simple groups of Lie type. \[lem:equivalence\] Let ${{\tt{x}}}\in \PL_n(q)$. 1. ${{\tt{x}}}$ is $\theta$-semisimple if and only if there is a $g\in\PL_n({\Bbbk})$ such that $g\cdot_\theta {{\tt{x}}}$ lies in a $\theta$-stable torus $\Tt_0$ in $\PL_n({\Bbbk})$. 2. ${{\tt{x}}}$ is $\theta$-semisimple if and only if there is a $g'\in\SL_n({\Bbbk})\subset \PL_n({\Bbbk})$ such that $g'\cdot_\theta {{\tt{x}}}\in \Tt$. \(1) is [@mohr Proposition 3.4]. Following the construction in [@mohr page 382] we can make sure that $\Tt_0$ is $F$-stable and that it is contained in $\Tt$. For (2), let ${{\mathcal Z}}:={{\mathcal Z}}(\PL_n({\Bbbk}))$, hence $\PL_n({\Bbbk})={{\mathcal Z}}{{\mathbb G}}$ and $\theta$ acts as inversion on ${{\mathcal Z}}$. Therefore, if $z\in {{\mathcal Z}}$, then $z\cdot_\theta x=xz^2$. Let $g=zg'\in {{\mathcal Z}}\,{{\mathbb G}}$ be such that $g\cdot_\theta x=t\in \Tt$. Then $g'\cdot x= t z^{-2}\in \Tt$, as ${{\mathcal Z}}$ is contained in every maximal torus. The lemma above motivates the following definition. We say that an element $x\in{\operatorname{PGL}}_n(q)$ is $\theta$-semisimple if it is $(\theta,p)$-regular. $F$-stable maximal tori {#subsec:weyl} ----------------------- In this section we collect preparatory material in order to find suitable representatives of ${\boldsymbol{G}}$-classes of $\theta$-semisimple elements in ${\operatorname{PGL}}_n(q)$. Unless otherwise stated, $p$ is arbitrary. Let $H$ denote either ${{\mathbb G}}$, ${\overline{{\mathbb G}}}$ or $\PL_n({\Bbbk})$ and, consequently, set $K={{\mathbb{T}}}$, ${\overline\Tt}$ or $\Tt$ ($={{\mathbb{T}}}{{\mathcal Z}}(\PL_n({\Bbbk}))$). Let $w \in W$, $\w\in wK$ and $g=g_w\in H$ be such that $g^{-1}F(g)=\w$ (Lang-Steinberg’s Theorem). We set $$K_w:=gKg^{-1}.$$ Then $K_w$ is an $F$-stable maximal torus of $H$ and all $F$-stable maximal tori in $H$ are obtained this way [@MT Proposition 25.1]. Two tori $K_w$ and $K_\sigma$ are $H^F$-conjugate if and only if $\sigma$ and $w$ are $W$-conjugate. We will provide a $\theta$-invariant version of this fact in Lemma \[lem:invt\] for $K=\Tt$ and ${\overline\Tt}$. We set $$\begin{aligned} \label{eq:invariance} &F_w:={\operatorname{Ad}}(\w)\circ F, \text{ so } (K_w)^F=g K^{F_w}g^{-1}. \end{aligned}$$ The automorphisms $\theta$ and $F$ preserve ${{\mathbb{T}}}$, hence they induce automorphisms on $W$ which we denote by the same symbol. The action of $F$ on $W$ is trivial, whereas the action of $\theta$ is conjugation by the longest element $w_0\in W$, so $W^\theta=W_{w_0}$. Observe that $$\begin{aligned} w_0=\begin{cases}(1,\,n)(2\,n-1)\dots(h,\,h+1)&\mbox{ if }n=2h,\\ (1,\,n)(2,\,n-1)\dots(h,\,h+2)&\mbox{ if }n=2h+1. \end{cases}\end{aligned}$$ Any $\sigma\in W^\theta$ can be written as $\sigma=\omega\tau$ where $\omega$ permutes the $2$-cycles in $w_0$ and $\tau$ is a product of transpositions occurring in the cyclic decomposition of $w_0$. In fact, $W^\theta\simeq \sym_h\rtimes{\mathbb Z}_2^h$, where $h=\left[\frac{n}{2}\right]$, the elements in $\sym_h$ correspond to products $c \theta(c)$ where $c$ is a cycle in $\sym_{{{\mathbb I}}_h}\leq\sym_n$, $\theta(c)=w_0 c w_0$ and the elements in ${\mathbb Z}_2^h$ are products of transpositions of the form $(i, n+1-i)$. \[rem:representatives\] There is a set of representatives $\{\sigmad\}\subset N_{{{\mathbb G}}}({{\mathbb{T}}})$ of $W$ such that $\sigmad\in N_{{{\mathbb G}}}({{\mathbb{T}}})^\theta$ if $\sigma\in W^\theta$, [@steinberg-endo 8.2, 8.3 (b)]. In addition, ${{\mathbb G}}^\theta=\Sp_n({\Bbbk})$ if $n$ is even, ${{\mathbb G}}^\theta=\SO_n({\Bbbk})$ if $n$ is odd and $W^\theta$ is the corresponding Weyl group. \[lem:invt\] Let $w,\sigma\in W^\theta$. Then $\Tt_w$ and $\Tt_\sigma$ are $\SL_n(q)^\theta$-conjugate if and only if $\sigma\in{{\mathcal O}}_w^{W^\theta}$ if and only if $\overline{\Tt}_w$ and $\overline{\Tt}_\sigma$ are $\pi(\SL_n(q)^\theta)$-conjugate. Since ${\rm Ker}(\pi)$ consists of central elements, it is enough to prove the first equivalence. By Remark \[rem:representatives\] there are representatives $\w$, $\sigmad$ of $w$ and $\sigma$ in ${{\mathbb G}}^\theta\cap N(\Tt)$. By Lang-Steinberg’s Theorem applied to ${{\mathbb G}}^\theta$ we may find $y,z\in{{\mathbb G}}^\theta$ such that $y^{-1}F(y)=\w$, $z^{-1}F(z)=\sigmad$. Assume there is $x\in \SL_n(q)^\theta$ such that $x\Tt_wx^{-1}=\Tt_\sigma$. Then, $\taud:=z^{-1}xy\in N(\Tt)\cap{{\mathbb G}}^\theta$ and $\taud\w \taud^{-1}\Tt=\taud\w F(\taud^{-1})\Tt=\sigmad \Tt$. Conversely, assume there is $\tau\in W^\theta$ such that $\tau w\tau^{-1}=\sigma$. Let $\taud\in {{\mathbb G}}^\theta\cap \tau \Tt$. Then there exist $h,k\in {{\mathbb G}}^\theta\cap \Tt=\Tt^\theta=\Tt^{\theta,\circ}$ such that $F(\taud)=\taud h$ and $\sigmad=\taud \w\taud^{-1}k$. For $t\in \Tt^{\theta}$ we set $x_t=z\taud t y^{-1}\in {{\mathbb G}}^\theta$. Now, $x_t \Tt_w x_t^{-1}=\Tt_\sigma$. In addition, $x_t\in \SL_n(q)^\theta$ if and only if $t=\w(\taud^{-1}k\taud)h F(t)\w^{-1}$. This happens if and only if $t^{-1} ({\operatorname{Ad}}(\w)\circ F)(t)=\w h^{-1}(\taud^{-1}k^{-1}\taud)\w^{-1}$. By Lang-Steinberg’s Theorem applied to the Steinberg endomorphism ${\operatorname{Ad}}(\w)\circ F$ on $\Tt^{\theta}$, there is $t\in \Tt^{\theta}$ satisfying this condition. \[lem:representatives\]Let $w\in W^\theta$, $v\in W$, $\w\in N_{{{\mathbb G}}^\theta}({{\mathbb{T}}})$ and $\v\in N_{{{\mathbb G}}}({{\mathbb{T}}})$ be representatives of $w$ and $v$, respectively. Let $y\in{{\mathbb G}}^\theta$ such that $y^{-1}F(y)=\w$. Then 1. $\v {{\mathbb{T}}}\cap {{\mathbb G}}^\theta\cap {{\mathbb G}}^{F_w}\neq\emptyset$ if and only if $v\in W^\theta_w$. 2. An element $v$ in $W=N_{{\overline{{\mathbb G}}}}({\overline\Tt})/{\overline\Tt}$ has a representative in ${\overline{{\mathbb G}}}^{\theta,\circ}\cap \pi({{\mathbb G}}^{F_w})$ if and only if $v\in W^\theta_w$. \(1) If $\v {{\mathbb{T}}}\cap {{\mathbb G}}^\theta\neq\emptyset$, then, $\theta(\v)\in \v{{\mathbb{T}}}$, so $v\in W^\theta$ and we may assume $\v\in{{\mathbb G}}^\theta$. If $\v{{\mathbb{T}}}\cap{{\mathbb G}}^{F_w}\neq\emptyset$, then $F_w(\v)\in \v{{\mathbb{T}}}$, that is ${\operatorname{Ad}}(\w)(\v)\in\v{{\mathbb{T}}}$, [*i. e.*]{} $wv=vw$. Conversely, assume $v\in W^\theta_w$. Now $W^\theta$ is the Weyl group of ${{\mathbb G}}^\theta$ and $F_w$ is a Steinberg endomorphism of ${{\mathbb G}}^\theta$ preserving its maximal torus ${{\mathbb{T}}}^\theta$. By [@MT Proposition 23.2 ff], $$(W^\theta)^{F_w}=\left(N_{{{\mathbb G}}^\theta}({{\mathbb{T}}}^\theta)/{{\mathbb{T}}}^\theta\right)^{F_w}\simeq N_{{{\mathbb G}}^\theta\cap{{\mathbb G}}^{F_w}}({{\mathbb{T}}}^\theta)/({{\mathbb{T}}}^\theta\cap{{\mathbb{T}}}^{F_w})$$ so any $v\in W_w^\theta=(W^\theta)^{F_w}$ has a representative in $$N_{{{\mathbb G}}^\theta\cap{{\mathbb G}}^{F_w}}({{\mathbb{T}}}^\theta)=N_{{{\mathbb G}}^\theta}({{\mathbb{T}}}^\theta)\cap {{\mathbb G}}^{F_w}=N_{{{\mathbb G}}^\theta}({{\mathbb{T}}})\cap {{\mathbb G}}^{F_w}=N_{{{\mathbb G}}}({{\mathbb{T}}})\cap {{\mathbb G}}^\theta\cap {{\mathbb G}}^{F_w}.$$ \(2) Follows from (1) recalling that $\pi({{\mathbb G}}^\theta)={\overline{{\mathbb G}}}^{\theta,\circ}$. We end the section with a lemma that shows how some some of the results on the Weyl group apply to the quest of preferred representatives in a twisted class. \[lem:intersections\] Let $t\in \Tt$ be such that ${{\mathcal O}}_t^{\theta,{{\mathbb G}}}\cap\PL_n(q)\neq\emptyset$. Then 1. There are $\sigma\in W^\theta$ and $\sigmad\in \sigma \Tt\cap {{\mathbb G}}^\theta$ such that ${{\mathcal O}}_t^{\theta,{{\mathbb G}}}\cap \Tt_\sigma^F \neq\emptyset$. 2. Let $\sigma$ be as in (1). Then ${{\mathcal O}}_t^{\theta,{{\mathbb G}}}\cap \Tt_w^F \neq\emptyset$ for every $w\in{{\mathcal O}}_{\sigma}^{W^\theta}$. 3. Fix $p$ odd and $x\in {{\mathcal O}}_t^{\theta,{{\mathbb G}}}\cap\PL_n(q)$. Then ${{\mathcal O}}_t^{\theta,{{\mathbb G}}}\cap \PL_n(q)={{\mathcal O}}_x^{\theta,{{\mathbb G}}^F}$. \(1) Pick a set of representatives $\{\dot{\tau},\;\tau\in W\}\subset N_{{{\mathbb G}}}(\Tt)$ as in Remark \[rem:representatives\]. Let $g\in{{\mathbb G}}$ be such that $F(t)=gt\theta(g^{-1})$, see Remark \[rem:stable-orbit\] (1). Let $u\in{\mathbb{U}}\cap \tau^{-1} {\mathbb{U}}^-\tau$, $\dot{\tau}\in N_{{{\mathbb G}}}(\Tt)\cap\tau {{\mathbb{T}}}$, $s\in{{\mathbb{T}}}$, $v\in {\mathbb{U}}$ such that $g=u\dot{\tau} sv$. Then $$\begin{aligned} F(t)\theta(g)=\left(F(t)\theta(u)F(t^{-1})\right)\cdot \left( F(t)\theta(\dot{\tau})\theta(s)\right)\cdot \theta(v)\in {{\mathbb{B}}}\theta\tau{{\mathbb{B}}}. \end{aligned}$$ On the other hand, $F(t)\theta(g)=gt=u\dot{\tau} svt=u (\dot{\tau} st )(t^{-1}vt)\in {{\mathbb{B}}}\tau{{\mathbb{B}}}$, which gives, by the uniqueness of the Bruhat decomposition, $\theta(\tau)=\tau\in W$ and, by construction, $\theta(\dot{\tau})=\dot{\tau}$. Also this yields $F(t)\theta(\dot{\tau})\theta(s)=\dot{\tau} st$, that is $F(t)=(\dot{\tau} s)\cdot_\theta t\in {{\mathcal O}}_t^{\theta,N_{{{\mathbb G}}}({{\mathbb{T}}})}$. Let $\sigmad:=\dot{\tau}^{-1}\in N_{{{\mathbb G}}^\theta}({{\mathbb{T}}})$. Then $F_{\sigma}={\operatorname{Ad}}(\sigmad)\circ F$ is again a Steinberg endomorphism for ${{\mathbb{T}}}$ and $F_{\sigma}(t)=ts\theta(s^{-1})\in {{\mathcal O}}_t^{\theta,{{\mathbb{T}}}}$. Let $r\in {{\mathbb{T}}}$ be such that $r^{-1}F_{\sigma}(r)=s$. Then $x=r^{-1}\cdot_\theta t\in {{\mathcal O}}_t^{\theta,{{\mathbb{T}}}}\cap \Tt^{F_{\sigma}}$. Indeed, $$\begin{aligned} F_{\sigma}(x)=F_{\sigma}(r^{-1}) F_{\sigma}(t)\theta(F_{\sigma}(r))=F_{\sigma}(r^{-1}) st\theta(s^{-1}F_{\sigma}(r))=r^{-1}t\theta(r)=x.\end{aligned}$$ Let $y\in{{\mathbb G}}^\theta$ be such that $y^{-1}F(y)=\sigmad$ and set $z=y\cdot_\theta x=yxy^{-1}$. Then $z\in y\Tt^{F_\sigma}y^{-1}=(y\Tt y^{-1})^F\cap {{\mathcal O}}_t^{\theta,{{\mathbb G}}}$, by and (1) follows. \(2) By Lemma \[lem:invt\] there is $g\in \SL_n(q)^\theta$ such that $g\Tt^F_\sigma g^{-1}=\Tt_w^F$. Hence, for $x\in {{\mathcal O}}_t^{\theta,{{\mathbb G}}}\cap \Tt_\sigma^F$ we have $g\cdot_\theta x\in {{\mathcal O}}_t^{\theta,{{\mathbb G}}}\cap \Tt_w^F$. \(3) The group ${{\mathbb G}}_{\theta}(t)={{\mathbb G}}^{{\operatorname{Ad}}(t^{-1})\circ\theta}$ is connected by [@steinberg-endo Theorem 8.1] since ${\operatorname{Ad}}(t^{-1})\circ\theta$ is a semisimple automorphism as defined in [@steinberg-endo p. 51]. The result follows from [@MT Theorem 21.11]. Twisted classes of $\theta$-semisimple elements {#sec:regular} =============================================== We assume from now on that $p$ is odd. Recall the notation from §\[sec:Gb\], specially in , . Strategy -------- Next theorem is the first main result of the paper and a key step to apply the strategy in Section \[subsec:strategy\]. \[thm:x-in-calT\] Let $x\in{\operatorname{PGL}}_n(q)$ be $\theta$-semisimple. Then there are $w\in W^\theta$ and $z\in {\overline\Tt}^F_w$ such that ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}={{\mathcal O}}_z^{\theta,{\boldsymbol{G}}}$. Let ${{\tt{x}}}\in\PL_n(q)$ be such that $x=\pi({{\tt{x}}})$. By Lemma \[lem:equivalence\] (3), there is $g\in {{\mathbb G}}$ such that $g\cdot_\theta {{\tt{x}}}=t\in \Tt$. Then there is $w\in W^\theta$ and ${{\tt{z}}}\in {{\mathcal O}}_t^{\theta,{{\mathbb G}}}\cap \Tt_w^F$ such that ${{\mathcal O}}_{{\tt{x}}}^{\theta,{{\mathbb G}}}={{\mathcal O}}_{{\tt{z}}}^{\theta,{{\mathbb G}}}$, by Lemma \[lem:intersections\] (1). On the other hand, we have that ${{\mathcal O}}_{{\tt{z}}}^{\theta,{{\mathbb G}}}\cap \PL_n(q)={{\mathcal O}}_{{\tt{z}}}^{\theta,\SL_n(q)}$, by Lemma \[lem:intersections\] (3). The statement now follows applying $\pi$, for $z=\pi({{\tt{z}}})$, as $\pi\left(\Tt^F_w\right)\subset {\overline\Tt}^F_w$ and $\pi\left({{\mathcal O}}_{{\tt{z}}}^{\theta,\SL_n(q)}\right)={{\mathcal O}}_z^{\theta,{\boldsymbol{G}}}$. ### The strategy {#subsec:strategy} Let $x$ be a $\theta$-semisimple element in ${\operatorname{PGL}}_n(q)$. By Theorem \[thm:x-in-calT\] we may assume $x\in{\overline\Tt}_w^F$ for some $w\in W^\theta$. We have the following inclusions of subracks: $$\label{eqn:strategy-inclusions} {{\mathcal O}}_x^{\psi,{\boldsymbol{G}}}\supseteq {{\mathcal O}}_x^{\psi,{\boldsymbol{G}}}\cap{\overline\Tt}_w^F\supseteq {{\mathcal O}}_x^{\psi,\pi({{\mathbb{T}}}_w^F)}\simeq{{\mathcal O}}_1^{\psi,\pi({{\mathbb{T}}}_w^F)}.$$ We will establish sufficient conditions ensuring ${{\mathcal O}}_1^{\psi,\pi({{\mathbb{T}}}_w^F)}$ is of type D. If the conditions are not satisfied and ${{\mathcal O}}_x^{\psi,{\boldsymbol{G}}}\cap{\overline\Tt}_w^F\neq {{\mathcal O}}_x^{\psi,\pi({{\mathbb{T}}}_w^F)}$, we will establish sufficient conditions ensuring ${{\mathcal O}}_x^{\psi,{\boldsymbol{G}}}$ is of type D. We look at the subracks ${{\mathcal O}}_x^{\theta,\pi({{\mathbb{T}}}_w^F)}\simeq {{\mathcal O}}_1^{\theta,\pi({{\mathbb{T}}}_w^F)}$ as in . Thus we investigate the abelian subgroups $\pi({{\mathbb{T}}}_w^F)$ and $\pi({{\mathbb{T}}}_w^F)\cap {\boldsymbol{G}}^\theta$. Let $\w\in w{{{\mathbb{T}}}}\cap {{\mathbb G}}^\theta$ and let $y\in {{\mathbb G}}^\theta$ be such that $y^{-1}F(y)=\w$. We have $${{\mathcal O}}_1^{\theta,\pi({{\mathbb{T}}}_w^F)}\simeq \pi({{\mathbb{T}}}_w^F)/(\pi({{\mathbb{T}}}_w^F)\cap{\boldsymbol{G}}^\theta)\simeq {{\mathbb{T}}}_w^F/K$$ for $K=\{t\in {{\mathbb{T}}}_w^F~|~\theta(t)\in t{{\mathcal Z}}(\SL_n(q))\}$. Let us set $$\begin{aligned} {{\tt{K}}}_w=\{s\in {{\mathbb{T}}}^{F_w}~|~\theta(s)\in s{{\mathcal Z}}(\SL_n(q))\}.\end{aligned}$$ \[lem:zeta\] Assume ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}\cap {\overline\Tt}_w^F\neq\emptyset$. If there is $s\in {{\mathbb{T}}}^{F_w}/{{\tt{K}}}_w$ such that $|s|$ is even and $>4$, then ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}$ is of type D. It follows from Remark \[rem:reductions\] (\[item:abelian2\]) and Lemma \[lem:vendra\], as conjugation by $y$ gives the group isomorphism ${{\mathbb{T}}}^{F_w} /{{\tt{K}}}_w\simeq {{\mathcal O}}_1^{\theta,\pi({{\mathbb{T}}}_w^F)}$. When conditions in Lemma \[lem:zeta\] do not hold, we will use the following lemma. \[lem:two-orbits\] Let $x\in{\overline\Tt}_w^F$ for some $w\in W^\theta$, and assume ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}\cap{\overline\Tt}_w^F\neq{{\mathcal O}}_x^{\theta,\pi({{\mathbb{T}}}_w^F)}$. If there is $z\in \mO_1^{\theta,\pi({{\mathbb{T}}}_w^F)}\simeq{{\mathbb{T}}}^{F_w}/{{\tt{K}}}_w$ such that $z^4\neq1$, then ${{\mathcal O}}_x^{\psi,{\boldsymbol{G}}}$ is of type D. The subrack $X={{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}\cap{\overline\Tt}_w^F$ is a disjoint union of orbits under the $\theta$-conjugation by $\pi({{\mathbb{T}}}_w^F)$, one of which is $R={{\mathcal O}}_x^{\theta,\pi({{\mathbb{T}}}_w^F)}=x{{\mathcal O}}_1^{\theta,\pi({{\mathbb{T}}}_w^F)}$. Let $S={{\mathcal O}}_t^{\theta,\pi({{\mathbb{T}}}_w^F)}=t{{\mathcal O}}_1^{\theta,\pi({{\mathbb{T}}}_w^F)}\subset X$, $S\neq R$. As ${\overline\Tt}_w^F$ is abelian and $\theta^2=1$, becomes $$\label{eqn:invo}(r\theta(r)^{-1})^2\neq(s\theta(s^{-1}))^2.$$ If holds for $r:=x$, $s:=t$, we are done. Otherwise, we replace $s$ by $s'=sz\in S$, obtaining the desired inequality. Conjugacy classes in $W^\theta$ {#subsec:conj classes in W} ------------------------------- We need to describe ${{\tt{S}}}_w$ and ${{\tt{K}}}_w$, $w\in W^\theta$. We will use the identification of $W^\theta$ with $\sym_h\rtimes{\mathbb Z}_2^h$, for $h=\left[\frac{n}{2}\right]$. Set $\{{\mathbf{e}}_i:1\leq i\leq h\}$ the canonical ${\mathbb Z}_2$-basis of ${\mathbb Z}_2^h$. Also, for $\lambda=(\lambda_1,\ldots,\lambda_r)$ $\lambda_j\geq\lambda_{j+1}$ a partition of $h$, consider the set ${\mathcal E}(\lambda)$ consisting of all vectors $\varepsilon\in {\mathbb Z}_2^r$ such that if $\lambda_j=\lambda_{j+1}$, then $\varepsilon_j=0$ implies $\varepsilon_{j+1}=0$. By Lemma \[lem:intersections\] (2) it is enough to look at a set representatives of each $W^\theta$-conjugacy class. According to [@carter-cc Proposition 24] such a set is given by all $$\sigma_{\lambda,\varepsilon}:=(1,2,\ldots,i_1){\mathbf{e}}_{i_1}^{\varepsilon_1}(i_1+1, i_1+2,\ldots,i_2){\mathbf{e}}_{i_2}^{\varepsilon_2}\cdots (i_{r-1},i_{r-1}+1,\ldots,h){\mathbf{e}}_{h}^{\varepsilon_r}.$$ with $i_j=\sum_{l\leq j}\lambda_j$ and $\varepsilon\in {\mathcal E}(\lambda)$. To simplify the exposition, let $\vartheta:{{\mathbb I}}_n\to {{\mathbb I}}_n$ be the permutation $i\mapsto n+1-i$. Let us denote by ${\mathbf{s}}_{p,q}$ the permutation $(p,q)$. As an element in $\symm_{n}$, $w$ becomes a product of cycles as follows: $$\label{eqn:w-in-W} w=\left({\mathbf{c}}_1\theta({\mathbf{c}}_1){\mathbf{s}}_{i_1,\vartheta(i_1)}^{\varepsilon_1}\right) \dots \left({\mathbf{c}}_h\theta({\mathbf{c}}_h){\mathbf{s}}_{i_h,\vartheta(i_h)}^{\varepsilon_h}\right),$$ ${\mathbf{c}}_j=(i_{j-1},i_{j-1}+1,\ldots,i_j)$, $1\leq j\leq h$, $i_{-1}=0$. We set $w_j:={\mathbf{c}}_j\theta({\mathbf{c}}_j){\mathbf{s}}_{i_j,\vartheta(i_j)}^{\varepsilon_j}$. We analyze cases $n$ odd and even separately and apply the results in Lemma \[lem:Tw\]. ### $n$ odd {#subsubsec:n-odd} Let $n=2h+1$ and $w=\sigma_{\lambda,\varepsilon}$. Let, for $j=1,\ldots, r$: $${{\mathbb F}}({j}):=\begin{cases}{{\mathbb F}}_{q^{\lambda_j}}^\times\times {{\mathbb F}}_{q^{\lambda_j}}^\times,& \text{ if }\varepsilon_j=0,\\ {{\mathbb F}}_{q^{2\lambda_j}}^\times, & \text{ if }\varepsilon_j=1. \end{cases}$$ Direct computation shows that $\Tt^{F_w}\simeq {{\mathbb F}}_q^\times\times \prod_{j=1}^r{{\mathbb F}}(j)$. For $j\in {{\mathbb I}}_r$, $z_j\in {{\mathbb F}}(j)$, we set: $$\begin{aligned} &\overline{z}_j:=\begin{cases}x_jy_j,& \text{ if }\varepsilon_j=0 \text{ and }z_j=(x_j,y_j),\\ z_j,& \text{ if }\varepsilon_j=1, \end{cases} &&\text{ and }&&{{\tt{z}}}_j:=\overline{z}_j^{1+\varepsilon_jq^{\lambda_j}}\in{{\mathbb F}}_{q^{\lambda_j}}.\end{aligned}$$ Observe that as $z_j$ runs in ${{\mathbb F}}(j)$ then ${{\tt{z}}}_j$ covers ${{\mathbb F}}_{q^{\lambda_j}}^\times$ and ${{\tt{z}}}_j^{(\lambda_j)_q}$ covers ${{\mathbb F}}_q^\times$. We have $${{\mathbb{T}}}^{F_w}:=\{(z,z_1,\ldots,z_r)\in {{\mathbb F}}_q^\times\times \prod_{j=1}^r{{\mathbb F}}(j)~|~z\prod_j{{\tt{z}}}_j^{(\lambda_j)_q}=1\}\simeq \prod_{j=1}^r{{\mathbb F}}(j).$$ It follows from direct computation that $$\begin{aligned} {{\tt{K}}}_w\simeq \{(z_1,\ldots,z_r)\in \prod_{j=1}^r{{\mathbb F}}(j)~|~{{\tt{z}}}_j=\zeta, 1\leq j\leq r, \ \zeta\in \mu_n({{\mathbb F}}_q)\}. \end{aligned}$$ Hence, if $\gamma:{{\mathbb{T}}}^{F_w}\to {{\mathbb F}}_{q^{\lambda_1}}^\times \times {{\mathbb F}}_{q^{\lambda_1\lambda_2}}^\times \times \dots \times {{\mathbb F}}_{q^{\lambda_{r-1}\lambda_r}}^\times $ is given by $$\begin{aligned} \label{eq:gamma} (z_1,\ldots,z_r)\mapsto ({{\tt{z}}}_1^{\mathbf{d}}, {{\tt{z}}}_1{{\tt{z}}}_2^{-1}, \dots, {{\tt{z}}}_{r-1}{{\tt{z}}}_r^{-1}),\end{aligned}$$ then ${{\mathbb{T}}}^{F_w}/{{\tt{K}}}_w\simeq {\operatorname{Im}}\gamma$. ### $n$ even {#subsec:n-even} Let $n=2h$, $w=\sigma_{\lambda,\varepsilon}$. With notation as in §\[subsubsec:n-odd\], we have: $${{\mathbb{T}}}^{F_w}=\{(z_1,\ldots,z_r)\in \prod_{j=1}^r{{\mathbb F}}(j)~|~\prod_j{{\tt{z}}}_j^{(\lambda_j)_q}=1\}.$$ It follows from direct computation that $$\begin{aligned} {{\tt{K}}}_w\simeq \{(z_1\ldots,z_r)\in{{\mathbb{T}}}^{F_w}~|~{{\tt{z}}}_j=\zeta, 1\leq j\leq r, \ \zeta\in \mu_n({{\mathbb F}}_q)\},\end{aligned}$$ hence ${{\mathbb{T}}}^{F_w}/{{\tt{K}}}_w\simeq {\operatorname{Im}}\gamma$, for $\gamma:{{\mathbb{T}}}^{F_w}\to {{\mathbb F}}_{q^{\lambda_1}}^\times \times {{\mathbb F}}_{q^{\lambda_1\lambda_2}}^\times \times \dots \times {{\mathbb F}}_{q^{\lambda_{r-1}\lambda_r}}^\times$ as in . Applying the strategy --------------------- We will deal with classes ${{\mathcal O}}_x^{\theta,\pi({{\mathbb{T}}}_w^F)}$ for $x\in {\overline\Tt}_w^F$. Observe that as ${\overline\Tt}_w={{\tt{y}}}{\overline\Tt}{{\tt{y}}}^{-1}$, $x$ is represented by an element in $\Tt^{F_w}$ up to multiplication by matrices in ${{\mathcal Z}}(\PL_n(q))$, [*i. e.*]{} , up to a scalar factor in ${{\mathbb F}}_q^\times$. We apply Lemma \[lem:zeta\] and the description of ${{\mathbb{T}}}^{F_w}/{{\tt{K}}}_w$ from Section \[subsec:conj classes in W\] on each case to detect classes of type D. Let ${\boldsymbol{1}}$ denote the partition $(1,\dots,1)$. \[lem:Tw\] Let $\lambda=(\lambda_1,\dots,\lambda_r)$ be a partition of $h$, $\varepsilon\in{\mathcal E}(\lambda)$ and let $w=\sigma_{\lambda,\varepsilon}\in W^\theta$. Let $x\in{\overline\Tt}_w^F$. Then ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}$ is of type D provided any of the following conditions hold. 1. $n$ is odd, $\lambda\neq {\boldsymbol{1}}$. 2. $n$ is even, $\lambda\neq {\boldsymbol{1}}$, and $r>2$. 3. $\lambda ={\boldsymbol{1}}$, $n\neq3,4$ and $q>5$. 4. If $\lambda ={\boldsymbol{1}}$, $n=3$ and $q=9,11$ or $q>13$. 5. If $\lambda ={\boldsymbol{1}}$, $n=4$ and $q>9$ and $q\equiv 1\mod(4)$. In all cases we will provide a suitable element in the image of the map $\gamma$ from and apply Lemma \[lem:zeta\]. \(1) Assume $r>1$. If $j$ is such that $\varepsilon_j= 0$ and $\lambda_j>1$, consider $\tilde z_j=(x_j,1)$, for a generator $x_j\in {{\mathbb F}}_{q^{\lambda_j}}^\times$. If $$\gamma_j:=\gamma(1,\ldots,\tilde z_j,\ldots,1)=(x_j^{\delta_{1,j}{\mathbf{d}}},\dots,x_j,x_j^{-1},\dots,1),$$ then $\mid\gamma_j\mid=\mid x_j\mid= q^{\lambda_j}-1>4$ and even. Similarly, if $r>1$ and $j$ is such that $\varepsilon_j= 1$ and $\lambda_j>1$, then it follows that if $$\gamma_j:=\gamma(1,\ldots,z_j,\ldots,1)= (z_j^{\delta_{1,j}{\mathbf{d}}(1+q^{\lambda_j})},\dots,z_j^{1+q^{\lambda_j}},z_j^{-1-q^{\lambda_j}},\dots, 1)$$ for a generator $z_j$ of ${{\mathbb F}}_{q^{2\lambda_j}}^\times$, then $$\mid\gamma_j\mid=\mid z_j^{1+q^{\lambda_j}}\mid=\frac{q^{2\lambda_j}-1}{(q^{2\lambda_j}-1,1+q^{\lambda_j})}=q^{\lambda_j}-1>4.$$ Now, if $r=1$, then $\lambda=(h)$, $h>1$. Pick $\overline{z}$ such that ${{\tt{z}}}$ is a generator of ${{\mathbb F}}_{q^h}^\times$. Then $$\mid\gamma(z)\mid=\mid{{\tt{z}}}^{\mathbf{d}}\mid=\frac{q^h-1}{({\mathbf{d}},q^h-1)}=\frac{q-1}{{\mathbf{d}}}(h)_q>(h)_q\geq 4.$$ Observe that $\frac{q-1}{{\mathbf{d}}}$ is always even, whence the first inequality. Moreover, $(h)_q=4$ only if $q=3$, $n=5$ in which case $\frac{q-1}{{\mathbf{d}}}(h)_q=2(h)_q>4$. \(2) Assume now that $n$ is even. We distinguish the following cases: [*Case $r>2$, $\lambda\neq{\boldsymbol{1}}$.*]{} Let us choose $\overline{z}_1$ such that ${{\tt{z}}}_1$ is a generator of ${{\mathbb F}}_{q^{\lambda_1}}^\times$. Choose ${{\tt{z}}}_2=\dots={{\tt{z}}}_{r-1}=1$ and $z_r$ such that ${{\tt{z}}}_1^{(\lambda_1)_q}{{\tt{z}}}_r^{(\lambda_r)_q}=1$. Then $(z_1,\dots,z_r)\in{{\mathbb{T}}}^{F_w}$ and $$\mid\gamma(z_1,\dots,z_r)\mid\geq\mid{{\tt{z}}}_1\mid=q^{\lambda_1}-1>4 \text{ and even.}$$ (3), (4), (5) If $n$ is odd, $n\neq3$ the computation in (1) shows that we can find $x\in{\operatorname{Im}}\gamma$ with $\mid x\mid=q-1>4$ for $q>5$. If $n=3$, then ${\operatorname{Im}}\gamma$ is cyclic of order $\frac{q-1}{{\mathbf{d}}}>4$ for $q\geq9$, $q\neq 13$ and always even. If $n$ is even, then $h=r\ge2$. If $r>2$ we may choose ${{\tt{z}}}_1$ as a generator of ${{\mathbb F}}_{q}^\times$, ${{\tt{z}}}_2={{\tt{z}}}_1^{-1}$ and ${{\tt{z}}}_j=1$ for $j\geq3$ and proceed as before. If $r=2$ then $n=4$. We need ${{\tt{z}}}_2={{\tt{z}}}_1^{-1}$ and, choosing ${{\tt{z}}}_1$ as above we have $|({{\tt{z}}}_1^{\mathbf{d}},{{\tt{z}}}_1^2)|=\frac{q-1}{2}$. \[lem:weyl\] Let $w\in W^\theta$ and $x\in {\overline\Tt}_w^F$. If ${{\mathcal O}}_{w_0}^{W_w}$ is of type D, then ${{\mathcal O}}_{x}^{\theta,{\boldsymbol{G}}}$ is so. Let $\w$ be a representative of $w$ in ${\overline{{\mathbb G}}}^\theta\cap N({\overline\Tt})$, see Remark \[rem:representatives\], and let $y\in ({\overline{{\mathbb G}}}^\theta)^\circ=\pi({{\mathbb G}}^\theta)$ be such that $y^{-1}F(y)=\w$, so $x=yty^{-1}$ for some $t\in {\overline\Tt}^{F_w}$. Since ${\boldsymbol{G}}=[{\overline{{\mathbb G}}}^F,{\overline{{\mathbb G}}}^F]=y[{\overline{{\mathbb G}}}^{F_w},{\overline{{\mathbb G}}}^{F_w}]y^{-1}$ we have ${{\mathcal O}}_{x}^{\theta,{\boldsymbol{G}}}\simeq {{\mathcal O}}_{t}^{\theta, [{\overline{{\mathbb G}}}^{F_w},{\overline{{\mathbb G}}}^{F_w}]}$. Now, ${F_w}$ is again a Steinberg endomorphism of ${\overline{{\mathbb G}}}$, and ${\overline\Tt}$ is $F_w$-stable. Hence, [@MT Proposition 23.2] applies and by [@MT Exercise 30.13] there is a group epimorphism $N_{{\overline{{\mathbb G}}}}({\overline\Tt})\cap [{\overline{{\mathbb G}}}^{F_w},{\overline{{\mathbb G}}}^{F_w}]\twoheadrightarrow W^{F_w}=W_w$ inducing a rack epimorphism ${{\mathcal O}}_{x}^{\theta,{\boldsymbol{G}}}\twoheadrightarrow {{\mathcal O}}_1^{\theta,W_w}$. The statement follows from Lemma \[lem:inner\] (2). For $\lambda={\boldsymbol{1}}$ and $j=0,\,\ldots,\,h$ we set $\varepsilon^j:=(\underbrace{1,\ldots,1}_\textrm{$(h-j)$ times},\underbrace{0,\ldots,0}_\textrm{$j$ times})\in {\mathcal E}(\lambda)$. \[lem:weyl-j\]Let $w=\sigma_{{\boldsymbol{1}},\varepsilon^j}$ and let $x\in {\overline\Tt}_w^F$. If $n$ is even and $j\geq3$, or if $n$ is odd and $j>3$, then ${{\mathcal O}}_{x}^{\theta,{\boldsymbol{G}}}$ is of type D. In particular, if $x\in {\overline\Tt}^F$, then ${{\mathcal O}}_{x}^{\theta,{\boldsymbol{G}}}$ is of type D provided $n\geq 6$, $n\neq 7$. By Lemma \[lem:weyl\] it is enough to prove that ${{\mathcal O}}_{w_0}^{W_w}$ is of type D. Now $w\in W$ is the permutation $(1, n)\cdots(h-j, n+1-h+j)\in W'\times 1\leq W'\times W''$ where $$W'\times W''\:=\sym_{\{1,\ldots,h-j,n+1-h+j,\ldots,n\}}\times\sym_{\{h-j+1,\ldots,n-h+j\}}\simeq \sym_{2(h-j)}\times \sym_{n-2(h-j)}$$ and $W_w=W'_w\times W''$, so ${{\mathcal O}}_{w_0}^{W_w}\simeq {{\mathcal O}}_{w}^{W'_w}\times {{\mathcal O}}_{ww_0}^{W''}\simeq {{\mathcal O}}_{ww_0}^{W''}$. The latter is of type D by [@AFGaV Theorem 4.1]. \[lem:blocks\] Assume $n=2h$ and let $\lambda=(\lambda_1,\dots,\lambda_h)$ be a partition of $h$. 1. If $w=\sigma_{\lambda,\varepsilon}=w_1\dots w_j\in W^\theta$ as in , then there is a block matrix $y={\operatorname{Diag}}(y_1,\dots,y_h)\in {{\mathbb G}}^{\theta}$ such that $\w=y^{-1}F(y)$ is a representative of $w$ in $N_{{{\mathbb G}}^\theta}({{\mathbb{T}}})\cap \SL_n(q)$, each block $y_j\in \Sp_{2\lambda_j}({\Bbbk})$ and $\w_j=y_j^{-1}F(y_j)\in w_j {{\mathbb{T}}}$. 2. If $\lambda=(\lambda_1)$ and $w=\sigma_{\lambda,0}$, then there are $y_1\in \SL_{\lambda_1}({\Bbbk})$ and $\w\in w{{\mathbb{T}}}\cap N_{{{\mathbb G}}^\theta}({{\mathbb{T}}})\cap \SL_n(q)$ such that $\w=y^{-1}F(y)$, $y={\operatorname{Diag}}(y_1, {{\tt{J}}}_{\lambda_1}\,^t\!y_1^{-1}{{\tt{J}}}_{\lambda_1}^{-1})\in {{\mathbb G}}^{\theta}$. \(1) Set $i_j=\sum_{l\leq j}\lambda_j$, $i_{-1}:=0$, $\Lambda_j=\{i_{j-1}+1,\dots,i_j\}$, $1\leq j\leq h$. Recall from that $w\in \symm_{2h}$ can be viewed as an element in $\symm_{2\lambda_1}\times \dots\times \symm_{2\lambda_h}$, if we identify $\symm_{2\lambda_j}$ with the permutation group of $\Lambda_j\cup\vartheta(\Lambda_j)$, for $1\leq j\leq h$. Notice that $w_j={\mathbf{c}}_j\theta({\mathbf{c}}_j){\mathbf{s}}_{i_j,\vartheta(i_j)}\in \sym_{2\lambda_j}^{\theta_j}$ for each $1\leq j\leq h$. Hence each $w_j$ lies in the Weyl group of a $\theta$-invariant subgroup ${{\mathbb G}}_j\simeq \Sp_{2\lambda_j}({\Bbbk})$ of ${{\mathbb G}}$, namely the subgroup of matrices of the shape $$\left(\begin{smallmatrix} {\operatorname{Id}}& & & & \\ & A& & B& \\ & &{\operatorname{Id}}& & \\ & C & & D& \\ & & & &{\operatorname{Id}}\\ \end{smallmatrix} \right), \qquad \left(\begin{smallmatrix} A & B \\ C & D \end{smallmatrix}\right)\in \Sp_{2\lambda_j}({\Bbbk})$$ and the non-zero entries outside the diagonal are indexed by integers in $\Lambda_j\cup\vartheta(\Lambda_j)$. Let us denote by $\theta_j$ the graph automorphism for ${{\mathbb G}}_j$. There exists a representative $\w_j$ of $w_j$ in ${{\mathbb G}}_j^{\theta_j}\simeq \Sp_{2\lambda_j}({\Bbbk})$, as $n$ is even. Therefore, there exists $y_j\in {{\mathbb G}}_j^{\theta_j}\simeq \Sp_{2\lambda_j}({\Bbbk})$ such that $y_j^{-1}F(y_j)=\w_j$. We remark that $[{{\mathbb G}}_i,{{\mathbb G}}_j]=1$ for $i\neq j$ and thus $y$ can be chosen as $y=y_1\dots y_h$. \(2) If $\varepsilon=0$ then $w$ lies in $\sym_{\lambda_1}$ and it is represented by block matrices of the form $\w={\operatorname{Diag}}(A, {{\tt{J}}}_{\lambda_1}\,^t\!A^{-1}{{\tt{J}}}_{\lambda_1}^{-1})\in {{\mathbb G}}^{\theta}$. As we can always make sure that $A\in\SL_{\lambda_1}(q)$ [@MT Proposition 23.2], we can apply Lang-Steinberg’s Theorem to the connected group $\SL_{\lambda_1}({\Bbbk})$. \[lem:n=2h,r=1,e=0\]Let $n=2h$ for $h>1$, $\lambda=(h)$, $\varepsilon=(0)$ and $w=\sigma_{\lambda,\varepsilon}$. Let $x\in {\overline\Tt}_w^F$. Then ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}$ is of type D provided one of the following holds: 1. $x\theta$ is not an involution and ${\boldsymbol{G}}\neq \PSL_4(3), \PSL_4(7)$. 2. $n\geq 6$. 3. $x\theta$ is an involution, $n=4$ and $q\equiv 1(4)$, $q\neq5,\,9$. We have $w=(1,2,\cdots,h)(n,n-1,\cdots,h+1)$. Let $y\in \Sp_n({\Bbbk})$ satisfy $\w=\pi(y^{-1}F(y))$. Set ${{\tt{y}}}=\pi(y)$. Thus we may assume $$x={{\tt{y}}}\pi(t){{\tt{y}}}^{-1}, \quad \text{ for } t={\operatorname{diag}}(a,a^q,\dots,a^{q^{h-1}},b^{q^{h-1}},\dots,b),$$ for some $a,b\in{{\mathbb F}}_{q^h}^\times$. We set, for $\xi\in {\Bbbk}, \xi^{(h)_q}=1$: $$t_\xi={\operatorname{diag}}(a\xi,(a\xi)^q,\dots,(a\xi)^{q^{h-1}},(b\xi)^{q^{h-1}},\dots, b\xi)\in {\overline\Tt}_w^F.$$ It follows that ${{\mathcal O}}^{\theta,\pi({{\mathbb{T}}}_w^F)}_x={{\tt{y}}}\{\pi(t_\xi):\xi\in {\Bbbk}, \xi^{(h)_q}=1\}{{\tt{y}}}^{-1}$. Set $\dag=\dag_n:=\pi({\operatorname{diag}}(-{\operatorname{id}}_h,{\operatorname{id}}_h))\in{\operatorname{PGL}}_n(q)$. Notice that $x\theta$ is an involution if and only if $\theta x=x^{-1}$ which happens only if $x\in {{\mathcal O}}_1^{\theta,{\overline\Tt}_w^F}\cup {{\mathcal O}}_\dag^{\theta,{\overline\Tt}_w^F}$. We claim that if $x\not\in {{\mathcal O}}_1^{\theta,{\overline\Tt}_w^F}\cup {{\mathcal O}}_\dag^{\theta,{\overline\Tt}_w^F}$, then ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}\cap {\overline\Tt}_w^F\neq{{\mathcal O}}_x^{\theta,\pi({{\mathbb{T}}}_w^F)}$. Let us compute the (twisted) action of $w_0\in W_w^\theta=\langle w,w_0\rangle$ on $\pi(t)$. We have $$\begin{aligned} w_0\cdot_\theta \pi(t)t&={\operatorname{diag}}(b,b^q,\dots,b^{q^{h-1}},a^{q^{h-1}},\dots,a).\end{aligned}$$ Hence, $w_0\cdot_\theta x\in {{\mathcal O}}^{\theta,\pi({{\mathbb{T}}}_w^F)}_x$ only if $ab^{-1}=ba^{-1}$. This gives the claim. \(1) We apply Lemma \[lem:two-orbits\]: we search for $z\in \mO_1^{\theta,\pi({{\mathbb{T}}}_w^F)}$ such that $\mid z\mid\neq 1,2,4$. According to the discussion in §\[subsec:n-even\], $\mO_1^{\theta,\pi({{\mathbb{T}}}_w^F)}$ is a cyclic group of order $\ell$, for $\ell=\frac{(h)_q}{({\mathbf{d}},(h)_q)}=\frac{(h)_q}{(q-1,h)}$, as ${\mathbf{d}}=(q-1,2h)$ and $(q-1,(h)_q)=(q-1,h)$. If $h=2$, so $n=4$, we have $\ell=\frac{1+q}{2}$, so $q\neq 3,7$ is enough. If $h$ is odd, then $\ell$ is odd and $\ell>1$ since $\ell>\frac{1+q}{q-1}$. Then we can find such a $z$. From now we shall assume that $h\geq 4$ is even. We distinguish three cases, according to $h>q-1$, $h=q-1$ or $h<q-1$. If $h>q-1$ then $ \ell>\frac{1+q(h-1)}{q-1}=h+\frac{h-(q-1)}{q-1}>4 $ and we are done. The same computation proves the claim if $h=q-1>4$. If $h=q-1=4$ a direct computation gives the claim. Finally, if $h<q-1$, then $\ell>\frac{(h)_{h+1}}{h}\geq \frac{h+h(h-2)_h}{h}>6$. \(2) If $x\theta$ is not an involution, then we apply (1). If $x\theta$ is an involution, then we have that either ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}\simeq {{\mathcal O}}_1^{\theta,{\boldsymbol{G}}}$ or ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}\simeq {{\mathcal O}}_{\dag}^{\theta,{\boldsymbol{G}}}$, by Lemma \[lem:inner\] (3). Now, $1,\kappa\in {\overline\Tt}^F$ by Lemma \[lem:blocks\] (2), as we may assume $y={\operatorname{Diag}}(A,{{\tt{J}}}\,^tA^{-1}{{\tt{J}}}^{-1})$ for some there $A\in\PL_h({\Bbbk})$. Thus, by Lemma \[lem:weyl-j\], ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}$ is of type D if $h\geq 3$. \(3) Since $1,\kappa\in {\overline\Tt}^F$, we apply Lemma \[lem:Tw\] (5). \[pro:h2\] Let $q\equiv 3(4)$, $q\neq 3,7$, ${\boldsymbol{G}}=PSL_4(q)$. Let $t$ be either $1$ or $\kappa=\left(\begin{smallmatrix} {\operatorname{id}}_2&0\\0&-{\operatorname{id}}_2 \end{smallmatrix}\right)$. Then ${{\mathcal O}}_t^{\theta,{\boldsymbol{G}}}$ is of type D. We will apply Lemma \[lem:vendra\]. It is enough to find $x\in {\boldsymbol{G}}$ such that the order of $xt\theta(x)^{-1}t$ in ${\boldsymbol{G}}$ is even and $>4$. Set ${\bf u}_t:{\boldsymbol{G}}\to {\boldsymbol{G}}$, ${\bf u}_t(x)=-xt\theta(x)^{-1}t=xt{{\tt{J}}}\,^t x {{\tt{J}}}t$. For each $e,f\in{{\mathbb F}}_q$, $A,E,F\in{{\mathbb F}}_q^{2\times 2}$, with $E,F$ traceless, let us set $$\begin{aligned} &{\bf m}(A,e,f)=\left(\begin{matrix}A&e\,{\operatorname{id}}_2\\ f\,{\operatorname{id}}_2&{{\tt{J}}}_2 \,^tA{{\tt{J}}}_2^{-1} \end{matrix}\right),&& {\bf n}(A,E,F)=\left(\begin{matrix}A&E\\ F&{{\tt{J}}}_2 \,^tA{{\tt{J}}}_2^{-1}\end{matrix}\right).\end{aligned}$$ We have that ${\bf u}_1({\bf m}(A,e,f))={\bf m}(A,e,f)^2$, ${\bf u}_\kappa({\bf n}(A,E,F))={\bf n}(A,E,F)^2$. Moreover, for any $x\in{\boldsymbol{G}}$ there are $e,f\in{{\mathbb F}}_q$, $A,E,F\in{{\mathbb F}}_q^{2\times 2}$, $E,F$ traceless such that ${\bf u}_1(x)={\bf m}(A,e,f)$ and ${\bf u}_\kappa(x)={\bf m}(A,E,F)$. We shall exhibit a matrix ${\bf m}(A,0,0)={\bf n}(A,0,0)$ whose projective order is a multiple of $4$ and it is bigger than $8$. This will prove the statement for both $t=1,\kappa$. Let ${{\mathbb F}}^\times_{q^2}=\langle \xi \rangle$ and consider the matrix $z={\operatorname{diag}}(\xi^{\frac{q-1}{2}},-\xi^{\frac{1-q}{2}},-\xi^{\frac{1-q}{2}},\xi^{ \frac{q-1}{2}} )$ in $\SL_4({{\mathbb F}}_{q^2})$. The order of $z$ is $2({q+1})$ and $z^{\frac{q+1}{2}}={\operatorname{diag}}(\omega,\omega^{-1},\omega^{-1},\omega)$ for $\omega$ a primitive fourth root of $1$, hence the projective order of $z$ is $q+1$. We claim that $z$ is ${\operatorname{PGL}}_4(\overline{\Fq})$-conjugate to $x={\bf m}(\left(\begin{smallmatrix}\Tr(z)/2&1\\ 1&0\\ \end{smallmatrix}\right),0,0)$ and that $\Tr(z)\in{{\mathbb F}}_q^\times$. If this is the case, ${\bf u}_1(x)=x^2$ and its projective order is $\frac{q+1}{2}$ which is even as $q\equiv 3(4)$ and bigger than 4 since $q\geq11$. The claim is proved if the following conditions hold, namely $$\begin{aligned} &\det z=1; && \Tr z=2(\xi^{\frac{q-1}{2}}-\xi^{\frac{1-q}{2}})\in{{\mathbb F}}_q; && \xi^{\frac{q-1}2}\neq -\xi^{\frac{1-q}2}.\end{aligned}$$ Indeed, in this case, the matrix ${\bf m}(\left(\begin{smallmatrix}\Tr(z)/2&1\\ 1&0\\ \end{smallmatrix}\right),0,0)$ is diagonalizable and it is necessarily $\PL_4(\overline{\Fq})$-conjugate to the matrix $z$. The first and third conditions are immediate. For the second, let $\sigma$ be the (involutive) generator of the Galois group $\Gal({{\mathbb F}}_{q^2},{{\mathbb F}}_q)$ of the extension ${{\mathbb F}}_q\subset {{\mathbb F}}_{q^2}$. We need $\sigma(\Tr x)=\Tr x$. But $\sigma$ coincides with ${\operatorname{Fr}}_p^m$, that is $\sigma(\xi)=\xi^q$ and thus the equality holds. \[lem:n=2h,r=1,e=1\]Let $n=2h$, $h>1$, $\lambda=(h)$ and $\varepsilon=(1)$, $w=\sigma_{\lambda,\varepsilon}$. Let $x\in {\overline\Tt}_w^F$. Then ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}$ is of type D provided that one of the following holds: 1. $x\theta$ is not an involution and ${\boldsymbol{G}}\neq \PSL_4(3), \PSL_4(7)$. 2. $x\in {{\mathcal O}}_1^{\theta,{\operatorname{PGL}}_n(q)}$, $n\geq6$. 3. $x\in {{\mathcal O}}_1^{\theta,{\operatorname{PGL}}_n(q)}$, $n=4$, $q>9$. 4. $x\theta$ is an involution, $x\not\in{{\mathcal O}}_1^{\theta,{\operatorname{PGL}}_n(q)}$ and $h$ is even. In this case $w=(1,2,\ldots,h-1,h,n,n-1,\ldots,h+2,h+1)$ as a permutation in $\sym_n$. Arguing as in Lemma \[lem:n=2h,r=1,e=0\] we may assume that, for some $a\in{{\mathbb F}}_{q^{n}}^\times$, $$x={{\tt{y}}}\pi(t){{\tt{y}}}^{-1}, \quad \text{ for } t=t_a={\operatorname{diag}}(a,a^q,\dots,a^{q^{h-1}},a^{q^{2h-1}},\dots,a^{q^h})$$ and ${{\tt{y}}}$ such that ${{\tt{y}}}^{-1}F({{\tt{y}}})=\w$. \(1) Notice that $x\theta$ is an involution if and only if $a^2$ lies in ${{\mathbb F}}_{q^h}^\times$. Now, set, for $\xi\in {{\mathbb F}}_{q^n}^\times$ such that $\xi^{(n)_q}=1$ and $z=\xi^{1+q^h}$ in $C_{(h)_q}\subset {{\mathbb F}}_{q^h}^\times$: $$t_{az}={\operatorname{diag}}(az,(az)^q,\dots,(az)^{q^{h-1}},\dots, (az)^{q^h})\in \Tt^{F_w}.$$ It follows that ${{\mathcal O}}^{\theta,\pi({{\mathbb{T}}}_w^F)}_x={{\tt{y}}}\{\pi(t_{az}):\xi\in {\Bbbk}, \xi^{(n)_q}=1\}{{\tt{y}}}^{-1}$. Observe that $(yw_0^{-1}y^{-1})\cdot_\theta x=y \pi(t^{q^h}) y^{-1}$ lies in ${{\mathcal O}}_x^{\theta,\pi({{\mathbb{T}}}_w^F)}$ if and only if $a^2\in {{\mathbb F}}_{q^h}^\times$. In other words, if $w_0\cdot_\theta t$ lies in ${{\mathcal O}}_t^{\theta, \pi({{\mathbb{T}}}^{F_w})}$ only if $x\theta$ is an involution. If this is not the case, we can proceed as in Lemma \[lem:n=2h,r=1,e=0\] and obtain that if ${\boldsymbol{G}}\neq \PSL_4(3), \PSL_4(7)$, then ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}$ is of type D. (2), (3) Assume $a\in{{\mathbb F}}_{q^h}^\times$. Then ${{\mathcal O}}_x^{\theta,{\overline\Tt}_w^F}={{\mathcal O}}_1^{\theta,{\overline\Tt}_w^F}$ and ${{\mathcal O}}_x^{\theta,\pi({{\mathbb{T}}}_w^F)}\cap{\overline\Tt}_w^F={{\mathcal O}}_x^{\theta,\pi({{\mathbb{T}}}_w^F)}$. In this case $x\in{{\mathcal O}}_1^{\theta, ({{\tt{y}}}{\overline\Tt}{{\tt{y}}}^{-1})^F}\subset {{\mathcal O}}_1^{\theta,{\operatorname{PGL}}_n(q)}$ and we may assume $a=1$, $t={\operatorname{id}}$ by Proposition \[pro:x-in-PGL\]. If $n\geq6$ this class is of type D by Lemma \[lem:weyl-j\]. Assume $n=4$. If $q\equiv1(4)$, $q>9$, then we may apply Lemma \[lem:Tw\] (5). For $q\equiv 3(4)$, $q>7$ we apply Proposition \[pro:h2\] \(4) Assume $a^{q^h}=-a$ and moreover that $h$ is even. We apply Lemma \[lem:vendra\]: We search for $r\in{{\mathcal O}}_{x\theta}^{{\boldsymbol{G}}\ltimes\langle\theta\rangle}$ such that $\mid rx\theta\mid$ is even and bigger than 4. Equivalently, we look for $z\in {\boldsymbol{G}}^{F_w}$ such that the order of $(z\cdot_{\theta} t)\theta \, t\theta=(z\cdot_{\theta} t) t^{-1}$ is even and bigger than 4. If $h$ is even, then this is achieved by taking $z=t$, as $t\cdot_{\theta} t=t^3$ and thus $\mid t^2\mid=(h)_q$ which is even and bigger than 4, as $h>1$. We are missing the case $r=2$, $\lambda=(\lambda_1,\lambda_2)\neq {\boldsymbol{1}}$. That is, the case in which $n=2(\lambda_1+\lambda_2)$, with $\lambda_2\geq 1$, $\lambda_1\geq 2$. This is the content of Lemmas \[lem:r=2,varepsilon=0\] (when $\varepsilon_1=0$) and \[lem:r=2,varepsilon=1\] (when $\varepsilon_1=1$). Let us set $$w=c_1\theta(c_1)s_{\lambda_1,n-\lambda_1+1}^{\varepsilon_1}c_2\theta(c_2)s_{ \lambda_1+\lambda_2,\lambda_1+\lambda_2+1}^{\varepsilon_2}$$ where $c_1=(1,2,\ldots,\lambda_1)$ and $c_2=(\lambda_1+1,\ldots,\lambda_1+\lambda_2)$. We write $$w_1:=c_1\theta(c_1)s_{\lambda_1,n-\lambda_1+1}^{\varepsilon_1},\quad w_2=c_2\theta(c_2)s_{\lambda_1+\lambda_2,\lambda_1+\lambda_2+1}^{\varepsilon_2} .$$ The group $W^\theta_w$ always contains $w_1$, $w_2$ and the elements $$w_{0}^{(1)}=(1n)\cdots (\lambda_1,n-\lambda_1+1),\quad w_{0}^{(2)}=(\lambda_1+1,n-\lambda_1)\cdots (\lambda_1+\lambda_2,\lambda_1+\lambda_2+1),$$ which correspond to the longest elements in each block. \[lem:r=2,varepsilon=0\] Let $n=2(\lambda_1+\lambda_2)$, $\lambda=(\lambda_1,\lambda_2)$ with $\lambda_1\geq2$, $\lambda_2\geq1$, and $\varepsilon_1=0$. Let $x\in {\overline\Tt}_w^F$. If $q>5$, then ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}$ is of type D. We have $x=\pi({{\tt{y}}})\pi(t)\pi({{\tt{y}}})^{-1}$ where ${{\tt{y}}}^{-1}F({{\tt{y}}})=\w$, $t={\operatorname{diag}}(t_{11},t_2,t_{12})$. Here $t_{11}$ and $t_{12}$ are diagonal matrices of size $\lambda_1$ and $t_2\in \PL_{2\lambda_2}({\Bbbk})^{F_{w_2}}$ is also diagonal. Also, $t_1:={\operatorname{diag}}(t_{11},t_{12})\in \PL_{2\lambda_1}({\Bbbk})^{F_{w_1}}$. Assume first $W_w^\theta\cdot {{\mathcal O}}_{\pi(t)}^{\theta,\pi({{\mathbb{T}}}^{F_w})}\neq {{\mathcal O}}_{\pi(t)}^{\theta,\pi({{\mathbb{T}}}^{F_w})}$. With notation as in §\[subsec:n-even\], the abelian group ${{\mathcal O}}_1^{\theta,\pi({{\mathbb{T}}}_w^F)}$ is isomorphic to the image of the map $\gamma$. Since $\lambda_1>1$ we may choose $z_1$ so that $|\gamma(z_1,1)|=|({{\tt{z}}}_1^{\bf d}, {{\tt{z}}}_1)|=(\lambda_1)_q>4$ and Lemma \[lem:two-orbits\] applies. We now determine when $W_w^\theta\cdot {{\mathcal O}}_{\pi(t)}^{\theta,\pi({{\mathbb{T}}}^{F_w})}= {{\mathcal O}}_{\pi(t)}^{\theta,\pi({{\mathbb{T}}}^{F_w})}$, acting by $w_0^{(1)}$. Arguing as in Lemma \[lem:n=2h,r=1,e=0\], we see that $w_0^{(1)}\cdot_\theta \pi(t)\in{{\mathcal O}}_{\pi(t)}^{\theta,\pi({{\mathbb{T}}}^{F_w})}$ only if ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}\simeq {{\mathcal O}}_{x'}^{\theta,{\boldsymbol{G}}}$ where $x'$ has the form $x'=\pi({{\tt{y}}}{\operatorname{diag}}({\operatorname{id}}_{\lambda_1},t_2',\pm {\operatorname{id}}_{\lambda_1}){{\tt{y}}}^{-1})$ with $t_2'$ a diagonal element in $\PL_{2\lambda_2}({\Bbbk})^{F_{w_2}}$. By Lemma \[lem:blocks\] (2), $x'$ lies in ${\overline\Tt}_{w_2}^F$. So, if $\lambda_2=1$, then the partition associated with $w_2$ is $\lambda'={\boldsymbol{1}}$ and Lemma \[lem:Tw\] (3) applies. If $\lambda_2>1$, then the partition associated with $w_2$ is $\lambda'\neq{\boldsymbol{1}}$ so $r=\lambda_1+1>2$ and Lemma \[lem:Tw\] (2) applies. \[rem:r=2\] It follows from the proof of Lemma \[lem:r=2,varepsilon=0\] that even in the case $q=5$ the class ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}$ is of type D, provided $\lambda_2>1$. Also, if $q=3$ then this class is of type D provided $\lambda_1>1$ and $\lambda_2>2$. \[lem:r=2,varepsilon=1\] Let $n=2(\lambda_1+\lambda_2)$, $\lambda=(\lambda_1,\lambda_2)$ with $\lambda_1\geq2$, $\lambda_2\geq1$, and $\varepsilon_1=1$. Let $x\in {\overline\Tt}_w^F$. Then ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}$ is of type D. We follow the strategy and notation from Lemma \[lem:r=2,varepsilon=0\]. In this case $x=\pi({{\tt{y}}})\pi(t)\pi({{\tt{y}}})^{-1}$ for $t={\operatorname{diag}}(t_{11},t_2,t_{12})$, and $$t_1={\operatorname{diag}}(t_{11},t_{12})=(a,a^q,\ldots,a^{q^{\lambda_1-1}},a^{q^{2\lambda_1-1} } , \ldots , a^{q^{\lambda_1}})\in\PL_{2\lambda_1}({\Bbbk})^{F_{w_1}}.$$ Applying $w_0^{(1)}$ and arguing as in the proof of Lemma \[lem:n=2h,r=1,e=1\], we see that $\w_0^{(1)}\cdot_\theta {{\mathcal O}}_{\pi(t)}^{\theta,\pi({{\mathbb{T}}}^{F_{w}})}\neq {{\mathcal O}}_{\pi(t)}^{\theta,\pi({{\mathbb{T}}}^{F_{w}})}$ with the possible exception of the case in which $a^2\in{{\mathbb F}}_{q^\lambda_1}^\times$. If $a\in{{\mathbb F}}^\times_{q^{\lambda_1}}$, then there are diagonal elements $d_{11}$, $d_{12}$ in $\PL_{\lambda_1}({\Bbbk})$ and $d_2\in \PL_{2\lambda_2}({\Bbbk})^{F_{w_2}}$ such that $d_1:={\operatorname{diag}}(d_{11},d_{12})\in \PL_{2\lambda_1}({\Bbbk})^{F_{w_1}}$, $\det d=1$ for $d:={\operatorname{diag}}(d_{11},d_2,d_{12})$ and $d\cdot_\theta t=({\operatorname{id}}_{\lambda_1},t_2',{\operatorname{id}}_{\lambda_1})$. By Lemma \[lem:blocks\] the latter lies in the $F$-stable maximal torus associated with $w_2$. In this case, $r>2$ and we apply Lemma \[lem:Tw\] (2). We assume from now on that $a^2\in{{\mathbb F}}_{q^\lambda_1}^\times$ and $a\not\in{{\mathbb F}}_{q^\lambda_1}^\times$, [ *i. e.*]{}, $a^{q^{\lambda_1}-1}=-1$. Hence if $\varepsilon_2=0$, then, the element $t$ is: $$\begin{aligned} t=t_{a,b,c}=(a,a^q,\ldots,a^{q^{\lambda_1-1}},b,b^q,\ldots,b^{q^{\lambda_2-1}}, c^ { q^{\lambda_2-1}},\ldots,c, -a^{q^{\lambda_1-1}},\ldots,-a),\end{aligned}$$ for $b,c\in{{\mathbb F}}_{q^{\lambda_2}}^\times$, while if $\varepsilon_2=1$, then $$\begin{aligned} t=t_{a,b}=(a,a^q,\ldots,a^{q^{\lambda_1-1}},b,\ldots,b^{q^{\lambda_2-1}},b^{ q^ {2\lambda_2-1}},\ldots,b^{q^\lambda_2}, -a^{q^{\lambda_1-1}},\ldots,-a),\end{aligned}$$ for $b\in {{\mathbb F}}_{q^{2\lambda_2}}^\times$. The ${{\mathbb{T}}}^{F_w}$-orbit consists of elements of the form $t_{a{{\tt{z}}}_1,b{{\tt{z}}}_2,c{{\tt{z}}}_2}$ ($t_{a{{\tt{z}}}_1,b{{\tt{z}}}_2}$, respectively) for ${{\tt{z}}}_1\in {{\mathbb F}}_{q^{\lambda_1}}^\times$ and ${{\tt{z}}}_2\in{{\mathbb F}}_{q^{\lambda_2}}^\times$ satisfying ${{\tt{z}}}_1^{(\lambda_1)_q}{{\tt{z}}}_2^{(\lambda_2)_q}=1$. Since all elements in ${{\mathbb F}}_{q^{2\lambda_1}}^\times$ satisfying $a^{q^{\lambda_1}-1}=-1$ lie in the same ${{\mathbb F}}_{q^{\lambda_1}}^\times$-coset, we may assume that $|a|=2(q^{\lambda_1}-1)$. We consider $w^{-1}_1\cdot \pi(t_{a,b})$ ($w^{-1}_1\cdot \pi(t_{a,b, c})$, respectively). If it lies in a different $\pi({{\mathbb{T}}}^{F_w})$-orbit than $\pi(t_{a,b})$ ($\pi(t_{a,b, c})$, respectively), we apply Lemma \[lem:two-orbits\]. Otherwise, there are ${{\tt{z}}}_1\in {{\mathbb F}}_{q^{\lambda_1}}^\times$, $\ell\in{{\mathbb F}}_q^\times$, and ${{\tt{z}}}_2\in {{\mathbb F}}_{q^{\lambda_2}}^\times$ such that $\ell{{\tt{z}}}_2=1$, $a^{q-1}=\ell {{\tt{z}}}_1$, and ${{\tt{z}}}_1^{(\lambda_1)_q}{{\tt{z}}}_2^{(\lambda_2)_q}=1$. If this is the case, then $|a^{q-1}|=|{{\tt{z}}}_2^{-1} {{\tt{z}}}_1|=2(\lambda_1)_q$. Thus, there is an element in the image of the map $\gamma$ in of even order $>4$ and Lemma \[lem:zeta\] applies. \[rem:missing\] Lemma \[lem:n=2h,r=1,e=1\] does not cover the case $h$ odd, $W_w^\theta\cdot_\theta {{\mathcal O}}_{\pi(t)}^{\theta,\pi({{\mathbb{T}}}^{F_w})}={{\mathcal O}}_{\pi(t)}^{\theta,\pi({{\mathbb{T}}}^{F_w})}$, and $x\neq 1$. This actually amounts to at most a single class for each group, up to rack isomorphism: Keep the notation from the lemma, let $\zeta$ be a generator of ${{\mathbb F}}_{q^n}^\times$, $\eta=\zeta^{\frac{1+q^h}{2}}$ and set $$\begin{aligned} \label{eqn:missing} {\nu}={{\tt{y}}}\pi(t_\eta){{\tt{y}}}^{-1}.\end{aligned}$$ Then $x\in {{\mathcal O}}_{\nu}^{\theta, {\operatorname{PGL}}_n(q)}$ and ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}\simeq {{\mathcal O}}_{\nu}^{\theta,{\boldsymbol{G}}}$, by Proposition \[pro:x-in-PGL\]. \(1) This class seems to be difficult. For instance, the subrack $${{\mathcal O}}_{{\nu}}^{\theta, \pi({{\mathbb{T}}}_w^F)}\simeq {{\mathcal O}}_{\pi(t_\eta)}^{\theta, \pi({{\mathbb{T}}}^{F_w})}= {{\mathcal O}}_{\pi(t_\eta)}^{\theta, N_{\pi({{\mathbb G}}^{F_w,\theta})}({{\mathbb{T}}})}$$ is not of type D. Indeed, $|\pi(t_\eta) \theta|=2$ and ${{\mathcal O}}_{t_\eta}^{\theta, \pi(\Tt^{F_w})}=\{\pi(t_{az}):\xi\in {\Bbbk}, \xi^{(n)_q}=1,z=\xi^{1+q^h}\}$. Then for every $r\in {{\mathcal O}}_{t_\eta}^{\theta, \pi(\Tt^{F_w})}$ the order $|r\theta \pi(t_\eta)\theta|=|r\pi(t_\eta^{-1})|$ divides $(h)_q$ and hence it is odd. \(2) This class does not occur when dealing with $\theta$-conjugacy classes in ${\boldsymbol{G}}$ instead of ${\operatorname{PGL}}_n(q)$, if $q\equiv1(4)$. Let $s=t_a$ as in the proof of Lemma \[lem:n=2h,r=1,e=1\]. Then $s$ lies in $\SL_n({\Bbbk})$ only if $q\equiv 3(4)$. Indeed $\det(s)=a^{(n)_q}=a^{(1+q^h)(h)_q}=-a^{2(h)_q}=1$ gives $a^{2(h)_q}=-1$. Also, $-1=a^{(1-q)(h)_q}=a^{\frac{q-1}{2} 2(h)_q}=(-1)^{\frac{q-1}{2}}$ so $\frac{q-1}{2}$ is odd. In group-theoretical terms, the class of ${\nu}$ is of type D if and only if the following question has an affirmative answer: \[question\] Let $\eta$ and $s=t_\eta$ be as above, recall the matrix ${{\tt{J}}}$ from . Is there a matrix in $A\in\SL_n(\overline{{{\mathbb F}}_q})^{F_w}$ such that the projective order of $$\begin{aligned} \label{eqn:m(A)} m(A):={{\tt{J}}}A{{\tt{J}}}\,^t\!A s\end{aligned}$$ is even and bigger than 4? ### A class not of of type D So far, we have seen that most $\theta$-semisimple classes in $\PSL_n(\Fq)$ are of type D. Next proposition shows that there exist classes that are not of type D. This shows that the condition $q\neq3$ in Proposition \[pro:h2\] is necessary. \[pro:not\] The class ${{\mathcal O}}_1^{\theta,\PSL_4(3)}$ is not of type D. We adopt the notation from the proof of Proposition \[pro:h2\] to show that the projective order of any matrix $Y$ of the form ${\bf m}(A,e,f)$ is at most equal to $4$. For such $Y$, we verify that $\det Y=(\det A-ef)^2$, that $\Tr(Y)=2\Tr(A)$, and that the matrix $Y$ annihilates the polynomial $X^2-\Tr(A)X+(\det A-ef)$. Since $Y\in{\rm SL}_4({{\mathbb F}}_3)$ we have $\delta:=\det A-ef=\pm1$. Therefore, whenever $\Tr(A)=0$, the matrix $Y$ is an involution in $\PSL_4(3)$. Let us assume $\Tr(A)\neq0$. We have $Y^2=\Tr(A)Y-\delta$ so $$Y^3=\Tr(A)Y^2-\delta Y=\Tr(A)^2 Y-\delta\Tr(A)-\delta Y=(1-\delta)Y-\delta\Tr(A).$$ If $\delta=1$, then $Y^3=\pm1$. If $\delta=-1$, then $Y^4=-Y^2+\Tr(A)Y=-1$. Proof of Theorem \[thm:one\] {#the-proof} ---------------------------- We cite the rows in the table according to their position in the last column. We make the convention that the head row is the 0th one. Let us consider the class of an element $x$ that might possibly be not of type D. If $\lambda\neq{\boldsymbol{1}}$ then by Lemma \[lem:Tw\] (1) and (2), $n$ has to be even and $r\leq 2$. By Lemmata \[lem:r=2,varepsilon=0\] and \[lem:r=2,varepsilon=1\], the case $r=2$ and either $\varepsilon_1=1$ or $q>5$ is ruled out, yielding the first row. If $r=1$ and $\varepsilon=(0)$, then by Lemma \[lem:n=2h,r=1,e=0\] and Proposition \[pro:h2\], the classes that are not of type D may occur only for $n=4$ and either $q=3,7$, or $\theta(x)=x^{-1}$ and $q=5,9$. This gives the second and the third row. If $r=1$ and $\varepsilon=(1)$, then by Lemma \[lem:n=2h,r=1,e=1\], the classes that are not of type D may occur only in the following situations: - $n=4$; $q=3,7$ and $\theta(x)\neq x^{-1}$, which is the fourth row; - $n=4$, $q=3,5, 7,9$ and $x\in{{\mathcal O}}_1^{\theta,{\operatorname{PGL}}_n(q)}$. This case is considered in the last row, as, up to rack isomorphism, this class is represented by an element in an $F$-stable maximal torus with associated partition ${\boldsymbol{1}}$. - $n$ twice an odd number, $x\in{{\mathcal O}}_\nu^{\theta,{\operatorname{PGL}}_n(q)}$, which is the fifth row. Assume now $\lambda={\boldsymbol{1}}$. Then, by Lemma \[lem:Tw\] the only classes that could occur in the table are those for $q=3,5$, or for $n=3$ and $q=7,13$, or else $n=4$ and $q\equiv 3(4)$, or $q=9$. This gives the sixth and the seventh row. Twisted classes of elements with trivial $\theta$-semisimple part {#sec:unipotent} ================================================================= Recall that for $x\in{\operatorname{PGL}}_n(q)$ there is a unique decomposition $x=us$ with $u$ unipotent, $s$ a $\theta$-semisimple element and $us=s\theta(u)$. Then we have the rack inclusions, see : $${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}\supset{{\mathcal O}}_u^{{\boldsymbol{G}}_{\theta}(s)}.$$ Assume $s=\pi(s')$, $s'\in \Tt$. According to [@steinberg-endo 8.1], ${{\mathbb G}}_{\theta}(s')$ is a connected reductive group. By [@mohr-tesis Theorem 1.1, Proposition 3.1], any simple component of ${{\mathbb G}}_{\theta}(s')$ is isomorphic either to $Sp_{2a}({\Bbbk})$ or to $\SL_a({\Bbbk})$, $a\in{{\mathbb N}}$, if $n$ is even; and either to $Sp_{2a}({\Bbbk})$, $\SL_a({\Bbbk})$ or $SO_{2a+1}({\Bbbk})$, $a\in {{\mathbb N}}$, if $n$ is odd. Taking $F$-invariants and arguing as in [@carter-centralizers §3], one sees that ${{\mathbb G}}_{\theta}(s')^F$ contains a product of finite classical groups: unitary, special linear and symplectic if $n$ is even, and orthogonal, unitary, special linear and symplectic if $n$ is odd. Then ${{\mathcal O}}_x^{\theta,{\boldsymbol{G}}}$ is of type D whenever the conjugacy class of some component of $u$ in one of these factors is so. The group ${\boldsymbol{G}}_{\theta}(s)=\pi({{\mathbb G}}^F)^{Ad(s)\circ \theta}$ might properly contain $\pi(({{\mathbb G}}^{Ad(s')\circ \theta})^F)=\pi({{\mathbb G}}_{\theta}(s')^F)$ although the latter already contains all unipotent elements. So, even if ${{\mathcal O}}_u^{\theta, \pi({{\mathbb G}}_{\theta}(s')^F)}$ fails to be of type D, it is still possible that ${{\mathcal O}}_u^{\theta,{\boldsymbol{G}}_{\theta}(s)}$ is so. The unipotent classes in $\PSL_n(q)$ and $Sp_{2n}(q)$ are studied in [@ACGa; @ACGa2], whereas a similar analysis for unitary groups and orthogonal groups is in preparation. This enables us to draw conclusions in case $s=1$. \[pro:trivial-even\] Assume $n=2h$ is even and $q>3$. Let ${{\mathcal O}}$ be a $\theta$-twisted conjugacy class with trivial $\theta$-semisimple part and non-trivial unipotent part. Then ${{\mathcal O}}$ is of type D. By the discussion in Subsection \[subsec:semis\], a representative of the class is a unipotent element $u\in{\boldsymbol{G}}^\theta$ and ${{\mathcal O}}$ has a subrack isomorphic to ${{\mathcal O}}_u^{{\operatorname{id}},({\boldsymbol{G}}^\theta)^\circ}\simeq{{\mathcal O}}_u^{{\operatorname{id}},Sp_n(q)}$ (see the isogeny argument in [@ACGa 1.2]). This rack is of type D with the exception of the classes with Jordan form corresponding to the partition $(2,1,\ldots,1)$, for $q$ either $9$ or not a square. The reader should be alert that the form used for defining $Sp_n({\Bbbk})$ in [@ACGa2] differs from the one considered here. Explicitly, they are related by the change of basis: $$e_i\mapsto\begin{cases}e_i& i\le h \mbox{ odd } \mbox { or } i> h \mbox{ even;} \\ e_{n-i+1}& i\le h \mbox{ even } \mbox{ or } i> h \mbox{ odd.}\\ \end{cases}$$ Hence, if $M$ is the matrix that gives this basis change, elements in ${{\mathbb G}}^\theta$ are obtained from matrices therein conjugating by $M$. Let us consider the partition $(2,1,\ldots,1)$. There are two unipotent classes in $\Sp_n(q)$ associated with it. They are represented by $u_1=1+\eta_1e_{1,n}$ and by $u_2=1+\eta_2 e_{1,n}$ for $\eta_1$ a square and $\eta_2$ not a square in ${{\mathbb F}}_q^\times$, respectively. Consider $g=\pi({\operatorname{diag}}(-{\operatorname{id}}_{2},{\operatorname{id}}_{n-2}))\in{\boldsymbol{G}}$ and set $$v_i:=g\cdot_\theta u_i=\pi\left(\sum_{\begin{smallmatrix}j=1,2,\\ \,n-1,n\end{smallmatrix}}e_{jj}-\sum_{j=3}^{n-3}e_{jj}-\eta_i e_{1n}\right)\in {\boldsymbol{G}}^\theta, \qquad i=1,2.$$ Let us consider the matrix $\sigma=\left(\begin{smallmatrix} 0&&1\\ &{\operatorname{id}}_{n-2}\\ -1&&0\\ \end{smallmatrix}\right)$; in particular $\pi(\sigma)\in {\boldsymbol{G}}^\theta$. Now, set $r_i=u_i$, $s_i=\pi(\sigma)\cdot_\theta v_i$ and $R_i={{\mathcal O}}_{u_i}^{{\operatorname{id}},{\boldsymbol{G}}^\theta}$, $S_i={{\mathcal O}}_{s_i}^{{\operatorname{id}},{\boldsymbol{G}}^\theta}$, $i=1,2$. It follows that the elements $r_i, s_i$ satisfy . Unless $q\equiv 1(4)$ and $n=4$, the subracks $R$ and $S$ are disjoint as one is a unipotent class and the other is not. Assume now $n=4$, $q\equiv1(4)$. Then, for $i=1,2$ there is $\xi_i\in {{\mathbb F}}_q^\times$ such that $\xi_i^2\neq1$ and $(\xi\eta_i)^2\neq2$. Indeed, this excludes at most 4 elements, hence the case of $q>5$ follows, whereas if $q=5$, then 2 is not a square and we can take $\xi_i=2\in{{\mathbb F}}_5$. In this case we take $g=\pi({\operatorname{diag}}(1,\xi_i,1,\xi_i^{-1}))\in{\boldsymbol{G}}$ and $$v_i:=g\cdot_\theta u_i=\pi\left(\begin{smallmatrix} 1&&\xi_i\eta_i\\ &\xi_i^2{\operatorname{id}}_2\\ &&1\end{smallmatrix}\right), \qquad i=1,2.$$ Then we consider $r=u_i$, $s=\pi(s)\cdot_\theta v_i$ with $s_i$ as above, $R={{\mathcal O}}_{u_i}^{{\operatorname{id}},{\boldsymbol{G}}^\theta}$ and $S={{\mathcal O}}_{s_i}^{{\operatorname{id}},{\boldsymbol{G}}^\theta}$ and the proposition follows. If $n$ is odd, less is known about unipotent conjugacy classes in ${\boldsymbol{G}}^\theta$. We can still obtain the following. Recall that in this case ${{\mathbb G}}^\theta\simeq\SO_n({\Bbbk})$. \[pro:trivial-odd\] Assume $n>3$ is odd. Let ${{\mathcal O}}$ be a $\theta$-twisted conjugacy class with trivial $\theta$-semisimple part. If the Jordan form of its $p$-part in ${\boldsymbol{G}}^\theta$ corresponds to the partition $(n)$, then ${{\mathcal O}}$ is of type D. As above, a representative of the class is $u\in{\boldsymbol{G}}^\theta$ and ${{\mathcal O}}$ has a subrack isomorphic to ${{\mathcal O}}_u^{{\operatorname{id}},{\boldsymbol{G}}^\theta}\simeq{{\mathcal O}}_u^{{\operatorname{id}},SO_n(q)}$. We apply [@ACGa2 3.7]. Acknowledgements {#acknowledgements .unnumbered} ================ The authors are grateful to Nicolás Andruskiewitsch, Kei-Yuen Chan, Fernando Fantino, Gastón García, Donna Testerman and Leandro Vendramin for useful discussions and important e-mail exchanges. We are also indebted to Ernest Vinberg for the suggestion in Remark \[rem:vinberg\] and to the referee for useful comments on the exposition of this manuscript. [AAAaA1]{} J. of Algebra, doi:10.1016/j.jalgebra.2014.06.019. , Communications in Contemporary Mathematics, doi: 10.1142/S0219199715500534. Rev. Un. Mat. Argentina [**51**]{} (2), 1–16 (2010). , Contemporary Mathematics [**537**]{}, Amer. Math. Soc., Providence, RI, 31–56 (2011). Ann. Mat. Pura Appl. (4) Volume [**190**]{}, Number 2, 225-245 (2011). J. Algebra [**325**]{} 305-320, (2011). Adv. in Math. [**178**]{} (2), 177–243 (2003). , *The Nichols algebra of a semisimple Yetter-Drinfeld module*, Amer. J. Math., vol. **132**, no. 6, 1493–1547. Compositio Math. vol. **25**, 1–59, (1972). , Proc. London Math. Soc. **(3)42** 1–41 (1981). , Hopf Algebras and Tensor Categories, Contemp. Math. [**585**]{} 247–259 (2013). J. Algebra **79**, 307–318, (1982). , *Root systems and Weyl groupoids for semisimple Nichols algebras*, Proc. London Math. Soc. **101**, 623–654, (2010). *Methods of Algebraic Geometry, Vol I* Cambridge University Press (1968). Cambridge studies in advanced mathematics [**133**]{}, Cambridge University Press (2011). Ph.D thesis, Univ. Hamburg (2000). Transformation Groups, 8(4), 377–395 (2003). Memoirs of the AMS, (1968). [^1]: Recall that $\PSL_2(2)\simeq \mathbb{S}_3$, $\PSL_2(3)\simeq \mathbb{A}_4\leq\mathbb{S}_4$. [^2]: Indeed, this is not the choice made in [@ACGa] but it is, however, more adequate for our setting.

--- abstract: 'In a recent paper, we considered integers $n$ for which the polynomial $x^n - 1$ has a divisor in ${{\mathbb Z}}[x]$ of every degree up to $n$, and we gave upper and lower bounds for their distribution. In this paper, we consider those $n$ for which the polynomial $x^n-1$ has a divisor in ${{\mathbb F}}_p[x]$ of every degree up to $n$, where $p$ is a rational prime. Assuming the validity of the Generalized Riemann Hypothesis, we show that such integers $n$ have asymptotic density $0$.' address: | Department of Mathematics\ 6188 Kemeny Hall\ Dartmouth College\ Hanover, NH 03755, USA author: - Lola Thompson title: 'On the divisors of $x^n-1$ in ${{\mathbb F}}_p[x]$' --- Introduction and statement of results ===================================== In a recent paper [@thompson], we examined the question “How often does $x^n-1$ have a divisor in ${{\mathbb Z}}[x]$ of every degree between $1$ and $n$?” We called an integer $n$ with this property *$\varphi$-practical* and showed that $$\#\{n \leq X: n \ \hbox{is} \ \varphi\hbox{-practical}\} \asymp \frac{X}{\log X}.$$ We examined variants of this question over other polynomial rings in [@pollackthompson] and [@thompson3]. In [@pollackthompson], Pollack and I extended the notion of $\varphi$-practical by defining an integer $n$ to be $F$-practical if $x^n-1$ has a divisor of every degree between $1$ and $n$ over a number field $F$. We showed that, for any number field $F$, $$\#\{n \leq X : n \ \hbox{is} \ F\hbox{-practical}\} \asymp_F \frac{X}{\log X}.$$ We shifted our focus to fields with positive characteristic in [@thompson3]. For each rational prime $p$, we defined an integer $n$ to be *$p$-practical* if $x^n-1$ has a divisor in ${{\mathbb F}}_[x]$ of every degree between $1$ and $n$. Since every $\varphi$-practical number is $p$-practical for all $p$, our work from [@thompson] immediately implies that $\#\{n \leq X: n \ \hbox{is} \ p\hbox{-practical}\}$ is at least a positive constant times $\frac{X}{\log X}$. Moreover, we showed in [@thompson3] that $$\#\{n \leq X : n \ \hbox{is} \ p\hbox{-practical for all} \ p\} \ll \frac{X}{\log X}$$ and that, for any fixed $p$, $$\#\{n \leq X: n \ \hbox{is} \ p\hbox{-practical but not} \ \varphi\hbox{-practical}\} \gg \frac{X}{\log X}.$$ The difficulty lies in finding an upper bound for the count of integers up to $X$ that are $p$-practical for an arbitrary but fixed prime $p$. This will be the subject of our present investigation. For each fixed prime $p$, we define $$F_p(X): = \# \{n \leq X : n \ \hbox{is} \ p\hbox{-practical}\}.$$ Computational data seem to suggest an estimate for the order of magnitude of $F_p(X)$. For example, when $p = 2$, we can use Sage to compute a table of ratios of $F_2(X)/\frac{X}{\log X}.$ $X$ $F_2(X)$ $F_2(X)/(X/\log X)$ -------- ---------- --------------------- $10^2$ 34 1.565758 $10^3$ 243 1.678585 $10^4$ 1790 1.648651 $10^5$ 14703 1.692745 $10^6$ 120276 1.661674 $10^7$ 1030279 1.660614 : Ratios for $2$-practicals[]{data-label="table:f2"} The table looks similar for other small values of $p$. For example, when $p = 3, 5$ we have: $X$ $F_3(X)$ $F_3(X)/(X/\log X)$ -------- ---------- --------------------- $10^2$ 41 1.888120 $10^3$ 258 1.782201 $10^4$ 1881 1.732465 $10^5$ 15069 1.734883 $10^6$ 127350 1.759405 $10^7$ 1080749 1.741962 : Ratios for $3$-practicals[]{data-label="table:f3"} $X$ $F_5(X)$ $F_5(X)/(X/\log X)$ -------- ---------- --------------------- $10^2$ 46 2.118378 $10^3$ 286 1.975618 $10^4$ 2179 2.006933 $10^5$ 16847 1.939583 $10^6$ 141446 1.954149 $10^7$ 1223577 1.972173 : Ratios for $5$-practicals[]{data-label="table:f5"} The fact that the sequences of ratios appear to vary slowly suggests the following conjecture: For each prime $p$, $\lim_{X \rightarrow \infty} F_p(X)/\frac{X}{\log X}$ exists. The strongest bound that we have been able to prove in this vein is as follows: \[GRH20\] Let $p$ be a prime number. Assuming that the Generalized Riemann Hypothesis holds, we have $F_p(X) = O\left(X\sqrt{\frac{\log \log X}{\log X}}\right).$ Here we use a version of the Generalized Riemann Hypothesis for Kummerian fields. The dependence on the GRH arises from a lemma of Li and Pomerance that we will use in Section 2. For ease of reference, we compile a list of the common notation that will be used throughout this paper. Let $n$ always represent a positive integer. Let $p$ and $q$, as well as any subscripted variations, be primes. Let $P(n)$ denote the largest prime factor of $n$, with $P(1) = 1$. We say that an integer $n$ is $B$-smooth if $P(n) \leq B$. We will use $P^-(n)$ to denote the smallest prime factor of $n$, with $P^-(1) = + \infty$. We will use several common arithmetic functions in this body of work. Let $\tau(n)$ denote the number of positive divisors of $n$. We use $\Omega(n)$ to denote the number of prime factors of $n$ counting multiplicity. Lastly, let $\lambda(n)$ denote the Carmichael $\lambda$-function, which represents the exponent of the multiplicative group of integers modulo $n$. Preliminary lemmas ================== In this section, we provide some preliminary lemmas that will be used in the proof of Theorem \[GRH20\]. We begin by discussing multiplicative orders and their connection to the $p$-practical numbers. Let $\ell_a(n)$ denote the multiplicative order of $a$ modulo $n$ for integers $a$ with $(a, n) = 1$. If $(a, n) > 1$, let $n_{(a)}$ represent the largest divisor of $n$ that is coprime to $a$, and let $\ell_a^*(n) = \ell_a(n_{(a)}).$ In particular, if $(a, n) = 1$, then $\ell_a^*(n) = \ell_a(n).$ In [@thompson3], we gave an alternative characterization of the $p$-practical numbers in terms of the function $\ell_p^*(n)$, which we state here as a lemma: \[pcondsum\] An integer $n$ is *$p$-practical* if and only if every $m$ with $1 \leq m \leq n$ can be written as $m = \sum_{d \mid n} \ell_p^*(d) n_d$, where $n_d$ is an integer with $0 \leq n_d \leq \frac{\varphi(d)}{\ell_p^*(d)}.$ Throughout the remainder of this section, let $a > 1$ be an integer and let $A_q$ denote the set of primes $p \equiv 1 \pmod{q}$ with $a^{\frac{p-1}{q}} \equiv 1 \pmod{p}$. We will make use of several lemmas from [@lp], which we state here for the sake of completeness. \[lp1\] Let $\psi(X)$ be an arbitrary function for which $\psi(X) = o(X)$ and $\psi(X) \geq \log \log X.$ The number of integers $n \leq X$ divisible by a prime $p > \psi(X)$ with $\ell^*_a(p) < \frac{p^{1/2}}{\log p}$ is $O(\frac{X}{\log \psi(X)})$. \[lp2\] The number of integers $n \leq X$ divisible by a prime $p \equiv 1 \pmod{q}$ with $$\frac{q^2}{4 \log^2 q} < p \leq q^2 \log^4 q$$ is $O(\frac{X \log \log q}{q \log q}).$ \[lp3\](GRH) Suppose that $q$ is an odd prime and that $a$ is not a $q^{th}$ power. The number of integers $n \leq X$ divisible by a prime $p \in A_q$ with $p \geq q^2 \log^4 q$ is $O\left(\frac{X}{q \log q} + \frac{X \log \log X}{q^2}\right).$ Next, we present a version of Proposition 1 from Li and Pomerance’s paper [@lp], which will play an important role in obtaining the bound stated in Theorem \[GRH20\]. As in [@lp], our lemma will make use of Lemma \[lp3\]; thus, it will depend on the validity of the Generalized Riemann Hypothesis. \[prop1\](GRH) Let $a$ be a positive integer. Let $\psi(X)$ be defined as in Lemma \[lp1\]. The number of integers $n \leq X$ with $P(\frac{\lambda(n)}{\ell^*_a(n)}) \geq \psi(X)$ is $O(\frac{X \log \log \psi(X)}{\log \psi(X)}).$ Suppose that $n \leq X$ and $q = P(\lambda(n)/\ell^*_a(n)) \geq \psi(X).$ We may assume that $X$ is large, so $a$ is not a $q^{th}$ power and $\psi(X) > a$. Moreover, as we will now show, it must be the case that either $q^2 \mid n$ or $p \mid n$ for some $p \in A_q$. Observe that $$q \mid \frac{\lambda(n)}{\ell^*_a(n)} \mid \frac{\mathrm{lcm}_{p^{e} \mid \mid n} \left[\lambda(p^{e})\right]}{\mathrm{lcm}_{p^{e} \mid \mid n} \left[\ell^*_a(p^e)\right]} \mid \mathrm{lcm}_{p^{e} \mid \mid n} \left[\frac{\lambda(p^e)}{\ell^*_a(p^e)}\right].$$ In particular, $q$ must divide $\frac{\lambda(p^e)}{\ell^*_a(p^e)}$ for some prime $p$. If $q = p,$ then $q \mid \lambda(p^{e})$ implies that $e \geq 2,$ so $q^2 \mid n.$ If $q \neq p$, then $q \mid \frac{\lambda(p)}{\ell^*_a(p)}$, so $p > q > \psi(X) > a$. Thus, $\ell^*_a(p) = \ell_a(p) \mid \frac{p-1}{q}$, so $p \mid a^{\frac{p-1}{q}} - 1,$ which implies that $p \in A_q.$ To handle the case where $q^2 \mid n$, we observe that $$\begin{aligned} \#\{n \leq X: q^2 \mid n \ \hbox{for some prime} \ q \geq \psi(X)\} & \leq \sum_{\substack{q \geq \psi(X) \\ q \ \mathrm{prime}}} \left\lfloor \frac{X}{q^2} \right\rfloor \\ & \leq X \sum_{\substack{q \geq \psi(X) \\ q \ \mathrm{prime}}} \frac{1}{q^2} \ll \frac{X}{\psi(X)}.\end{aligned}$$ Thus, we may assume that $n$ is divisible by a prime $p \in A_q$ with $p > a$. By Lemma \[lp1\], we may assume that $\ell^*_a(p) \geq p^{1/2}/\log p.$ However, since $p \in A_q$ implies that $a^{\frac{p-1}{q}} \equiv 1 \pmod{p}$, then $\ell_a(p) \leq \frac{p-1}{q}$, so $p > \frac{q^2}{(4 \log^2 q)}.$ Thus, we can use Lemmas \[lp2\] and \[lp3\] to deal with the remaining values of $n \leq X$. In particular, we have $$\begin{aligned} \notag & \#\{n \leq X: p \mid n \ \hbox{for some} \ p \in A_q \ \hbox{with} \ p > q^2/(4 \log^2 q)\} \\ \notag & \leq \#\{n \leq X: p \mid n \ \hbox{for some} \ p \equiv 1 \pmod{q} \ \hbox{with} \ p \in (\frac{q^2}{4 \log^2 q}, q^2 \log^4 q]\} \\ \notag & + \#\{n \leq X: p \mid n \ \hbox{for some} \ p \in A_q \ \hbox{with} \ p \geq q^2 \log^4 q\} \\ & \label{lp23} \ll \frac{X \log \log q}{q \log q} + \frac{X}{q \log q} + \frac{X \log \log X}{q^2}, \end{aligned}$$ where the final inequality follows from Lemmas \[lp2\] and \[lp3\]. Since our hypotheses specify that $q \geq \psi(X)$, then the bound given in implies $$\begin{aligned} & \#\{n \leq X: q \geq \psi(X) \ \hbox{and} \ p \mid n \ \hbox{for some} \ p \in A_q\} \\ & \ll X \sum_{q \geq \psi(X)} \left(\frac{\log \log q}{q \log q} + \frac{\log \log X}{q^2}\right) \\ & \ll \frac{X \log \log \psi(X)}{\log \psi(X)}.\end{aligned}$$ Key lemma ========= The key to proving Theorem \[GRH20\] rests in showing that $\ell_p^*(n)$ is usually not too small. We make this statement precise with the following lemma: \[cor2\](GRH) Let $\theta$ be a constant satisfying $\frac{1}{10} \leq \theta \leq \frac{9}{10}$. Let $Y = e^{110(\log X)^\theta (\log \log X)^2}.$ For all $a > 1$ and $X$ sufficiently large, uniformly in $\theta$, we have $$\label{GRHppractical}\#\{n \leq X: \ell^*_a(n) \leq \frac{X}{Ye^{(\log X)^\theta}}\} \ll \frac{X}{(\log X)^{\theta} \log \log X}.$$ Before we prove Lemma \[cor2\], we will introduce three additional results, the first of which is due to Friedlander, Pomerance and Shparlinski [@fps] and the last of which is due to Luca and Pollack [@lucapollack]. \[originalfrposh\] For sufficiently large numbers $X$ and for $\Delta \geq (\log \log X)^3$, the number of positive integers $n \leq X$ with $$\lambda(n) \leq n \exp(-\Delta)$$ is at most $X \exp (-0.69 (\Delta \log \Delta)^{1/3}).$ \[frposh\] Let $\theta$ be as in Lemma \[cor2\]. For sufficiently large $X$, the number of positive integers $n \leq X$ with $$\lambda(n) \leq \frac{X}{e^{(\log X)^\theta}}$$ is at most $X/e^{(\log X)^{\theta/3}}.$ Trivially, there are at most $X/\exp((\log X)^{\theta/2})$ values of $n \leq X/\exp((\log X)^{\theta/2})$ with $\lambda(n) \leq X/\exp((\log X)^\theta).$ On the other hand, if $X/\exp((\log X)^{\theta/2}) < n \leq X$, then $X \leq n \exp((\log X)^{\theta/2}).$ Thus, for large $X$, we have $$\begin{aligned} \#\left\{\frac{X}{e^{(\log X)^{\theta/2}}} < n \leq X : \lambda(n) \leq \frac{X}{e^{(\log X)^\theta}}\right\} & \leq \#\left\{ n \leq X : \lambda(n) \leq \frac{n e^{(\log X)^{\theta/2}}}{e^{(\log X)^\theta}}\right\} \\ & < \#\left\{n \leq X : \lambda(n) \leq \frac{n}{e^{\frac{1}{2}(\log X)^\theta}}\right\}.\end{aligned}$$ Applying Lemma \[originalfrposh\] with $\Delta = \frac{1}{2}(\log X)^\theta$, we see that this is at most $X/\exp(2(\log X)^{\theta/3}).$ Therefore, $$\#\left\{n \leq X: \lambda(n) \leq \frac{X}{e^{(\log X)^\theta}} \right\} \leq \frac{X}{e^{(\log X)^{\theta/2}}} + \frac{X}{e^{2(\log X)^{\theta/3}}} \leq \frac{X}{e^{(\log X)^{\theta/3}}}.$$ \[lucapollack\] But for $O(\frac{X}{(\log X)^3})$ choices of $n \leq X$, we have $$\Omega(\varphi(n)) < 110(\log \log X)^2.$$ We will use these results in the proof of Lemma \[cor2\], which we present below. Let $\theta$ be such that $\frac{1}{10} \leq \theta \leq \frac{9}{10}$, let $B = e^{(\log X)^{\theta}}$ and let $u(n)$ denote the $B$-smooth part of $\lambda(n)$. Let $Y$ be defined as in the statement of Lemma \[cor2\]. If $\lambda(n)$ has a large $B$-smooth part, say $u(n) > Y$, then so does $\varphi(n)$, since $u(n)$ must divide $\varphi(n)$ as well. First, we will estimate the number of $n \leq X$ for which $u(n) > Y.$ Let $\Omega(u(n)) = k$. By definition, all prime factors of $u(n)$ are at most $e^{(\log X)^{\theta}}$. Thus, we have $$Y < u(n) \leq (e^{(\log X)^{\theta}})^k.$$ Solving for $k$, we obtain $k \geq 110(\log \log X)^2$. However, Lemma \[lucapollack\] implies that $k < 110 (\log \log X)^2$ except for $O(\frac{X}{(\log X)^3})$ values of $n \leq X.$ Hence, we can conclude that there are at most $O(\frac{X}{(\log X)^3})$ values of $n$ for which the $B$-smooth part of $\lambda(n)$ is larger than $Y.$ Thus, using Lemma \[frposh\], we have $$\begin{aligned} \#\{n \leq X: \frac{\lambda(n)}{u(n)} \leq \frac{X}{Ye^{(\log X)^\theta}}\} & \leq \#\{n \leq X: \lambda(n) \leq \frac{X}{e^{(\log X)^\theta}}\} + \#\{n\leq X: u(n) > Y\} \\ & \ll \frac{X}{e^{(\log X)^{\theta/3}}} + \frac{X}{(\log X)^{3}}.\end{aligned}$$ However, if we take $\psi(X) = Y \exp\{(\log X)^{\theta}\}$ then we can use Lemma \[prop1\] to show that, for all but $O(\frac{X}{(\log X)^{\theta} \log \log X})$ choices of $n \leq X$, we have $\frac{\lambda(n)}{u(n)} \mid \ell^*_a(n).$ Therefore, we have $$\ell^*_a(n) \geq \frac{\lambda(n)}{u(n)} > \frac{X}{Ye^{(\log X)^\theta}},$$ except for at most $O(\frac{X}{(\log X)^{\theta} \log \log X})$ values of $n \leq X$. Proof of Theorem \[GRH20\] ========================== In this section, we present the proof of our main theorem. We begin by discussing the remaining lemmas that we will need in order to complete the argument. Let $n$ be a positive integer, with $d_1 < d_2 < \cdots < d_{\tau(n)}$ its increasing sequence of divisors. Let $Z \geq 2.$ We say that $n$ is $Z$-dense if $\max_{1 \leq i \leq \tau(n)} \frac{d_{i+1}}{d_i} \leq Z$ holds. The following lemma, due to Saias (cf. [@saias Theorem 1]), describes the count of integers with $Z$-dense divisors. \[saias1\] For $X \geq Z \geq 2$, we have $$\begin{aligned} \label{saiasppractical}\# \{n \leq X: n \ \hbox{is} \ Z\hbox{-dense}\} \ll \frac{X \log Z}{\log X}.\end{aligned}$$ The next lemma is due essentially to Friedlander, Pomerance and Shparlinski (cf. [@fps Lemma 2]). \[l2\] Let $n$ and $d$ be positive integers with $d \mid n$. Then, for any rational prime $p$, we have $\frac{d}{\ell^*_p(d)} \leq \frac{n}{\ell^*_p(n)}.$ The result is proven in [@fps] when $(p, n) = 1$. In the case where $(p, n) > 1$, let $n_{(p)}$ and $d_{(p)}$ represent the largest divisors of $n$ and $d$ that are coprime to $p$, respectively. Then $$\frac{d}{d_{(p)}} \leq \frac{n}{n_{(p)}},$$ since the highest power of $p$ dividing $d$ is at most the highest power of $p$ dividing $n$. After a rearrangement, we have $$\frac{d}{n} \leq \frac{d_{(p)}}{n_{(p)}} \leq \frac{\ell^*_p(d)}{\ell^*_p(n)},$$ where the final inequality follows from the coprime case. We will also use the following elementary lemma: \[kappa\] Let $X \geq 2$ and let $\kappa \geq 1.$ Then, we have $$\#\{n \leq X : \tau(n) \geq \kappa\} \ll \frac{1}{\kappa} X \log X.$$ We observe that $$\sum_{n \leq X} \tau(n) = \sum_{n \leq X} \sum_{d \mid n} 1 \leq X \sum_{d \leq X} \frac{1}{d} \ll X \log X.$$ The number of terms in the sum on the left-hand side of the equation that are $\geq \kappa$ is $\ll \frac{1}{\kappa} X \log X.$ Now we have all of the tools needed to prove Theorem \[GRH20\]. Below, we present its proof. Let $n$ be a positive integer with divisors $d_1 < d_2 < \cdots < d_{\tau(n)}.$ Let $p$ be a rational prime with $p \nmid n$. Let $\theta$ and $Y$ be as in Lemma \[cor2\]. In , set $Z = Y^2$. Assume that $n$ is not in the set of size $O(X \log Y^2/ \log X)$ of integers with $Y^2$-dense divisors. Then there exists an index $j$ with $$\label{zdense} \frac{d_{j+1}}{d_j} > Y^2.$$ Moreover, we can use Lemma \[kappa\] to show that $$\label{taulogx} \# \{n \leq X: \tau(n) > Y/e^{(\log X)^\theta}\} \ll \frac{X e^{(\log X)^\theta} \log X }{Y}.$$ As a result, we will assume hereafter that $\tau(n) \leq Y/e^{(\log X)^\theta}.$ Examining the ratios $\frac{d_{k+1}}{d_k}$, we remark that it is always the case that $d_1 = 1$ and $d_2 = P^-(n);$ hence, we have $$\#\{n \leq X : \frac{d_2}{d_1} > Y^2\} = \sum_{\substack{n \leq X \\ P^-(n) > Y^2}} 1 \ll X \prod_{q \leq Y^2} \left(1 - \frac{1}{q}\right),$$ where the final inequality follows from applying Brun’s Sieve (cf. [@hr Theorem 2.2]). By Mertens’ Theorem (cf. [@pollack Theorem 3.15]), we have $$\label{brunexcept} X \prod_{q \leq Y^2} \left(1 - \frac{1}{q}\right) \ll \frac{X}{\log Y}.$$ Now, suppose that $k > 1$. On one hand, for all $k > 1$, we have $$\label{lpeq1} 1 + \sum_{l \leq k} \ell^*_p(d_l) \frac{\varphi(d_l)}{\ell^*_p(d_l)} = 1 + \sum_{l \leq k} \varphi(d_l) \leq k d_k \leq Ye^{-(\log X)^\theta} d_k.$$ On the other hand, Lemma \[cor2\] implies that $\ell^*_p(n) > \frac{X}{Ye^{(\log X)^\theta}}$ but for $$\begin{aligned} \label{bigohppracexcept} O\left(\frac{X}{(\log X)^{\theta} \log \log X}\right)\end{aligned}$$ integers $n \leq X$. For such numbers $n$, for all $i \geq 1$, we have $$\begin{aligned} \label{lpeq2} \ell^*_p(d_{j+i}) \geq \frac{\ell^*_p(n) d_{j+i}}{n} > \frac{d_{j+i}}{Y e^{(\log X)^\theta}} > \frac{d_j Y^2}{Y e^{(\log X)^\theta}} = Ye^{-(\log X)^\theta} d_j \end{aligned}$$ where the inequalities follow, respectively, from Lemma \[l2\], Lemma \[cor2\] and the assumption that there exists an index $j$ for which holds. As a result, we can combine the inequality from applied with $k = j$ with to show that $$1 + \sum_{l \leq j} \ell^*_p(d_l) \frac{\varphi(d_l)}{\ell^*_p(d_l)} < \ell^*_p(d_{j+i})$$ holds for all $i \geq 1$. Thus, $x^n-1$ has no divisor of degree $1 + \sum_{l \leq j} \varphi(d_l)$ in ${{\mathbb F}}_p[x]$, so $n$ is not $p$-practical. Therefore, by , , and , we have $$\begin{aligned} \label{finalfpineq} F_p(X) \ll \frac{X \log Y}{\log X} + \frac{X e^{(\log X)^\theta} \log X}{Y} + \frac{X}{\log Y} + \frac{X}{(\log X)^\theta \log \log X}.\end{aligned}$$ Now, the only significant terms in are $\frac{X}{(\log X)^\theta \log \log X}$ and $\frac{X \log Y}{\log X}$. Equating these expressions and using the fact that $Y = e^{110(\log X)^\theta(\log \log X)^2}$, we obtain $\theta = \frac{1}{2} - \frac{3 \log_3 X}{2 \log_2 X}$ as a good choice for $\theta$. Plugging this value of $\theta$ into the bound $\frac{X}{(\log X)^\theta \log \log X}$ yields a bound of $O\left(X \sqrt{\frac{\log \log X}{\log X}}\right)$ for the size of the set of $p$-practicals up to $X$. *Acknowledgements.* The work contained in this paper comprises a portion of my Ph.D. thesis [@thompsonthesis]. I would like to thank my adviser, Carl Pomerance, for his guidance throughout the process of completing this work. I would also like to thank Paul Pollack for pointing out the relevant results in [@lucapollack] and for his help with simplifying a few of my arguments. [10]{} P. Erdős, C. Pomerance, and E. Schmutz, *Carmichael’s lambda function*, Acta Arith. **58** no. 4 (1991), 363 – 385. J. Friedlander, C. Pomerance, and I. E. Shparlinski, *Period of the power generator and small values of the Carmichael function*, Math. Comp. **70** (2001), 1591 – 1605. H. Halberstam and H.-E. Richert, *Sieve methods*. Academic Press, London, 1974. P. Kurlberg and C. Pomerance, *On a problem of Arnold: the average multiplicative order of a given integer*, Algebra Number Theory (to appear). S. Li and C. Pomerance, *On generalizing Artin’s conjecture on primitive roots to composite moduli*, J. Reine Angew. Math. **556** (2003), 205 – 224. F. Luca and P. Pollack, *An arithmetic function arising from Carmichael’s conjecture*, J. Théorie des Nombres de Bordeaux **23** (2011), 697 – 714. P. Pollack, *Not always buried deep: a second course in elementary number theory*. Amer. Math. Soc., Providence, 2009. P. Pollack and L. Thompson, *On the degrees of divisors of $T^n-1$*, submitted, e-print available at <http://arxiv.org/abs/1206.2084>. E. Saias, *Entiers à diviseurs denses. I.*, J. Number Theory **62** (1997), 163 – 191. L. Thompson, *Polynomials with divisors of every degree*, J. Number Theory **132** (2012), p. 1038 - 1053. , *Variations on a question concerning the degrees of divisors of $x^n-1$*, submitted, e-print available at <http://math.dartmouth.edu/~thompson/notes.html>. , *Products of distinct cyclotomic polynomials*, Ph.D. thesis, Dartmouth College, 2012.

--- abstract: 'In the $\mathcal{D}$-modules theory, Gauss-Manin systems are defined by the direct image of the structure sheaf $\mathcal{O}$ by a morphism. A major theorem says that these systems have only regular singularities. This paper examines the irregularity of an analogue of the Gauss-Manin systems. It consists in the direct image complex of a $\mathcal{D}$-module twisted by the exponential of a polynomial $g$ by another polynomial $f$, $f_+(\mathcal{O}e^g)$, where $f$ and $g$ are two polynomials in two variables. The analogue of the Gauss-Manin systems can have irregular singularities (at finite distance and at infinity). We express an invariant associated with the irregularity of these systems at $c\in \mathbb{P}^1$ by the geometry of the map $(f,g)$.' address: | Laboratoire J. A. Dieudonné, Université de Nice\ parc Valrose, 06108 Nice cedex 02 author: - Céline Roucairol title: 'Irregularity of an analogue of the Gauss-Manin systems' --- Introduction ============ We denote by $\mathcal{O}_{\mathbb{C}^n}$ the sheaf of regular functions on $\mathbb{C}^n$ and by $\mathcal{D}_{\mathbb{C}^n}$ the sheaf of algebraic differential operators on $\mathbb{C}^n$. If $f:\mathbb{C}^n\to\mathbb{C}$ is a polynomial, we define the Gauss-Manin connection as the extension of the flat bundle $\underset{t\in\mathbb{C}\setminus\Sigma}{\cup}H^{k+n-1}(f^{-1}(t)^{an},\mathbb{C})$, where $\Sigma\subset\mathbb{C}$ is a finite subset such that $f:f^{-1}(\mathbb{C}\setminus\Sigma)\to\mathbb{C}\setminus\Sigma$ is a locally trivial fibration. A major theorem says that these connections are regular. In the $\mathcal{D}$-module theory, we study this connection with the help of a complex of $\mathcal{D}_{\mathbb{C}}$-modules, it being the direct image complex $f_+(\mathcal{O}_{\mathbb{C}^n})$. Their cohomology modules are called Gauss-Manin systems. They are holonomic and regular. Now, let $g:\mathbb{C}^n\to\mathbb{C}$ be another polynomial. We denote by $\mathcal{O}_{\mathbb{C}^n}e^g$ the $\mathcal{D}_{\mathbb{C}^n}$-module obtained from $\mathcal{O}_{\mathbb{C}^n}$ by twisting by $e^g$. We are interested in an analogue of the Gauss-Manin systems, it being the direct image complex $f_+(\mathcal{O}_{\mathbb{C}^n}e^g)$. In [@Ma], F. Maaref calculates the generic fibre of the sheaf of horizontal analytic sections of the systems $\mathcal{H}^k(f_+(\mathcal{O}_{\mathbb{C}^n}e^g))$. It consists in a relative version of a result of C. Sabbah in [@Sa]. Indeed, the generic fiber of the sheaf of horizontal analytic sections of $\mathcal{H}^k(f_+(\mathcal{O}_{\mathbb{C}^n}e^g))$ is canonically isomorphic to the cohomology group with closed support $H^{k+n-1}_{\Phi_t}(f^{-1}(t)^{an},\mathbb{C})$, where $\Phi_t$ is a family of closed subsets of $f^{-1}(t)$, on which $e^{-g}$ is rapidly decreasing. More precisely, this family is defined as follow. Let $\pi:\widetilde{\mathbb{P}^1}\to\mathbb{P}^1$ be the oriented real blow-up of $\mathbb{P}^1$ at infinity. $\widetilde{\mathbb{P}^1}$ is diffeomorphic to $\mathbb{C}\cup S^1$, where $S^1$ is the circle of directions at infinity. $A$ is in $\Phi_t$ if $A$ is a closed subset of $f^{-1}(t)$ and the closure of $g(A)$ in $\mathbb{C}\cup S^1$ intersects $S^1$ in $]-\frac{\pi}{2},\frac{\pi}{2}[$. This isomorphism can be better understood using relative cohomology group. F. Maaref shows that for all $t\notin\Sigma$ and for all $\rho$, such that $Re(-\rho)$ is sufficiently large, the fibre at $t$ of the sheaf of horizontal analytic sections of $\mathcal{H}^k(f_+(\mathcal{O}_{\mathbb{C}^n}e^g))$ is isomorphic to the relative cohomology group $H^{k+n-1}(f^{-1}(t)^{an}, (f^{-1}(t)\cap g^{-1}(\rho))^{an},\mathbb{C})$. Finally, he proves the quasi-unipotence of the corresponding local monodromy. The Gauss-Manin systems have only regular singularities. In our case, the complex $f_+(\mathcal{O}_{\mathbb{C}^n}e^g)$ can have irregular singularities. The aim of this paper is to characterize this irregularity in terms of the geometry of the map $(f,g)$, when $f$ and $g$ are two polynomials in two variables. In $f$ and $g$ are algebraically independant, we will prove that the complex $f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$ is essentially concentrated in degree zero. Then, we can associate to this complex a system of differential equations in one variable. We want to calculate the irregularity number of this system at a point at finite distance and at infinity. Let $\mathbb{X}$ be a smooth projective compactification of $\mathbb{C}^2$ such that there exists $F,G:\mathbb{X}\to\mathbb{P}^1$, two meromorphic maps, which extend $f$ and $g$. Let us denote by $D$ the divisor $\mathbb{X}\setminus\mathbb{C}^2$. In the following, we identify $\mathbb{P}^1$ with $\mathbb{C}\cup\{\infty\}$. Let $\Gamma$ be the critical locus of $(F,G)$. We denote by $\Delta_1$ the cycle in $\mathbb{P}^1\times\mathbb{P}^1$ which is the closure in $\mathbb{P}^1\times\mathbb{P}^1$ of $(F,G)(\Gamma)\cap(\mathbb{C}^2\setminus\{c\}\times\mathbb{C})$ where the image is counted with multiplicity and by $\Delta_2$ the cycle in $\mathbb{P}^1\times\mathbb{P}^1$ which is the closure in $\mathbb{P}^1\times\mathbb{P}^1$ of $(F,G)(D)\cap(\mathbb{C}^2\setminus\{c\}\times\mathbb{C})$ where the image is counted with multiplicity. For all $c\in\mathbb{P}^1$, the germs at $(c,\infty)$ of the support of $\Delta_1$ and $\Delta_2$ are some germs of curves or are empty. Then, we denote by $I_{(c,\infty)}(\Delta_i,\mathbb{P}^1\times\{\infty\})$ the intersection number of the cycles $\Delta_i$ and $\mathbb{P}^1\times\{\infty\}$. If the germ at $(c,\infty)$ of $\Delta_i$ is empty, this number is equal to $0$. Let $f,g\in\mathbb{C}[x,y]$ be algebraically independant. Let $c\in\mathbb{P}^1$.\ Then, the irregularity number at $c$ of the system $\mathcal{H}^0(f_+(\mathcal{O}_{\mathbb{C}^2}e^g))$ is equal to $I_{(c,\infty)}(\Delta_1,\mathbb{P}^1\times\{\infty\})+I_{(c,\infty)}(\Delta_2,\mathbb{P}^1\times\{\infty\})$. When $c\in\mathbb{C}$, we can prove that the germ at $(c,\infty)$ of $\Delta_2$ is empty. Moreover, the germ at $(c,\infty)$ of $\Delta_1$ coincide with the one of the closure in $\mathbb{P}^1\times\mathbb{P}^1$ of $(f,g)(\tilde{\Gamma})\setminus\{c\}\times\mathbb{C}$, where $\tilde{\Gamma}$ is the critical locus of $(f,g)$. In general, we do not know how to calculate directly the irregularity number of a system associated with $f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$. The notion of irregularity complex along an hypersurface defined by Z. Mebkhout (see [@Me1] and [@Me2]) is the appropriate tool to express the irregularity of $f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$ (see paragraph \[irr\]). Indeed, this irregularity complex along an hypersurface is a generalization of the irregularity number at a point of a system of differential equations in one variable. Moreover, Z. Mebkhout proves a theorem of commutation between the direct image functor and the irregularity functor (see theorem \[commute\]). Then, the irregularity number at $c\in\mathbb{P}^1$ of the system of differential equations associated with $f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$ can be expressed with the help of an irregularity complex of a $\mathcal{D}$-module in two variables along a curve. In the general case where $f$ and $g$ are not necessarily algebraically independant, the complex $f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$ is not necessarily concentrated in degree $0$. Then, we want to calculate the alternative sum of the irregularity number at $c\in\mathbb{P}^1$ of the systems $\mathcal{H}^k(f_+(\mathcal{O}_{\mathbb{C}^2}e^g))$. This irregularity number $IR_c$ is equal to the Euler characteristic of a complex of vector spaces over $\mathbb{C}$, it being the irregularity complex of $f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$ at $c\in\mathbb{P}^1$. When $f$ and $g$ are algebraically independant, this number coincide with the irregularity number of the system $\mathcal{H}^0(f_+(\mathcal{O}_{\mathbb{C}^2}e^g))$. Then, we can prove that the irregularity number $IR_c$ is equal to $-\chi(\mathbb{R}\Gamma(F^{-1}(c)\cap G^{-1}(\infty), IR_{F^{-1}(c)}(\mathcal{O}_{\mathbb{X}}[\ast D]e^G)))$, where $IR_{F^{-1}(c)}(\mathcal{O}_{\mathbb{X}}[\ast D]e^G)$ is the irregularity complex of $\mathcal{O}_{\mathbb{X}}[\ast D]e^G$ along $F^{-1}(c)$. Then, according to a result of C. Sabbah in [@Sa], we know that, for $x\in F^{-1}(c)\cap G^{-1}(\infty)$, the Euler characteristic of $(IR_{F^{-1}(c)}(\mathcal{O}_{\mathbb{X}}[\ast D]e^G))_x$ is equal to the Euler characteristic of the fiber $f^{-1}(D^{\ast}(c,\eta))\cap g^{-1}(\rho)\cap B(x,\epsilon)$, where $\epsilon$ and $\eta$ are small enough and $|\rho|$ is big enough. This result is stated in theorem \[Sab\] paragraph \[exp\] in terms of complex of nearby cycles. Then, we have to globalize the situation (see paragraph \[topo\]). First of all, we prove that for $\eta$ small enough and $R$ big enough, $g:f^{-1}(D^{\ast}(c,\eta))\cap g^{-1}(\{|\rho|>R\})\to \{|\rho|>R\}$ is a locally trivial fibration. Then, the irregularity number $IR_c$ is equal to the opposite of the Euler characteristic of its fiber $f^{-1}(D^{\ast}(c,\eta))\cap g^{-1}(\rho)$. This result hold in the general case where $f$ and $g$ are not necessarily algebraically independant. Then, we have to study the topology of this fiber. We have to distinguished the case where $f$ and $g$ are algebraically independant (see paragraph \[ind\]) and the one where they are algebraically dependant (see paragraph \[dep\]). Irregularity complex along an hypersurface {#irr} ========================================== We will use the definition of regularity given by Z. Mebkhout (see [@Me1] and [@Me2]). First of all, we recall the definition of irregularity complex of analytic $\mathcal{D}$-modules. Then, we define the notion of irregularity complex for algebraic $\mathcal{D}$-modules. Here, we have to take into account the behaviour of these modules at infinity. Moreover, we state major theorems on irregularity: the positivity theorem, the stability of the category of complex of regular holonomic $\mathcal{D}$-modules (analytic) by direct image by a proper map and the comparison theorem of Grothendieck. The analytic case ----------------- Let $X$ be a smooth analytic variety over $\mathbb{C}$. In this section, $\mathcal{D}_X$ denotes the sheaf of analytic differential operators on $X$. Let $Z$ be an analytic closed subset of $X$. Denote by $i$ the canonical inclusion of $X\setminus Z$ in $X$. Let $\mathcal{M}^{\bullet}$ be a bounded complex of analytic $\mathcal{D}_X$-modules with holonomic cohomology. \[irran\] We define the irregularity complex of $\mathcal{M}^{\bullet}$ along $Z$ as the complex : $$\begin{array}{lll} IR_Z(\mathcal{M}^{\bullet})&:=& R\Gamma_Z(DR(\mathcal{M}^{\bullet}[\ast Z]))[+1]\\ &:=&cone\big(DR(\mathcal{M}[\ast Z])\to Ri_{\ast}i^{-1}(DR(\mathcal{M}^{\bullet}[\ast Z]))\big). \end{array}$$ According to the constructibility theorem (c.f. [@K] and [@Me3]), this complex is a bounded complex of constructible sheaves on $X$ with support in $Z$. Then, we can define the covariant exact functor $IR_Z$ between the category of bounded complexes of $\mathcal{D}_X$-modules with holonomic cohomology and the category of bounded complexes of constructible sheaves on $X$ with support in $Z$. $\mathcal{M}^{\bullet}$ is said to be regular if its irregularity complex along all hypersurfaces of $X$ is zero. In one variable, the previous definition of regularity generalises the notion of regular singular point of a differential equation which is characterized by the annulation of the irregularity number (Fuchs theorem). Indeed, irregularity complex along an hypersurface generalizes irregularity number in the case of one variable. According to Z. Mebkhout (see [@Me1], [@Me2]), the characteristic cycle of the irregularity complex of a holonomic $\mathcal{D}$-module along an hypersurface is positive. \[posit\] If $Z$ is an hypersurface of $X$ and $\mathcal{M}$ is a holonomic $\mathcal{D}_X$-module, the complex $IR_Z(\mathcal{M})$ is perverse on $Z$. The category of complexes of $\mathcal{D}$-modules with regular holonomic cohomology is stable by proper direct image. Let us state the theorem which proves this stability (see [@Me1] and Prop. 3.6-4 of [@Me2]). It will be a major tool in this paper. Let $\pi:X\to Y$ be a proper morphism of smooth analytic varieties over $\mathbb{C}$. Let $T$ be a hypersurface of $Y$. \[commute\] Let $\mathcal{M}^{\bullet}$ be a bounded complex of analytic $\mathcal{D}_X$-modules with holonomic cohomology. We have an isomorphism: $$IR_T(\pi_+(\mathcal{M}^{\bullet}))[\dim Y]\simeq R\pi_{\ast}(IR_{\pi^{-1}(T)}(\mathcal{M}^{\bullet}))[\dim X].$$ The algebraic case ------------------ Let $X$ be a smooth affine variety over $\mathbb{C}$. In this section, $\mathcal{D}_X$ denotes the sheaf of algebraic differential operators on $X$. Denote by $j:X\to\mathbb{P}^n$ an immersion of $X$ in a projective space. Let $Z$ be a locally closed subvariety of $\mathbb{P}^n$ and $\mathcal{M}^{\bullet}$ a bounded complex of algebraic $\mathcal{D}_X$-modules with holonomic cohomology. \[irral\] We define the irregularity complex of $\mathcal{M}^{\bullet}$ along $Z$ as the complex : $$IR_Z(j_+(\mathcal{M}^{\bullet})):=IR_{Z^{an}}(j_+(\mathcal{M}^{\bullet})^{an}),$$ where $Z^{an}$ denotes the analytic variety associated with $Z$ and $j_+(\mathcal{M}^{\bullet})^{an}$ denotes the complex of analytic $\mathcal{D}$-modules associated with $j_+\mathcal{M}$. \[regulier\] $\mathcal{M}^{\bullet}$ is said to be regular if its irregularity complex along all subvariety of $\mathbb{P}^n$ is zero. This condition of regularity does not depend on the choice of the immersion $j$ (see proposition 9.0-4 in [@Me2]). The definition \[regulier\] of regular holonomic complex was motivated by the comparison theorem of Grothendieck (see [@Gro]) and the comparison theorem of Deligne (see [@Del]). As is shown in [@Me4], the comparison theorem of Grothendieck is a consequence of the following theorem: \[groth\] The structure sheaf $\mathcal{O}_X$ is regular in the sense of definition \[regulier\]. Concerning to the stability of regularity under direct image, Theorem \[commute\] allows to prove the following theorem (see Theorem 9.0-7 of [@Me2]): The category of complexes of $\mathcal{D}$-modules with regular holonomic cohomology is stable under direct image. [Notation]{} We denote by $IR_Z^k(j_+(\mathcal{M}^{\bullet}))$ the $k$-th space of cohomology of the complex $IR_Z(j_+(\mathcal{M}^{\bullet}))$. We are interested in the irregularity of the direct image complex $f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$. This is a complex of $\mathcal{D}_{\mathbb{C}}$-modules in one variable, with holonomic cohomology. According to the definition \[irral\] of irregularity complex, we have to consider an immersion $j:\mathbb{C}\to\mathbb{P}^1$. Let $c\in\mathbb{P}^1$. We want to examine the complex $IR_{c}(j_+f_+(\mathcal{O}_{\mathbb{C}^n}e^g))$. As this complex has its support in $c$, we want to compute its Euler characteristic $\chi(IR_{c}(j_+f_+(\mathcal{O}_{\mathbb{C}^n}e^g))_c)$. In the following, we will denote this number by $IR_c$. Regular holonomic $\mathcal{D}$-modules twisted by an exponential {#exp} ================================================================= Definitions ----------- Let $X$ be an algebraic variety over $\mathbb{C}$. We denote by $\mathcal{O}_X$ the sheaf of regular functions on $X$. We identify $\mathbb{P}^1$ to $\mathbb{C}\cup\{\infty\}$. Let $g:X\to\mathbb{P}^1$ be a meromorphic function on $X$. We define the $\mathcal{D}_X$-module $\mathcal{O}_X[\ast g^{-1}(\infty)]e^g$ as a $\mathcal{D}_X$-module which is isomorphic to $\mathcal{O}_{X}[\ast g^{-1}(\infty)]$ as $\mathcal{O}_X$-module; the action of $\xi$, vector field on an open subset of $X$, on a section $he^g$ of $\mathcal{O}_X[\ast g^{-1}(\infty)]e^g$ is defined by $\xi(he^g)=\xi(h)e^g+h\xi(g)e^g$. Let $\mathcal{M}$ be a holonomic $\mathcal{D}_X$-module. We define the $\mathcal{D}_X$-module $\mathcal{M}[\ast g^{-1}(\infty)]e^g$ as the $\mathcal{D}_X$-module $\mathcal{M}\otimes_{\mathcal{O}_X}\mathcal{O}_X[\ast g^{-1}(\infty)]e^g$. $\mathcal{O}_X[\ast g^{-1}(\infty)]e^g$ is the direct image by an open immersion of a vector bundle with integrable connection. Then, it is a holonomic $\mathcal{D}_X$-module as algebraic direct image of a holonomic $\mathcal{D}$-module. $\mathcal{M}[\ast g^{-1}(\infty)]e^g$ is a holonomic left $\mathcal{D}_X$-module as tensor product of two holonomic left $\mathcal{D}_X$-modules. We have analogous definitions in the analytic case. We just have to transpose in the analytic setting. On irregularity of regular holonomic $\mathcal{D}$-modules twisted by an exponential ------------------------------------------------------------------------------------ Let $X$ be a complex analytic manifold and let $f,g:X\to\mathbb{C}$ be two analytic functions. Assume that $\mathcal{M}$ is a regular holonomic $\mathcal{D}_X$-module (analytic). Generally, $\mathcal{M}[\frac{1}{g}]e^{\frac{1}{g}}$ is an irregular $\mathcal{D}_X$-module. We want to relate the irregularity complex of this module along $f^{-1}(0)$ with some topological data. \[Sab\] The complex $IR_{f=0}(\mathcal{M}[\frac{1}{g}]e^{\frac{1}{g}})$ and the complex of nearby cycles $\Psi_g(DR(\mathcal{M}[\frac{1}{f}]))$ have the same characteristic function on $f^{-1}(0)\cap g^{-1}(0)$. According to corollary 5.2 of [@Sa], this lemma is true in the case where $f$ and $g$ are the same function. Assume that $f$ and $g$ are not equal. Then, using the case where the two functions are equal, we remark that it is sufficient to prove that the complex $IR_{f=0}(\mathcal{M}[\frac{1}{g}]e^{\frac{1}{g}})$ and the complex $IR_{g=0}(\mathcal{M}[\frac{1}{fg}]e^{\frac{1}{g}})$ have the same characteristic function on $f^{-1}(0)\cap g^{-1}(0)$. - Let us first prove that $IR_{f=0}(\mathcal{M}[\frac{1}{g}]e^{\frac{1}{g}}) =R\Gamma_{f=0}(IR_{g=0}(\mathcal{M}[\frac{1}{fg}]e^{\frac{1}{g}}))$. Let $X^{\ast}$ denote $X\setminus g^{-1}(0)$ and $\eta$ be the inclusion of $X^{\ast}$ in $X$. By definition, we have : $$\begin{array}{lll} IR_{g=0}(\mathcal{M}[\frac{1}{fg}]e^{\frac{1}{g}})&=& cone(DR(\mathcal{M}[\frac{1}{fg}]e^{\frac{1}{g}})\to R\eta_{\ast}\eta^{-1}DR(\mathcal{M}[\frac{1}{fg}]e^{\frac{1}{g}}))\\ &=&cone(DR(\mathcal{M}[\frac{1}{fg}]e^{\frac{1}{g}})\to R\eta_{\ast}(DR(\mathcal{M}[\frac{1}{f}])_{|X^{\ast}})). \end{array}$$ Now, consider the following diagram : $$\xymatrix{X^{\ast}\ar@{^(->}[r]^{\eta}&X\\ X^{\ast}\setminus f^{-1}(0)\ar@{^(->}[r]^{\eta^{'}}\ar@{^(->}[u]_j& X\setminus f^{-1}(0).\ar@{^(->}[u]_{j^{'}}}$$ Then, since $\mathcal{M}$ is regular, according to the definition \[regulier\] of regularity, we have : $$\begin{array}{lll} R\eta_{\ast}(DR(\mathcal{M}[\frac{1}{f}])_{|X^{\ast}}))&=& R\eta_{\ast}Rj_{\ast}(DR(\mathcal{M})_{|X^{\ast}\setminus f^{-1}(0)})\\ &=&Rj_{\ast}^{'}R\eta_{\ast}^{'} (DR(\mathcal{M})_{|X^{\ast}\setminus f^{-1}(0)}). \end{array}$$ As $R\Gamma_{f=0}Rj_{\ast}^{'}=0$, we obtain : $$\begin{array}{lll} R\Gamma_{f=0}(IR_{g=0}(\mathcal{M}[\frac{1}{fg}]e^{\frac{1}{g}})) &=&R\Gamma_{f=0}(DR(\mathcal{M}[\frac{1}{fg}]e^{\frac{1}{g}}))[+1]\\ &=&IR_{f=0}(\mathcal{M}[\frac{1}{g}]e^{\frac{1}{g}}). \end{array}$$ - Then, we are led to show that the complex $R\Gamma_{f=0}(IR_{g=0}(\mathcal{M}[\frac{1}{fg}]e^{\frac{1}{g}}))$ and the complex $IR_{g=0}(\mathcal{M}[\frac{1}{fg}]e^{\frac{1}{g}})$ have the same characteristic function on $f^{-1}(0)\cap g^{-1}(0)$. Using the following distinguished triangle, $$\xymatrix{& Rj_{\ast}^{'}j^{'-1}(IR_{g=0}(\mathcal{M}[\frac{1}{fg}]e^{\frac{1}{g}})) \ar[dl]_{[+1]}&\\ R\Gamma_{f=0}(IR_{g=0}(\mathcal{M}[\frac{1}{fg}]e^{\frac{1}{g}}))\ar[rr]&& IR_{g=0}(\mathcal{M}[\frac{1}{fg}]e^{\frac{1}{g}}),\ar[ul]}$$ it is sufficient to show that the characteristic function of the complex $Rj_{\ast}^{'}j^{'-1}(IR_{g=0}(\mathcal{M}[\frac{1}{fg}]e^{\frac{1}{g}}))$ is zero on $f^{-1}(0)\cap g^{-1}(0)$. Now, if $\mathcal{F}$ is a constructible sheaf on $X$ and $x\in f^{-1}(0)$, $\chi((Rj_{\ast}^{'}j^{'-1}\mathcal{F})_x)=\chi((D(j_!^{'}j^{'-1}D\mathcal{F}))_x)=\chi((j_!^{'}j^{'-1}D\mathcal{F})_x)=0$ ($D$ is the Verdier duality (see [@duality])). Topological interpretation of the irregularity of $f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$ {#topo} ====================================================================================== Notations --------- Let $f,g:\mathbb{C}^2\to \mathbb{C}$ be two polynomials. Let $\mathbb{X}$ be a smooth projective compactification of $\mathbb{C}^2$ such that there exists $F,G:\mathbb{X}\to\mathbb{P}^1$ two meromorphic maps which extend $f$ and $g$. In view to construct $\mathbb{X}$, $F$ and $G$, we consider an immersion of $\mathbb{C}^2$ in $\mathbb{P}^2$ and we define a rational map $(\tilde{f},\tilde{g})$ on $\mathbb{P}^2$ which extends the map $(f,g)$. Then, after a finite number of blowing ups, we lift the indeterminacies of the rational map $(\tilde{f},\tilde{g})$. In the following, we fixe such a compactification and use the following notations: $$\xymatrix{\mathbb{C}^2\ar[r]^f\ar@{^(->}[d]^i&\mathbb{C}\ar@{^(->}[d]^j&&\mathbb{C}^2\ar[r]^g\ar@{^(->}[d]^i&\mathbb{C}\ar@{^(->}[d]^j\\ \mathbb{X}\ar[r]^F&\mathbb{P}^1&,&\mathbb{X}\ar[r]^G&\mathbb{P}^1.}$$ Two fibration theorems ---------------------- \[fibre1\] Let $c\in\mathbb{C}$. There exists $R>0$ big enough such that $$g:g^{-1}(\{|\rho|>R\})\setminus (f^{-1}(c)\cap g^{-1}(\{|\rho|>R\}))\to \{|\rho|>R\}$$ is a locally trivial fibration. Let $\mathcal{S}$ be an algebraic Whitney stratification of $\mathbb{X}$ such that $D$ and $F^{-1}(c)$ are union of strata. According to the Sard’s theorem, there exists $U$, a dense Zariski open subset of $\mathbb{P}^1$, such that $G:G^{-1}(U)\to U$ is transverse to the Whitney stratification $\mathcal{S}^{'}$ of $G^{-1}(U)$ induced by $\mathcal{S}$. According to the first isotopy lemma of Thom-Mather, $G:G^{-1}(U)\to U$ is a locally trivial fibration with respect to $\mathcal{S}^{'}$. Then, we choose $R>0$ big enough such that $\{|\rho|>R\}\subset U$. Using the inclusion $j:\mathbb{C}\to\mathbb{P}^1$, we identify $P^1$ to $\mathbb{C}\cup\{\infty\}$. If $c\in\mathbb{C}$, we denote by $D(c,\eta)$ the open disc in $\mathbb{C}$ centered at $c$ of radius $\eta$. Let $D(\infty,\eta)=\{z\in\mathbb{C}~|~|z|>\frac{1}{\eta}\}\cup\{\infty\}$. If $c\in\mathbb{P}^1$, we denote by $D^{\ast}(c,\eta)\subset\mathbb{C}$ the set $D(c,\eta)\setminus\{c\}$. \[fibre\] Let $c\in\mathbb{P}^1$. There exists $\eta$ small enough and $R$ big enough such that: $$g:f^{-1}(D^{\ast}(c,\eta))\cap g^{-1}(\{|\rho|>R\})\to\{|\rho|>R\}$$ is a locally trivial fibration. According to the first isotopy lemma of Thom-Mather, we want to find a Whitney stratification $\mathcal{S}^{'}$ of $F^{-1}(\overline{D(c,\eta)})$ such that $D\cap F^{-1}(\overline{D(c,\eta)})$, $F^{-1}(c)$ and $F^{-1}(S(c,\eta))$ are union of strata and such that the morphism $G:F^{-1}(\overline{D(c,\eta)})\cap G^{-1}(\{|\rho|>R\})\to \{|\rho|>R\}$ is transverse to $\mathcal{S}^{'}$. Let $\mathcal{S}$ be an algebraic Whitney stratification of $\mathbb{X}$ such that $F^{-1}(c)$ and $D$ are union of strata. For $\eta>0$, we denote by $\mathcal{T}$ the real analytic stratification $\{F^{-1}(S(c,\eta)), F^{-1}(D(c,\eta))\}$. For $\eta>0$ small enough, $\mathcal{S}$ and $\mathcal{T}$ are transverse. Let $\mathcal{S}^{'}$ be the real analytic stratification $\mathcal{S}\cap\mathcal{T}$. Now, let us prove that for $\eta$ small enough and $R$ big enough, the map $G:F^{-1}(\overline{D(c,\eta)})\cap G^{-1}(\{|\rho|>R\})\to \{|\rho|>R\}$ is transverse to $\mathcal{S}^{'}$. - If $S^{'}=S\cap F^{-1}(S(c,\eta))$, for a $S\in\mathcal{S}$, we have to prove that for $\eta$ small enough and $R$ big enough, $G_{|S\cap F^{-1}(S(c,\eta))}$ is a submersion. It is sufficient to prove that for $\eta$ small enough and $R$ big enough, $F^{-1}(c^{'})$ is transverse to $G_{|S}^{-1}(\rho)$, where $|\rho|>R$ and $c^{'}\in S(c,\eta)$. Let $\Gamma_S$ be the critical locus of $(F,G)_{|S}$ and $\Delta_S=(F,G)(\Gamma_S)$ be the discriminant variety of $F_{|S}$ and $G_{|S}$. We denote by $\Delta_S^{'}$ the closure in $\mathbb{P}^1\times\mathbb{P}^1$ of $\Delta_S\cap\mathbb{C}^2$. In our case, the dimension of $\Delta_S^{'}$ is always less than $1$. Then we argue by the absurd. - If $S^{'}=S\cap F^{-1}(D(c,\eta))$, for a $S\in\mathcal{S}$, as for $R$ big enough, the map $G:S\cap G^{-1}(\{|\rho|>R\})\to\{|\rho|>R\}$ is a submersion, the map $G:S^{'}\cap G^{-1}(\{|\rho|>R\})\to\{|\rho|>R\}$ is also a submersion. Topological interpretation of the irregularity of $f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$ {#topological-interpretation-of-the-irregularity-of-f_mathcalo_mathbbc2eg} -------------------------------------------------------------------------------------- In this section, we describe the relation between the irregularity of the complex $f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$ and the fibre $f^{-1}(D^{\ast}(c,\eta))\cap g^{-1}(\rho)$ given by the lemma \[fibre\]. \[theo1\] Let $c\in\mathbb{P}^1$. For $\eta$ small enough and $|\rho|$ big enough, $$IR_c=-\chi(f^{-1}(D^{\ast}(c,\eta))\cap g^{-1}(\rho)).$$ This theorem can be proved in two steps. \[L1\] $$IR_c=-\chi(\mathbb{R}\Gamma(F^{-1}(c)\cap G^{-1}(\infty),\psi_{\frac{1}{G}}(DR(\mathcal{O}_{\mathbb{X}}[\ast D\cup F^{-1}(c)])))).$$ \[L2\] For $\eta$ small enough, $|\rho|$ big enough, $\chi(f^{-1}(D^{\ast}(c,\eta))\cap g^{-1}(\rho))$ is equal to $\chi(\mathbb{R}\Gamma(F^{-1}(c)\cap G^{-1}(\infty),\psi_{\frac{1}{G}}(DR(\mathcal{O}_{\mathbb{X}}[\ast D\cup F^{-1}(c)])))).$ This proof consists in applying the lemma \[Sab\] on irregularity of regular holonomic $\mathcal{D}$-modules twisted by an exponential and in globalizing the situation. - First, we want to prove that $$IR_c=-\chi(\mathbb{R}\Gamma(F^{-1}(c)\cap G^{-1}(\infty), IR_{F^{-1}(c)}(\mathcal{O}_{\mathbb{X}}[\ast D]e^G)^{an})).$$ According to the definition \[irral\] and the theorem \[commute\], we have: $$\begin{array}{lll} IR_c(j_+f_+(\mathcal{O}_{\mathbb{C}^2}e^g))&=& IR_c(F_+(\mathcal{O}_{\mathbb{X}}[\ast D]e^G))\\ &=&RF_{\ast}(IR_{F^{-1}(c)}(\mathcal{O}_{\mathbb{X}}[\ast D]e^G))[+1] \end{array}$$ Then, $IR_c=-\chi(\mathbb{R}\Gamma(F^{-1}(c), IR_{F^{-1}(c)}(\mathcal{O}_{\mathbb{X}}[\ast D]e^G)^{an}))$. So we have to prove that the support of $IR_{F^{-1}(c)}(\mathcal{O}_{\mathbb{X}}[\ast D]e^G)^{an}$ is included in $F^{-1}(c)\cap G^{-1}(\infty)$. Let $x\notin G^{-1}(\infty)$. Then, $G$ is holomorphic in a neighbourhood of $x$ and $(\mathcal{O}_{\mathbb{X}}[\ast D]e^G)_x^{an}$ is isomorphic to $(\mathcal{O}_{\mathbb{X}}[\ast D])_x^{an}$. According to the theorem \[groth\], $\mathcal{O}_{\mathbb{X}}$ is regular. Then $\mathcal{O}_{\mathbb{X}}[\ast D]$ is also regular and $(IR_{F^{-1}(c)}(\mathcal{O}_{\mathbb{X}}[\ast D]e^G)^{an})_x=0$. - According to the theorem \[Sab\], the complexes $IR_{F^{-1}(c)}(\mathcal{O}_{\mathbb{X}}[\ast D]e^G)^{an}$ and $\psi_{\frac{1}{G}}(DR(\mathcal{O}_{\mathbb{X}}[\ast D\cup F^{-1}(c)]))$ have the same characteristic function on $F^{-1}(c)\cap G^{-1}(\infty)$. We conclude using the following lemma: Let $X$ be an algebraic variety over $\mathbb{C}$. Let $\mathcal{F}^{\bullet}_1$ and $\mathcal{F}^{\bullet}_2$ be two constructible complexes on $X$ which have the same characteristic function on $X$. Then, $\chi(\mathbb{R}\Gamma(X,\mathcal{F}_1^{\bullet}))= \chi(\mathbb{R}\Gamma(X,\mathcal{F}_2^{\bullet}))$. We argue by induction on the dimension of $X$. 1. If the dimension of $X$ is $0$, the result is clear. 2. Assume that the lemma is true for all $X$ of dimension $<n$. Let $X$ be a complex algebraic manifold of dimension $n$ and $Z$ be a closed complex algebraic submanifold of $X$, proper to $X$ ($dim ~Z<n$) such that $\mathcal{F}_{1|X\setminus Z}$ and $\mathcal{F}_{2|X\setminus Z}$ are some local systems $\mathcal{L}_1$ and $\mathcal{L}_2$ on $X\setminus Z$. For $i=1,2$, we have the following distinguished triangle: $$\xymatrix{\mathbb{R}\Gamma_Z(\mathcal{F}^{\bullet}_i)\ar[r]& \mathcal{F}^{\bullet}_i\ar[r]& \mathbb{R}j_{\ast}j^{-1}(\mathcal{F}^{\bullet}_i)\ar[r]^-{[+1]}&},$$ where $j$ is the inclusion of $X\setminus Z$ in $X$. Then: $$\xymatrix{\mathbb{R}\Gamma(X,\mathbb{R}\Gamma_Z(\mathcal{F}^{\bullet}_i)) \ar[r]&\mathbb{R}\Gamma(X,\mathcal{F}^{\bullet}_i)\ar[r]& \mathbb{R}\Gamma(X,\mathbb{R}j_{\ast}j^{-1}(\mathcal{F}^{\bullet}_i)) \ar[r]^-{[+1]}&}.$$ - $\chi(\mathbb{R}\Gamma(X,Rj_{\ast}j^{-1}(\mathcal{F}^{\bullet}_i)))= \chi(\mathbb{R}\Gamma(X\setminus Z, \mathcal{L}_i))$. As $X$ is an algebraic variety, $X\setminus Z$ is a finite union of connected open subsets $U_j$, $j=1,\ldots,k$, of $X$. Then, $$\chi(\mathbb{R}\Gamma(X,Rj_{\ast}j^{-1}(\mathcal{F}^{\bullet}_i)))= \sum_{j=1}^k\chi(U_j)rk(\mathcal{L}_i).$$ As $\mathcal{F}^{\bullet}_1$ and $\mathcal{F}^{\bullet}_2$ have the same characteristic function, $rk(\mathcal{L}_1)= rk(\mathcal{L}_2)$. Then, $$\chi(\mathbb{R}\Gamma(X,Rj_{\ast}j^{-1}(\mathcal{F}^{\bullet}_1)))= \chi(\mathbb{R}\Gamma(X,Rj_{\ast}j^{-1}(\mathcal{F}^{\bullet}_2))).$$ - We have $\mathbb{R}\Gamma(X,\mathbb{R}\Gamma_Z(\mathcal{F}^{\bullet}_i))= \mathbb{R}\Gamma(Z,\mathbb{R}\Gamma_Z(\mathcal{F}^{\bullet}_i))$. As at the end of the proof of lemma \[Sab\], using the Verdier duality ([@duality]), we can prove that $\mathbb{R}\Gamma(Z,\mathbb{R}\Gamma_Z(\mathcal{F}^{\bullet}_i))$ and $\mathbb{R}\Gamma(X,\mathcal{F}^{\bullet}_i)$ have the same characteristic function on $Z$. Then, $\mathbb{R}\Gamma(Z,\mathbb{R}\Gamma_Z(\mathcal{F}^{\bullet}_1))$ and $\mathbb{R}\Gamma(Z,\mathbb{R}\Gamma_Z(\mathcal{F}^{\bullet}_2))$ have the same characteristic function on $Z$. As $dim~Z<n$, we apply the inductive hypothesis to obtain: $$\chi(\mathbb{R}\Gamma(Y,\mathbb{R}\Gamma_Z(\mathcal{F}^{\bullet}_1)))= \chi(\mathbb{R}\Gamma(Y,\mathbb{R}\Gamma_Z(\mathcal{F}^{\bullet}_2))).$$ Then, $$\chi(\mathbb{R}\Gamma(Y,\mathcal{F}^{\bullet}_1))= \chi(\mathbb{R}\Gamma(Y,\mathcal{F}^{\bullet}_2)).$$ Denote by $\mathcal{F}^{\bullet}$ the complex $\psi_{\frac{1}{G}}(DR(\mathcal{O}_{\mathbb{X}}[\ast D\cup F^{-1}(c)]))$. - Let $\eta_2$ small enough such that $G:G^{-1}(D^{\ast}(\infty,\eta_2))\to D^{\ast}(\infty,\eta_2)$ is a locally trivial fibration. We denote by $\widetilde{D^{\ast}(\infty,\eta_2)}$ the universal covering of $D^{\ast}(\infty,\eta_2)$. Let $(E,\pi,\widetilde{G})$ be the fiber product over $D^{\ast}(\infty,\eta_2)$ of $G^{-1}(D^{\ast}(\infty,\eta_2))$ and $\widetilde{D^{\ast}(\infty,\eta_2)}$. Then, we have the following diagram: $$\xymatrix{G^{-1}(\infty)\ar@{^(->}[r]^-j&\mathbb{X}&G^{-1}(D^{\ast}(\infty,\eta_2))\ar@{_(->}[l]_-i\ar[d]_G&E\ar[l]_-{\pi}\ar[d]\\ &&D^{\ast}(\infty,\eta_2)&\widetilde{D^{\ast}(\infty,\eta_2)}\ar[l]}$$ By definition, $\mathcal{F}^{\bullet}=j^{-1}R(i\circ\pi)_{\ast}(i\circ\pi)^{-1}(DR(\mathcal{O}_{\mathbb{X}}[\ast D\cup F^{-1}(c)]))$. - Let $\alpha:\mathbb{X}\setminus(F^{-1}(c)\cap D)\to \mathbb{X}$ open inclusion. As $\mathcal{O}_{\mathbb{X}}$ is regular in the sense of definition \[irral\] (theorem \[groth\]), $$DR(\mathcal{O}_{\mathbb{X}}[\ast D\cup F^{-1}(c)])=R\alpha_{\ast}\alpha^{-1}(\underline{\mathbb{C}}_{\mathbb{X}}).$$ - Let $Z=F^{-1}(c)\cap G^{-1}(\infty)$, $Z_{\eta_1,\eta_2}=F^{-1}(D(c,\eta_1))\cap G^{-1}(D(\infty,\eta_2))$ and $Z_{\eta_1,\rho}=F^{-1}(D(c,\eta_1))\cap G^{-1}(\rho)$. $$\begin{array}{rl} \mathbb{R}\Gamma(Z,\mathcal{F}^{\bullet})=&\underset{\eta_1,\eta_2>0}{indlim}\mathbb{R}\Gamma(Z_{\eta_1,\eta_2},R(i\circ\pi)_{\ast}(i\circ\pi)^{-1}(R\alpha_{\ast}\alpha^{-1}(\mathbb{C}_{\mathbb{X}}))\\ =& \underset{\eta_1,\eta_2>0}{indlim}\mathbb{R}\Gamma((i\circ\pi)^{-1}(Z_{\eta_1,\eta_2}),(i\circ\pi)^{-1}(R\alpha_{\ast}\alpha^{-1}(\mathbb{C}_{\mathbb{X}})) \end{array}$$ Let $\Sigma$ be a Whitney stratification associated with the constructible sheaf $R\alpha_{\ast}\alpha^{-1}(\underline{\mathbb{C}}_{\mathbb{X}})$. Then, $F^{-1}(c)$ and $D$ are union of strata. According to the proof of lemma \[fibre\], for $\eta_1$ and $\eta_2$ small enough, $G:F^{-1}(D(c,\eta))\cap G^{-1}(D^{\ast}(\infty,\eta_2))\to D^{\ast}(\infty,\eta_2)$ is a locally trivial fibration with respect to $\Sigma$. Then, there exists a homotopy equivalence $p:(i\circ\pi)^{-1}(Z_{\eta_1,\eta_2})\to Z_{\eta_1,\rho}$ compatible with $\Sigma$. Thus, while adapting the proposition I.3-4 of [@Me3] to constructible sheaves, $$\begin{array}{rl} \mathbb{R}\Gamma(Z,\mathcal{F}^{\bullet})=&\underset{\eta_1>0}{indlim}\mathbb{R}\Gamma(Z_{\eta_1,\rho},R\alpha_{\ast}\alpha^{-1}(\mathbb{C}_{\mathbb{X}}))\\ =&\underset{\eta_1>0}{indlim}\mathbb{R}\Gamma(\alpha^{-1}(Z_{\eta_1,\rho}),\alpha^{-1}(\mathbb{C}_{\mathbb{X}}))\\ =&\underset{\eta_1>0}{indlim}\mathbb{R}\Gamma(f^{-1}(D^{\ast}(c,\eta_1))\cap g^{-1}(\rho),\alpha^{-1}(\mathbb{C}_{\mathbb{X}}))\\ \end{array}$$ Then, $\chi(\mathbb{R}\Gamma(Z,\mathcal{F}^{\bullet}))=\chi(f^{-1}(D^{\ast}(c,\eta_1))\cap g^{-1}(\rho))$. When $f$ and $g$ are algebraically independant {#ind} ============================================== Let $f,g\in\mathbb{C}[x,y]$ be two polynomials which are algebraically independant. In this section, we will prove that the complex $f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$ is essentially concentrated in degree $0$. Finally we obtain a formula for the irregularity number at $c\in\mathbb{P}^1$ in terms of some geometric data associated with $f$ and $g$. The complex $f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$ is essentially concentrated in degree zero ------------------------------------------------------------------------------------------- The complex $f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$ is concentrated in degree zero except at a finite number of points. - First of all, we recall the result of F. Maaref [@Ma] about the generic fibre of the sheaf of horizontal analytic sections of $\mathcal{H}^{k-1}(f_+(\mathcal{O}_{\mathbb{C}^2}e^g))$. \[theoma\] There exists a finite subset $\Sigma$ of $\mathbb{C}$ such that for all $c\in\mathbb{C}\setminus\Sigma$ and all $\rho\in\mathbb{C}$, such that $Re(-\rho)$ is big enough, $$i_c^+\mathcal{H}^{k-1}(f_+(\mathcal{O}_{\mathbb{C}^2}e^g))\simeq H^k(f^{-1}(c),(f,g)^{-1}(c,\rho),\mathbb{C}),$$ where $i_c$ is the inclusion of $\{c\}$ in $\mathbb{C}$. - For all $c,\rho\in\mathbb{C}$, we have the long exact sequence of relative cohomology: $$\xymatrix{0\ar[r]& H^0(f^{-1}(c),(f,g)^{-1}(c,\rho),\mathbb{C})\ar[r]& H^0(f^{-1}(c),\mathbb{C})\ar[r]^-{\alpha}& H^0((f,g)^{-1}(c,\rho),\mathbb{C})\ar[dll] \\ &H^1(f^{-1}(c),(f,g)^{-1}(c,\rho),\mathbb{C})\ar[r]& H^1(f^{-1}(c),\mathbb{C})\ar[r]^-{\beta}& H^1((f,g)^{-1}(c,\rho),\mathbb{C})\ar[dll]\\ &H^2(f^{-1}(c),(f,g)^{-1}(c,\rho)),\mathbb{C})\ar[r]& 0.}$$ - We want to prove that $H^k(f^{-1}(c),(f,g)^{-1}(c,\rho))=0$ for all $k\neq 1$. As $H^1((f,g)^{-1}(c,\rho),\mathbb{C})=0$, it is enough to prove that $\alpha$ is injective. Then, it is sufficient to prove that the fibre $g^{-1}(\rho)$ intersects all the connected components of $f^{-1}(c)$. Let $(F,G):\mathbb{X}\to\mathbb{P}^1\times\mathbb{P}^1$ be a compactification of $(f,g):\mathbb{C}^2\to\mathbb{C}^2$. As $(F,G):\mathbb{X}\to\mathbb{P}^1\times\mathbb{P}^1$ is proper, we know that its image is closed in $\mathbb{P}^1\times\mathbb{P}^1$. Moreover, as $f$ and $g$ are algebraically independant, $(F,G)$ is necessarily surjective. According to the theorem of Stein factorization ([@Ha] corollary 11.5 p. 280), there exists $F^{'}:\mathbb{X}\to Y$, surjective morphism of projective varieties with connected fibres and a finite morphism $\Psi:Y\to\mathbb{P}^1$, such that $F=\psi\circ F^{'}$. As $(F,G)$ is surjective, $(F^{'},G)$ is also surjective. Then, for all $(c,\rho)\in\mathbb{P}^1\times\mathbb{P}^1$, $G^{-1}(\rho)$ intersects all the connected components of $F^{-1}(c)$. Furthermore, there exists $\Sigma\subset\mathbb{C}$ finite subset such that for all $c\in\mathbb{C}\setminus\Sigma$, the fibre $F^{-1}(c)$ is the union of $f^{-1}(c)$ with a finite number of points. Then, for a such $c$, there exists $\Sigma_c\subset\mathbb{C}$ finite subset such that for all $\rho\in \mathbb{C}\setminus\Sigma_c$, $g^{-1}(\rho)$ intersects all the connected components of $f^{-1}(c)$. Then, for all $(c,\rho)\in\mathbb{C}^2$ except a finite number, the fibre $g^{-1}(\rho)$ intersects all the connected components of $f^{-1}(c)$. - Then, according to the theorem \[theoma\], for all $c\in\mathbb{C}\setminus\Sigma$, for all $k\neq 0$, $i_c^+(\mathcal{H}^{k}(f_+(\mathcal{O}_{\mathbb{C}^2}e^g))=0$. As $\mathcal{H}^{k}(f_+(\mathcal{O}_{\mathbb{C}^2}e^g))$ is an integrable connection except at a finite number of points, we have that, for all $c\in\mathbb{C}$ except a finite number, $\mathcal{H}^{k}(f_+(\mathcal{O}_{\mathbb{C}^2}e^g))_c=0$, if $k\neq 0$. Thus, $f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$ is essentially concentrated in degree $0$. For all $c\in\mathbb{P}^1$, the complex $IR_c(j_+f_+(\mathcal{O}_{\mathbb{C}^2}e^g))$ is concentrated in degree $0$. Let $c\in\mathbb{P}^1$. Denote by $\mathcal{M}^{\bullet}$ the complex $j_+f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$. For $k\neq 0$, $\mathcal{H}^{k}(\mathcal{M}^{\bullet})$ has punctual support. Let $\eta$ be small enough such that for all $k\neq 0$, $\mathcal{H}^k(\mathcal{M}^{\bullet})_{|D(c,\eta)^{\ast}}=0$. Then, $(\mathcal{M}^{\bullet}[\ast \{c\}])_{|D(c,\eta)}=(\mathcal{H}^0(\mathcal{M}^{\bullet})[\ast\{c\}])_{|D(c,\eta)}$. Then, $IR_c(\mathcal{M}^{\bullet})=IR_c(\mathcal{H}^0(\mathcal{M}^{\bullet})[\ast\{c\}])$. Then, according to the positivity theorem \[posit\], this complex is just a vector space over $\mathbb{C}$. So, for all $k\neq 0$, $\mathcal{H}^k(IR_c(j_+f_+(\mathcal{O}_{\mathbb{C}^2}e^g)))=0$. [Remark]{} According to this corollary, the complex $IR_c(j_+f_+(\mathcal{O}_{\mathbb{C}^2}e^g))$ is entirely determined by its Euler characteristic $IR_c$. Geometrical interpretation of the irregularity ---------------------------------------------- [Notation]{} - Let $\Gamma$ be the critical locus of $(F,G)$. In the case where $f$ and $g$ are algebraically independant, this variety has dimension $1$. - Let $\Delta$ be the discriminant variety of $F$ and $G$. $\Delta$ is the image by $(F,G)$ of the curve $\Gamma$ (counted with multiplicity). $\Delta$ is an algebraic closed subset of $\mathbb{P}^1\times\mathbb{P}^1$. - Denote by $\Delta_1$ the cycle in $\mathbb{P}^1\times\mathbb{P}^1$ which is the closure in $\mathbb{P}^1\times\mathbb{P}^1$ of $\Delta\cap(\mathbb{C}^2\setminus\{c\}\times\mathbb{C})$, where $\Delta$ is counted with multiplicity. - Denote by $\Delta_2$ the cycle in $\mathbb{P}^1\times\mathbb{P}^1$ which is the closure in $\mathbb{P}^1\times\mathbb{P}^1$ of $(F,G)(D)\cap(\mathbb{C}^2\setminus\{c\}\times\mathbb{C})$, where the image is counted with multiplicity. - (the supports of $\Delta_1$ and $\Delta_2$ are two algebraic closed subsets in $\mathbb{P}^1\times\mathbb{P}^1$. They are some union of curves and points.) - For all $c\in\mathbb{P}^1$, the germs at $(c,\infty)$ of the supports of $\Delta_1$ and $\Delta_2$ are some germs of curves or are empty. We denote by $I_{(c,\infty)}(\Delta_i,\mathbb{P}^1\times\{\infty\})$ the intersection number of the cycles $\Delta_i$ and $\mathbb{P}^1\times\{\infty\}$. If the germ at $(c,\infty)$ of $\Delta_i$ is empty, this number is equal to $0$. Let $f,g\in\mathbb{C}[x,y]$ be two polynomials algebraically independants. Let $c\in\mathbb{P}^1$. $$IR_c=I_{(c,\infty)}(\Delta_1,\mathbb{P}^1\times\{\infty\})+ I_{(c,\infty)}(\Delta_2,\mathbb{P}^1\times\{\infty\}).$$ According to theorem \[theo1\], $$IR_c=-\chi(f^{-1}(D^{\ast}(c,\eta))\cap g^{-1}(\rho)).$$ We want to study the topology of the fibre $f^{-1}(D^{\ast}(c,\eta))\cap g^{-1}(\rho)$. For $|\rho|$ big enough, $G^{-1}(\rho)$ is smooth and cut transversally $D$. Then, $$\begin{array}{ll} \chi(f^{-1}(D^{\ast}(c,\eta))\cap g^{-1}(\rho))=& \chi(F^{-1}(D^{\ast}(c,\eta))\cap G^{-1}(\rho))\\ &-\chi(F^{-1}(D^{\ast}(c,\eta))\cap G^{-1}(\rho)\cap D). \end{array}$$ - We want to prove that $\chi(F^{-1}(D^{\ast}(c,\eta))\cap G^{-1}(\rho))=I_{(c,\infty)}(\Delta_1,\mathbb{P}^1\times\{\infty\})$. As $\Delta_1$ is a union of curves and points, for $\eta$ small enough and $|\rho|$ big enough, $\Delta_1\cap(D^{\ast}(c,\eta)\times\{\rho\})$ is a finite set $\{(c_1,\rho),\ldots,(c_r,\rho)\}$. Moreover, $F:F^{-1}(D^{\ast}(c,\eta))\cap G^{-1}(\rho)\to D^{\ast}(c,\eta)$ is a ramified covering. The ramified points are the points $P\in F^{-1}(D^{\ast}(c,\eta))\cap G^{-1}(\rho)\cap\Gamma$ and the ramification index at $P$ is $I_P(F^{-1}(F(P)),G^{-1}(\rho))=I_P(\Gamma, G^{-1}(\rho))+1$. Then, $$\chi(F^{-1}(D^{\ast}(c,\eta))\cap G^{-1}(\rho))= -\displaystyle\sum_{P\in F^{-1}(D^{\ast}(c,\eta))\cap G^{-1}(\rho)\cap\Gamma}I_P(\Gamma, G^{-1}(\rho)).$$ According to the projection formula for a proper map, $$\begin{array}{lll} \chi(F^{-1}(D^{\ast}(c,\eta))\cap G^{-1}(\rho))&=& \displaystyle\sum_{i=1}^rI_{(c_i,\rho)}(\Delta,\mathbb{P}^1\times\{\rho\})\\ &=&I_{(c,\infty)}(\Delta_1,\mathbb{P}^1\times\{\infty\}) \end{array}$$ - We want to prove that $\chi(F^{-1}(D^{\ast}(c,\eta))\cap G^{-1}(\rho)\cap D)=I_{(c,\infty)}(\Delta_2,\mathbb{P}^1\times\{\infty\})$. As $G^{-1}(\rho)$ is smooth and cut transversally $D$, $$\begin{array}{lll} \chi(F^{-1}(D^{\ast}(c,\eta))\cap G^{-1}(\rho)\cap D)&=& Card(F^{-1}(D^{\ast}(c,\eta))\cap G^{-1}(\rho)\cap D)\\ &=&\displaystyle\sum_{Q\in F^{-1}(D^{\ast}(c,\eta))\cap G^{-1}(\rho)\cap D}I_Q(D,G^{-1}(\rho)) \end{array}$$ According to the projection formula for a proper map, $$\begin{array}{lll} \chi(F^{-1}(D^{\ast}(c,\eta))\cap G^{-1}(\rho)\cap D)&=& \displaystyle \sum_{(c^{'},\rho)\in \Delta_2\cap(D^{\ast}(c,\eta)\times\{\rho\})}I_{(c^{'},\rho)}((F,G)(D),\mathbb{P}^1\times\{\rho\})\\ &=&I_{(c,\infty)}(\Delta_2,\mathbb{P}^1\times\{\infty\}) \end{array}$$ [Remark]{} Denote by $\tilde{\Gamma}$ the critical locus of $(f,g)$. Let $\tilde{\Delta}$ be the closure of $(f,g)(\Gamma)\setminus (\{c\}\times\mathbb{C})$ in $\mathbb{P}^1\times\mathbb{P}^1$. If $c\in\mathbb{C}$, the germ at $(c,\infty)$ of $\Delta_1$ is the germ at $(c,\infty)$ of $\tilde{\Delta}$ and the one at $(c,\infty)$ of $\Delta_2$ is empty. Then, if $c\in\mathbb{C}$, $$IR_c=I_{(c,\infty)}(\tilde{\Delta},\mathbb{P}^1\times\{\infty\}).$$ We deduce this remark from the following lemma: Let $c\in\mathbb{C}$. For $\eta>0$ small enough and $R>0$ big enough, $F^{-1}(D^{\ast}(c,\eta))\cap G^{-1}(\rho)$ does not intersect $D$, for $|\rho|>R$. We recall that $F,G:\mathbb{X}\to\mathbb{P}^1$ are constructed in the following way. We consider an immersion of $\mathbb{C}^2$ in $\mathbb{P}^2$. We construct a rational map $(\tilde{f},\tilde{g}):\mathbb{P}^2-->\mathbb{P}^1$ which extend $(f,g)$. On $\mathbb{P}^2\setminus\mathbb{C}^2$, $(\tilde{f},\tilde{g})$ takes the value $(\infty,\infty)$ or is not well defined. Then, we lift the indeterminacies of $(\tilde{f},\tilde{g})$ after a finite number of blowing ups. Then, according to [@LeWe], we know that $F^{-1}(\infty)$ and $G^{-1}(\infty)$ are connected. As $F^{-1}(\infty)$ and $G^{-1}(\infty)$ have a non empty intersection, $F^{-1}(\infty)\cup G^{-1}(\infty)$ is connected. Now let $Z$ be an irreducible component of $D$. We want to prove that for $\eta$ small enough and $R$ big enough, $F^{-1}(D^{\ast}(c,\eta))\cap G^{-1}(\rho)\cap Z=\emptyset$. If it is not true, we can construct a sequence $(x_n)_{n\in\mathbb{N}}$ such that $x_n\in Z$, $0<|F(x_n)-c|<\frac{1}{n}$ and $|G(x_n)|=\rho_n$, with $\lim_{n\to +\infty}\rho_n=\infty$. Then there exists a point $x\in F^{-1}(c)\cap G^{-1}(\infty)\cap Z$. As $Z\simeq \mathbb{P}^1$, $F_{|Z}$ and $G_{|Z}$ are necessarily surjective or constant. The existence of the sequence $(x_n)_{n\in\mathbb{N}}$ allows us to conclude that $F_{|Z}$ and $G_{|Z}$ are necessarily surjective. Then there exists another point $y\in Z\cap F^{-1}(\infty)$, $y\neq x$. This contradicts the facts that $F^{-1}(\infty)\cup G^{-1}(\infty)$ is connected and $F_{|Z}$ is surjective. When $f$ and $g$ are algebraically dependant {#dep} ============================================ Let $f,g\in\mathbb{C}[x,y]$ be two polynomials which are algebraically dependant. Then, the complex $f_+(\mathcal{O}_{\mathbb{C}^2}e^g)$ is not necessarily concentrated in degree $0$. However, we give a formula for the irregularity number $IR_c$ at $c\in\mathbb{P}^1$. Let $(F,G):\mathbb{X}\to\mathbb{P}^1\times\mathbb{P}^1$ be a compactification of the map $(f,g):\mathbb{C}^2\to\mathbb{C}^2$. As $f$ and $g$ are algebraically dependant, $\tilde{\Delta}=im~(F,G)$ is a closed subvariety of $\mathbb{P}^1\times\mathbb{P}^1$. - Let $\Delta$ be the cycle which is the closure in $\mathbb{P}^1\times\mathbb{P}^1$ of $\tilde{\Delta}\cap(\mathbb{C}^2\setminus\{c\}\times\mathbb{C})$. In a neighbourhood of $(c,\infty)$, $\Delta$ is a curve or is empty. We denote by $I_{(c,\infty)}(\Delta,\mathbb{P}^1\times\{\infty\})$ the intersection number of the cycles $\Delta$ and $\mathbb{P}^1\times\{\infty\}$. If the germ at $(c,\infty)$ of $\Delta$ is empty, this number is equal to $0$. - Let $F$ be the generic fiber of $(f,g):\mathbb{C}^2\to im~(f,g)$. Let $f,g\in\mathbb{C}[x,y]$ be two polynomials algebraically dependants. Let $c\in\mathbb{P}^1$. $$IR_c=-\chi(F)*I_{(c,\infty)}(\Delta,\mathbb{P}^1\times\{\infty\}).$$ According to the theorem \[theo1\], $IR_c=-\chi(f^{-1}(D^{\ast}(c,\eta))\cap g^{-1}(\rho))$, where $\eta$ is small enough and $|\rho|$ is big enough. As $\Delta$ is a curve or is empty in a neighbouhood of $(c,\infty)$, $\Delta\cap(D^{\ast}(c,\eta)\times\{\rho\})$ is a finite union of points $(c_1,\rho),\ldots,(c_r,\rho)$. Then, $f^{-1}(D^{\ast}(c,\eta))\cap g^{-1}(\rho)=\bigcup_{i=1}^r(f,g)^{-1}(c_i,\rho)$. We conclude that $\chi(f^{-1}(D^{\ast}(c,\eta))\cap g^{-1}(\rho))=\chi(F)*I_{(c,\infty)}(\Delta,\mathbb{P}^1\times\{\infty\})$. [99]{} P. Deligne, *Équations différentielles à points singuliers réguliers*, Lecture Notes in Math. **163**, Springer-Verlag, (1970). A. Grothendieck, *On the De Rham Cohomology of Algebraic varieties*, Publ. Math. I.H.É.S. **29**, (1966), 93-103. R. Hartshorne, *Algebraic geometry*, Graduate Texts in Math., **52**, Springer-Verlag, New York, (1977). M. Kashiwara, *On the maximally overdetermined systems of differential equations*, Publ. R.I.M.S. **10**, (1975), p. 563-579. Lê Dũng Trãng, C. Weber, *Polynômes à fibres rationnelles et conjecture jacobienne à $2$ variables*, C. R. Acad. Sci. Paris, **320 I**, (1995), 581-584. F. Maaref, *Sur un analogue irrégulier de la connexion de Gauss-Manin*, Ann. Fac. Sc. Toul., **VIII(1)**, (1999), 117-124. Z. Mebkhout, *Le théorème de comparaison entre cohomologies de De Rham d’une variété algébrique complexe et le théorème d’existence de Riemann*, Publ. Math. I.H.É.S. **69**, (1989), p. 47-89. Z. Mebkhout, *Le théorème de positivité de l’irrégularité pour les $\mathcal{D}_X$-modules*, Grothendieck festschrift **III**, Progress in Math. **88**, (1990), 84-131. Z. Mebkhout, L. Narváez-Macarro, *Le Théorème de Constructibilité de Kashiwara*, in images directes et constructibilité, Les cours du CIMPA, (1993), p. 47-98. Z. Mebkhout, *Le théorème de positivité, le théorème de comparaison, le théorème d’existence de Riemann*, séminaire et congrès **8**, (2004), 165-307. C. Sabbah, *On the comparison theorem for elementary irregular $\mathcal{D}$-modules*, Nagoya J. Math., **141** (1996), 107-124. *Dualité de Poincaré*, Séminaire Hedelberg-Strasbourg 1966-67, Publ. I.R.M.A. **3**, Strasbourg, (1969).

--- abstract: | We derive necessary and sufficient conditions for periodic and for elliptic periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlining elliptic curve. We provide several examples of periodic and elliptic periodic trajectories with small periods. We observe relationship between Cayley-type conditions and discriminantly separable and factorizable polynomials. Equivalent conditions for periodicity and elliptic periodicity are derived in terms of polynomial-functional equations as well. The corresponding polynomials are related to the classical extremal polynomials. In particular, the light-like periodic trajectories are related to the classical Chebyshev polynomials. The similarities and differences with respect to previously studied Euclidean case are indicated. <span style="font-variant:small-caps;">MSC2010 numbers</span>: `14H70, 41A10, 70H06, 37J35, 26C05` <span style="font-variant:small-caps;">Keywords</span>: Minkowski plane, relativistic ellipses and hyperbolas, elliptic billiards, periodic and elliptic periodic trajectories, extremal polynomials, Chebyshev polynomials, Akhiezer polynomials, discriminantly separable polynomials author: - Anani Komla Adabrah - Vladimir Dragović - Milena Radnović title: Periodic billiards within conics in the Minkowski plane and Akhiezer polynomials --- Introduction {#sec:intro} ============ Billiards within quadrics in pseudo-Euclidean spaces were studied in [@KhTab2009; @DragRadn2012adv; @DragRadn2013publ]. In [@DragRadn2018; @DragRadn2019rcd], the relationship between the billiards within quadrics in the Euclidean spaces and extremal polynomials has been studied. The aim of this paper is to develop the connection between extremal polynomials and billiards in the Minkowski plane. This paper is organised as follows. In Section \[sec:confocal\], we recall the basic notions connected with the Minkowski plane, confocal families of conics, relativistic ellipses and hyperbolas, and billiards. In Section \[sec:periodic\], we give a complete description of the periodic billiard trajectories in algebro-geometric terms. In Section \[sec:small\], we use the conditions obtained in the previous section to study examples of periodic trajectories with small periods. We also emphasize intriguing connection between the Cayley-type conditions and discriminantly separable polynomials. The notion of relativistic ellipses and hyperbolas enables definition of Jacobi-type elliptic coordinates in the Minkowski setting. Since the correspondence between Cartesian and elliptic coordinates is not one-to-one, there is a notion of elliptic periodicity which refers to a weaker assumption that a trajectory is periodic in elliptic coordinates. In Section \[sec:elliptic\], we provide algebro-geometric characterization of trajectories to be $n$-elliptic periodic without being $n$-periodic. Section \[sec:examples-elliptic\] provides examples and connections with discriminantly separable polynomials. In Section \[sec:polynomial\], we derive a characterisation of elliptic periodic trajectories using polynomial equations. In the last Section \[sec:extremal\], we establish the connection between characteristics of periodic billiard trajectories and extremal polynomials: the Zolotarev polynomials, the Akhiezer polynomials on symmetric intervals, and the general Akhiezer polynomials on two intervals. We conclude our study of the relationship of billiards in the Minkowski plane with the extremal polynomials by relating the case of light-like trajectories to the classical Chebyshev polynomials, see Section \[sec:ll\]. Apart from similarities with previously studied Euclidean spaces, see [@DragRadn2018; @DragRadn2019rcd], there are also significant differences: for example, among the obtained extremal polynomials are such with winding numbers $(3,1)$, which was never the case in the Euclidean setting. Confocal families of conics and billiards {#sec:confocal} ========================================= #### The Minkowski plane {#the-minkowski-plane .unnumbered} is $\mathbf{R}^2$ with *the Minkowski scalar product*: $\langle X,Y\rangle=X_1Y_1-X_2Y_2$. *The Minkowski distance* between points $X$, $Y$ is $ \dist(X,Y)=\sqrt{\langle{X-Y,X-Y}\rangle}. $ Since the scalar product can be negative, notice that the Minkowski distance can have imaginary values as well. In that case, we choose the value of the square root with the positive imaginary part. Let $\ell$ be a line in the Minkowski plane, and $v$ its vector. $\ell$ is called *space-like* if $\langle{v,v}\rangle>0$; *time-like* if $\langle{v,v}\rangle<0$; and *light-like* if $\langle{v,v}\rangle=0$. Two vectors $x$, $y$ are *orthogonal* in the Minkowski plane if $\langle x,y \rangle=0$. Note that a light-like vector is orthogonal to itself. #### Confocal families. {#confocal-families. .unnumbered} Denote by $$\label{eq:ellipse} {\pazocal{E}}\ :\ \frac{\mathsf{x}^2}{a}+\frac{\mathsf{y}^2}{b}=1$$ an ellipse in the plane, with $a$, $b$ being fixed positive numbers. The associated family of confocal conics is: $$\label{eq:confocal.conics} {\pazocal{C}}_{\lambda}\ :\ \frac{\mathsf{x}^2}{a-\lambda}+\frac{\mathsf{y}^2}{b+\lambda}=1, \quad \lambda\in\mathbf{R}.$$ The family is shown on Figure \[fig:confocal.conics\]. (.7,0) arc (0:60:0.7cm and 1.2cm); (.35,1.05) arc (60:120:0.7cm and 1.2cm); (-.35,1.05) arc (120:180:0.7cm and 1.2cm); (-.7,0) arc (180:240:0.7cm and 1.2cm); (-.35,-1.05) arc (240:300:0.7cm and 1.2cm); (.35,-1.05) arc (300:360:0.7cm and 1.2cm); (1.325,-.01) arc (0:30:1.32287565553cm and .5cm); (1.14,.25) arc (30:150:1.32287565553cm and .5cm); (-1.14,.25) arc (150:210:1.32287565553cm and .5cm); (-1.14,-.25) arc (210:330:1.32287565553cm and .5cm); (1.14,-.25) arc (330:360:1.32287565553cm and .5cm); (1.13,0) arc (0:30:1.1401754251cm and 0.83666002653 cm); (.99,.41) arc (30:150:1.1401754251cm and 0.83666002653 cm); (-.99,.41) arc (150:210:1.1401754251cm and 0.83666002653 cm); (-.99,-.41) arc (210:330:1.1401754251cm and 0.83666002653 cm); (.99,-.41) arc (330:357:1.1401754251cm and 0.83666002653 cm); (2.61,-1.2) – (-1.2,2.61); (2.61,1.2) – (-1.2,-2.61); (-2.61,1.2) – (1.2,-2.61); (-2.61,-1.2) – (1.2,2.61); plot (,[sqrt(1+2.5)\*sqrt(1-/(1-2.5))]{}); plot (,[sqrt(1+2.5)\*sqrt(1-/(1-2.5))]{}); plot (,[sqrt(1+2.5)\*sqrt(1-/(1-2.5))]{}); plot (,[-sqrt(1+2.5)\*sqrt(1-/(1-2.5))]{}); plot (,[-sqrt(1+2.5)\*sqrt(1-/(1-2.5))]{}); plot (,[-sqrt(1+2.5)\*sqrt(1-/(1-2.5))]{}); plot (,[sqrt(1+1.5)\*sqrt(1-/(1-1.5))]{}); plot (,[sqrt(1+1.5)\*sqrt(1-/(1-1.5))]{}); plot (,[sqrt(1+1.5)\*sqrt(1-/(1-1.5))]{}); plot (,[-sqrt(1+1.5)\*sqrt(1-/(1-1.5))]{}); plot (,[-sqrt(1+1.5)\*sqrt(1-/(1-1.5))]{}); plot (,[-sqrt(1+1.5)\*sqrt(1-/(1-1.5))]{}); plot ([sqrt(1+2.5)\*sqrt(1-/(1-2.5))]{},); plot ([sqrt(1+2.5)\*sqrt(1-/(1-2.5))]{},); plot ([sqrt(1+2.5)\*sqrt(1-/(1-2.5))]{},); plot ([-sqrt(1+2.5)\*sqrt(1-/(1-2.5))]{},); plot ([-sqrt(1+2.5)\*sqrt(1-/(1-2.5))]{},); plot ([-sqrt(1+2.5)\*sqrt(1-/(1-2.5))]{},); plot ([sqrt(1+1.5)\*sqrt(1-/(1-1.5))]{},); plot ([sqrt(1+1.5)\*sqrt(1-/(1-1.5))]{},); plot ([sqrt(1+1.5)\*sqrt(1-/(1-1.5))]{},); plot ([-sqrt(1+1.5)\*sqrt(1-/(1-1.5))]{},); plot ([-sqrt(1+1.5)\*sqrt(1-/(1-1.5))]{},); plot ([-sqrt(1+1.5)\*sqrt(1-/(1-1.5))]{},); We may distinguish the following three subfamilies in the family [eq:confocal.conics]{}: for $\lambda\in(-b,a)$, conic ${\pazocal{C}}_{\lambda}$ is an ellipse; for $\lambda<-b$, conic ${\pazocal{C}}_{\lambda}$ is a hyperbola with $\mathsf{x}$-axis as the major one; for $\lambda>a$, it is a hyperbola again, but now its major axis is $\mathsf{y}$-axis. In addition, there are three degenerated quadrics: ${\pazocal{C}}_{a}$, ${\pazocal{C}}_{b}$, ${\pazocal{C}}_{\infty}$ corresponding to $\mathsf{y}$-axis, $\mathsf{x}$-axis, and the line at the infinity respectively. The confocal family has three pairs of foci: $F_1(\sqrt{a+b},0)$, $F_2(-\sqrt{a+b},0)$; $G_1(0,\sqrt{a+b})$, $G_2(0,-\sqrt{a+b})$; and $H_1(1:-1:0)$, $H_2(1:1:0)$ on the line at the infinity. We notice four distinguished lines: $$\begin{aligned} &\mathsf{x}+\mathsf{y}=\sqrt{a+b},\quad \mathsf{x}+\mathsf{y}=-\sqrt{a+b},\\ &\mathsf{x}-\mathsf{y}=\sqrt{a+b},\quad \mathsf{x}-\mathsf{y}=-\sqrt{a+b}.\end{aligned}$$ These lines are common tangents to all conics from the family. Conics in the Minkowski plane have geometric properties analogous to the conics in the Euclidean plane. Namely, for each point on conic ${\pazocal{C}}_{\lambda}$, either sum or difference of its Minkowski distances from the foci $F_1$ and $F_2$ is equal to $2\sqrt{a-\lambda}$; either sum or difference of the distances from the other pair of foci $G_1$, $G_2$ is equal to $2\sqrt{-b-\lambda}$ [@DragRadn2012adv]. In the Minkowkski plane, it is natural to consider relativistic conics, which are suggested in [@BirkM1962]. In this section, we give a brief account of the related analysis. Consider points $F_1(\sqrt{a+b},0)$ and $F_2(-\sqrt{a+b},0)$. For a given constant $c\in\mathbf{R}^{+}\cup i\mathbf{R}^{+}$, *a relativistic ellipse* is the set of points $X$ satisfying $ \dist(F_1,X)+\dist(F_2,X)=2c, $ while *a relativistic hyperbola* is the union of the sets given by the following equations: $$\begin{gathered} \dist(F_1,X)-\dist(F_2,X)=2c,\\ \dist(F_2,X)-\dist(F_1,X)=2c.\end{gathered}$$ Relativistic conics can be described as follows. $0<c<\sqrt{a+b}$ : The corresponding relativistic conics lie on ellipse ${\pazocal{C}}_{a-c^2}$ from family [eq:confocal.conics]{}. The ellipse ${\pazocal{C}}_{a-c^2}$ is split into four arcs by touching points with the four common tangent lines; thus, the relativistic ellipse is the union of the two arcs intersecting the $\mathsf{y}$-axis, while the relativistic hyperbola is the union of the other two arcs. $c>\sqrt{a+b}$ : The relativistic conics lie on ${\pazocal{C}}_{a-c^2}$ – a hyperbola with $\mathsf{x}$-axis as the major one. Each branch of the hyperbola is split into three arcs by touching points with the common tangents; thus, the relativistic ellipse is the union of the two finite arcs, while the relativistic hyperbola is the union of the four infinite ones. $c$ is imaginary : The relativistic conics lie on hyperbola ${\pazocal{C}}_{a-c^2}$ – a hyperbola with $\mathsf{y}$-axis as the major one. As in the previous case, the branches are split into six arcs in total by common points with the four tangents. The relativistic ellipse is the union of the four infinite arcs, while the relativistic hyperbola is the union of the two finite ones. Notice that all relativistic ellipses are disjoint with each other, as well as all relativistic hyperbolas, see Figure \[fig:confocal.conics\]. Moreover, at the intersection point of a relativistic ellipse which is a part of the geometric conic ${\pazocal{C}}_{\lambda_1}$ from the confocal family [eq:confocal.conics]{} and a relativistic hyperbola belonging to ${\pazocal{C}}_{\lambda_2}$, it is always $\lambda_1<\lambda_2$. #### Elliptic coordinates. {#elliptic-coordinates. .unnumbered} Each point inside ellipse ${\pazocal{E}}$ has elliptic coordinates $(\lambda_1,\lambda_2)$, such that $-b<\lambda_1<0<\lambda_2<a$. The differential equation of the lines touching a given conic ${\pazocal{C}}_{\gamma}$ is: $$\label{eq:diff-eq} \frac{d\lambda_1}{\sqrt{(a-\lambda_1)(b+\lambda_1)(\gamma-\lambda_1)}} + \frac{d\lambda_2}{\sqrt{(a-\lambda_2)(b+\lambda_2)(\gamma-\lambda_2)}}=0.$$ #### Billiards. {#billiards. .unnumbered} Let $v$ be a vector and $p$ a line in the Minkowski plane. Decompose vector $v$ into the sum $v=a+n_{p}$ of a vector $n_{p}$ orthogonal to $p$ and $a$ belonging to $p$. Then vector $v'=a-n_{p}$ is *the billiard reflection* of $v$ on $p$. It is easy to see that $v$ is also the billiard reflection of $v'$ with respect to $p$. Moreover, since $\langle{v,v}\rangle=\langle{v',v'}\rangle$, vectors $v$, $v'$ are of the same type. Note that $v=v'$ if $v$ is contained in $p$ and $v'=-v$ if it is orthogonal to $p$. If $n_{p}$ is light-like, which means that it belongs to $p$, then the reflection is not defined. Line $\ell'$ is *the billiard reflection* of $\ell$ off ellipse ${\pazocal{E}}$ if their intersection point $\ell\cap\ell'$ belongs to ${\pazocal{E}}$ and the vectors of $\ell$, $\ell'$ are reflections of each other with respect to the tangent line of ${\pazocal{E}}$ at this point. The lines containing segments of a given billiard trajectory within ${\pazocal{E}}$ are all of the same type: they are all either space-like, time-like, or light-like. For the detailed explanation, see [@KhTab2009]. Billiard trajectories within ellipses in the Minkowski plane have caustic properties: each segment of a given trajectory will be tangent to the same conic confocal with the boundary, see [@DragRadn2012adv]. More about Minkowski plane or higher-dimensional pseudo-Euclidean spaces and related integrable systems can be found in [@BirkM1962; @GKT2007; @WFSWZZ2009; @JJ1; @JJ2]. Periodic trajectories {#sec:periodic} ===================== Sections \[sec:periodic\]–\[sec:extremal\] deal with the trajectories with non-degenerate caustic ${\pazocal{C}}_{\gamma}$, which will mean that $\gamma\in\mathbf{R}\setminus\{-b,a\}$. Such trajectories are either space-like or time-like. The case of light-like trajectories, which correspond to the degenerate caustic ${\pazocal{C}}_{\infty}$ is considered separately, in Section \[sec:ll\]. The periodic trajectories of elliptical billiards in the Minkowski plane can be characterized in algebro-geometric terms using the underlying elliptic curve: \[th:curve-billiard\] The billiard trajectories within ${\pazocal{E}}$ with non-degenerate caustic ${\pazocal{C}}_{\gamma}$ are $n$-periodic if and only if $nQ_{0}\sim nQ_{\gamma}$ on the elliptic curve: $$\label{eq:billiard-cubic} {\mathcal{C}}\ :\ y^2=\varepsilon(a-x)(b+x)(\gamma-x),$$ with $Q_0$ being a point of ${\mathcal{C}}$ corresponding to $x=0$, $Q_{\gamma}$ to $x=\gamma$, and $\varepsilon=\sign\gamma$. Along a billiard trajectory within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\gamma}$, the elliptic coordinate $\lambda_1$ traces the segment $[\alpha_1,0]$, and $\lambda_2$ the segment $[0,\beta_1]$, where $\alpha_1$ is the largest negative and $\beta_1$ the smallest positive member of the set $\{a,-b,\gamma\}$. *Case 1.* If ${\pazocal{C}}_{\gamma}$ is an ellipse and $\gamma<0$, then $\alpha_1=\gamma$, $\beta_1=a$. The coordinate $\lambda_1$ takes value $\lambda_1=\gamma$ at the touching points with the caustic and value $\lambda_1=0$ at the reflection points off the arcs of ${\pazocal{E}}$ where the restricted metric is time-like. On the other hand, $\lambda_2$ takes value $\lambda_2=a$ at the intersections with $\mathsf{y}$-axis, and $\lambda_2=0$ at the reflection points off the arcs of ${\pazocal{E}}$ where the restricted metric is space-like. *Case 2.* If ${\pazocal{C}}_{\gamma}$ is an ellipse and $\gamma>0$, then $\alpha_1=-b$, $\beta_1=\gamma$. The coordinate $\lambda_1$ takes value $\lambda_1=-b$ at the intersections with $\mathsf{x}$-axis and value $\lambda_1=0$ at the reflection points off the arcs of ${\pazocal{E}}$ where the restricted metric is time-like. On the other hand, $\lambda_2$ takes value $\lambda_2=\gamma$ at the touching points with the caustic, and $\lambda_2=0$ at the reflection points off the arcs of ${\pazocal{E}}$ where the restrictes metric is space-like. *Case 3.* If ${\pazocal{C}}_{\gamma}$ is a hyperbola, then $\alpha_1=-b$, $\beta_1=a$. The coordinate $\lambda_1$ takes value $\lambda_1=-b$ at the intersections with $\mathsf{x}$-axis and value $\lambda_1=0$ at the reflection points off the arcs of ${\pazocal{E}}$ where the restricted metric is time-like. On the other hand, $\lambda_2$ takes value $\lambda_2=a$ at the intersections with $\mathsf{y}$-axis, and $\lambda_2=0$ at the reflection points off the arcs of ${\pazocal{E}}$ where the restricted metric is space-like. In each case, the elliptic coordinates change monotonously between their extreme values. Consider an $n$-periodic billiard trajectory and denote by $n_1$ the number of reflections off time-like arcs, i.e. off relativistic ellipses, and by $n_2$ the number of reflections off space-like arcs, i.e. relativistic hyperbolas. Obviously, $n_1+n_2=n$. Integrating along the trajectory, we get: $$\label{InteEqn} n_1\int_{\alpha_1}^{0}\frac{d\lambda_1}{\sqrt{\varepsilon(a-\lambda_1)(b+\lambda_1)(\gamma-\lambda_1)}} + n_2\int_{\beta_1}^{0}\frac{d\lambda_2}{\sqrt{\varepsilon(a-\lambda_2)(b+\lambda_2)(\gamma-\lambda_2)}} =0,$$ i.e. $$n_1(Q_0-Q_{\alpha_1})+n_2(Q_0-Q_{\beta_1})\sim0.$$ In Case 1, this is equivalent to $$n_1(Q_0-Q_{\gamma})+n_2(Q_0-Q_{a})\sim n(Q_0-Q_{\gamma}),$$ since a closed trajectory crosses the $\mathsf{y}$-axis even number of times, i.e $n_2$ must be even, and $2Q_{a}\sim2Q_{\gamma}$. Similarly, in Case 2, it follows since $n_1$ is even, and in Case 3 both $n_1$ and $n_2$ need to be even. From the proof of Theorem \[th:curve-billiard\], we have: The period of a closed trajectory with hyperbola as caustic is even. \[th:cayley-billiard\] The billiard trajectories within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\gamma}$ are $n$-periodic if and only if: $$\begin{gathered} C_2=0, \quad \left| \begin{array}{cc} C_2 & C_3 \\ C_3 & C_4 \end{array} \right|=0, \quad \left| \begin{array}{ccc} C_2 & C_3 & C_4 \\ C_3 & C_4 & C_5 \\ C_4 & C_5 & C_6 \end{array} \right|=0, \dots \quad\text{for}\quad n=3,5,7,\dots \\ B_3=0, \quad \left| \begin{array}{cc} B_3 & B_4 \\ B_4 & B_5 \end{array} \right|=0, \quad \left| \begin{array}{ccc} B_3 & B_4 & B_5 \\ B_4 & B_5 & B_6 \\ B_5 & B_6 & B_7 \end{array} \right|=0, \dots \quad\text{for}\quad n=4,6,8,\dots. \end{gathered}$$ Here, we denoted: $$\begin{gathered} \sqrt{\varepsilon(a-x)(b+x)(\gamma-x)}=B_0+B_1x+B_2x^2+\dots, \\ \frac{\sqrt{\varepsilon(a-x)(b+x)(\gamma-x)}}{\gamma-x}=C_0+C_1x+C_2x^2+\dots, \end{gathered}$$ the Taylor expansions around $x=0$. Denote by $Q_{\infty}$ the point of ${\mathcal{C}}$ [eq:billiard-cubic]{} corresponding to $x=\infty$ and notice that $$\label{eq:2Q} 2Q_{\gamma}\sim2 Q_{\infty}.$$ Consider first $n$ even. Because of [eq:2Q]{}, the condition $nQ_{0}\sim nQ_{\gamma}$ is equivalent to $nQ_{0}\sim nQ_{\infty}$, which is equivalent to the existence of a meromorphic function of ${\mathcal{C}}$ with the unique pole at $Q_{\infty}$ and unique zero at $Q_{0}$, such that the pole and the zero are both of multiplicity $n$. The basis of $\mathcal{L}(nQ_{\infty})$ is: $$\label{eq:basis-even} 1,x,x^2,\dots,x^{n/2},y,xy, x^{n/2-2}y,$$ thus a non-trivial linear combination of those functions with a zero of order $n$ at $x=0$ exists if and only if: $$\left| \begin{array}{llll} B_{n/2+1} & B_{n/2} & \dots & B_3\\ B_{n/2+2} & B_{n/2+1} &\dots & B_4\\ \dots\\ B_{n-1} & B_n &\dots & B_{n/2+1} \end{array} \right|=0.$$ Now, suppose $n$ is odd. Because of [eq:2Q]{}, the condition $nQ_{0}\sim nQ_{\gamma}$ is equivalent to $nQ_{0}\sim (n-1)Q_{\infty}+Q_{\gamma}$, which is equivalent to the existence of a meromorphic function of ${\mathcal{C}}$ with only two poles: of order $n-1$ at $Q_{\infty}$ and a simple pole at $Q_{\gamma}$, and unique zero at $Q_{0}$. The basis $\mathcal{L}( (n-1)Q_{\infty}+Q_{\gamma})$ is: $$\label{eq:basis-odd} 1,x,x^2,\dots,x^{(n-1)/2},\frac{y}{\gamma-x}, \frac{xy}{\gamma-x}, \dots, \frac{x^{(n-1)/2-1}y}{\gamma-x},$$ thus a non-trivial linear combination of those functions with a zero of order $n$ at $x=0$ exists if and only if: $$\left| \begin{array}{llll} C_{(n-1)/2+1} & C_{(n-1)/2} & \dots & C_2\\ C_{(n-1)/2+2} & C_{(n-1)/2+1} &\dots & C_3\\ \dots\\ C_{n-1} & C_n &\dots & C_{(n-1)/2+1} \end{array} \right|=0.$$ Trajectories with small periods and discriminantly separable polynomials {#sec:small} ======================================================================== Examples of periodic trajectories: $3\le n\le8$ {#sec:examples-table} ----------------------------------------------- ### 3-periodic trajectories {#periodic-trajectories .unnumbered} There is a $3$-periodic trajectory of the billiard within [eq:ellipse]{}, with a non-degenerate caustic ${\pazocal{C}}_{\gamma}$ in the Minkowski plane if and only if, according to Theorem [th:cayley-billiard]{}, the caustic is an ellipse, i.e. $\gamma \in (-b,a)$ and $C_2=0$. We solve the equation $$\label{eqn:3-periodic} C_2 = \dfrac{3a^{2}b^{2}+2ab(a-b)\gamma-(a+b)^{2}\gamma^{2} } {8(ab)^{3/2}\gamma^{5/2}}=0,$$ which yields the following two solutions for the parameter $\gamma$ for the caustic: $$\label{CaleyEq1} {\gamma}_{1}= \dfrac{ab}{(a+b)^{2}}(a-b+2\sqrt{a^{2}+ab+b^{2}}), \quad {\gamma}_{2}= -\dfrac{ab}{(a+b)^{2}}(-a+b+2\sqrt{a^{2}+ab+b^{2}}).$$ Notice that both caustics ${\pazocal{C}}_{{\gamma}_{2}}$ and ${\pazocal{C}}_{{\gamma}_{1}}$ are ellipses since $-b<{\gamma}_{2}<0<{\gamma}_{1}<a$. Two examples of a $3$-periodic trajectories are shown in Figure \[fig:3-periodic\]. (0,0) circle \[x radius=[sqrt(3)]{}, y radius=[sqrt(2)]{}\]; (0,0) circle \[x radius=[sqrt(3-2.332)]{}, y radius=[sqrt(2+2.332)]{}\]; ([sqrt(5)+0.5]{},-0.5) – (-0.5,[sqrt(5)+0.5]{}); ([-sqrt(5)-0.5]{},0.5) – (0.5,[-sqrt(5)-0.5]{}); (0.5,[sqrt(5)+0.5]{}) – ([-sqrt(5)-0.5]{},-0.5); (-0.5,[-sqrt(5)-0.5]{}) – ([sqrt(5)+0.5]{},0.5); (.9, -1.208) – (.7499, 1.27) – (1.709, -.229) – (.9, -1.208); (0,0) circle \[x radius=[sqrt(7)]{}, y radius=[sqrt(5)]{}\]; (0,0) circle \[x radius=[sqrt(7+4.589)]{}, y radius=[sqrt(5-4.589)]{}\]; ([sqrt(12)+0.5]{},-0.5) – (-0.5,[sqrt(12)+0.5]{}); ([-sqrt(12)-0.5]{},0.5) – (0.5,[-sqrt(12)-0.5]{}); (0.5,[sqrt(12)+0.5]{}) – ([-sqrt(12)-0.5]{},-0.5); (-0.5,[-sqrt(12)-0.5]{}) – ([sqrt(12)+0.5]{},0.5); (1, -2.07) – (2.433, -.880) – (-2.540, -.4935) – (1, -2.07); ### 4-periodic trajectories {#periodic-trajectories-1 .unnumbered} There is a $4$-periodic trajectory of the billiard within [eq:ellipse]{}, with a non-degenerate caustic ${\pazocal{C}}_{\gamma}$ in the Minkowski plane if and only if $B_{3}=0$. We solve the equation $$\label{eqn:4-periodic} B_{3}=-\dfrac{(ab+a\gamma+b\gamma)(ab+a\gamma-b\gamma)(ab-a\gamma-b\gamma)}{16(ab\gamma)^{5/2}}=0,$$ which yields the following solutions for the parameter $\gamma$ for the caustic $$\label{4pCaustic} {\gamma}_{1}=-\dfrac{ab}{a+b},\qquad {\gamma}_{2}=-\dfrac{ab}{a-b}, \qquad {\gamma}_{3}=\dfrac{ab}{a+b}.$$ Since $\gamma_{1}\in(-b,0)$, ${\gamma}_{3} \in (0,a)$ and ${\gamma}_{2} \notin (-b,a)$, therefore conic ${\pazocal{C}}_{{\gamma}_{2}}$ is a hyperbola whereas conics ${\pazocal{C}}_{{\gamma}_{1}}$ and ${\pazocal{C}}_{{\gamma}_{3}}$ are ellipses. In Figures \[fig:4-periodic\] and \[fig:4-periodich\], examples of a $4$-periodic trajectories with each type of caustic are shown. (0,0) circle \[x radius=[sqrt(2)]{}, y radius=[sqrt(4)]{}\]; (0,0) circle \[x radius=[sqrt(2-1.333)]{}, y radius=[sqrt(4+1.333)]{}\]; ([sqrt(6)+0.5]{},-0.5) – (-0.5,[sqrt(6)+0.5]{}); ([-sqrt(6)-0.5]{},0.5) – (0.5,[-sqrt(6)-0.5]{}); (0.5,[sqrt(6)+0.5]{}) – ([-sqrt(6)-0.5]{},-0.5); (-0.5,[-sqrt(6)-0.5]{}) – ([sqrt(6)+0.5]{},0.5); (1.1, -1.257) – (.6064, 1.807) – (1.099, 1.259) – (.6069, -1.807) – (1.1, -1.257); (0,0) circle \[x radius=[sqrt(9)]{}, y radius=[sqrt(3)]{}\]; (0,0) circle \[x radius=[sqrt(9+2.250)]{}, y radius=[sqrt(3-2.250)]{}\]; ([sqrt(12)+0.5]{},-0.5) – (-0.5,[sqrt(12)+0.5]{}); ([-sqrt(12)-0.5]{},0.5) – (0.5,[-sqrt(12)-0.5]{}); (0.5,[sqrt(12)+0.5]{}) – ([-sqrt(12)-0.5]{},-0.5); (-0.5,[-sqrt(12)-0.5]{}) – ([sqrt(12)+0.5]{},0.5); (.5, -1.708) – (2.902, -.4391) – (-.4958, -1.708) – (-2.902, -.4385) – (.5, -1.708); (-7,-4) rectangle (7,4); (0,0) circle \[x radius=[sqrt(5)]{}, y radius=[sqrt(3)]{}\]; hyperbola lambda=-7.500 plot ([sqrt(5+7.500)\*sqrt(1-/(3-7.500))]{},); plot ([-sqrt(5+7.500)\*sqrt(1-/(3-7.500))]{},); ([sqrt(8)+2]{},-2) – (-2,[sqrt(8)+2]{}); ([-sqrt(8)-2]{},2) – (2,[-sqrt(8)-2]{}); (2,[sqrt(8)+2]{}) – ([-sqrt(8)-2]{},-2); (-2,[-sqrt(8)-2]{}) – ([sqrt(8)+2]{},2); (.9, -1.586) – (2.162, -.4427) – (-.9015, 1.585) – (-2.162, .4430) – (.9, -1.586); ### 5-periodic trajectories {#periodic-trajectories-2 .unnumbered} There is a $5$-periodic trajectory of the billiard within [eq:ellipse]{}, with a non-degenerate caustic ${\pazocal{C}}_{\gamma}$ if and only if, according to Theorem \[th:cayley-billiard\], the caustic is an ellipse, i.e. $\gamma \in (-b,a)$, and $C_{2}C_{4}-C^{2}_{3}=0$, which is equivalent to: $$\begin{aligned} \begin{split}\label{eqn:5-periodic} 0=& \left( a+b \right) ^{6}\gamma^{6}-2ab \left(a-b \right) \left( a-3b \right) \left( 3a-b \right) \left( a+b \right)^{2}\gamma^{5} -a^{2}b^{2} \left( 29a^{2}-54ab+29b^{2} \right) \left( a+b \right)^{2} \gamma^{4} \\ & -36{a}^{3}{b}^{3} \left(a -b \right) \left( a+b \right)^{2} \gamma^{3} -a^{4}b^{4} \left( 9a^{2}+34ab+9b^{2} \right) \gamma^{2}+10a^{5}b^{5} \left(a -b \right) \gamma+5a^{6}b^{6}. \end{split}\end{aligned}$$ Examples of $5$-periodic billiard trajectories are shown in Figures \[fig:5-periodicy\] and \[fig:5-periodicx\]. (0,0) circle \[x radius=[sqrt(5)]{}, y radius=[sqrt(2)]{}\]; (0,0) circle \[x radius=[sqrt(5-4.7375)]{}, y radius=[sqrt(2+4.7375)]{}\]; ([sqrt(7)+0.5]{},-0.5) – (-0.5,[sqrt(7)+0.5]{}); ([-sqrt(7)-0.5]{},0.5) – (0.5,[-sqrt(7)-0.5]{}); (0.5,[sqrt(7)+0.5]{}) – ([-sqrt(7)-0.5]{},-0.5); (-0.5,[-sqrt(7)-0.5]{}) – ([sqrt(7)+0.5]{},0.5); (1, -1.265) – (.4548, 1.385) – (.6076, -1.362) – (1.548, 1.021) – (2.151, .386) – (1, -1.265); (0,0) circle \[x radius=[sqrt(6)]{}, y radius=[sqrt(4)]{}\]; (0,0) circle \[x radius=[sqrt(6-1.4205)]{}, y radius=[sqrt(4+1.4205)]{}\]; ([sqrt(10)+0.5]{},-0.5) – (-0.5,[sqrt(10)+0.5]{}); ([-sqrt(10)-0.5]{},0.5) – (0.5,[-sqrt(10)-0.5]{}); (0.5,[sqrt(10)+0.5]{}) – ([-sqrt(10)-0.5]{},-0.5); (-0.5,[-sqrt(10)-0.5]{}) – ([sqrt(10)+0.5]{},0.5); (2.15, -.9585) – (2.130, 1.0) – (1.745, 1.403) – (2.447, -0.97e-1) – (1.701, -1.439) – (2.15, -.9585); (0,0) circle \[x radius=[sqrt(6)]{}, y radius=[sqrt(4)]{}\]; (0,0) circle \[x radius=[sqrt(6+3.9947)]{}, y radius=[sqrt(4-3.9947)]{}\]; ([sqrt(10)+0.5]{},-0.5) – (-0.5,[sqrt(10)+0.5]{}); ([-sqrt(10)-0.5]{},0.5) – (0.5,[-sqrt(10)-0.5]{}); (0.5,[sqrt(10)+0.5]{}) – ([-sqrt(10)-0.5]{},-0.5); (-0.5,[-sqrt(10)-0.5]{}) – ([sqrt(10)+0.5]{},0.5); (2.3, -.6879) – (.7881, -1.894) – (-2.409, -.3624) – (2.448, -0.0571) – (-2.447, -0.09091) – (2.3, -.6879); (0,0) circle \[x radius=[sqrt(6)]{}, y radius=[sqrt(4)]{}\]; (0,0) circle \[x radius=[sqrt(6+1.5413)]{}, y radius=[sqrt(4-1.5413)]{}\]; ([sqrt(10)+0.5]{},-0.5) – (-0.5,[sqrt(10)+0.5]{}); ([-sqrt(10)-0.5]{},0.5) – (0.5,[-sqrt(10)-0.5]{}); (0.5,[sqrt(10)+0.5]{}) – ([-sqrt(10)-0.5]{},-0.5); (-0.5,[-sqrt(10)-0.5]{}) – ([sqrt(10)+0.5]{},0.5); (1.5, -1.581) – (2.080, -1.056) – (0.04168, -2.000) – (-2.071, -1.068) – (-1.545, -1.552) – (1.5, -1.581); ### 6-periodic trajectories {#periodic-trajectories-3 .unnumbered} There is a $6$-periodic trajectory of the billiard within [eq:ellipse]{}, with a non-degenerate caustic ${\pazocal{C}}_{\gamma}$ if and only if $B_{3}B_{5}-B^{2}_{4}=0$, which is equivalent to: $$\begin{aligned} \begin{split}\label{eqn:6-periodic} 0=&\Big({-}(a+b)^{2}\gamma^{2}+2ab ( a-b) \gamma+3a^{2}b^{2}\Big) \Big( ( a+b)( a-3b) \gamma^{2}+2ab ( a+b) \gamma+a^{2}b^{2}\Big) \\ &\times \Big( ( a+b)^{2}\gamma^{2}+2ab ( a-b) \gamma + a^{2}b^{2}\Big) \Big ( -( a+b)( 3a-b) \gamma^{2}-2ab( a+b) \gamma+a^{2}b^{2}\Big) . \end{split}\end{aligned}$$ The first factor, $-(a+b)^{2}\gamma^{2}+2ab ( a-b)\gamma+3a^2b^2$, is a constant multiple of $C_2$ (see Equation [eqn:3-periodic]{}), thus it produces $3$-periodic trajectories, which have already been studied. The discriminant of the third factor $( a+b)^{2}\gamma^{2}+2ab ( a-b) \gamma + a^{2}b^{2}$ is $-16a^3b^3$, which is negative, therefore the expression has no real roots in $\gamma$. Next, we consider the second factor: $ ( a+b)( a-3b) \gamma^{2}+2ab ( a+b) \gamma+a^{2}b^{2}=0, $ which has two real solutions: $$\gamma= \frac{ab}{(a+b)(a-3b)}\Big(-a-b\pm 2\sqrt{ab+b^{2}}\Big)$$ Finally we consider the fourth factor: $ -( a+b)( 3a-b) \gamma^{2}-2ab( a+b) \gamma+a^{2}b^{2}=0, $ which yields two real solutions: $$\gamma= \frac{ab}{(a+b)(3a-b)}\Big(-a-b\pm 2\sqrt{ab+a^{2}}\Big).$$ Examples of $6$-periodic trajectories with hyperbolas as caustics are shown in Figure \[fig:6-periodich\]. (0,0) circle \[x radius=[sqrt(5)]{}, y radius=[sqrt(3)]{}\]; hyperbola lambda=-3.2264 plot ([sqrt(5+3.2264)\*sqrt(1-/(3-3.2264))]{},); plot ([-sqrt(5+3.2264)\*sqrt(1-/(3-3.2264))]{},); ([sqrt(8)+2]{},-2) – (-2,[sqrt(8)+2]{}); ([-sqrt(8)-2]{},2) – (2,[-sqrt(8)-2]{}); (2,[sqrt(8)+2]{}) – ([-sqrt(8)-2]{},-2); (-2,[-sqrt(8)-2]{}) – ([sqrt(8)+2]{},2); (2, -0.7746) – (1.366, -1.372) – (-2.232, -.1051) – (2.001, .7732) –(1.364, 1.372)–(-2.232, .1055)– (2, -0.7746); (0,0) circle \[x radius=[sqrt(3)]{}, y radius=[sqrt(7)]{}\]; hyperbola lambda=3.1189 plot (,[sqrt(7+3.1189)\*sqrt(1-/(3-3.1189)]{}); plot (,[-sqrt(7+3.1189)\*sqrt(1-/(3-3.1189)]{}); ([sqrt(10)+2]{},-2) – (-2,[sqrt(10)+2]{}); ([-sqrt(10)-2]{},2) – (2,[-sqrt(10)-2]{}); (2,[sqrt(10)+2]{}) – ([-sqrt(10)-2]{},-2); (-2,[-sqrt(10)-2]{}) – ([sqrt(10)+2]{},2); (.4, -2.574) – (-.1910, 2.630) – (-1.653, -.7911) – (-.3996, -2.574)–(.1911, 2.629)–(1.652, -.7943)–(.4, -2.574); ### 7-periodic trajectories {#periodic-trajectories-4 .unnumbered} According to Theorem \[th:cayley-billiard\], there is a $7$-periodic trajectory of the billiard within [eq:ellipse]{}, with a non-degenerate caustic ${\pazocal{C}}_{\gamma}$ if and only if the caustic is an ellipse, i.e. $\gamma \in (-b,a)$, and $$\left| \begin{array}{ccc} C_2 & C_3 & C_4 \\ C_3 & C_4 & C_5 \\ C_4 & C_5 & C_6 \end{array} \right|=0,$$ which is equivalent to: $$\begin{aligned} \begin{split}\label{eqn:7-periodic} 0 =& - \left( a+b \right) ^{12}\gamma^{12}+4ab \left( a-b \right) \left( a-3b \right) \left( 3a-b \right) \left( {a}^{2}-6ab+{b}^{2} \right)\left( a+b \right) ^{6}{\gamma}^{11} \\ & +2{a}^{2}{b}^{2} \left( 59{a}^{4}-332{a}^{3}b+626{a}^{2}{b}^{2}-332a{b}^{3}+59{b}^{4} \right) \left( a+b \right) ^{6}{\gamma}^{10}\\ &+28{a}^{3}{b}^{3} \left( a-b \right) \left( 13{a}^{2}-38ab+13{b}^{2} \right) \left( a+b \right)^{6}{\gamma}^{9}\\ & +{a}^{4}{b}^{4} \left( 7{a}^{2}+30ab+7{b}^{2} \right) \left( 63{a}^{4}-84{a}^{3}b-38{a}^{2}{b}^{2}-84a{b}^{3}+63{b}^{4} \right)\left( a+b \right) ^{2}{\gamma}^{8} \\ & -8{a}^{5}{b}^{5} \left( a-b \right) \left( 21{a}^{4}-420{a}^{3}b-50{a}^{2}{b}^{2}-420a{b}^{3}+21{b}^{4} \right) \left( a+b \right) ^{2}{\gamma}^{7}\\ &-12{a}^{6}{b}^{6} \left( 105{a}^{4}-420{a}^{3}b+422{a}^{2}{b}^{2}-420a{b}^{3}+105{b}^{4} \right) \left( a+b \right) ^{2}{\gamma}^{6}\\ & -24{a}^{7}{b}^{7} \left( a-b \right) \left( 75{a}^{2}-106ab+75{b}^{2} \right) \left( a+b \right) ^{2}{\gamma}^{5} \\ & -3{a}^{8}{b}^{8} \left( 437{a}^{2}-726ab+437{b}^{2} \right) \left( a+b \right)^{2}{\gamma}^{4} -4{a}^{9}{b}^{9} \left( a-b \right) \left( 121{a}^{2}+250ab+121{b}^{2} \right) {\gamma}^{3} \\& -14{a}^{10}{b}^{10} \left( 3{a}^{2}+14ab+3{b}^{2} \right) {\gamma}^{2} +28{a}^{11}{b}^{11} \left( a-b \right) \gamma+7{a}^{12}{b}^{12}. \end{split}\end{aligned}$$ Examples of $7$-periodic trajectories are shown in Figure \[fig:7-periodic\]. (0,0) circle \[x radius=[sqrt(3)]{}, y radius=[sqrt(7)]{}\]; (0,0) circle \[x radius=[sqrt(3+6.9712)]{}, y radius=[sqrt(7-6.9712)]{}\]; ([sqrt(10)+0.5]{},-0.5) – (-0.5,[sqrt(10)+0.5]{}); ([-sqrt(10)-0.5]{},0.5) – (0.5,[-sqrt(10)-0.5]{}); (0.5,[sqrt(10)+0.5]{}) – ([-sqrt(10)-0.5]{},-0.5); (-0.5,[-sqrt(10)-0.5]{}) – ([sqrt(10)+0.5]{},0.5); (0, -2.64575) – (1.45903, -1.42579) – (-1.70584, -.458533) – (1.72848, -.169774) – (-1.72849, -.169638)–(1.70591, -.457905) –(-1.45976, -1.42403)– (0, -2.64575); (0,0) circle \[x radius=[sqrt(7)]{}, y radius=[sqrt(3)]{}\]; (0,0) circle \[x radius=[sqrt(7-6.9712)]{}, y radius=[sqrt(3+6.9712)]{}\]; ([sqrt(10)+0.5]{},-0.5) – (-0.5,[sqrt(10)+0.5]{}); ([-sqrt(10)-0.5]{},0.5) – (0.5,[-sqrt(10)-0.5]{}); (0.5,[sqrt(10)+0.5]{}) – ([-sqrt(10)-0.5]{},-0.5); (-0.5,[-sqrt(10)-0.5]{}) – ([sqrt(10)+0.5]{},0.5); (1.5, -1.427) – (.4857, 1.703) – (.1761, -1.728) – (.1651, 1.73) – (.4356, -1.708)–(1.360, 1.484)–(2.641, -0.099)– (1.5, -1.427); ### 8-periodic trajectories {#periodic-trajectories-5 .unnumbered} There is an $8$-periodic trajectory of the billiard within ellipse [eq:ellipse]{}, with a non-degenerate caustic ${\pazocal{C}}_{\gamma}$ if and only if $$\left| \begin{array}{ccc} B_3 & B_4 & B_5 \\ B_4 & B_5 & B_6 \\ B_5 & B_6 & B_7 \end{array} \right|=0,$$ which is equivalent to: $$\begin{aligned} \begin{split}\label{eqn:8-periodic} 0= & \left( ab-a\gamma-b\gamma \right) \left( ab+a\gamma+b\gamma \right) \left( ab+a\gamma-b\gamma \right) \\ & \Big( \left( a+b \right)^{4}{\gamma}^{4}-4ab \left( a+b \right) \left( -b+a \right) ^{2}{\gamma}^{3}-2{a}^{2}{b}^{2} \left( a+b \right) \left( 5a-3b \right) {\gamma}^{2}-4{a}^{3}{b}^{3} \left( a+b \right) \gamma\\ & +{a}^{4}{b}^{4} \Big) \Big( \left( a+b \right) ^{4}{\gamma}^{4}+4ab \left( a+b \right) \left( -b+a \right) ^{2}{\gamma}^{3}+2{a}^{2}{b}^{2} \left( a+b \right) \left( 3a-5b \right) {\gamma}^{2}\\ & +4{a}^{3}{b}^{3} \left( a+b \right) \gamma +{a}^{4}{b}^{4} \Big) \Big( \left( {a}^{2}-6ab+{b}^{2} \right) \left( a+b \right) ^{2}{\lambda}^{4}+4ab \left( -b+a \right) \left( a+b \right) ^{2}{\gamma}^{3}\\ & +2{a}^{2}{b}^{2} \left( 3{a}^{2}+2ab+3{b}^{2} \right) {\gamma}^{2} +4{a}^{3}{b}^{3} \left( -b+a \right) \gamma+{a}^{4}{b}^{4} \Big). \end{split}\end{aligned}$$ In Figures \[fig:8-periodich\] and \[fig:8-periodice\], three examples of $8$-periodic trajectories are shown. (0,0) circle \[x radius=[sqrt(6)]{}, y radius=[sqrt(3)]{}\]; hyperbola lambda=-3.0151, plot ([sqrt(6+3.0151)\*sqrt(1-/(3-3.0151))]{},); plot ([-sqrt(6+3.0151)\*sqrt(1-/(3-3.0151))]{},); ([sqrt(9)+2]{},-2) – (-2,[sqrt(9)+2]{}); ([-sqrt(9)-2]{},2) – (2,[-sqrt(9)-2]{}); (2,[sqrt(9)+2]{}) – ([-sqrt(9)-2]{},-2); (-2,[-sqrt(9)-2]{}) – ([sqrt(9)+2]{},2); (1.5, -1.369) – (2.250, -.685) – (-2.448, -0.0550) – (2.439, .1630)–(-1.510, 1.364)–(-2.247, .689)–(2.448, 0.0557)–(-2.439, -.1615) – (1.5, -1.369); (0,0) circle \[x radius=[sqrt(6)]{}, y radius=[sqrt(3)]{}\]; hyperbola lambda=6.9168 plot (,[sqrt(3+6.9168)\*sqrt(1-/(6-6.9168)]{}); plot (,[-sqrt(3+6.9168)\*sqrt(1-/(6-6.9168)]{}); ([sqrt(9)+2]{},-2) – (-2,[sqrt(9)+2]{}); ([-sqrt(9)-2]{},2) – (2,[-sqrt(9)-2]{}); (2,[sqrt(9)+2]{}) – ([-sqrt(9)-2]{},-2); (-2,[-sqrt(9)-2]{}) – ([sqrt(9)+2]{},2); (1.5, -1.369) – (.1935, 1.727) – (-.8948, -1.612) – (-2.370, .437)–(-1.501, 1.369)–(-.1936, -1.727)–(.8948, 1.612)–(2.370, -.437)– (1.5, -1.369); (0,0) circle \[x radius=[sqrt(6)]{}, y radius=[sqrt(3)]{}\]; (0,0) circle \[x radius=[sqrt(6-5.3707)]{}, y radius=[sqrt(3+5.3707)]{}\]; ([sqrt(9)+0.5]{},-0.5) – (-0.5,[sqrt(9)+0.5]{}); ([-sqrt(9)-0.5]{},0.5) – (0.5,[-sqrt(9)-0.5]{}); (0.5,[sqrt(9)+0.5]{}) – ([-sqrt(9)-0.5]{},-0.5); (-0.5,[-sqrt(9)-0.5]{}) – ([sqrt(9)+0.5]{},0.5); (1.5, -1.369) – (.6650, 1.667) – (1.057, -1.564) – (2.386, .392) – (1.498, 1.370)–(.6646, -1.667) –(1.059, 1.561)–(2.386, -.392)– (1.5, -1.369); ### Summary of numbers of touching points with relativistic ellipses and hyperbolas {#sec:table .unnumbered} In the table below, we summarise the examples given in this section. Here, $n_1$ and $n_2$ represent the numbers of bouncing points off relativistic ellipses and relativistic hyperbolas respectively. Period $n_1+n_2$ Caustic $n_{1}$ $n_{2}$ ------------------ ------------------------------------- --------- --------- $n=3$ [Ellipse along $\mathsf{y}$-axis]{} 2 1 [Ellipse along $\mathsf{x}$-axis]{} 1 2 $n=4$ [Ellipse along x-axis]{} 2 2 [Ellipse along y-axis]{} 2 2 [Hyperbola along x-axis]{} 2 2 $n=5$ [Ellipse along y-axis]{} 2 3 [Ellipse along x-axis]{} 3 2 [Ellipse along y-axis]{} 4 1 [Ellipse along x-axis]{} 1 4 $n=6$ [Hyperbola along x-axis]{} 2 4 [Ellipse along y-axis]{} 4 2 $n=7$ [Ellipse along x-axis]{} 1 6 [Ellipse along y-axis]{} 6 1 $n=8$ [Hyperbola along x-axis]{} 2 6 [Hyperbola along y-axis]{} 6 2 [Ellipse along x-axis]{} 6 2 Cayley-type conditions and discriminantly separable polynomials --------------------------------------------------------------- Similarly to the case of Euclidean plane [@DragRadn2019rcd], the Cayley-type conditions obtained above have a very interesting algebraic structure. Namely, the numerators of the corresponding expressions are polynomials in $3$ variables. As examples below show, those polynomials have factorizable discriminants which, after a change of varibles, lead to discriminantly separable polynomials in the sense of the following definition. A polynomial $F(x_{1},\dots, x_{n})$ is *discriminantly separable* if there exist polynomials $f_{1}(x_1),\dots , f_{n}(x_n)$ such that the discriminant $\mathcal{D}_{x_i}F$ of F with respect to $x_{i}$ satisfies: $$\mathcal{D}_{x_i}F(x_{1},\dots,\hat{x}_i,\dots, x_{n})= \prod_{j\neq i}^{}f_{j}(x_{j}),$$ for each $i=1,\dots,n$. Discriminantly factorizable polynomials were introduced in [@Drag2012] in connection with $n$-valued groups. Various applications of discriminantly separable polynomials in continuous and discrete integrable systems were presented in [@DragKuk2014jgm; @DragKuk2014rcd; @DragKuk2017]. The connection between Cayley-type conditions in the Euclidean setting and discriminantly factorizable and separable polynomials has been observed in [@DragRadn2019rcd]. As examples below show, the Cayley conditions in the Minkowski plane provide examples of discriminantly factorisable polynomials which, after a change of variables, have separable discriminants. It would be interesting to establish this relationship as a general statement. \[ex:G2\] The expression [eqn:3-periodic]{} is: $$\mathsf{G}_{2}(\gamma,a,b)= - \left( a+b \right) ^{2}{\gamma}^{2}+2\,ab \left( a-b \right) \gamma+3\,{a}^{2}{b}^{2},$$ and its discriminant with respect to $\gamma$: $$\mathcal{D}_{\gamma}\mathsf{G}_{2}=2^4 \left( {a}^{2}+ab+{b}^{2} \right) {a}^{2}{b}^{2},$$ which is obviously factorizable. The expression [eqn:4-periodic]{} is: $$\mathsf{G}_{3}(\gamma,a,b) =-(ab+a\gamma+b\gamma)(ab+a\gamma-b\gamma)(ab-a\gamma-b\gamma)$$ and its discriminant with respect to $\gamma$ is factored as: $$\mathcal{D}_{\gamma}\mathsf{G}_{3}=2^6{a}^{8}{b}^{8} \left( a+b \right) ^{2}.$$ The expression [eqn:5-periodic]{} is: $$\begin{aligned} \mathsf{G}_{6}(\gamma,a,b)=& \left( a+b \right) ^{6}\gamma^{6}-2ab \left( a-b \right) \left( a-3b \right) \left( 3a-b \right) \left( a+b \right)^{2}\gamma^{5}\\ & -a^{2}b^{2} \left( 29a^{2}-54ab+29b^{2} \right) \left( a+b \right)^{2} \gamma^{4}-36{a}^{3}{b}^{3} \left(a -b \right) \left( a+b \right)^{2} \gamma^{3}\\ & -a^{4}b^{4} \left( 9a^{2}+34ab+9b^{2} \right) \gamma^{2}+10a^{5}b^{5} \left(a -b \right) \gamma+5a^{6}b^{6}.\end{aligned}$$ is discriminantly factorizable since its discriminant with respect to $\gamma$ is: $$\mathcal{D}_{\gamma}\mathsf{G}_{6}=-5\cdot 2^{44} ( 27{a}^{6}+81{a}^{5}b+322\,{a}^{4}{b}^{2}+509{a}^{3}{b}^{3}+322{a}^{2}{b}^{4} +81a{b}^{5}+27{b}^{6} ) \left( a+b \right) ^{8}{b}^{38}{a}^{38}.$$ Let us denote the expression [eqn:6-periodic]{} as: $$\begin{aligned} \mathsf{G}_{8}(\gamma,a,b)=&( 3a-b ) ( a-3b ) ( a+b ) ^{6}{\gamma}^{8} +8ab ( a-b ) ( a+b ) ^{6}{\gamma}^{7} \\& -4\,{a}^{2}{b}^{2} ( 3\,{a}^{4}-24\,{a}^{3}b+10\,{a}^{2}{b}^{2} -24\,a{b}^{3}+3\,{b}^{4} ) \left( a+b \right) ^{2} {\gamma}^{6} \\& -8\,{a}^{3}{b}^{3} \left( a-b \right) \left( 9\,{a}^{2}-14\,ab+9\,{b}^{2} \right) \left( a+b \right) ^{2}{\gamma}^{5}\\ & -10\,{a}^{4}{b}^{4} \left( 11\,{a}^{2}-18\,ab+11\,{b}^{2} \right) \left( a+b \right) ^{2}{\gamma}^{4}-72\,{a}^{5}{b}^{5} \left( a-b \right) \left( a+b \right) ^{2}{\gamma}^{3}\\ & -4\,{a}^{6}{b}^{6} \left( a+3\,b \right) \left( 3\,a+b \right) {\gamma}^{2}+8\,{a}^{7}{b}^{7} ( a-b ) \gamma+3\,{a}^{8}{b}^{8} \end{aligned}$$ We find that the discriminant of $\mathsf{G}_{8}$ with respect to $\gamma$ factors as: $$\mathcal{D}_{\gamma}\mathsf{G}_{8}=-2^{88}\, \left( {a}^{2}+ab+{b}^{2} \right) \left( a+b \right) ^{18}\\ \mbox{}{b}^{74}{a}^{74}.$$ The discriminant $\mathcal{D}_{\gamma}\mathsf{G}_{12}$ of the expression in [eqn:7-periodic]{} is: $$\begin{aligned} \mathcal{D}_{\gamma}\mathsf{G}_{12}=&-2^{184}\cdot7^{2} \left( a+b \right) ^{40}{(ab)}^{172}\times \\&\times ( 84375{a}^{12}+506250{a}^{11}b +4266243\,{a}^{10}{b}^{2}+16690590\,{a}^{9}{b}^{3}+34989622\,{a}^{8}{b}^{4} \\&\quad +45383698\,{a}^{7}{b}^{5}+46564971\,{a}^{6}{b}^{6}+ 45383698\,{a}^{5}{b}^{7}+34989622\,{a}^{4}{b}^{8}+16690590\,{a}^{3}{b}^{9} \\&\quad +4266243\,{a}^{2}{b}^{10}+506250\,a{b}^{11} +84375\,{b}^{12} ) , \end{aligned}$$ thus $\mathsf{G}_{12}$ is a discriminantly factorizable polynomial. \[ex:G15\] The discriminant $\mathcal{D}_{\gamma}\mathsf{G}_{15}$ of the expression [eqn:8-periodic]{} is: $$\begin{aligned} \mathcal{D}_{\gamma}\mathsf{G}_{15}=&-2^{246} (ab)^{278}\left( 27\,{a}^{2}+46\,ab+27\,{b}^{2} \right) \left( a+b \right) ^{8} \times\\&\times \left( {a}^{5}+5\,{a}^{4}b+10\,{a}^{3}{b}^{2}+10\,{a}^{2}{b}^{3}+5\,a{b}^{4}+{b}^{5} \right) \times\\&\times ( {a}^{7}+7\,{a}^{6}b+21\,{a}^{5}{b}^{2}+35\,{a}^{4}{b}^{3}+35\,{a}^{3}{b}^{4} +21\,{a}^{2}{b}^{5}+7\,a{b}^{6}+{b}^{7} ) \times\\&\times ( 8\,{a}^{26}+27\,{b}^{26}+200\,{a}^{25}b+2427\,{a}^{24}{b}^{2}+19048\,{a}^{23}{b}^{3}+108652\,{a}^{22}{b}^{4} +479688\,{a}^{21}{b}^{5} \\&\quad +1703702\,{a}^{20}{b}^{6}+4993208\,{a}^{19}{b}^{7}+12286692\,{a}^{18}{b}^{8}+25688608\,{a}^{17}{b}^{9} +46007797\,{a}^{16}{b}^{10} \\&\quad +70961808\,{a}^{15}{b}^{11}+94556312\,{a}^{14}{b}^{12}+108998288\,{a}^{13}{b}^{13} +108671412\,{a}^{12}{b}^{14}\\ &\quad +93545968\,{a}^{11}{b}^{15}+69297712\,{a}^{10}{b}^{16}+43955208\,{a}^{9}{b}^{17} +23703317\,{a}^{8}{b}^{18}+10761608\,{a}^{7}{b}^{19}\\ &\quad +4059132\,{a}^{6}{b}^{20}+1248808\,{a}^{5}{b}^{21} +305302\,{a}^{4}{b}^{22}+57048\,{a}^{3}{b}^{23}+7652\,{a}^{2}{b}^{24}+656\,a{b}^{25} ) \\&\times ( 27\,{a}^{26}+8\,{b}^{26}+656\,{a}^{25}b+7652\,{a}^{24}{b}^{2}+57048\,{a}^{23}{b}^{3}+305302\,{a}^{22}{b}^{4} +1248808\,{a}^{21}{b}^{5} \\&\quad +4059132\,{a}^{20}{b}^{6}+10761608\,{a}^{19}{b}^{7}+23703317\,{a}^{18}{b}^{8}+43955208\,{a}^{17}{b}^{9} +69297712\,{a}^{16}{b}^{10} \\&\quad +93545968\,{a}^{15}{b}^{11}+108671412\,{a}^{14}{b}^{12}+108998288\,{a}^{13}{b}^{13} +94556312\,{a}^{12}{b}^{14}\\ &\quad +70961808\,{a}^{11}{b}^{15}+46007797\,{a}^{10}{b}^{16}+25688608\,{a}^{9}{b}^{17} +12286692\,{a}^{8}{b}^{18}+4993208\,{a}^{7}{b}^{19}\\ &\quad +1703702\,{a}^{6}{b}^{20}+479688\,{a}^{5}{b}^{21}+108652\,{a}^{4}{b}^{22} +19048\,{a}^{3}{b}^{23}+2427\,{a}^{2}{b}^{24}+200\,a{b}^{25} ) , \end{aligned}$$ so $\mathsf{G}_{15}$ is a discriminantly factorizable polynomial. Since the determinants obtained in Theorem \[th:cayley-billiard\] are symmetric in $a$, $-b$, and $\gamma$, the discriminants with respect to $a$ and $b$ of the polynomials in Examples \[ex:G2\]–\[ex:G15\] will be also factorizable. We observed in the Examples \[ex:G2\]–\[ex:G15\] that all polynomials are discriminantly factorizable. However, it is important to note that their factors are homogeneous, thus, by a change of variables $(a,b)\mapsto (a,\hat{b})$, with $\hat{b}=\dfrac{b}{a}$, transforms the polynomials into discriminantly separable polynomials in new variables $(a, \hat{b})$: $$\begin{gathered} \mathcal{D}_{\gamma}\mathsf{G}_{2}=2^4\,a^{8} \hat{b}^{2}\left( 1+\hat{b}+\hat{b}^{2} \right), \\ \mathcal{D}_{\gamma}\mathsf{G}_{3}=2^6\,{a}^{18}{\hat{b}}^{8} \left( 1+\hat{b} \right) ^{2}, \\ \mathcal{D}_{\gamma}\mathsf{G}_{6}=-52^{44}a^{90} \hat{b}^{38}( 27+81\hat{b}+322\hat{b}^{2}+509\hat{b}^{3}+322\hat{b}^{4} +81\hat{b}^{5}+27{b}^{6} ) \left( a+b \right) ^{8}, \\ \mathcal{D}_{\gamma}\mathsf{G}_{8}=-2^{88}\, a^{168} \hat{b}^{74}\left( 1+\hat{b}+\hat{b}^{2} \right) \left( 1+\hat{b} \right) ^{18},\end{gathered}$$ Elliptic periodic trajectories {#sec:elliptic} ============================== Points of the plane which are symmetric with respect to the coordinate axes share the same elliptic coordinates, thus there is no bijection between the elliptic and the Cartesian coordinates. Thus, we introduce a separate notion of periodicity in elliptic coordinates. A billiard trajectory is *$n$-elliptic periodic* if it is $n$-periodic in elliptic coordinates joined to the confocal family [eq:confocal.conics]{}. Now, we will derive algebro-geometric conditions for elliptic periodic trajectories. \[th:elliptic-periodic\] A billiard trajectory within ${\pazocal{E}}$ with the caustic ${\pazocal{C}}_{\gamma}$ is $n$-elliptic periodic without being $n$-periodic if and only if one of the following conditions is satisfied on ${\mathcal{C}}$: - ${\pazocal{C}}_{\gamma}$ is an ellipse, $0<\gamma<a$, and $nQ_{0}-(n-1)Q_{\gamma}-Q_{-b}\sim0$; - ${\pazocal{C}}_{\gamma}$ is an ellipse, $-b<\gamma<0$, and $nQ_{0}-(n-1)Q_{\gamma}-Q_{a}\sim0$; - ${\pazocal{C}}_{\gamma}$ is a hyperbola, $n$ is even and $nQ_{0}-(n-2)Q_{\gamma}-Q_{-b}-Q_{a}\sim0$; - ${\pazocal{C}}_{\gamma}$ is a hyperbola, $n$ is odd, and $nQ_{0}-(n-1)Q_{\gamma}-Q_a\sim0$; - ${\pazocal{C}}_{\gamma}$ is a hyperbola, $n$ is odd, and $nQ_{0}-(n-1)Q_{\gamma}-Q_{-b}\sim0$. Moreover, such trajectories are always symmetric with respect to the origin in Case (c). They are symmetric with respect to the $\mathsf{x}$-axis in Cases (b) and (d), and with respect to the $\mathsf{y}$-axis in Cases (a) and (e). Let $M_0$ be the initial point of a given $n$-elliptic periodic trajectory, and $M_1$ the next point on the trajectory with the same elliptic coordinates. Then, integrating $M_0$ to $M_1$ along the trajectory, we get: $$n_1(Q_0-Q_{\alpha_1})+n_2(Q_{0}-Q_{\beta_1})\sim0,$$ where $n=n_1+n_2$, and $n_1$ is the number of times that the particle hit the arcs of ${\pazocal{E}}$ with time-like metrics, and $n_2$ the number of times it hit the arcs with space-like metrics. We denoted by $\alpha_1$ the largest negative member of the set $\{a,-b,\gamma\}$, and by $\beta_1$ its smallest positive member. The trajectory is not $n$-periodic if and only if at least one of $n_1$, $n_2$ is odd, which then leads to the stated conclusions. The explicit Cayley-type conditions for elliptic periodic trajectories are: \[th:elliptic-cayley\] A billiard trajectory within ${\pazocal{E}}$ with the caustic ${\pazocal{Q}}_{\gamma}$ is $n$-elliptic periodic without being $n$-periodic if and only if one of the following conditions is satisfied: - ${\pazocal{C}}_{\gamma}$ is an ellipse, $0<\gamma<a$, and $$\begin{gathered} D_1=0, \quad \left| \begin{array}{cc} D_1 & D_2 \\ D_2 & D_3 \end{array} \right|=0, \quad \left| \begin{array}{ccc} D_1 & D_2 & D_3 \\ D_2 & D_3 & D_4 \\ D_3 & D_4 & D_5 \end{array} \right|=0, \dots \quad\text{for}\quad n=2,4,6,\dots \\ E_2=0, \quad \left| \begin{array}{cc} E_2 & E_3 \\ E_3 & E_4 \end{array} \right|=0, \quad \left| \begin{array}{ccc} E_2 & E_3 & E_4 \\ E_3 & E_4 & E_5 \\ E_4 & E_5 & E_6 \end{array} \right|=0, \dots \quad\text{for}\quad n=3,5,7,\dots; \end{gathered}$$ - ${\pazocal{C}}_{\gamma}$ is an ellipse, $-b<\gamma<0$, and $$\begin{gathered} E_1=0, \quad \left| \begin{array}{cc} E_1 & E_2 \\ E_2 & E_3 \end{array} \right|=0, \quad \left| \begin{array}{ccc} E_1 & E_2 & E_3 \\ E_2 & E_3 & E_4 \\ E_3 & E_4 & E_5 \end{array} \right|=0, \dots \quad\text{for}\quad n=2,4,6,\dots \\ D_2=0, \quad \left| \begin{array}{cc} D_2 & D_3 \\ D_3 & D_4 \end{array} \right|=0, \quad \left| \begin{array}{ccc} D_2 & D_3 & D_4 \\ D_3 & D_4 & D_5 \\ D_4 & D_5 & D_6 \end{array} \right|=0, \dots \quad\text{for}\quad n=3,5,7,\dots;\end{gathered}$$ - ${\pazocal{Q}}_{\gamma}$ is a hyperbola, $n$ even and $$\begin{gathered} C_1=0, \quad \left| \begin{array}{cc} C_1 & C_2 \\ C_2 & C_3 \end{array} \right|=0, \quad \left| \begin{array}{ccc} C_1 & C_2 & C_3 \\ C_2 & C_3 & C_4 \\ C_3 & C_4 & C_5 \end{array} \right|=0, \dots \quad\text{for}\quad n=2,4,6,\dots \end{gathered}$$ - ${\pazocal{Q}}_{\gamma}$ is a hyperbola, $n$ is odd, and $$D_2=0, \quad \left| \begin{array}{cc} D_2 & D_3 \\ D_3 & D_4 \end{array} \right|=0, \quad \left| \begin{array}{ccc} D_2 & D_3 & D_4 \\ D_3 & D_4 & D_5 \\ D_4 & D_5 & D_6 \end{array} \right|=0, \dots \quad\text{for}\quad n=3,5,7,\dots.$$ - ${\pazocal{Q}}_{\gamma}$ is a hyperbola, $n$ is odd, and $$E_2=0, \quad \left| \begin{array}{cc} E_2 & E_3 \\ E_3 & E_4 \end{array} \right|=0, \quad \left| \begin{array}{ccc} E_2 & E_3 & E_4 \\ E_3 & E_4 & E_5 \\ E_4 & E_5 & E_6 \end{array} \right|=0, \dots \quad\text{for}\quad n=3,5,7,\dots.$$ Here, we denoted: $$\begin{gathered} \frac{\sqrt{\varepsilon(a-x)(b+x)(\gamma-x)}}{a-x}=D_0+D_1x+D_2x^2+\dots, \\ \frac{\sqrt{\varepsilon(a-x)(b+x)(\gamma-x)}}{b+x}=E_0+E_1x+E_2x^2+\dots, \end{gathered}$$ the Taylor expansion around $x=0$, while $B$s and $C$s are as in Theorem \[th:cayley-billiard\]. \(a) Take first $n$ even. Using Theorem \[th:elliptic-periodic\], we have: $$nQ_0 \sim (n-1)Q_{\gamma}+Q_{-b}\sim (n-2)Q_{\infty}+Q_{-b}+Q_{\gamma} \sim (n-2) Q_{\infty}+Q_{\infty}+Q_{a} \sim (n-1)Q_{\infty}+Q_{a}.$$ The basis of $\mathcal{L}((n-1)Q_{\infty}+Q_{a})$ is: $$1,x,x^2,\dots,x^{n/2-1},\frac{y}{x-a},\frac{xy}{x-a}, \frac{x^{n/2-1}y}{x-a},$$ thus a non-trivial linear combination of these functions with a zero of order $n$ at $x=0$ exists if and only if: $$\left| \begin{array}{llll} D_{n/2} & D_{n/2-1} & \dots & D_1\\ D_{n/2+1} & D_{n/2} & \dots & D_2\\ \dots\\ D_{n-1} & D_{n-2} & \dots & D_{n/2} \end{array} \right| =0.$$ For odd $n$, we have: $$nQ_0 \sim (n-1)Q_{\gamma}+Q_{-b} \sim (n-1)Q_{\infty}+Q_{-b}.$$ The basis of $\mathcal{L}((n-1)Q_{\infty}+Q_{-b})$ is: $$1,x,x^2,\dots,x^{(n-1)/2},\frac{y}{x+b},\frac{xy}{x+b}, \frac{x^{(n-1)/2-1}y}{x+b},$$ thus a non-trivial linear combination of these functions with a zero of order $n$ at $x=0$ exists if and only if: $$\left| \begin{array}{llll} E_{(n-1)/2+1} & E_{(n-1)/2} & \dots & E_2\\ E_{(n-1)/2+2} & E_{(n-1)/2+1} & \dots & E_3\\ \dots\\ E_{n-1} & E_{n-2} & \dots & E_{(n-1)/2+1} \end{array} \right| =0.$$ Case (b) is done similarly as (a). \(c) We have $ nQ_0\sim(n-2)Q_{\gamma}+Q_{-b}+Q_a\sim(n-1)Q_{\infty}+Q_{\gamma}. $ \(d) We have $nQ_0\sim (n-1)Q_{\gamma}+Q_a\sim(n-1)Q_{\infty}+Q_a$. \(e) We have $nQ_0\sim (n-1)Q_{\gamma}+Q_{-b}\sim(n-1)Q_{\infty}+Q_{-b}$. Examples of elliptic periodic trajectories: $2\le n\le5$ {#sec:examples-elliptic} ======================================================== ### 2-elliptic periodic trajectories {#elliptic-periodic-trajectories .unnumbered} There is a $2$-elliptic periodic trajectory without being $2$-periodic of the billiard within [eq:ellipse]{}, with a non-degenerate caustic ${\pazocal{C}}_{\gamma}$ if and only if, according to Theorem \[th:elliptic-cayley\] one of the following is satisfied: - the caustic is an ellipse, with $\gamma \in (0,a)$ and $D_1=0$; - the caustic is an ellipse, with $\gamma \in (-b,0)$ and $E_1=0$; - the caustic is a hyperbola, $n$ is even, and $C_1=0$. We consider the following equations: $$\begin{aligned} D_1=\dfrac{(a+b)\gamma-ab}{2\sqrt{a^3b\gamma}}=0, \quad E_1=-\dfrac{(a+b)\gamma+ab}{2\sqrt{ab^3\gamma}}=0, \quad C_1= \dfrac{(a-b)\gamma+ab}{2\sqrt{ab\gamma^3}}=0,\end{aligned}$$ which respectively yield the solutions for the parameter $\gamma$ of the caustic: $$\begin{aligned} \gamma=\dfrac{ab}{a+b}, \quad \gamma=-\dfrac{ab}{a+b}, \quad \gamma=-\dfrac{ab}{a-b}.\end{aligned}$$ Some examples of $2$-elliptic periodic trajectories without being $2$-periodic are shown in Figures \[fig:2-elliptic-periodice\] and \[fig:2-elliptic-periodich\]. (0,0) circle \[x radius=[sqrt(5)]{}, y radius=[sqrt(3)]{}\]; (0,0) circle \[x radius=[sqrt(5+1.875)]{}, y radius=[sqrt(3-1.875)]{}\]; ([sqrt(8)+0.5]{},-0.5) – (-0.5,[sqrt(8)+0.5]{}); ([-sqrt(8)-0.5]{},0.5) – (0.5,[-sqrt(8)-0.5]{}); (0.5,[sqrt(8)+0.5]{}) – ([-sqrt(8)-0.5]{},-0.5); (-0.5,[-sqrt(8)-0.5]{}) – ([sqrt(8)+0.5]{},0.5); (.3, 1.7163916) – (-2.0697794, .65544485) – (-.3, 1.716391); (0,0) circle \[x radius=[sqrt(5)]{}, y radius=[sqrt(7)]{}\]; (0,0) circle \[x radius=[sqrt(5-2.91667)]{}, y radius=[sqrt(7+2.91667)]{}\]; ([sqrt(12)+0.5]{},-0.5) – (-0.5,[sqrt(12)+0.5]{}); ([-sqrt(12)-0.5]{},0.5) – (0.5,[-sqrt(12)-0.5]{}); (0.5,[sqrt(12)+0.5]{}) – ([-sqrt(12)-0.5]{},-0.5); (-0.5,[-sqrt(12)-0.5]{}) – ([sqrt(12)+0.5]{},0.5); (2, -1.18322) – (1.04167, 2.34112) – (2, 1.18322); (-6,-3) rectangle (6,3); (0,0) circle \[x radius=[sqrt(7)]{}, y radius=[sqrt(3)]{}\]; hyperbola lambda=-5.25 plot ([sqrt(7+5.25)\*sqrt(1-/(3-5.25))]{},); plot ([-sqrt(7+5.25)\*sqrt(1-/(3-5.25))]{},); ([sqrt(10)+2]{},-2) – (-2,[sqrt(10)+2]{}); ([-sqrt(10)-2]{},2) – (2,[-sqrt(10)-2]{}); (2,[sqrt(10)+2]{}) – ([-sqrt(10)-2]{},-2); (-2,[-sqrt(10)-2]{}) – ([sqrt(10)+2]{},2); (1.5, 1.4267846) – (-2.5376610, -.49001897) – (-1.5, -1.4267846); ### 3-elliptic periodic trajectories {#elliptic-periodic-trajectories-1 .unnumbered} There is a $3$-elliptic periodic trajectory without being $3$-periodic of the billiard within [eq:ellipse]{}, with a non-degenerate caustic ${\pazocal{C}}_{\gamma}$ if and only if one of the following is satisfied: - $E_2=0$ and either the caustic is an ellipse with $\gamma \in (0,a)$ or the caustic is a hyperbola with $n$ even; - $D_2=0$ and either the caustic is an ellipse with $\gamma \in (-b,0)$ or the caustic is a hyperbola. The equations $E_2=0$ and $D_2=0$ are respectively equivalent to: $$\begin{gathered} -(a+b)(3a-b)\gamma^2-2ab(a+b)\gamma+a^2b^2=0,\label{eq:3-elliptic-periodic1} \\ (a+b)(a-3b)\gamma^2+2ab(a+b)\gamma+a^2b^2 =0,\label{eq:3-elliptic-periodic2}\end{gathered}$$ which respectively yield the pairs of solutions for the parameter $\gamma$ of the caustic: $$\begin{aligned} \gamma=\dfrac{(-a-b\pm2\sqrt{a^2+ab})ba}{(a+b)(3a-b)} , \quad \gamma= \dfrac{(-a-b\pm2\sqrt{b^2+ab})ba}{(a+b)(a-3b)}.\end{aligned}$$ Examples of $3$-elliptic periodic trajectories which are not $3$-periodic are shown in Figures \[fig:3-elliptic-periodich\], \[fig:3-elliptic-periodicex\], \[fig:3-elliptic-periodicey\]. (-4,-3.5) rectangle (4,3.5); (0,0) circle \[x radius=[sqrt(6)]{}, y radius=[sqrt(3)]{}\]; hyperbola lambda=-3.1595918 plot ([sqrt(6+3.1595918)\*sqrt(1-/(3-3.1595918))]{},); plot ([-sqrt(6+3.1595918)\*sqrt(1-/(3-3.1595918))]{},); ([sqrt(9)+2]{},-2) – (-2,[sqrt(9)+2]{}); ([-sqrt(9)-2]{},2) – (2,[-sqrt(9)-2]{}); (2,[sqrt(9)+2]{}) – ([-sqrt(9)-2]{},-2); (-2,[-sqrt(9)-2]{}) – ([sqrt(9)+2]{},2); (1,1.5811388) – (-2.4369583, .1749772) – (2.3389687, -.51440518) – (1,-1.5811388); (-3.5,-3.5) rectangle (3.5,3.5); (0,0) circle \[x radius=[sqrt(3)]{}, y radius=[sqrt(5)]{}\]; hyperbola lambda=3.2264236 plot (,[sqrt(5+3.2264236)\*sqrt(1-/(3-3.2264236)]{}); plot (,[-sqrt(5+3.2264236)\*sqrt(1-/(3-3.2264236)]{}); ([sqrt(8)+2]{},-2) – (-2,[sqrt(8)+2]{}); ([-sqrt(8)-2]{},2) – (2,[-sqrt(8)-2]{}); (2,[sqrt(8)+2]{}) – ([-sqrt(8)-2]{},-2); (-2,[-sqrt(8)-2]{}) – ([sqrt(8)+2]{},2); (.4, 2.1756225) – (1.7264240, .18009374) – (.32752004, -2.1957271) – (-.4, 2.1756225); (-3.5,0.4) rectangle (3.5,1.5); (0,0) circle \[x radius=[sqrt(9)]{}, y radius=[sqrt(2)]{}\]; (0,0) circle \[x radius=[sqrt(9+.8831827)]{}, y radius=[sqrt(2-.8831827)]{}\]; ([sqrt(11)+0.5]{},-0.5) – (-0.5,[sqrt(11)+0.5]{}); ([-sqrt(11)-0.5]{},0.5) – (0.5,[-sqrt(11)-0.5]{}); (0.5,[sqrt(11)+0.5]{}) – ([-sqrt(11)-0.5]{},-0.5); (-0.5,[-sqrt(11)-0.5]{}) – ([sqrt(11)+0.5]{},0.5); (2.4, .84852814) – (-1.3599411, 1.2605607) – (-2.8265056, .47395874) – (-2.4, .84852814); (0,0) circle \[x radius=[sqrt(4)]{}, y radius=[sqrt(9)]{}\]; (0,0) circle \[x radius=[sqrt(4-1.312805)]{}, y radius=[sqrt(9+1.312805)]{}\]; ([sqrt(13)+0.5]{},-0.5) – (-0.5,[sqrt(13)+0.5]{}); ([-sqrt(13)-0.5]{},0.5) – (0.5,[-sqrt(13)-0.5]{}); (0.5,[sqrt(13)+0.5]{}) – ([-sqrt(13)-0.5]{},-0.5); (-0.5,[-sqrt(13)-0.5]{}) – ([sqrt(13)+0.5]{},0.5); (1.4, 2.1424285) – (.94282998, 2.6457345) – (1.8560494, 1.1175561) – (1.4, -2.1424285); ### 4-elliptic periodic trajectories {#elliptic-periodic-trajectories-2 .unnumbered} There is a $4$-elliptic periodic trajectory without being $4$-periodic of the billiard within [eq:ellipse]{}, with a non-degenerate caustic ${\pazocal{C}}_{\gamma}$ if and only if, according to Theorem \[th:elliptic-cayley\], one of the following is satisfied: - the caustic is an ellipse, with $\gamma \in (0,a)$ and $D_3D_1-D^{2}_2=0$, i.e. $$(a+b)^4\gamma^4-4ab(a+b)(a-b)^2\gamma^3-2a^2b^2(a+b)(5a-3b)\gamma^2-4a^3b^3(a+b)\gamma+a^4b^4=0;$$ - the caustic is an ellipse, with $\gamma \in (-b,0)$ and $E_3E_1-E^{2}_2=0$, i.e. $$(a+b)^4\gamma^4+4ab(a+b)(a-b)^2\gamma^3+2a^2b^2(a+b)(3a-5b)\gamma^2+4a^3b^3(a+b)\gamma+a^4b^4=0;$$ - the caustic is a hyperbola and $C_3C_1-C^{2}_2=0$, i.e. $$(a^2-6ab+b^2)(a+b)^2\gamma^4+4ab(a-b)(a+b)^2\gamma^3 +2a^2b^2(3a^2+2ab+3b^2)\gamma^2+4a^3b^3(a-b)\gamma+a^4b^4 =0.$$ Each real solution $\gamma$ for the above equations for some fixed values of $a$ and $b$ will produce a $4$-elliptic periodic trajectory which is not $4$-periodic. Some examples are shown in Figure \[fig:4-elliptic-periodic\]. (-3.5,-3.5) rectangle (3.5,3.5); (0,0) circle \[x radius=[sqrt(5)]{}, y radius=[sqrt(3)]{}\]; (0,0) circle \[x radius=[sqrt(5-4.6212)]{}, y radius=[sqrt(3+4.6212)]{}\]; ([sqrt(8)+2]{},-2) – (-2,[sqrt(8)+2]{}); ([-sqrt(8)-2]{},2) – (2,[-sqrt(8)-2]{}); (2,[sqrt(8)+2]{}) – ([-sqrt(8)-2]{},-2); (-2,[-sqrt(8)-2]{}) – ([sqrt(8)+2]{},2); (2, -0.774596) – (.718617, 1.64017) – (.544007, -1.6800) – (1.51403, 1.27460) – (2, 0.774596); (-3.5,-3.5) rectangle (3.5,3.5); (0,0) circle \[x radius=[sqrt(5)]{}, y radius=[sqrt(3)]{}\]; hyperbola lambda=-3.0243 plot ([sqrt(5+3.0243)\*sqrt(1-/(3-3.0243)]{},); plot ([-sqrt(5+3.0243)\*sqrt(1-/(3-3.0243)]{},); ([sqrt(8)+2]{},-2) – (-2,[sqrt(8)+2]{}); ([-sqrt(8)-2]{},2) – (2,[-sqrt(8)-2]{}); (2,[sqrt(8)+2]{}) – ([-sqrt(8)-2]{},-2); (-2,[-sqrt(8)-2]{}) – ([sqrt(8)+2]{},2); (2, -0.774596) – (1.35590, -1.37729) – (-2.18, -.17) – (2.1, 0.072622) – (-2, 0.774596); ### 5-elliptic periodic trajectories {#elliptic-periodic-trajectories-3 .unnumbered} According to Theorem \[th:elliptic-cayley\], there is a $5$-elliptic periodic trajectory without being $5$-periodic of the billiard within [eq:ellipse]{}, with a non-degenerate caustic ${\pazocal{C}}_{\gamma}$ if and only if one of the following is satisfied: - the caustic is an ellipse, with $\gamma \in (0,a)$ or a hyperbola and $E_2E_4-E^{2}_3=0$, i.e. $$\begin{aligned} & (5a^2-10ab+b^2)(a+b)^4\gamma^6+2ab(5a-3b)(a+b)^4\gamma^5\\ &-a^2b^2(a+b)(9a^3-45a^2b-5ab^2-15b^3)\gamma^4-4a^3b^3(a+b)(9a^2-10ab+5b^2)\gamma^3\\ &-a^4b^4(a+b)(29a-15b)\gamma^2-6a^5b^5(a+b)\gamma+a^6b^6=0;\end{aligned}$$ - the caustic is an ellipse, with $\gamma \in (-b,0)$ or a hyperbola and $D_2D_4-D^{2}_3=0$, i.e. $$\begin{aligned} &(a^2-10ab+5b^2)(a+b)^4\gamma^6+2ab(3a-5b)(a+b)^4\gamma^5\\ &+a^2b^2(a+b)(15a^3+5a^2b+45ab^2-9b^3)\gamma^4+4a^3b^3(a+b)(5a^2-10ab+9b^2)\gamma^3\\ &+a^4b^4(a+b)(15a-29b)\gamma^2+6a^5b^5(a+b)\gamma+a^6b^6=0.\end{aligned}$$ Each real solution $\gamma$ for the above equations for some fixed values of $a$ and $b$ will produce a $5$-elliptic periodic trajectory which is not $5$-periodic. Some examples are shown in Figure \[fig:5-elliptic-periodic\]. (-4,-4) rectangle (4,4); (0,0) circle \[x radius=[sqrt(7)]{}, y radius=[sqrt(4)]{}\]; (0,0) circle \[x radius=[sqrt(7+3.3848)]{}, y radius=[sqrt(4-3.3848)]{}\]; ([sqrt(11)+2]{},-2) – (-2,[sqrt(11)+2]{}); ([-sqrt(11)-2]{},2) – (2,[-sqrt(11)-2]{}); (2,[sqrt(11)+2]{}) – ([-sqrt(11)-2]{},-2); (-2,[-sqrt(11)-2]{}) – ([sqrt(11)+2]{},2); (1, -1.852) – (2.479, -.699) – (-2.373, -.8846) – (-1.613, -1.585) – (2.569, -.4789) – (-1, -1.852); (-4,-4) rectangle (4,4); (0,0) circle \[x radius=[sqrt(3)]{}, y radius=[sqrt(7)]{}\]; hyperbola lambda=3.4462 plot (,[sqrt(7+3.4462)\*sqrt(1-/(3-3.4462)]{}); plot (,[-sqrt(7+3.4462)\*sqrt(1-/(3-3.4462)]{}); ([sqrt(10)+2]{},-2) – (-2,[sqrt(10)+2]{}); ([-sqrt(10)-2]{},2) – (2,[-sqrt(10)-2]{}); (2,[sqrt(10)+2]{}) – ([-sqrt(10)-2]{},-2); (-2,[-sqrt(10)-2]{}) – ([sqrt(10)+2]{},2); (.5, -2.53311) – (-.543390, 2.51218) – (-1.43296, 1.48620) –(0.0130805, -2.64568) – (1.49069, 1.34723) – (.5, 2.53311); ### Discriminantly separable polynomials and elliptic periodicity {#discriminantly-separable-polynomials-and-elliptic-periodicity .unnumbered} Since the case $n=2$ is trivial, we start with the case $n=3$. From [eq:3-elliptic-periodic1]{} and [eq:3-elliptic-periodic2]{}, we have: $$\begin{gathered} \mathsf{G}_{1}(a,b,\gamma)=-(a+b)(3a-b)\gamma^2-2ab(a+b)\gamma+a^2b^2, \\ \mathsf{G}_{2}(a,b,\gamma)= (a+b)(a-3b)\gamma^2+2ab(a+b)\gamma+a^2b^2,\end{gathered}$$ and we calculate the discriminants, which factorize as follows: $$\mathcal{D_{\gamma}}\mathsf{G}_{1}=16a^3b^2(a+b), \quad \mathcal{D_{\gamma}}\mathsf{G}_{2}=16b^3a^2(a+b).$$ Similarly, for $n=4$, we have: $$\begin{aligned} \mathsf{G}_{3}(a,b,\gamma)=&(a+b)^4\gamma^4-4ab(a+b)(a-b)^2\gamma^3-2a^2b^2(a+b)(5a-3b)\gamma^2-4a^3b^3(a+b)\gamma+a^4b^4, \\ \mathsf{G}_{4}(a,b,\gamma)=&(a+b)^4\gamma^4+4ab(a+b)(a-b)^2\gamma^3+2a^2b^2(a+b)(3a-5b)\gamma^2+4a^3b^3(a+b)\gamma+a^4b^4, \\ \mathsf{G}_{5}(a,b,\gamma)=&(a^2-6ab+b^2)(a+b)^2\gamma^4+4ab(a-b)(a+b)^2\gamma^3 +2a^2b^2(3a^2+2ab+3b^2)\gamma^2 \\& +4a^3b^3(a-b)\gamma+a^4b^4.\end{aligned}$$ The discriminants of these polynomials factorize as follows: $$\begin{gathered} \mathcal{D_{\gamma}}\mathsf{G}_{3}=-2^{16}a^{16}b^{14}(8a^2+8ab+27b^2)(a+b)^4, \\ \mathcal{D_{\gamma}}\mathsf{G}_{4}=-2^{16}a^{14}b^{16}(27a^2+8ab+8b^2)(a+b)^4, \\ \mathcal{D_{\gamma}}\mathsf{G}_{5}=2^{12}(32a^6-491a^5b-439a^4b^2+194a^3b^3-62a^2b^4-39ab^5+5b^6)(a+b)^3b^{15}a^{12}.\end{gathered}$$ Using the transformation $(a,b)\mapsto (a,\hat{b})$, where $\hat{b}=\dfrac{b}{a}$, we get: $$\begin{aligned} &\mathcal{D_{\gamma}}\mathsf{G}_{1}=16a^6\hat{b}^2(1+\hat{b}), \\ &\mathcal{D_{\gamma}}\mathsf{G}_{2}=16a^5\hat{b}^3(1+\hat{b}), \\ &\mathcal{D_{\gamma}}\mathsf{G}_{3}=-2^{16}a^{36}\hat{b}^{14}(8+8\hat{b}+27\hat{b}^2)(1+\hat{b})^4, \\ &\mathcal{D_{\gamma}}\mathsf{G}_{4}=-2^{16}a^{36}\hat{b}^{16}(27+8\hat{b}+8\hat{b}^2)(1+\hat{b})^4, \\ &\mathcal{D_{\gamma}}\mathsf{G}_{5}=2^{12}a^{36}\hat{b}^{15}(32-491\hat{b}-439\hat{b}^2+194\hat{b}^3-62\hat{b}^4-39\hat{b}^5+5\hat{b}^6)(1+\hat{b})^3.\end{aligned}$$ Polynomial equations {#sec:polynomial} ==================== Now we want to express the periodicity conditions for billiard trajectories in the Minkowski plane in terms of polynomial functions equations. \[th:polynomial\] The billiard trajectories within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\gamma}$ are $n$-periodic if and only if there exists a pair of real polynomials $p_{d_1}$, $q_{d_2}$ of degrees $d_1$, $d_2$ respectively, and satisfying the following: - if $n=2m$ is even, then $d_1=m$, $d_2=m-2$, and $$p_{m}^2(s) - s\left(s-\frac1a\right)\left(s+\frac1b\right)\left( s-\frac1{\gamma}\right) {q}_{m-2}^2(s)=1;$$ - if $n=2m+1$ is odd, then $d_1=m$, $d_2=m-1$, and $$\left( s-\frac1{\gamma}\right)p_m^2(s) - s\left(s-\frac1a\right)\left(s+\frac1b\right)q_{m-1}^2(s)=-\sign\gamma.$$ We note first that the proof of Theorem \[th:cayley-billiard\] implies that there is a non-trivial linear combination of the bases for $n$ even, or for $n$ odd, with the zero of order $n$ at $x=0$. \(a) For $n=2m$, from there we get that there are real polynomials $p_m^*(x)$ and $q_{m-2}^*(x)$ of degrees $m$ and $m-2$ respectively, such that the expression $$p_{m}^*(x)-q_{m-2}^*(x)\sqrt{\varepsilon(a-x)(b+x)(\gamma-x)}$$ has a zero of order $2m$ at $x=0$. Multiplying that expression by $$p_{m}^*(x)+q_{m-2}^*(x)\sqrt{\varepsilon(a-x)(b+x)(\gamma-x)},$$ we get that the polynomial $(p_{m}^*(x))^2-\varepsilon(a-x)(b+x)(\gamma-x)(q_{m-2}^*(x))^{2}$ has a zero of order $2m$ at $x=0$. Since the degree of that polynomial is $2m$, is follows that: $$(p_{m}^*(x))^2-\varepsilon(a-x)(b+x)(\gamma-x)(q_{m-2}^*(x))^{2}=cx^{2m},$$ for some constant $c$. Notice that $c$ is positive, since it equals the square of the leading coefficient of $p_m^*$. Dividing the last relation by $cx^{2m}$ and introducing $s=1/x$, we get the requested relation. \(b) On the other hand, for $n=2m+1$, we get that there are real polynomials $p_m^*(x)$ and $q_{m-1}^*(x)$ of degrees $m$ and $m-1$ respectively, such that the expression $$p_{m}^*(x)-q_{m-1}^*(x)\frac{\sqrt{\varepsilon(a-x)(b+x)(\gamma-x)}}{\gamma-x}$$ has a zero of order $2m+1$ at $x=0$. Multiplying that expression by $$(\gamma-x) \left( p_{m}^*(x)+q_{m-1}^*(x)\frac{\sqrt{\varepsilon(a-x)(b+x)(\gamma-x)}}{\gamma-x} \right) ,$$ we get that the polynomial $(\gamma-x)(p_{m}^*(x))^2-\varepsilon(a-x)(b+x)(q_{m-1}^*(x))^{2}$ has a zero of order $2m+1$ at $x=0$. Since the degree of that polynomial is $2m+1$, is follows that: $$(\gamma-x)(p_{m}^*(x))^2-\varepsilon(a-x)(b+x)(q_{m-1}^*(x))^{2}=cx^{2m+1},$$ for some constant $c$. Notice that $c$ is negative, since it equals the opposite of the square of the leading coefficient of $p_m^*$. Dividing the last relation by $-\varepsilon cx^{2m+1}$ and introducing $s=1/x$, we get the requested relation. \[cor:pell-periodic\] If the billiard trajectories within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\gamma}$ are $n$-periodic, then there exist real polynomials $\hat{p}_n$ and $\hat{q}_{n-2}$ of degrees $n$ and $n-2$ respectively, which satisfy the Pell equation: $$\label{eq:pell} \hat{p}_{n}^2(s)-s\left(s-\frac1a\right)\left(s+\frac1b\right)\left( s-\frac1{\gamma}\right)\hat{q}_{n-2}^2(s)=1.$$ For $n=2m$, take $\hat{p}_n=2p_{m}^2-1$ and $\hat{q}_{n-2}=2p_mq_{m-2}$. For $n=2m+1$, we set $\hat{p}_n=2\left(\gamma s-1\right)p_{m}^2+\sign\gamma$ and $\hat{q}_{n-2}=2p_mq_{m-1}$. \[th:polynomial-elliptic\] The billiard trajectories within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\gamma}$ are elliptic $n$-periodic without being $n$-periodic if and only if there exists a pair of real polynomials $p_{d_1}$, $q_{d_2}$ of degrees $d_1$, $d_2$ respectively, and satisfying the following: - ${\pazocal{C}}_{\gamma}$ is an ellipse, $0<\gamma<a$, and - $n=2m$ is even, $d_1=d_2=m-1$, $$s\left(s-\frac1a\right)p_{m-1}^2(s) -\left(s+\frac1b\right)\left( s-\frac1{\gamma}\right)q_{m-1}^2(s)=1;$$ - $n=2m+1$ is odd, $ d_1=m$, $d_2=m-1$, $$\left(s+\frac1b\right) p_{m}^2(s) -s\left(s-\frac1a\right)\left( s-\frac1{\gamma}\right)q_{m-1}^2(s)=1;$$ - ${\pazocal{C}}_{\gamma}$ is an ellipse, $-b<\gamma<0$, and - $n=2m$ is even, $d_1=d_2=m-1$, $$s\left(s+\frac1b\right)p_{m-1}^2(s) -\left(s-\frac1a\right)\left( s-\frac1{\gamma}\right)q_{m-1}^2(s)=1;$$ - $n=2m+1$ is odd, $ d_1=m$, $d_2=m-1$, $$\left(s-\frac1a\right) p_{m}^2(s) -s\left(s+\frac1b\right)\left(s-\frac{1}{\gamma}\right)q_{m-1}^2(s)=-1;$$ - ${\pazocal{C}}_{\gamma}$ is a hyperbola and $n=2m$ is even, $d_1=d_2=m-1$, $$\left( s-\frac{1}{\gamma}\right)p_{m-1}^2(s) -s\left(s-\frac1a\right)\left(s+\frac1b\right)q_{m-1}^2(s)=-\sign\gamma;$$ - ${\pazocal{C}}_{\gamma}$ is a hyperbola, $n=2m+1$ is odd, $d_1=m$, $d_2=m-1$, $$\left(s-\frac1a\right) p_{m}^2(s) -s\left(s+\frac1b\right)\left(s-\frac1{\gamma}\right)q_{m-1}^2(s)=-1;$$ - ${\pazocal{C}}_{\gamma}$ is a hyperbola, $n=2m+1$ is odd, $d_1=m$, $d_2=m-1$, $$\left(s+\frac1b\right) p_{m}^2(s) -s\left(s-\frac1a\right)\left(s-\frac1{\gamma}\right)q_{m-1}^2(s)=1.$$ \(a) For $n=2m$, the proof of Theorem \[th:elliptic-cayley\] implies that there are polynomials $p_{m-1}^*(x)$ and $q_{m-1}^*(x)$ of degrees $m-1$, such that the expression $$p_{m-1}^*(x)-q_{m-1}^*(x)\frac{\sqrt{(a-x)(b+x)(\gamma-x)}}{a-x}$$ has a zero of order $2m$ at $x=0$. Multiplying that expression by $$(a-x)\left(p_{m-1}^*(x)+q_{m-1}^*(x)\frac{\sqrt{(a-x)(b+x)(\gamma-x)}}{a-x}\right),$$ we get that the polynomial $(a-x)(p_{m-1}^*(x))^2-(a-x)(b+x)(q_{m-1}^*(x))^2$ has a zero of order $2m$ at $x=0$. Since the degree of that polynomial is $2m$, is follows that: $$(a-x)(p_{m-1}^*(x))^2-(b+x)(\gamma-x)(q_{m-1}^*(x))^2=cx^{2m},$$ for some constant $c$. Notice that $c$ is positive, since it equals the square of the leading coefficient of $q_{m-1}^*$. Dividing the last relation by $cx^{2m}$ and introducing $s=1/x$, we get the requested relation. For $n=2m+1$, the proof of Theorem \[th:elliptic-cayley\] implies that there are polynomials $p_{m}^*(x)$ and $q_{m-1}^*(x)$ of degrees $m$ and $m-1$, such that the expression $$p_{m}^*(x)-q_{m-1}^*(x)\frac{\sqrt{(a-x)(b+x)(\gamma-x)}}{b+x}$$ has a zero of order $2m+1$ at $x=0$. Multiplying that expression by $$(b+x)\left(p_{m}^*(x)+q_{m-1}^*(x)\frac{\sqrt{(a-x)(b+x)(\gamma-x)}}{b+x}\right) ,$$ we get that the polynomial $(b+x)(p_{m}^*(x))^2-(a-x)(\gamma-x)(q_{m-1}^*(x))^2$ has a zero of order $2m+1$ at $x=0$. Since the degree of that polynomial is $2m+1$, is follows that: $$(b+x)(p_{m}^*(x))^2-(a-x)(\gamma-x)(q_{m-1}^*(x))^2=cx^{2m+1}$$ for some constant $c$. Notice that $c$ is positive, since it equals the square of the leading coefficient of $p_{m}^*$. Dividing the last relation by $cx^{2m+1}$ and introducing $s=1/x$, we get the requested relation. \(b) For $n=2m$, the proof of Theorem \[th:elliptic-cayley\] implies that there are real polynomials $p_{m-1}^*(x)$ and $q_{m-1}^*(x)$ of degrees $m-1$, such that the expression $$p_{m-1}^*(x)-q_{m-1}^*(x)\frac{\sqrt{-(a-x)(b+x)(\gamma-x)}}{b+x}$$ has a zero of order $2m$ at $x=0$. Multiplying that expression by $$(b+x)\left(p_{m-1}^*(x)+q_{m-1}^*(x)\frac{\sqrt{-(a-x)(b+x)(\gamma-x)}}{b+x}\right),$$ we get that the polynomial $(b+x)(p_{m-1}^*(x))^2+(a-x)(\gamma-x)(q_{m-1}^*(x))^2$ has a zero of order $2m$ at $x=0$. Since the degree of that polynomial is $2m$, is follows that: $$(b+x)(p_{m-1}^*(x))^2+(a-x)(\gamma-x)(q_{m-1}^*(x))^2=cx^{2m},$$ for some constant $c$. Notice that $c$ is positive, since it equals to the square of the leading coefficient of $q_{m-1}^*$. Dividing the last relation by $cx^{2m}$ and introducing $s=1/x$, we get the requested relation. For $n=2m+1$, the proof of Theorem \[th:elliptic-cayley\] implies that there are polynomials $p_{m}^*(x)$ and $q_{m-1}^*(x)$ of degrees $m$ and $m-1$, such that the expression $$p_{m}^*(x)-q_{m-1}^*(x)\frac{\sqrt{-(a-x)(b+x)(\gamma-x)}}{a-x}$$ has a zero of order $2m+1$ at $x=0$. Multiplying that expression by $$(a-x)\left(p_{m}^*(x)+q_{m-1}^*(x)\frac{\sqrt{-(a-x)(b+x)(\gamma-x)}}{a-x}\right) ,$$ we get that the polynomial $(a-x)(p_{m}^*(x))^2+(b+x)(\gamma-x)(q_{m-1}^*(x))^2$ has a zero of order $2m+1$ at $x=0$. Since the degree of that polynomial is $2m+1$, is follows that: $$(a-x)(p_{m}^*(x))^2+(b+x)(\gamma-x)(q_{m-1}^*(x))^2=cx^{2m+1}$$ for some constant $c$. Notice that $c$ is negative, since it is opposite to the square of the leading coefficient of $p_{m}^*$. Dividing the last relation by $-cx^{2m+1}$ and introducing $s=1/x$, we get the requested relation. For (c), the proof of Theorem \[th:elliptic-cayley\] implies that there are polynomials real $p_{m}^*(x)$ and $q_{m-1}^*(x)$ of degrees $m$ and $m-1$, such that the expression $$p_{m}^*(x)-q_{m-1}^*(x)\frac{\sqrt{\varepsilon(a-x)(b+x)(\gamma-x)}}{\gamma-x}$$ has a zero of order $2m+1$ at $x=0$. Multiplying that expression by $$(\gamma-x)\left(p_{m}^*(x)+q_{m-1}^*(x)\frac{\sqrt{\varepsilon(a-x)(b+x)(\gamma-x)}}{\gamma-x}\right) ,$$ we get that the polynomial $(\gamma-x)(p_{m}^*(x))^2-\varepsilon(a-x)(b+x)(q_{m-1}^*(x))^2$ has a zero of order $2m+1$ at $x=0$. Since the degree of that polynomial is $2m+1$, is follows that: $$(\gamma-x)(p_{m}^*(x))^2-\varepsilon(a-x)(b+x)(q_{m-1}^*(x))^2=cx^{2m+1}$$ for some constant $c$. Notice that $c$ is negative, since it is opposite to the square of the leading coefficient of $p_{m}^*$. Dividing the last relation by $-\varepsilon cx^{2m+1}$ and introducing $s=1/x$, we get the requested relation. \(d) The proof of Theorem \[th:elliptic-cayley\] implies that there are real polynomials $p_{m}^*(x)$ and $q_{m-1}^*(x)$ of degrees $m$ and $m-1$, such that the expression $$p_{m}^*(x)-q_{m-1}^*(x)\frac{\sqrt{\varepsilon(a-x)(b+x)(\gamma-x)}}{a-x}$$ has a zero of order $2m+1$ at $x=0$. Multiplying that expression by $$(a-x)\left(p_{m}^*(x)+q_{m-1}^*(x)\frac{\sqrt{\varepsilon(a-x)(b+x)(\gamma-x)}}{a-x}\right) ,$$ we get that the polynomial $(a-x)(p_{m}^*(x))^2-\varepsilon(b+x)(\gamma-x)(q_{m-1}^*(x))^2$ has a zero of order $2m+1$ at $x=0$. Since the degree of that polynomial is $2m+1$, is follows that: $$(a-x)(p_{m}^*(x))^2-\varepsilon(b+x)(\gamma-x)(q_{m-1}^*(x))^2=cx^{2m+1}$$ for some constant $c$. Notice that $c$ is negative, since it is opposite to the square of the leading coefficient of $p_{m}^*$. Dividing the last relation by $-cx^{2m+1}$ and introducing $s=1/x$, we get the requested relation. \(e) For $n=2m+1$, the proof of Theorem \[th:elliptic-cayley\] implies that there are real polynomials $p_{m}^*(x)$ and $q_{m-1}^*(x)$ of degrees $m$ and $m-1$, such that the expression $$p_{m}^*(x)-q_{m-1}^*(x)\frac{\sqrt{\varepsilon(a-x)(b+x)(\gamma-x)}}{b+x}$$ has a zero of order $2m+1$ at $x=0$. Multiplying that expression by $$(b+x)\left(p_{m}^*(x)+q_{m-1}^*(x)\frac{\sqrt{\varepsilon(a-x)(b+x)(\gamma-x)}}{b+x}\right) ,$$ we get that the polynomial $(b+x)(p_{m}^*(x))^2-\varepsilon(a-x)(\gamma-x)(q_{m-1}^*(x))^2$ has a zero of order $2m+1$ at $x=0$. Since the degree of that polynomial is $2m+1$, is follows that: $$(b+x)(p_{m}^*(x))^2-\varepsilon(a-x)(\gamma-x)(q_{m-1}^*(x))^2=cx^{2m+1}$$ for some constant $c$. Notice that $c$ is positive, since it equals the square of the leading coefficient of $p_{m}^*$. Dividing the last relation by $cx^{2m+1}$ and introducing $s=1/x$, we get the requested relation. After corollary \[cor:pell-periodic\] and relation [eq:pell]{}, we see that the Pell equations arise as the functional polynomial conditions for periodicity. Let us recall some important properties of the solutions of the Pell equations. Classical Extremal Polynomials and Caustics {#sec:extremal} =========================================== Fundamental Properties of Extremal Polynomials ---------------------------------------------- From the previous section we know that the Pell equation plays a key role in functional-polynomial formulation of periodicity conditions in the Minkowski plane. The solutions of the Pell equation are so-called extremal polynomials. Denote $\{c_1,c_2,c_3,c_4\}=\{0, 1/a, -1/b, 1/\gamma\}$ with the ordering $c_{1}<c_{2}<c_{3}<c_{4}$. The polynomials $\hat {p}_n$ are so called *generalized Chebyshev polynomials* on two intervals $[c_1, c_2]\cup [c_3, c_4]$, with an appropriate normalization. Namely, one can consider the question of finding the monic polynomial of certain degree $n$ which minimizes the maximum norm on the union of two intervals. Denote such a polynomial as $\hat P_n$ and its norm $L_n$. The fact that polynomial $\hat {p}_n$ is a solution of the Pell equation on the union of intervals $[c_1, c_2]\cup [c_3, c_4]$ is equivalent to the following conditions: - $\hat {p}_n=\hat {P}_n/\pm L_n$ - the set $[c_1, c_2]\cup [c_3, c_4]$ is the maximal subset of $\mathbf R$ for which $\hat {P}_n$ is the minimal polynomial in the sense above. Chebyshev was the first who considered a similar problem on one interval, and this was how celebrated Chebyshev polynomials emerged in XIXth century. Let us recall a fundamental result about generalized Chebyshev polynomials [@AhiezerAPPROX; @Akh4]. A polynomial $P_{n}$ of degree $n$ satisfies a Pell equation on the union of intervals $[c_{1}, c_{2}]\cup[c_{3}, c_{4}]$ if and only if there exists an integer $n_{1}$ such that the equation holds: $$\label{eq:KLN} n_{1}\int_{c_{2}}^{c_{3}}\hat f(s)ds=n\int_{c_{4}}^{\infty}\hat f(s)ds, \quad \hat f(s)=\frac{C}{\sqrt{\prod_{i=1}^4(s-c_i)}}.$$ (Here $C$ is a nonessential constant.) The modulus of the polynomial reaches its maximal values $L_{n}$ at the points $c_{i}:$ $|P_{n}(c_{i})|=L_{n}$.\ In addition, there are exactly $\tau_{1}=n-n_{1}-1$ internal extremal points of the interval $[c_{3}, c_{4}]$ where $|P_{n}|$ reaches the value $L_{n}$, and there are $\tau_{2}=n_{1}-1$ internal extremal points of $[c_{1}, c_{2}]$ with the same property. We call the pair $(n, n_1)$ the partition and $(\tau_1, \tau_2)$ the signature of the generalized Chebyshev polynomial $P_n$. Now we are going to formulate and prove the main result of this Section, which relates $n_1, n_2$ the numbers of reflections off relativistic ellipses and off relativistic hyperbolas respectively with the partition and the signature of the related solution of a Pell equation. \[th:impactwinding\] Given a periodic billiard trajectory with period $n=n_{1}+n_{2}$, where $n_{1}$ is the number of reflections off relativistic ellipses, $n_{2}$ the number of reflections off the relativistic hyperbolas, then the partition corresponding to this trajectory is $(n, n_{1})$. The corresponding extremal polynomial $\hat{p}_{n}$ of degree $n$ has $n_{1}-1$ internal extremal points in the first interval and $n-n_{1}-1=n_{2}-1$ internal extremal points in the second interval. Recall that $c_{1}<c_{2}<c_{3}<c_{4}$. From the equation [InteEqn]{}, one has: $$\label{IntEq2} n_{1}\int_{\alpha_{1}}^{0}f(x)dx+n_{2}\int_{\beta_{1}}^{0}f(x)dx=0$$ where $\alpha_{1}$ is the largest negative value in $\{a, -b, \gamma\}$ and $\beta_{1}$ the smallest positive value in $\{a, -b, \gamma\}$. The proof decomposes to the following cases: - Case 1: ${\pazocal{C}}_{\gamma}$ is an ellipse and $\gamma <0$, shown in Figure \[fig:case1\]; (0,1)–(5,1) ; (0,0)–(5,0); (0,-1)–(5,-1); (0, 1) circle (2pt); (5, 1) circle (2pt); (1.5, 1) circle (2pt); (3.33, 1) circle (2pt); (0, 0) circle (2pt); (5, 0) circle (2pt); (1.5, 0) circle (2pt); (3.33, 0) circle (2pt); (0, -1) circle (2pt); (5, -1) circle (2pt); (1.5, -1) circle (2pt); (3.33, -1) circle (2pt); (0,.7) node [$-b$]{}; (5,.7) node [${a}$]{}; (1.5,.7) node [${\gamma}$]{}; (3.33,.7) node [${0}$]{}; (0,-.4) node [$\dfrac{1}{{\gamma}}$]{}; (5,-.4) node [$\dfrac{1}{{a}}$]{}; (1.5,-.4) node [$-\dfrac{1}{{b}}$]{}; (3.33,-.4) node [$0$]{}; (0,-1.3) node [${c}_{1}$]{}; (5,-1.3) node [${c}_{4}$]{}; (1.5,-1.3) node [${c}_{2}$]{}; (3.33,-1.3) node [${c}_{3}$]{}; - Case 2: ${\pazocal{C}}_{\gamma}$ is an ellipse and $\gamma >0$, shown in Figure \[fig:case2\]; (0,1)–(5,1) ; (0,0)–(5,0); (0,-1)–(5,-1); (0, 1) circle (2pt); (5, 1) circle (2pt); (1.5, 1) circle (2pt); (3.33, 1) circle (2pt); (0, 0) circle (2pt); (5, 0) circle (2pt); (1.5, 0) circle (2pt); (3.33, 0) circle (2pt); (0, -1) circle (2pt); (5, -1) circle (2pt); (1.5, -1) circle (2pt); (3.33, -1) circle (2pt); (0,.7) node [$-{b}$]{}; (5,.7) node [${a}$]{}; (1.5,.7) node [$0$]{}; (3.33,.7) node [${\gamma}$]{}; (0,-.4) node [$-\dfrac{1}{{b}}$]{}; (5,-.4) node [$\dfrac{1}{{\gamma}}$]{}; (1.5,-.4) node [$0$]{}; (3.33,-.4) node [$\dfrac{1}{{a}}$]{}; (0,-1.3) node [${c}_{1}$]{}; (5,-1.3) node [${c}_{4}$]{}; (1.5,-1.3) node [${c}_{2}$]{}; (3.33,-1.3) node [${c}_{3}$]{}; - Case 3(i): ${\pazocal{C}}_{\gamma}$ is a hyperbola and $\gamma <-b$, shown in Figure \[fig:case3i\]; (0,1)–(5,1) ; (0,0)–(5,0); (0,-1)–(5,-1); (0, 1) circle (2pt); (5, 1) circle (2pt); (1.5, 1) circle (2pt); (3.33, 1) circle (2pt); (0, 0) circle (2pt); (5, 0) circle (2pt); (1.5, 0) circle (2pt); (3.33, 0) circle (2pt); (0, -1) circle (2pt); (5, -1) circle (2pt); (1.5, -1) circle (2pt); (3.33, -1) circle (2pt); (0,.7) node [${\gamma}$]{}; (5,.7) node [${a}$]{}; (1.5,.7) node [$-{b}$]{}; (3.33,.7) node [$0$]{}; (0,-.4) node [$-\dfrac{1}{{b}}$]{}; (5,-.4) node [$\dfrac{1}{{a}}$]{}; (1.5,-.4) node [$\dfrac{1}{{\gamma}}$]{}; (3.33,-.4) node [$0$]{}; (0,-1.3) node [${c}_{1}$]{}; (5,-1.3) node [${c}_{4}$]{}; (1.5,-1.3) node [${c}_{2}$]{}; (3.33,-1.3) node [${c}_{3}$]{}; - Case 3(ii): ${\pazocal{C}}_{\gamma}$ is a hyperbola and $\gamma >a$, shown in Figure \[fig:case3ii\]. (0,1)–(5,1) ; (0,0)–(5,0); (0,-1)–(5,-1); (0, 1) circle (2pt); (5, 1) circle (2pt); (1.5, 1) circle (2pt); (3.33, 1) circle (2pt); (0, 0) circle (2pt); (5, 0) circle (2pt); (1.5, 0) circle (2pt); (3.33, 0) circle (2pt); (0, -1) circle (2pt); (5, -1) circle (2pt); (1.5, -1) circle (2pt); (3.33, -1) circle (2pt); (0,.7) node [$-{b}$]{}; (5,.7) node [${\gamma}$]{}; (1.5,.7) node [$0$]{}; (3.33,.7) node [${a}$]{}; (0,-.4) node [$-\dfrac{1}{{b}}$]{}; (5,-.4) node [$\dfrac{1}{{a}}$]{}; (1.5,-.4) node [$0$]{}; (3.33,-.4) node [$\dfrac{1}{{\gamma}}$]{}; (0,-1.3) node [${c}_{1}$]{}; (5,-1.3) node [${c}_{4}$]{}; (1.5,-1.3) node [${c}_{2}$]{}; (3.33,-1.3) node [${c}_{3}$]{}; We provide proof in the Case 1. The proofs for other cases are analogous. Equation [IntEq2]{} is equivalent to $$\begin{aligned} 0 &= n_{1}\int_{\gamma}^{0}f(x)dx + n_{1}\int_{0}^{a}f(x)dx+n_{2}\int_{a}^{0}f(x)dx-n_{1}\int_{0}^{a}f(x)dx \\& = n_{1}\int_{\gamma}^{a}f(x)dx+(n_{1}+n_{2})\int_{a}^{0}f(x)dx,\end{aligned}$$ thus $$n_{1}\int_{\gamma}^{a}f(x)dx=(n_{1}+n_{2})\int_{0}^{a}f(x)dx.$$ Since the cycles around the cuts on the elliptic curve are homologous: $$\int_{\gamma}^{a}f(x)dx=\int_{-\infty}^{-b}f(x)dx,$$ [IntEq2]{} is equivalent to: $$n_{1}\int_{-\infty}^{-b}f(x)dx=(n_{1}+n_{2})\int_{0}^{a}f(x)dx.$$ Substituting: $s=\dfrac{1}{x}$, $c_{1}=\dfrac{1}{\gamma}$, $c_{2}=-\dfrac{1}{b}$, $c_{3}=0$, $c_{4}=\dfrac{1}{a}$ (see Figure \[fig:case1\]), we get that $$n_{1}\int_{-1/b}^{0}\tilde{f}(s)ds=(n_{1}+n_{2})\int_{1/a}^{\infty}\tilde{f}(s)ds$$ is equivalent to $$\label{TagEq} n_{1}\int_{c_{2}}^{c_{3}}\tilde{f}(s)ds=(n_{1}+n_{2})\int_{c_{4}}^{\infty}\tilde{f}(s)ds,$$ where $\tilde f(s)ds$ is obtained from $f(x)dx$ by the substitution. In particular, for $n=3$, if the caustic ${\pazocal{C}}_{\gamma}$ is an ellipse with $\gamma<0$, then $n_{1}=1$. Such polynomials and corresponding partitions $(3,1)$ do not arise in the study of Euclidean billiard trajectories. On the other hand, if the caustic ${\pazocal{C}}_{\gamma}$ is an ellipse with $\gamma>0$, we have $n_{1}=2$. Such polynomials for $\gamma>0$ can be explicitly expressed in terms of the Zolatarev polynomials, see Proposition \[propZolot\]. Since their partition is $(3,2)$, they appeared before in the Euclidean case (see [@DragRadn2019rcd]). The corresponding extremal polynomials $\hat{p}_{3}$ in both cases $\gamma<0$ and $\gamma>0$ are shown in Figure \[fig:ana5\]. We will provide in Proposition \[prop\] the explicit formulae for such polynomials in terms of the general Akhiezer polynomials. (0,2)–(5.5,2); (0,0)–(5.5,0); (0,-2)–(5.5,-2); (0,2)–(0,-2); (1.22,-2)–(1.22,2); (2.65,-2)–(2.65,2); (5.5,-2)–(5.5,2); (0,-2)–(5.5,-2); plot \[smooth, tension=.8\] coordinates [ (0,-2) (2,2.98) (4,-2) (5.5,1.98)]{}; (2.65,-2) circle (2.5pt); (2.65,0) circle (2.5pt); (2.65,2) circle (2.5pt); (1.22,2) circle (2.5pt); (1.22,0) circle (2.5pt); (1.22,-2) circle (2.5pt); (0, 0) circle (2.5pt); (0,2) circle (2.5pt); (0,-2) circle (2.5pt); (5.5, 0) circle (2.5pt); (5.5,2) circle (2.5pt); (5.5,-2) circle (2.5pt); (-.4,2) node [$1$]{}; (-.5,-2) node [$-1$]{}; (-.2,-.2) node [${c}_{1}$]{}; (1,-.2) node [${c}_{2}$]{}; (2.45,-.2) node [${c}_{3}$]{}; (5.8,-.2) node [${c}_{4}$]{}; (0,2)–(6,2); (0,0)–(6,0); (0,-2)–(6,-2); (0,2)–(0,-2); (6,-2)–(6,2); (3.05,-2)–(3.05,2); (4.68,-2)–(4.68,2); (0,-2)–(6,-2); plot \[smooth, tension=.8\] coordinates [ (0,-2) (1.5,1.98) (3.8,-3) (6,1.98)]{}; (3.05,-2) circle (2.5pt); (3.05,0) circle (2.5pt); (3.05,2) circle (2.5pt); (4.68,-2) circle (2.5pt); (4.68,0) circle (2.5pt); (4.68,2) circle (2.5pt); (0, 0) circle (2.5pt); (0,2) circle (2.5pt); (0,-2) circle (2.5pt); (6, 0) circle (2.5pt); (6,2) circle (2.5pt); (6,-2) circle (2.5pt); (-.4,2) node [$1$]{}; (-.5,-2) node [$-1$]{}; (-.2,-.2) node [${c}_{1}$]{}; (2.9,-.2) node [${c}_{2}$]{}; (4.5,-.2) node [${c}_{3}$]{}; (5.8,-.2) node [${c}_{4}$]{}; Let us recall that the Chebyshev polynomials $T_n(x), n= 0, 1, 2,\dots$ defined by the recursion: $$\label{eq:cheb1} T_0(x)=1, \quad T_1(x)=x,\quad T_{n+1}(x)+T_{n-1}(x)=2xT_n(x),$$ for $n=1, 2\dots$ can be parameterized as $$\label{eq:cheb2} T_n(x)=\cos n\phi,\quad x=\cos\phi,$$ or, alternatively: $$\label{eq:cheb3} T_n(x)=\frac{1}{2}\left(v^n+\frac{1}{v^n}\right), \quad x=\frac{1}{2}\left(v+\frac{1}{v}\right).$$ Denote $L_0=1$ and $L_n=2^{1-n}, n=1, 2,\dots$. Then the Chebyshev Theorem states that the polynomials $L_nT_n(x)$ are characterized as the solutions of the following minmax problem: [*find the polynomial of degree $n$ with the leading coefficient equal 1 which minimizes the uniform norm on the interval $[-1, 1]$.*]{} Zolotarev polynomials --------------------- Following the ideas of Chebyshev, his student Zolotarev posed and solved a handful of problems, including the following ([@AhiezerAPPROX; @DragRadn2019rcd]): [*For the given real parameter $\sigma$ and all polynomials of degree $n$ of the form: $$\label{eq:zol1} p(x)=x^n-n\sigma x^{n-1} + p_2x^{n-2}+\dots p_n,$$ find the one with the minimal uniform norm on the interval $[-1, 1]$.*]{} Denote this minimal uniform norm as $L_n=L(\sigma, n)$. For $\sigma>\tan^2(\pi/2n)$, the solution $z_n$ has the following property ([@AhiezerAPPROX], p. 298, Fig. 9): [*$\Pi1$ – The equation $z_n(x)=L_n$ has $n-2$ double solutions in the open interval $(-1, 1)$ and simple solutions at $-1, 1, \alpha, \beta$, where $1<\alpha <\beta$, while in the union of the intervals $[-1,1]\cup [\alpha, \beta]$ the inequality $z_n^2\le L_n$ is satisfied and $z_n>L_n$ in the complement.*]{} The polynomials $z_n$ are given by the following explicit formulae: $$\label{eq:zn} z_n=\ell_n\left(v(u)^n+\frac1{v(u)^n}\right), \quad x=\frac{\sn^2u +\sn^2\frac{K}{n}}{\sn^2u -\sn^2\frac{K}{n}},$$ where $$\ell_n=\frac1{2^n}\left(\frac{\sqrt{\kappa}\theta_1^2(0)}{H_1\left(\frac{K}{n}\right)\theta_1\left(\frac{K}{n}\right)}\right)^{2n}, \quad v(u)=\frac{H\left(\frac{K}{n}-u\right)}{H\left(\frac{K}{n}+u\right)}$$ and $$\sigma=\frac{2\sn\frac{K}{n}}{\cn\frac{K}{n}\dn\frac{K}{n}}\left(\frac1{\sn\frac{2K}{n}}-\frac {\theta'\left(\frac{K}{n}\right)}{\theta\left(\frac{K}{n}\right)}\right)-1.$$ Formulae for the endpoints of the second interval are $$\label{eq:alphabetan} \alpha =\frac{1+\kappa^2\sn^2\frac{K}{n}}{\dn^2\frac{K}{n}}, \quad \beta =\frac{1+\sn^2\frac{K}{n}}{\cn^2\frac{K}{n}},$$ with $$\kappa^2=\frac{(\alpha-1)(\beta+1)}{(\alpha+1)(\beta-1)}.$$ According to Cayley’s condition for $n=3$ and $\gamma \in (0, a)$ we have $$\gamma=\frac{ab(a-b)+2ab\sqrt{a^2+ab+b^2}}{(a+b)^2}.$$ In order to derive the formulas for $\hat p_3$ in terms of $z_3$, let us construct an affine transformation: $$h:[-1, 1]\cup [\alpha, \beta]\rightarrow [-b^{-1}, 0]\cup[a^{-1}, \gamma^{-1}], \quad h(x) = \hat a x +\hat b.$$ We immediately get $$\hat a = -\hat b, \quad \hat a = \frac1{2b}$$ and $$\label{eq:alphabetacayley3} \alpha = 2t+1,$$ $$\label{eq:alphabetacayley4} \gamma=\frac{2b}{\beta -1} $$ where $t=b/a$. Now we get the following \[propZolot\] The polynomial $\hat p_3$ can be expressed through the Zolotarev polynomial $z_3$ up to a nonessential constant factor: $$\hat p_3 (s) \sim z_3( 2bs+1).$$ To verify the proposition, we should certify that the definition of $\alpha$ and $\beta$ from [eq:alphabetan]{} for $n=3$ and the relations [eq:alphabetacayley3]{}, [eq:alphabetacayley4]{} are compatible with the formula for $\gamma$ we got before as Cayley condition, see [CaleyEq1]{}. In order to do that we will use well-known identities for the Jacobi elliptic functions: $$\begin{gathered} \sn^2u+\cn^2u=1, \\ \kappa^2\sn^2u+\dn^2u=1, \\ \sn(u+v)=\frac{\sn\, u\cn\, v\dn \,v + \sn\, v\cn\, u\dn\, u}{1-\kappa^2\sn^2 u \sn ^2 v}, \\ \sn (K-u)=\frac{\cn\, u}{\dn\, u}.\end{gathered}$$ In particular, we get $$\begin{gathered} \sn \left(\frac {2K}{3}\right)= \frac{2\sn\, \frac{K}{3}\cn \,\frac{K}{3}\dn \,\frac{K}{3}}{1-\kappa^2\sn^4\frac{K}{3}}, \\ \sn \left(\frac {2}{3}K\right)= \sn \left(K-\frac {K}{3}\right)=\frac{\cn \,\frac{K}{3}}{\dn\, \frac{K}{3}}.\end{gathered}$$ Let us denote $$Y=\sn \left(\frac {K}{3}\right),$$ then from the previous two relations we get as in ([@DragRadn2019rcd]): $$1-2Y+2\kappa^2Y^3 -\kappa^2Y^4=0.$$ We can express $\kappa$ in terms of $Y$ and get: $$\label{Add F} \kappa^2=\frac{2Y-1}{Y^3(2-Y)}.$$ By plugging the last relation into [eq:alphabetan]{} for $n=3$ we get $$\alpha=\frac{Y^2-4Y+1}{Y^2-1}.$$ Since, at the same time from the Cayley condition we have $\alpha = 2t+1$, with $t=b/a$, we can express $Y$ in terms of $t$: $$tY^2+2Y-(t+1)=0,$$ and $$\label{eq:Yt} Y=\frac{-1\pm \sqrt{1+t+t^2}}{t}.$$ We plug the last relation into the formula for $\beta$ from [eq:alphabetan]{} for $n=3$ $$\beta=\frac{1+Y^2}{1-Y^2},$$ and we get another formula for $\beta$ in terms of $t$: $$\begin{gathered} \label{eq:betat} \beta=\frac{2t^2+t+2-\pm 2\sqrt{t^2+t+1}}{-t-2 \pm 2\sqrt{t^2+t+1}}.\end{gathered}$$ We see that the last formula with the choice of the $+$ sign corresponds to a formula for $\beta$ from [eq:alphabetacayley4]{}. This formula relates $\beta$ and $\gamma$ from the Caley condition [CaleyEq1]{}. From [eq:betat]{}, taking the positive sign in $\beta$ yields, $$\begin{gathered} \label{beta2} \beta=\frac{2t^2+t+2- 2\sqrt{t^2+t+1}}{-t-2 + 2\sqrt{t^2+t+1}}.\end{gathered}$$ Substituting [beta2]{} into [eq:alphabetacayley4]{} produces $$\label{P1} \gamma=\frac{2b}{\beta -1} = b\dfrac{-2-t+2\sqrt{1+t+t^2}}{2+t+t^2-2\sqrt{1+t+t^2}}.$$ On the other hand, from the Cayley formula [CaleyEq1]{}: $$\gamma= \dfrac{ab}{(a+b)^{2}}(a-b+2\sqrt{a^{2}+ab+b^{2}})$$ Knowing that $t=\frac{b}{a}$, the equation is equivalent to $$\label{P2} \gamma= b\dfrac{1-t+2\sqrt{1+t+t^2}}{(1+t)^2}$$ In order to show that the two expressions in [P1]{} and [P2]{} are identical, we simplify their difference that yields zero. This finalizes the verification. (One can observe that the $-$ sign option from the formula [eq:betat]{} would correspond to the $-$ sign in the formula for $\gamma$ [CaleyEq1]{}. Among the polynomials $\hat p_n$ the property of type $\Pi1$ can be attributed only to those with $n=2k+1$ and winding numbers $(2k+1, 2k)$, in other words to those with the signature $(0, 2k-1)$. Akhiezer polynomials on symmetric intervals $[-1, -\alpha]\cup[\alpha, 1]$ -------------------------------------------------------------------------- The problem of finding polynomials of degree $n$ with the leading coefficient 1 and minimizing the uniform norm on the union of two symmetric intervals $[-1, -\alpha]\cup[\alpha, 1]$, for given $0<\alpha <1$ appeared to be of a significant interest in radio-techniques applications. Following the ideas of Chebyshev and Zolotarev, Akhiezer derived in 1928 the explicit formulae for such polynomials $A_n(x;\alpha)$ with the deviation $L_n(\alpha)$ [@AhiezerAPPROX; @Akh4]. These formulas are especially simple in the case of even degrees $n=2m$, when Akhiezer polynomials $A_{2m}$ are obtained by a quadratic substitution from the Chebyshev polynomial $T_m$: $$\label{eq:A2m} A_{2m}(x;\alpha)=\frac{(1-\alpha^2)^m}{2^{2m-1}}T_m\left(\frac{2x^2-1-\alpha^2} {1-\alpha^2}\right),$$ with $$L_{2m}(\alpha)=\frac{(1-\alpha^2)^m}{2^{2m-1}}.$$ We are going to construct $\hat p_4(s)$ up to a nonessential constant factor as a composition of $A_4(x;\alpha)$ for certain $\alpha$ and an affine transformation. We are going to study the possibility to have an affine transformation $$g:[-1,-\alpha]\cup[\alpha, 1]\rightarrow [-b^{-1}, \gamma^{-1}]\cup [0, a^{-1}],\quad g(x)=\hat a x +\hat b,$$ which corresponds to the case when $\gamma<-b$ ie $a>b$. For $n=4$ such caustic is [4pCaustic]{}: $$\gamma=\frac{ab}{b-a}.$$ From $g(-1)=-b^{-1}$, $g(1)=a^{-1}$ we get $$\hat a = \frac{a+b}{2ab}, \quad \hat b =\frac{b-a}{2ab}.$$ Then, from $g(\alpha)=0$ we get $$\alpha =\frac {a-b}{a+b}.$$ Finally, we calculate: $$g(-\alpha)=\frac{a+b}{2ab}\frac {b-a}{a+b} + \frac{b-a}{2ab}=\frac{b-a}{ab}.$$ We recognize $\gamma^{-1}$ on the righthand side of the last relation.This proves the following: In this case the polynomial $\hat p_4(s)$ equals, up to a constant multiplier, to $$\label{eq:p4t2} \hat p_4(s) \sim T_2(2abs^2 +2(a-b)s+1),$$ where $T_2(x)=2x^2-1$ is the second Chebyshev polynomial and $x=\frac{1}{a+b}\big(2abs+a-b\big)$. Let us study the possibility to have an affine transformation $$f:[-1,-\alpha]\cup[\alpha, 1]\rightarrow [-b^{-1}, 0]\cup [\gamma^{-1}, a^{-1}],\quad f(x)=\hat a x +\hat b,$$ which corresponds to the case when $\gamma>a$ ie $a<b$. For $n=4$ such caustic is $$\gamma=\frac{ab}{b-a}.$$ From $f(-1)=-b^{-1}$, $f(1)=a^{-1}$ we get $$\hat a = \frac{a+b}{2ab}, \quad \hat b =\frac{b-a}{2ab}.$$ Then, from $f(-\alpha)=0$ we get $$\alpha =\frac {b-a}{a+b}.$$ Finally, we calculate: $$f(\alpha)=\frac{a+b}{2ab}\frac {b-a}{a+b} + \frac{b-a}{2ab}=\frac{b-a}{ab}.$$ We recognize $\gamma^{-1}$ on the righthand side of the last relation.\ This proves the following proposition which is the same as [eq:p4t2]{}. In this case the polynomial $\hat p_4(s)$ is equal up to a constant multiplier to $$\label{eq:p4t3} \hat p_4(s) \sim T_2(2abs^2 +2(a-b)s+1),$$ where $T_2(x)=2x^2-1$ is the second Chebyshev polynomial and $x=\dfrac{1}{a+b}\big(2abs+a-b\big)$. Let us study the possibility to have an affine transformation $$h:[-1,-\alpha]\cup[\alpha, 1]\rightarrow [\gamma^{-1}, -b^{-1}]\cup [0,a^{-1}],\quad h(x)=\hat a x +\hat b,$$ which corresponds to the case when $\gamma \in (-b,0)$. For $n=4$ such caustic is $$\gamma=-\frac{ab}{a+b}.$$ From $h(1)=a^{-1}$, $h(\alpha)=0$ we get $$\hat a = \frac{1}{1-\alpha}\frac{1}{a}, \quad \hat b =-\frac{\alpha}{1-\alpha}\frac{1}{a}.$$ Then, from $h(-\alpha)=-\frac{1}{b}$ we get $$\frac{\alpha}{1-\alpha} =\frac {a}{2b}.$$ ie $$\alpha =\frac {a}{a+2b}.$$ Finally, we calculate: $$h(-1)=-\big(1+\frac{\alpha}{1-\alpha}\big)\frac{1}{a}-\frac{\alpha}{1-\alpha}\frac{1}{a},$$ $$h(-1)=-\big(1+\frac {a}{2b}\big)\frac{1}{a}-\frac {a}{2b}\frac{1}{a}=-\frac{1}{a}-\frac{1}{b}= -\frac{a+b}{ab}.$$ We recognize $\gamma^{-1}$ on the righthand side of the last relation.\ This proves the following: In this case the polynomial $\hat p_4(s)$ equal,s up to a constant multiplier, to $$\label{eq:p4t4} \hat p_4(s) \sim T_2\left(\frac{2a^{2}bs^{2}+2a^{2}s-(a+b)}{a+b}\right),$$ where $T_2(x)=2x^2-1$ is the second Chebyshev polynomial. Let us study the possibility to have an affine transformation $$l:[-1,-\alpha]\cup[\alpha, 1]\rightarrow [-b^{-1},0]\cup [a^{-1},\gamma^{-1}],\quad l(x)=\hat a x +\hat b,$$ which corresponds to the case when $\gamma \in (0,a)$. For $n=4$ such caustic is $$\gamma=\frac{ab}{a+b}.$$ From $l(-1)=-b^{-1}$, $l(-\alpha)=0$ we get $$\hat a = \frac{1}{1-\alpha}\frac{1}{b}, \quad \hat b =\frac{\alpha}{1-\alpha}\frac{1}{b}.$$ Then, from $l(\alpha)=\frac{1}{a}$ we get $$\frac{\alpha}{1-\alpha} =\frac {b}{2a}.$$ ie $$\alpha =\frac {b}{b+2a}.$$ Finally, we calculate: $$l(1)=\big(1+\frac{\alpha}{1-\alpha}\big)\frac{1}{b}+\frac{\alpha}{1-\alpha}\frac{1}{b},$$ $$l(1)=\big(1+\frac {b}{2a}\big)\frac{1}{b}+\frac {b}{2a}\frac{1}{b}= \frac{1}{a}+\frac{1}{b}=\frac{a+b}{ab}.$$ We recognize $\gamma^{-1}$ on the righthand side of the last relation. This proves the following: In this case the polynomial $\hat p_4(s)$ equals, up to a constant multiplier, to $$\label{eq:p4t5} \hat p_4(s) \sim T_2\left(\frac{2ab^{2}s^{2}-2b^{2}s-(a+b)}{a+b}\right),$$ where $T_2(x)=2x^2-1$ is the second Chebyshev polynomial. General Akhiezer polynomials on unions of two intervals ------------------------------------------------------- Following Akhiezer [@Akh1; @Akh2; @Akh3], let us consider the union of two intervals $[-1,\alpha]\cup [\beta, 1]$, where $$\label{Akhiezer} \alpha = 1-2\sn^2(\frac{m}{n}K), \quad \beta = 2\sn^2\left(\frac{n-m}{n}K\right)-1.$$ Define $$\label{AKhTAn} TA_{n}(x,m,\kappa)=L\left(v^{n}(u)+\frac{1}{v^{n}(u)}\right),$$ where $$v(u)=\dfrac{\theta_1\big(u-\frac{m}{n}K\big)}{\theta_1\big(u+\frac{m}{n}K\big)}, \quad x=\dfrac{\sn^2(u)\cn^2(\frac{m}{n}K)+\cn^2(u)\sn^2(\frac{m}{n}K)}{\sn^2(u)-\sn^2(\frac{m}{n}K)},$$ and $$L=\frac{1}{2^{n-1}}\left(\dfrac{\theta_0(0)\theta_3(0)}{\theta_0(\frac{m}{n}K)\theta_3(\frac{m}{n}K)}\right), \quad \kappa^2=\frac{2(\beta -\alpha)}{(1 -\alpha)(1+\beta)}.$$ Here $\theta_i,\, i=0. 1, 2, 3,$ denote the standard Riemann theta functions, see for example [@Akh4] for more details. Akhiezer proved the following result: \[th:Akhiezer\] - The function $TA_{n}(x,m,\kappa)$ is a polynomial of degree $n$ in $x$ with the leading coefficient $1$ and the second coefficient equal to $-n\tau_{1}$, where $$\tau_{1}=-1+2\dfrac{\sn(\frac{m}{n}K)\cn(\frac{m}{n}K)}{\dn(\frac{m}{n}K)}\left(\frac{1}{sn(\frac{2m}{n}K)}-\frac{\theta_0^{\prime}(\frac{m}{n}K)}{\theta_0(\frac{m}{n}K)}\right).$$ - The maximum of the modulus of $TA_{n}$ on the union of the two intervals $[-1,\alpha]\cup [\beta, 1]$ is $L$. - The function $TA_{n}$ takes values $\pm L$ with alternating signs at $\mu=n-m+1$ consecutive points of the interval $ [- 1, \alpha]$ and at $\nu=m+1$ consecutive points of the interval $ [\beta, 1]$. In addition $$TA_{n}(\alpha,m,\kappa)=TA_{n}(\beta,m,\kappa)=(-1)^{m}L,$$ and for any $x\in (\alpha, \beta)$, it holds: $$(-1)^{m}TA_{n}(x,m,\kappa)>L.$$ - Let $F$ be a polynomial of degree $n$ in $x$ with the leading coefficient $1$, such that: <!-- --> - $max|F(x)|=L$ for $x\in [-1,\alpha]\cup [\beta, 1]$; - F(x) takes values $\pm L$ with alternating signs at n-m+1 consecutives points of the interval $[-1, \alpha]$ and at m+1 consecutive points of the interval $[\beta, 1]$. Then $F(x)=TA_{n}(x,m,\kappa).$ Let us determine the affine transformations when the caustic is an ellipse. #### Case $\gamma \in (-b,0)$. {#case-gamma-in--b0. .unnumbered} For $h:[-1,\alpha]\cup[\beta, 1]\rightarrow [\gamma^{-1},-b^{-1}]\cup [0,a^{-1}]$, $h(x)=\hat a x +\hat b$, we get $$\hat a = \frac{1}{\beta-\alpha}\frac{1}{b}, \quad \hat b =\frac{-\beta}{\beta-\alpha}\frac{1}{b}, \quad \frac{1-\beta}{\beta -\alpha}=\frac{b}{a}.$$ Thus: $$\label{GenAkhiezerTrans} \gamma=\frac{\beta-1}{1+\beta}a=\frac{ \alpha-\beta}{\beta+1}b$$ \[ex:n=3,m=2\] For $n=3$ and $m=2$. From [Akhiezer]{}, one gets: $$\alpha = 1-2\sn^2\frac{2}{3}K, \quad \beta = 2\sn^2\frac{K}{3}-1.$$ It follows that: $$\label{Eq1} \frac{b}{a}=t= \frac{1-\beta}{\beta-\alpha}= \frac{1-\sn^2\frac{K}{3}}{\sn^2\frac{2}{3}K +\sn^2\frac{K}{3}-1},$$ Thus $$\label{GenAkhiezerTrans1} \gamma=b\frac{\alpha -\beta}{\beta+1}=b\frac{1-\sn^2\frac{K}{3}-sn^2\frac{2}{3}K}{\sn^2\frac{K}{3}}.$$ From the addition formula: $$\sn\frac{2}{3}K=\sn(K-\frac{K}{3})=\frac{\sn K \cn\frac{-K}{3} \dn\frac{-K}{3}+\sn\frac{-K}{3}\cn K \dn K}{1-\kappa^2\sn^2\frac{-K}{3}\sn^2K}.$$ Hence $$\sn^2\frac{2}{3}K=\frac{1-\sn^2\frac{K}{3}}{1-\kappa^2\sn^2\frac{K}{3}} = \frac{\sn^2\frac{2}{3}K-1}{\kappa^2\sn^2\frac{2}{3}K-1}.$$ Let $sn\frac{K}{3}=Z$: $$\kappa^{2}=\frac{2Z-1}{Z^3(2-Z)}.$$ Also $$\alpha = 2Z^2-4Z+1, \quad \beta =2Z^2-1.$$ Equation [Eq1]{} implies that $$t=\frac{1-Z^2}{2Z-1}.$$ Thus, we have two expressions for $\gamma$. One is from the Cayley condition [CaleyEq1]{} and the other is from [GenAkhiezerTrans]{}. We want to show that these two expressions are identical, that is $$\label{EquaLambda} b\frac{\alpha -\beta}{\beta+1}=-\dfrac{ab}{(a+b)^{2}}(-a+b+2\sqrt{a^{2}+ab+b^{2}}).$$ In order to do so, we first express both the left hand side and the right hand side of the above equation in terms of $t=\dfrac{b}{a}$ and then transform both sides in terms of $Z$. We show that the left and the right handsides yield the same expression: $$\frac{Z^2-Z+1}{2Z-1},$$ therefore [EquaLambda]{} holds. #### Case $\gamma \in (0,a)$. {#case-gamma-in-0a. .unnumbered} For $ l:[-1,\alpha]\cup[\beta, 1]\rightarrow [-b^{-1},0]\cup [a^{-1}, \lambda^{-1}]$, $l(x)=\hat a x +\hat b$, we get $$\hat a = \frac{1}{\alpha + 1}\frac{1}{b}, \quad \hat b =\frac{-\alpha}{\alpha+1}\frac{1}{b}, \quad \frac{\alpha +1}{\beta -\alpha}=\frac{a}{b}.$$ Thus $$\label{Eq:LambaPositiv} \gamma =\frac{\alpha +1}{1-\alpha}b$$ For $n=3$, and $m=1$. From [Akhiezer]{}, one gets: $$\alpha = 1-2\sn^2\frac{K}{3}, \quad \beta = 2\sn^2\frac{2K}{3}-1.$$ $$\label{Eq2} \frac{b}{a}=t= \frac{\beta - \alpha}{\alpha+1}= \frac{\sn^2\frac{2}{3}K+\sn^2\frac{K}{3}-1}{1-\sn^2\frac{K}{3}}$$ Thus $$\gamma=\frac{1-\sn^2\frac{K}{3}}{\sn^2\frac{K}{3}}b.$$ From the addition formula: $$\sn\frac{2K}{3}=\sn(K-\frac{K}{3})=\frac{\sn K \cn\frac{-K}{3} \dn\frac{-K}{3}+\sn\frac{-K}{3}\cn K \dn K}{1-\kappa^2\sn^2\frac{-K}{3}\sn^2K}.$$ Hence $$\sn^2\frac{2}{3}K=\frac{1-\sn^2\frac{K}{3}}{1-\kappa^2\sn^2\frac{K}{3}}.$$ Let $\sn\frac{K}{3}=Z$: $$\kappa^{2}=\frac{2Z-1}{Z^3(2-Z)}.$$ Also $ \beta = -2Z^2+4Z-1, $ and $\alpha =1-2Z^2$. Equation [Eq2]{} implies that $$t=-\frac{2Z-1}{Z^2-1},$$ Thus, we have two expressions for $\gamma$. One is from the Cayley condition [CaleyEq1]{} and the other is from [Eq:LambaPositiv]{}. Similarly as in Example \[ex:n=3,m=2\], we can show that these two expressions are identical. \[prop\] For $n=3$ and $\gamma \in (-b,0)$, the polynomial $\hat{p}_{3}$ is up to a nonessential factor equal to: $$\hat{p}_{3} ~\sim TA_{3}\left(2a\left(1-\sn^2\frac{K}{3}\right)s+2\sn^2\frac{K}{3}-1 ; 2, \kappa \right),$$ For $n=3$ and $\gamma \in (0,a)$, the polynomial $\hat{p}_{3}$ is up to a nonessential factor equal to: $$\hat{p}_{3} ~\sim TA_{3}\left(2b\left(1-\sn^2\frac{K}{3}\right)s+1-2\sn^2\frac{K}{3} ; 1, \kappa \right)$$ Now, using the Akhiezer Theorem part (c), see Theorem \[th:Akhiezer\], one can compare and see that the number of internal extremal points coincides with $n_1-1$ and $n_2-1$ as proposed in Theorem \[th:impactwinding\]. These numbers match with Figure \[fig:ana5\] and the table from Section \[sec:examples-table\]. Periodic light-like trajectories and Chebyshev polynomials {#sec:ll} ========================================================== The light-like billiard trajectories, by definition, have at each point the velocity $v$ satisfying $\langle v,v\rangle=0$. Their caustic is the conic at infinity ${\pazocal{C}}_{\infty}$. Since successive segments of such trajectories are orthogonal to each other, the light-like trajectories can close only after an even number of reflections. In [@DragRadn2012adv]\*[Theorem 3.3]{}, it is proved that a light-like billiard trajectory within ${\pazocal{E}}$ is periodic with even period $n$ if and only if $$\label{eq:arc} \arccot\sqrt{\frac ab} \in \left\{ \frac{k\pi}n \mid 1\le k<\frac{n}2, \left(k,\frac{n}2\right)=1 \right\}.$$ For $k$ not being relatively prime with $n/2$, the corresponding light-like trajectories are also periodic, and their period is a divisor of $n$. Applyin the limit $\gamma\to+\infty$ in Corollary \[cor:pell-periodic\], we get the following proposition. \[prop:ll\] A light-like trajectory within ellipse ${\pazocal{E}}$ is periodic with period $n=2m$ if and only if there exist real polynomials $\hat{p}_m(s)$ and $\hat{q}_{m-1}(s)$ of degrees $m$ and $m-1$ respectively if and only if: - $\hat{p}_m^2(s)-\left(s-\dfrac1a\right)\left(s+\dfrac1b\right)\hat{q}_{m-1}^2(s)=1$; and - $\hat{q}_{m-1}(0)=0$. The first condition from Proposition \[prop:ll\] is the standard Pell equation describing extremal polynomials on one interval $[-1/b,1/a]$, thus the polynomials $\hat{p}_m$ can be obtained as Chebyshev polynomials composed with an affine transformation $[-1/b,1/a]\to[-1,1]$. The additional condition $\hat{q}_{m-1}(0)=0$, which is equivalent to $\hat{p}_m'(0)=0$ implies an additional constraint on parameters $a$ and $b$. We have the following The polynomials $\hat {p}_m$ and the parameters $a, b$ have the following properties: - $\hat{p}_m(s)=T_m\left(\dfrac{2ab}{a+b}s+\dfrac{a-b}{a+b}\right)$, where $T_m$ is defined by [eq:cheb2]{}; - the condition $\hat{q}_{m-1}(0)=0$ is equivalent to [eq:arc]{}. The increasing affine transformation $h:[-1/b, 1/a]\rightarrow [-1, 1]$ is given by the formula $h(s)=(2ab s + a-b)/(a+b)$. The internal extremal points of the Chebyshev polynomial $T_m$ of degree $m$ on the interval $[-1, 1]$ are given by $$x_k=\cos \left(\frac{k}{m}\pi\right), \quad k=1, \dots, m-1,$$ according to the formula [eq:cheb2]{}. The second item follows from $h(0)=x_k$. This is equivalent to $$\frac{a-b}{a+b}\in\left\{\cos \left(\frac{k}{m}\pi\right)\mid k=1, \dots, m-1\right\},$$ which is equivalent to [eq:arc]{}. Acknowledgment {#acknowledgment .unnumbered} -------------- The research of V. D. and M. R. was supported by the Discovery Project \#DP190101838 *Billiards within confocal quadrics and beyond* from the Australian Research Council and Project \#174020 *Geometry and Topology of Manifolds, Classical Mechanics and Integrable Systems* of the Serbian Ministry of Education, Technological Development and Science. V. D. would like to thank Sydney Mathematics Research Institute and their International Visitor Program for kind hospitality.

--- author: - | Hongyu Ren, Russell Stewart, Jiaming Song, Volodymyr Kuleshov, Stefano Ermon\ Department of Computer Science, Stanford University\ {hyren, stewartr, tsong, kuleshov, ermon}@cs.stanford.edu - | First Author$^1$, Second Author$^2$, Third Author$^3$,\ $^1$ First Affiliation\ $^2$ Second Affiliation\ $^3$ Third Affiliation\ first@email.address, second@email.address, third@email.address bibliography: - 'ijcai18.bib' title: Adversarial Constraint Learning for Structured Prediction --- Introduction ============ Background ========== In this section, we introduce structured prediction and constraint-based learning. The next section will expand upon these subjects to introduce the proposed adversarial constraint learning framework. Structured Prediction --------------------- Our work focuses on structured prediction, a form of supervised learning, in which the outputs ${{{\bf y}}}\in \mathcal{Y}$ can be a complex object such as a vector, a tree, or a graph [@koller2009probabilistic]. We capture the distribution of ${{{\bf y}}}$ using a conditional probabilistic model $p_\theta({{{\bf y}}}|{{{\bf x}}})$ parameterized by $\theta \in \Theta$. A model $p_\theta({{{\bf y}}}|{{{\bf x}}})$ maps each input ${{{\bf x}}}\in \mathcal{X}$ to the corresponding output distribution $p_\theta({{{\bf y}}}) \in \mathcal{P}(\mathcal{Y})$, where $\mathcal{P}(\mathcal{Y})$ denotes all the probability distributions over $\mathcal{Y}$. For example, we may take $p_\theta({{{\bf y}}}|{{{\bf x}}})$ to be a Gaussian distribution $\mathcal{N}({{{\bm \mu}}}_\theta({{{\bf x}}}),{{{\bf \Sigma}}}_\theta({{{\bf x}}}))$ with mean ${{{\bm \mu}}}_\theta({{{\bf x}}})$ and variance ${{{\bf \Sigma}}}_\theta({{{\bf x}}})$. A standard approach to learning $p_\theta({{{\bf y}}}|{{{\bf x}}})$ (or $p_\theta$ as an abbreviation) is to solve an optimization problem of the form $$\label{eq:supervise} \theta^{\ast}=\operatorname*{arg\,min}_{\theta \in {\Theta}}\sum_{i=1}^{n} \ell (p_\theta({{{\bf y}}}|{{{\bf x}}}_i),{{{\bf y}}}_i) + R(p_\theta)$$ over a labeled dataset ${{{\bf D}}}= \{({{{\bf x}}}_1,{{{\bf y}}}_1), \cdots, ({{{\bf x}}}_n, {{{\bf y}}}_n)\}$. A typical supervised learning objective is comprised of a loss function $\ell: \mathcal{P}(\mathcal{Y})\times{\mathcal{Y}}\to\mathbb{R}$ and a regularization term $R: \mathcal{P}(\mathcal{Y}) \to \mathbb{R}$ that encourages non-degenerate solutions or solutions that incorporate prior knowledge [@stewart2017label]. Constraint-Based Learning ------------------------- Collecting a large labeled dataset for supervised learning can often be tedious. Constraint-based learning is a form of weak supervision which instead asks users to specify high-level constraints over the output space, such as logical rules or physical laws [@shcherbatyi2016convexification; @stewart2017label; @richardson2006markov; @xu2017semantic]. For example, in an object tracking task where $\mathcal{Y}$ corresponds to the space of joint positions over time, we expect correct outputs to be consistent with the laws of physical mechanics. Let ${{{\bf X}}}= \{{{{\bf x}}}_{1}, \cdots, {{{\bf x}}}_m\}$ be an unlabeled dataset of inputs. Formally, constraints can be specified via a function $h: \mathcal{P}(\mathcal{Y})\to\mathbb{R}$, which penalizes conditional probabilistic models $p_\theta({{{\bf y}}}|{{{\bf x}}})$ that are inconsistent with known high-level structure of the label space. Learning from constraints proceeds by optimizing the following objective: $$\label{eq:CLobejctive} \hat{\theta}^{\ast}=\operatorname*{arg\,min}_{\theta \in {\Theta}}\sum_{i=1}^{m}h(p_\theta({{{\bf y}}}|{{{\bf x}}}_i))+R(p_\theta)$$ over ${{{\bf X}}}$. By solving this optimization problem, we look for a probabilistic model parameterized by $\hat{\theta}^{\ast}$ that satisfies known constraints when applied to the unlabeled dataset ${{{\bf X}}}$ (through the $h$ term), and is likely a priori (through the $R(p_\theta)$ term). Note that although the constraint $h$ is data-dependent, *it does not require explicit labels*. For example, in object tracking we could ask that when making predictions on ${{{\bf X}}}$, joint positions over time are consistent with known kinematic equations, with $h$ measuring how the output distribution from $p_\theta$ deviates from those equations. The regularization term can be used to avoid overly complex and/or degenerate solutions, and may include $L1$, $L2$, or entropy regularization terms. Stewart and Ermon [@stewart2017label] have shown that a model learned with the objective described in Eq. \[eq:CLobejctive\] can learn to track objects. Adversarial Constraint Learning =============================== The process of manually specifying high level constraints, $h$, can be time-consuming and may require significant domain expertise. Such is the case in pose estimation, where it is difficult to describe high dimensional rules for joints movements precisely; but the large availability of unpaired videos and motion capture data makes constraint learning attractive in spite of the difficulty of providing high dimensional constraints. In the sciences, discovering general invariants is often a data-driven approach; for example, the laws of physics are often discovered by validating hypotheses with experimental results. Motivated by this, we propose in this section a novel framework for learning constraints from data. Learning Constraints from Data ------------------------------ Suppose we have a dataset of inputs ${{{\bf X}}}=\{{{{\bf x}}}_1,\dots,{{{\bf x}}}_m\}$, a dataset of labels ${{{\bf Y}}}=\{{{{\bf y}}}_1,\dots,{{{\bf y}}}_k\}$, and a set ${{{\bf D}}}=\{({{{\bf x}}}_1,{{{\bf y}}}_1),\dots,({{{\bf x}}}_n,{{{\bf y}}}_n)\}$ that describes correspondence between some elements of ${{{\bf X}}}$ and ${{{\bf Y}}}$. We denote the empirical distributions of ${{{\bf X}}}$, ${{{\bf Y}}}$ and ${{{\bf D}}}$ as $p({{{\bf x}}})$, $p({{{\bf y}}})$ and $p({{{\bf x}}}, {{{\bf y}}})$ respectively. Note that ${{{\bf Y}}}$ can come from either a simulator (such as one based on physical rules), or from some other source of data (such as motion captures of people for which we have no corresponding videos). Let us first consider the setting where ${{{\bf D}}}= \varnothing$; i.e. there are inputs and labels but no correspondence between them. In spite of the lack of correspondences, we will see that constraints $h$ can be learned from the prior knowledge that the same underlying distribution generates both the empirical labels ${{{\bf Y}}}$ and the structured predictions obtained from applying our model to ${{{\bf X}}}$. These learned constraints can then be used for supervision. Let structured predictions be given by the following $\textit{implicit}$ sampling procedure: $$\begin{aligned} {{{\bf x}}}\sim p({{{\bf x}}}) \quad, \quad {{{\bf y}}}\sim p_\theta({{{\bf y}}}|{{{\bf x}}})\end{aligned}$$ where $p_\theta({{{\bf y}}}|{{{\bf x}}})$ is a (parameterized) conditional distribution of outputs given inputs. Discarding ${{{\bf x}}}$, the above procedure corresponds to sampling from the marginal distribution over $\mathcal{Y}$, $p_\theta({{{\bf y}}}) = \int p_\theta({{{\bf y}}}|{{{\bf x}}})p({{{\bf x}}}) \mathrm{d}{{{\bf x}}}$. Labels drawn from $p({{{\bf y}}})$ should have high likelihood values in $p_\theta({{{\bf y}}})$, but optimizing this objective directly is computationally infeasible; evaluating the marginal likelihood $p_\theta({{{\bf y}}})$ exactly is expensive due to the integration over $p({{{\bf x}}})$. Instead, we formulate the task of learning a constraint loss $h$ from $p({{{\bf y}}})$ through a likelihood-free approach using the framework of generative adversarial learning [@goodfellow2014generative], which only requires samples from $p_\theta({{{\bf y}}})$ and $p({{{\bf y}}})$. We introduce an auxiliary classifier $D_\phi$ (parametrized by $\phi$) called discriminator which scores outputs in the label space $\mathcal{Y}$. It is trained to assign high scores to representative output labels from $p({{{\bf y}}})$, while assigning low scores to samples from $p_\theta({{{\bf y}}})$. It learns to effectively extract latent constraints that hold over the output space and that are implicitly encoded in the samples from $p({{{\bf y}}})$. The goal of $p_\theta({{{\bf y}}}|{{{\bf x}}})$ is to produce outputs result in higher scores in the discriminator, satisfying the constraints imposed by $D_\phi$ in the process. For practical reasons, we consider $p_\theta({{{\bf y}}}|{{{\bf x}}})$ to be a Dirac-delta distribution $\delta({{{\bf y}}}-f_\theta({{{\bf x}}}))$, and thus we refer to the conditional probabilistic model as the mapping $f_\theta({{{\bf x}}}) : \mathcal{X} \to \mathcal{Y}$ in the experiment section for simplicity. We train $D_\phi$ and $p_\theta({{{\bf y}}}|{{{\bf x}}})$ for the following objective [@arjovsky2017wasserstein] $$\begin{gathered} \min_\theta \max_\phi \mathcal{L^A} \label{eq:probgan}\\ \mathcal{L^A} = \mathbb{E}_{{{{\bf y}}}\sim p({{{\bf y}}})}[D_\phi({{{\bf y}}})] - \mathbb{E}_{{{{\bf y}}}\sim p_\theta({{{\bf y}}}|{{{\bf x}}}), {{{\bf x}}}\sim p({{{\bf x}}})}[D_\phi({{{\bf y}}})] \nonumber\end{gathered}$$ Assuming infinite capacity, Theorem 1 of [@goodfellow2014generative] shows that at the optimal solution of Eq. \[eq:probgan\], $D_\phi$ cannot distinguish between the given set of labels and those predicted by the model $p_\theta$, suggesting that the latter satisfy the set of constraints defined by $D_\phi$. Unlike in constraint-based learning where a (possibly incomplete) set of constraints is manually specified, convergence in the adversarial setting implies that the label and output distributions match on all possible discriminator projections. Figure \[fig:subfig:b\] shows an overview of the adversarial constraint learning framework in the context of trajectory estimation. Constraint Learning via Matching Distributions ---------------------------------------------- Generative Adversarial Networks (GANs) are a prominent example of implicit probabilistic models [@mohamed2016learning] which are defined through a stochastic sampling procedure instead of an explicitly defined likelihood function. One advantage of implicit generative models is that they can be trained with methods that do not require likelihood evaluations. Hence, our approach to learning constraints for structured prediction can also be interpreted as learning an implicit generative model $p_\theta({{{\bf y}}})$ that matches the empirical label distribution $p({{{\bf y}}})$. Specifically, our adversarial constraint learning approach optimizes over an approximation to the optimal transport from $p_\theta({{{\bf y}}})$ to $p({{{\bf y}}})$ [@arjovsky2017wasserstein]; thus our constraint can be implicitly defined as “$\theta$ minimizes the optimal transport from $p_\theta({{{\bf y}}})$ to $p({{{\bf y}}})$”.   Semi-Supervised Structured Prediction ------------------------------------- Models $({{{\bf x}}},{{{\bf y}}})$ $({{{\bf x}}},)$ $(,{{{\bf y}}})$ -------- ----------------------------- ------------------ ------------------ SL $\surd$ SSL $\surd$ $\surd$ ACL $\surd$ $\surd$ SSACL $\surd$ $\surd$ $\surd$ : Settings in different learning paradigms. Supervised Learning (SL) requires a dataset with paired $({{{\bf x}}},{{{\bf y}}})$. Semi-Supervised Learning (SSL) utilizes additional unlabeled inputs $({{{\bf x}}},)$. Adversarial Constraint Learning (ACL) requires inputs $({{{\bf x}}},)$ and labels $(,{{{\bf y}}})$ but without correspondences between them. Semi-Supervised Adversarial Constraint Learning (SSACL) extends ACL by also considering labeled pairs $({{{\bf x}}},{{{\bf y}}})$. []{data-label="table:models"} Although our framework does not require datasets containing input-label pairs ${{{\bf D}}}\supsetneq \varnothing$, providing it with such data gives rise to a new semi-supervised structured prediction method. When given a set of labeled examples, we may extend our constraint learning objective (over both labeled and unlabeled data) with a standard classification loss term (over labeled data): $$\begin{gathered} \label{eq:advcl} \mathcal{L^{SS}} = \mathcal{L^A} + \alpha \mathbb{E}_{{{{\bf x}}}_i, {{{\bf y}}}_i \sim p({{{\bf x}}},{{{\bf y}}})}[\ell(p_\theta({{{\bf y}}}| {{{\bf x}}}_i),{{{\bf y}}}_i)]\end{gathered}$$ where $\mathcal{L^A}$ is the adversarial constraint learning objective defined in Eq. \[eq:probgan\], and $\alpha$ is a hyperparameter that balances between fitting to the general (implicit) label distribution (first term) and fitting to the explicit labeled dataset (second term). Our semi-supervised constraint learning framework is different from traditional semi-supervised learning approaches, as listed in Table \[table:models\]. In particular, traditional semi-supervised learning methods assume there is a large source of *inputs* and tend to impose regularization over $\mathcal{X}$, such as through latent variables [@kingma2014semi], through outputs [@miyato2017virtual], or through another network [@salimans2016improved]. We consider the case where there exists a source, e.g., a simulator that can provide abundant *samples from the label space that are not matched to particular inputs*, and impose regularization over $\mathcal{Y}$ by exploiting a discriminator that provides an implicit constraint over the predicted ${{{\bf y}}}$ values. Therefore, we can also utilize sample labels that are not associated with particular inputs, instead of merely restricting to standard labeled $({{{\bf x}}}, {{{\bf y}}})$ pairs. Moreover, our method can be easily combined with existing semi-supervised learning approaches [@kingma2014semi; @li2016max; @miyato2017virtual] to further boost performance. Experimental Results {#experiments} ==================== Conclusion ========== We have proposed adversarial constraint learning, a new framework for structured prediction that replaces hand-crafted, domain specific constraints with implicit, domain agnostic ones learned through adversarial methods. Experimental results on multiple structured prediction tasks demonstrate that adversarial constraint learning works across many real-world applications with limited data, and fits naturally into semi-supervised structured prediction problems. Our success with matching distributions of labeled and unlabeled model outputs motivates future work exploring analogous opportunities for adversarially matching labeled and unlabeled distributions of learned intermediate representations. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by a grant from the SAIL-Toyota Center for AI Research, TRI, Siemens, ONR, NSF grants \#1651565, \#1522054, \#1733686, and a Hellman Faculty Fellowship.

--- abstract: 'In this article we introduce an asymptotic preserving scheme designed to compute the solution of a two dimensional elliptic equation presenting large anisotropies. We focus on an anisotropy aligned with one direction, the dominant part of the elliptic operator being supplemented with Neumann boundary conditions. A new scheme is introduced which allows an accurate resolution of this elliptic equation for an arbitrary anisotropy ratio.' author: - Pierre Degond  - Fabrice Deluzet  - Claudia Negulescu title: An asymptotic preserving scheme for strongly anisotropic elliptic problems --- Introduction ============ The objective of this paper is to introduce an efficient and accurate numerical scheme to solve a strongly anisotropic elliptic problem of the form \[Aniso\] { [l]{} - ( ) = f ,\ = 0 \_D, \_z = 0 \_z, . where $\Omega \subset \RR^2$ or $\Omega \subset \RR^3$ is a domain, with boundary $\partial \Omega=\partial \Omega_D \cup \partial\Omega_z$ and the diffusion matrix $\mathbb{A}$ is given by $$\mathbb{A} =\left( \begin{array}[c]{cc} A_\perp & 0 \\ 0 & \frac{1}{\varepsilon}A_z \end{array}\right) \,.$$ The terms $A_\perp$ and $A_z$ are of the same order of magnitude, whereas the parameter $0<\eps<1$ can be very small, provoking thus the high anisotropy of the problem. In the present paper the considered anisotropy direction is fixed and is aligned with the $z$-axis of a Cartesian coordinate system. The method presented here is extended in some forthcoming works to more general anisotropies [@brull]. Anisotropic problems are common in mathematical modeling and numerical simulation. Indeed they occur in several fields of applications such as flows in porous media [@porous1; @TomHou], semiconductor modelling [@semicond], quasi-neutral plasma simulations [@Navoret], image processing [@Weickert; @image1], atmospheric or oceanic flows [@ocean], and so on, the list being not exhaustive. More specifically high anisotropy aligned with one direction may occur in shell problems or simulation in stretched media. The initial motivation for the present work is closely related to magnetized plasma simulations such as atmospheric [@Kelley2; @Kes_Oss] or inertial fusion plasmas [@Beer; @Sangam] or plasma thrusters [@SPT]. In this context, the medium is structured by the magnetic field. Indeed, the motion of charged particles in planes perpendicular to the magnetic field is governed by a fast gyration around the magnetic field lines. This explains the large number of collisions the particles encounter in the perpendicular plane, whereas the dynamic in the parallel direction is rather undisturbed. As a consequence the particle mobilities in the perpendicular and parallel directions differ by many orders of magnitude. In the context of ionospheric plasma modelling [@BCDDGT_4; @Hysell], the ratio of the aligned and transverse mobilities (denoted in this paper by $\eps^{-1}$) can be as huge as ten to the power ten. The relevant boundary conditions in many fields of application are periodic (for instance in simulations of tokamak plasmas on a torus) or Neumann boundary conditions (see for instance [@BCDDGT_1] for atmospheric plasmas). The system is thus a good model to elaborate a robust numerical method.\ The main difficulties with the resolution of problem (\[Aniso\]) are of numerical nature, as solving this singular perturbation problem for small $0<\eps \ll1$ is rather delicate. Indeed, replacing in the anisotropic elliptic equation $\eps$ by zero, yields an ill-posed problem, which has an infinite number of solutions (namely all functions which are constant in the $z$-direction). This feature is translated in the discrete case (after the discretization of the problem) into a linear system which is very ill-conditioned for $\eps \ll1$, due to the different order of magnitudes of the various terms. As a consequence standard numerical methods for the resolution of linear systems lead to important numerical costs and unacceptable numerical errors. [More generally, this numerical difficulty arises when the boundary conditions supplied to the dominant $O(1/\varepsilon)$ operator lead to an ill-posed problem with a multiplicity of solutions. This is the case for Neumann boundary conditions, but also of periodic boundary conditions. If instead, the boundary conditions are such that the dominant operator gives a well-posed problem with a unique solution, this difficulty vanishes as the leading operator alone will suffice to completely determine the limit solution. In this case, one can resort to standard methods. This is the case of Dirichlet or Robin boundary conditions. In spite of the fact that the problem addressed in the present paper arises only with specific boundary conditions, it has a considerable impact in many physics problem, such as plasmas, geophysical flows, plate and shells, etc. In this paper, we will focus on Neumann boundary conditions because they represent a larger range of physical applications, but we could address periodic boundary conditions in a similar way. ]{} [Numerical methods for anisotropic elliptic problems have been extensively investigated in the literature.]{} Depending on the underlying physics, distinct numerical methods are developed. For example domain decomposition (Schur complement) and multigrid techniques, using multiple coarse grid corrections are adapted to anisotropic equations in [@Giraud; @Koronskij] and [@Gee; @Notay]. For anisotropy aligned with one (or two directions), point (or plane) smoothers are shown to be very efficient [@ICASE]. A problem very similar to is addressed in [@Vladimir], treated via a parametrisation technique, and seems to give good results for rather large anisotropy ratios. However, these techniques are only developed in the context of an elliptic operator with a dominant part supplemented with Dirichlet boundary conditions.\ [An alternative approach]{} for dealing with highly anisotropic problems is based on a mathematical reformulation of the continuous problem, in order to obtain a more harmless problem, which can be solved numerically in an uncomplicated manner. In this category can be situated for example asymptotic models, describing for small values of the asymptotic parameter $\varepsilon$ the evolution of an approximation $\tilde{\phi}$ of the solution of (\[Aniso\]) [@BCDDGT_1; @Keskinen]. However, these asymptotic models are precise only for $\eps \ll 1$, and cannot be used on the whole range of values covered by the physical parameter $\varepsilon$. Thus model coupling methods have to be employed. In sub-domains where the limit model is no longer valid, the original model has to be used, which means that a model coupling strategy has to be developed. However the coupling strategy requires the existence of an area where both models are valid and still demands an accurate numerical method for the resolution of the original model (i.e. the anisotropic elliptic problem) with large anisotropies. This can be rather undesirable.\ In this paper, we present an original numerical algorithm belonging to the second approach. A reformulation of the continuous problem (\[Aniso\]) will permit us to solve this problem in an inexpensive way and accurately enough, independently of the parameter $\varepsilon$. This scheme is related to the [*Asymptotic Preserving*]{} numerical method introduced in [@ShiJin]. These techniques are designed to provide computations in various regimes without any restriction on the discretization meshes and with the additional property to converge towards the solution of the limit problem when the asymptotic parameter goes to zero. The derivation of such Asymptotic Preserving methods requires first the identification of the limit model. For singular perturbation problems, a reformulation of the problem is required in order to derive a set of equations containing both the initial and the limit model with a continuous transition from one regime to another, according to the values of the parameter $\varepsilon$. This reformulated system of equations sets the foundation of the AP-scheme. Other singular perturbations have already been explored in previous studies, for instance quasi-neutral or gyro-fluid limits [@Crispel; @Sangam]. These techniques have been first introduced for non-stationary systems of equations, for which the time discretization must be studied with care in order to guarantee the asymptotic preserving property. For the anisotropic elliptic equation investigated in this article, we only need to precise the reformulated system and provide a discretization of this one.\ The outline of this paper is the following. Section \[SEC2\] of this article presents first the initial anisotropic elliptic model. In the remainder of this paper, it will be referred to as the Singular-[*[Perturbation]{}*]{} model (P-model). The reformulated system (referred to as the [*Asymptotic Preserving*]{} formulation or AP-formulation) is then derived. It relates on a decomposition of the solution $\phi(x,z)$ according to its [*mean part*]{} $\bar \phi(x)$ along the $z$ coordinate and a [*fluctuation*]{} $\phi'(x,z)$ consisting of a correction to the mean part needed to recover the full solution. The mean part $\bar {\phi}(x)$ is solution of an $\eps$-independent elliptic problem, and the fluctuation $\phi'(x,z)=\phi(x,z)-\bar{\phi}(x)$ is given by a well-posed $\eps$-dependent elliptic problem. The advantage is that the $\eps$-dependent problem for the fluctuation is well-posed and solvable in an inexpensive way, and this uniformly in $\varepsilon$. In the limit $\eps \rightarrow 0$ the AP-formulation reduces to the so called [*Limit*]{} model ($L$-model), whose solution is an acceptable approximation of the P-model solution for $\eps \ll1$. The present derivation is carried out in the framework of an anisotropy aligned along one axis of a Cartesian coordinate system. In the context of magnetized plasma simulations, this initial work is extended in a forthcoming work for the three dimensional case in curvilinear coordinates, designed to fit a more complex magnetic field topology (i.e. anisotropy direction) [@BCDDGT_4]. The main constraints of this method reside in the construction of the mean part which necessitates the integration of the solution along the anisotropy direction. This operation is easily carried out in the context of coordinates adapted with the anisotropy direction. However, an extension of the techniques presented here is currently developed for non-adapted coordinates [@brull].\ Section \[SEC3\] is devoted to the numerical implementation of the AP-formulation. Numerical results are then presented for a test case, and the three approaches (AP-formulation, straight discretization and resolution of the P-model and L-model) are compared according to the precision of the approximation for different values of $\eps$. In section \[SEC4\] we shall rigorously analyse the convergence of the AP-scheme. Error estimates will be established which underline the advantages of the AP-scheme as compared to the initial Singular Perturbation model and the Limit model. [Current research directions are concerned with the adaptation of the present technique to the case of arbitrary spatially varying anisotropies, without adaptation of the coordinate system to the direction of the anisotropy. These developments will allow the treatment on nonlinear problems, when the diffusion tensor (and its principal directions) depend on the solution itself. This treatment will involve iterative methods which, at each iterate, will reduce the problem to the solution of a linear anisotropic diffusion problem. ]{} The asymptotic preserving formulation {#SEC2} ===================================== For simplicity we shall consider in this paper the two-dimensional problem, posed on a rectangular domain $\Omega= \Omega_x \times \Omega_z$, where $\Omega_x \subset \RR$ and $\Omega_z \subset \RR$ are intervals. The ideas exposed here can be extended without any problems to the more physical three-dimensional domain, with two transverse directions $(x,y)$ and an anisotropy direction aligned with the $z$-direction. In this section we introduce the Singular Perturbation Model, the Limit Model and the Asymptotic Preserving formulation. The Singular Perturbation Model (P-model) {#SEC21} ----------------------------------------- The main concern of this paper is the numerical resolution of the following anisotropic, elliptic problem, called in the sequel Singular Perturbation Model $$\label{eq:ellipti:original} (P)\,\,\, \left\{ \begin{array}{l} \displaystyle - \nabla \cdot \left ( \mathbb{A} \nabla \phi \right ) = f\,,\quad \text{in} \quad \Omega\,, \\[3mm] \displaystyle \frac{\partial \phi}{\partial z} = 0 \quad \text{on} \quad \Omega_x \times \partial \Omega_z\,,\qquad \displaystyle \phi = 0 \quad \text{on} \quad \partial \Omega_x \times \Omega_z \,. \end{array} \right.$$ The anisotropy of the media is modeled via the definition of the diffusion matrix $\mathbb{A}$ $$\mathbb{A} =\left( \begin{array}[c]{cc} A_\perp & 0 \\ 0 & \frac{1}{\varepsilon}A_z \end{array}\right) \,,$$ where $A_\perp(x,z)$ and $A_z(x,z)$ are given functions with comparable order of magnitudes. The source term $f(x,z)$ is given and the parameter $\varepsilon$ is small compared to both $A_\perp$ as well as $A_z$. The medium becomes more anisotropic as the value of $\varepsilon$ goes to zero. The limit regime (L-model) {#SEC22} -------------------------- In this section we establish that in the limit $\eps \rightarrow 0$ the solution of the perturbation model converges towards $\bar\phi$, solution of the L-model defined by $$\label{eq:ellipti:S} (L)\,\,\, \left\{ \begin{array}{l} \displaystyle- \frac{\partial}{\partial x} \left( \bar{A}_\perp \frac{\partial \bar{\phi}}{\partial x}\right) = \bar{f}(x) \,,\quad \text{in} \quad \Omega_x\,, \\[3mm] \displaystyle \bar{\phi} = 0 \quad \text{on} \quad \partial \Omega_x \,, \end{array} \right.$$ where overlined quantities designate averages over the z-coordinate : $$\bar f(x) = {\frac{1}{|\Omega_z|}\int_{\Omega_z}} f(x,z) \, dz.$$ First we can rewrite the P-model as $$\label{eq:ellipti:original_bis} (P)\,\,\, \left\{ \begin{array}{l} \displaystyle- \frac{\partial}{\partial x} \left( {A}_\perp \frac{\partial {\phi}}{\partial x}\right) - \frac{1}{\eps} \frac{\partial}{\partial z} \left( {A}_z \frac{\partial {\phi}}{\partial z}\right) = {f} \,,\quad \text{in} \quad \Omega\,, \\[3mm] \displaystyle \frac{\partial \phi}{\partial z} = 0 \quad \text{on} \quad \Omega_x \times \partial \Omega_z \,,\quad \phi = 0 \quad \text{on} \quad \partial \Omega_x \times \Omega_z \,, \end{array} \right.$$ and integrating along the $z$-coordinate gives $$\label{aniso:integrated} \frac{\partial}{\partial x} \left( \overline{A_\perp \frac{\partial \phi}{\partial x} } \right) = \bar f(x) \,.$$ This equation holds for any $\eps > 0$. Now, letting formally $\eps$ tend to zero in yields the reduced model (R-model) $$\label{redu} (R)\,\,\, \left\{ \begin{array}{l} \displaystyle - \frac{\partial }{\partial z} \left(A_z \frac{\partial \phi}{\partial z} \right) = 0 \,,\quad \text{in} \quad \Omega\,, \\[3mm] \displaystyle \frac{\partial \phi}{\partial z} = 0 \quad \text{on}\quad \Omega_x \times \partial \Omega_z \,, \qquad \displaystyle \phi = 0 \quad \text{on} \quad \partial \Omega_x \times \Omega_z\,. \end{array} \right.$$ The functions verifying this ill-posed R-model are constant along the $z$-coordinate. Thus including this asymptotic limit property into equation  gives rise to the L-model , verified by the solution of the Singular Perturbation model in the limit $\eps \rightarrow 0$. The L-model is the singular limit of the original P-model . It provides an accurate approximation of the P-solution only for small values of $\eps$. The P-model is valid for all $0<\eps <1$, but numerically impracticable for $\eps\ll 1$. Indeed working with a finite precision, the asymptotic model degenerates into the R-model defined by as $\eps$ vanishes. This R-model is ill-posed since it exhibits an infinite amount of solutions $\phi = \tilde \phi(x)$, depending only on the variable $x$. This implies that the discretization matrix derived from the P-model is very ill-conditioned for small $0<\eps \ll 1$. This point is addressed by the numerical experiments of section \[SEC33\]. Consequently, in a domain where $\eps$ varies significantly, a model coupling method has to be developed in order to exploit the validity of each model, the P- and L-model. This can be rather undesirable. In the next section we shall present an alternative approach, which is based on a reformulation of the Singular-Perturbation model providing a means of computing an accurate numerical approximation of the solution for all values $0<\eps<1$. [The asymptotics is totally different in the case of Dirichlet boundary conditions. In this case, the R-model is well posed, with a unique solution, and there is no difficulty anymore. Any standard numerical solution of the P-model will converge to that of the R-model. In other words, with Dirichlet boundary conditions, the perturbation becomes regular and the limit solution is fully determined by the formal limit system. The situation and the difficulty addressed in the present paper require that the R-model be ill-posed. This is the case with Neumann boundary conditions (which is the framework chosen here) but also with periodic boundary conditions, or any other boundary condition which would result in an ill-posed R-model. ]{} The Asymptotic Preserving reformulation (AP-formulation) {#SEC23} -------------------------------------------------------- In order to circumvent the just described numerical difficulties in handling the Singular Perturbation model, we introduce a reformulation, which permits a transition from the initial $P$-model to its singular limit (L-model), as $\eps \rightarrow 0$.\ For this, we shall decompose each quantity $f(x,z)$ into its mean value $\bar f(x)$ along the $z$ coordinate and a fluctuation part $f'(x,z)$. For simplicity reasons let in the following $\Omega_x:=(0,L_x)$ and $\Omega_z:=(0,L_z)$. Then $$\begin{aligned} \label{eq:def:decomp} f(x,z) &= \bar f(x) + f'(x,z) \,,\end{aligned}$$ with $$\begin{aligned} \label{eq:def:decomp:bis} \bar f(x) := \frac{1}{L_z} \int_{0}^{L_z} f(x,z) dz \,, \quad f'(x,z) := f(x,z) - \bar f(x) \,.\end{aligned}$$ Note that we have the following properties $$\begin{aligned} & \bar{f'} = 0 \,,& \overline{ \left( \nicefrac{\partial f}{\partial x} \right) }= \nicefrac{\partial \overline{f}}{\partial x}\,,\qquad & \displaystyle \overline{fg} = \bar f \bar g + \overline{f'g'} \,,\label{eq:prop:mean}\\ & \nicefrac{\partial f}{\partial z} = \nicefrac{\partial f'}{\partial z} \,, & \left( \nicefrac{\partial f}{\partial x} \right)' = \partial \nicefrac{f'}{\partial x}\,,\qquad & (fg)' = f'g' - \overline{f'g'}+ \bar f g' + f' \bar g\,. \label{eq:prop:fluct}\end{aligned}$$ Taking now the mean of the elliptic equation along the $z$-coordinate, we get thanks to  and , an equation for the evolution of the mean part $\bar{\phi}(x)$ $$\label{eq:compute:mean} (AP1)\,\,\, \left\{ \begin{array}{l} \displaystyle - \frac{\partial}{\partial x}\left( \bar A_\perp \frac{\partial \bar \phi}{\partial x} \right) = \bar f + \frac{\partial}{\partial x}\left( \overline{ A_\perp' \frac{\partial \phi'}{\partial x}} \right)\,,\quad \text{in} \quad \Omega_x\,, \\[3mm] \displaystyle \overline{\phi} = 0 \quad \text{on} \quad \partial \Omega_x\,. \end{array} \right.$$ Substracting from this mean equation , gives rise to the evolution equation for the fluctuation part $\phi'(x,z)$ $$\label{sys:phi:prime} \hspace*{-0.2cm}(AP2)\,\,\, \left\{ \begin{array}{l} \displaystyle - \frac{\partial }{\partial z} \left(A_z \frac{\partial \phi'}{\partial z} \right) - \varepsilon \frac{\partial}{\partial x}\left(A_\perp \frac{\partial \phi'}{\partial x} \right) + \varepsilon \frac{\partial}{\partial x}\left(\overline{A_{\perp}' \frac{\partial \phi'}{\partial x} }\right) = \\[3mm] \displaystyle \hspace{5.2cm} \varepsilon f' + \varepsilon \frac{\partial}{\partial x} \left(A_\perp' \frac{\partial \overline{\phi}}{\partial x} \right)\,,\quad \text{in} \,\,\, \Omega\,, \\[3mm] \displaystyle \frac{\partial \phi'}{\partial z} = 0 \quad \text{on}\quad \Omega_x \times \partial \Omega_z \,, \qquad \displaystyle \phi' = 0 \quad \text{on} \quad \partial \Omega_x \times \Omega_z\,, \\[3mm] \displaystyle \overline{ \phi'}= 0\,, \quad \text{in} \quad \Omega_x\,. \end{array} \right.$$ Thus we have replaced the resolution of the initial Singular Perturbation model (\[eq:ellipti:original\_bis\]) by the resolution of the system (\[eq:compute:mean\])-(\[sys:phi:prime\]), which will be done iteratively. Starting from a guess function $\phi'$, equation (\[eq:compute:mean\]) gives the mean value $\overline{\phi}(x)$, which inserted in (\[sys:phi:prime\]) shall give the fluctuation part $\phi'(x,z)$ and so on.\ The constraint $\overline{\phi'} =0$ in (\[sys:phi:prime\]) (which is automatic for $\eps>0$, as explained in Remark \[rem1\]) has the essential consequence that the conditioning of the discretized system becomes $\eps$-independent, because the problem (\[sys:phi:prime\]) reduces in the limit $\eps \rightarrow 0$ to the system \[NRNR\] { [l]{} - (A\_z ) = 0 , ,\ = 0 \_x \_z , ’ = 0 \_x \_z,\ |[’]{} = 0 \_x , . which is uniquely solvable, with the solution $\phi' \equiv 0$. Inserting this solution in (\[eq:compute:mean\]), we conclude that the solution of the AP formulation converges for $\eps \rightarrow 0$ towards the mean value part $\bar{\phi}(x)$, computed thanks to the Limit model $$(L)\,\, \left\{ \begin{array}{l} \displaystyle- \frac{\partial}{\partial x} \left( \bar{A}_\perp \frac{\partial \bar{\phi}}{\partial x}\right) = \bar{f}(x) \,,\quad \text{in} \quad \Omega_x\,, \\[3mm] \displaystyle \bar{\phi} = 0 \quad \text{on} \quad \partial \Omega_x \,. \end{array} \right.$$ The AP reformulation (\[eq:compute:mean\])-(\[sys:phi:prime\]) is equivalent to the Singular Perturbation problem (\[eq:ellipti:original\_bis\]) and is therefore valid for all $0<\varepsilon <1$. This new formulation guarantees that, working with a finite precision arithmetic, the computed solution converges in the limit $\eps \rightarrow 0$ towards the solution of the limit model (\[eq:ellipti:S\]). This is a huge difference with the original Singular Perturbation model which degenerates into an ill-posed problem. Thus, by using the AP-formulation, [we expect the computation of the numerical solution to be accurate]{}, uniformly in $\eps$.\ For the detailed mathematical proofs, we refer to the next section. \[rem1\] The condition $\overline{\phi'} =0$ in (\[sys:phi:prime\]) holds automatically for $\eps>0$, since the right-hand side has zero average along the $z$-coordinate. Indeed, let $\psi$ be the solution of \[neige\] { [l]{} - (A\_z ) - (A\_ ) + () = g’ , ,\ = 0 \_x \_z , = 0 \_x \_z, . with $\overline{g'}=0$. Taking the average along $z$, we get $$\left\{ \begin{array}{l} \displaystyle- \frac{\partial}{\partial x} \left( \bar{A}_\perp \frac{\partial \bar{\psi}}{\partial x}\right) = 0 \,,\quad \text{in} \quad \Omega_x\,, \\[3mm] \displaystyle \bar{\psi} = 0 \quad \text{on} \quad \partial \Omega_x \,, \end{array} \right.$$ and thus $\overline{\psi} \equiv 0$, which is nothing but the constraint added in (\[sys:phi:prime\]).\ The computations of the fluctuating part $\phi'$ via the equation requires the discretization of an integro-differential operator. This means that the discretization matrix will contain dense blocks. However, using the system (AP2) can be rewritten as $$\label{sys:phi:prime:prime} (AP2')\,\,\, \left\{ \begin{array}{l} \displaystyle - \frac{\partial }{\partial z} \left(A_z \frac{\partial \phi'}{\partial z} \right) - \varepsilon \frac{\partial}{\partial x}\left(A_\perp \frac{\partial \phi'}{\partial x} \right) = \\[3mm] \displaystyle \hspace{5.2cm} \varepsilon f + \varepsilon \frac{\partial}{\partial x} \left(A_\perp \frac{\partial \overline{\phi}}{\partial x} \right)\,,\quad \text{in} \quad \Omega\,, \\[3mm] \displaystyle \frac{\partial \phi'}{\partial z} = 0 \quad \text{on}\quad \Omega_x \times \partial \Omega_z \,, \qquad \displaystyle \phi' = 0 \quad \text{on} \quad \partial \Omega_x \times \Omega_z\,, \\[3mm] \displaystyle \overline{ \phi'}= 0\,, \quad \text{in} \quad \Omega_x\,. \end{array} \right.$$ In this expression the right-hand side has no longer zero mean value along the $z$-coordinate, but the integro-differential operator has disappeared. The associated discretization matrix is thus sparser than that obtained from the system . Systems (\[eq:compute:mean\])-(\[sys:phi:prime\]) and (\[eq:compute:mean\])-(\[sys:phi:prime:prime\]) are equivalent. Mathematical study of the AP-formulation {#SEC32} ---------------------------------------- We establish in this section the mathematical framework of the AP-formulation (\[eq:compute:mean\])-(\[sys:phi:prime\]) and study its mathematical properties. Let us thus introduce the two Hilbert-spaces $$\mathcal{V}:= \{ \psi(\cdot,\cdot) \in H^1(\Omega)\,\, / \,\, \psi=0 \,\, \textrm{on} \,\, \partial \Omega_x \times \Omega_z \}\,, \quad \mathcal{W}:= \{ \psi(\cdot) \in H^1(\Omega_x)\,\, / \,\, \psi=0 \,\, \textrm{on} \,\, \partial \Omega_x \}\,,$$ with the corresponding scalar-products $$\label{sc_VW} (\phi,\psi)_{\mathcal{V}}:=\varepsilon (\partial_x \phi, \partial_x \psi)_{L^2}+(\partial_z \phi,\partial_z \psi)_{L^2}\,, \quad (\phi,\psi)_{\mathcal{W}}:=(\partial_x \phi,\partial_x \psi)_{L^2}\,,$$ [ and the induced norms $||\cdot||_{\cal V}$, respectively $||\cdot||_{\cal W}$.]{} For simplicity reasons, we denote in the sequel the $L^2$ scalar-product simply by the bracket $(\cdot,\cdot)$. Defining the following bilinear forms $$\label{biVF} \begin{array}{ll} \displaystyle a_0 \left(\phi', \psi' \right) &:= \displaystyle \int_0^{L_z} \int_{0}^{L_x} A_z(x,z) \frac{\partial \phi'}{\partial z}(x,z) \frac{\partial \psi'}{\partial z}(x,z) dx dz \, , \\[3mm] \displaystyle a_1 \left(\phi', \psi' \right) &:=\displaystyle \int_0^{L_z} \int_{0}^{L_x} A_\perp(x,z) \frac{\partial \phi'}{\partial x}(x,z) \frac{\partial \psi'}{\partial x} (x,z) dx dz \, ,\\[3mm] \displaystyle a_2 \left(\bar \phi ,\bar \psi \right) &:=\displaystyle \int_{0}^{L_x} \bar A_\perp(x) \frac{\partial \bar \phi}{\partial x}(x) \frac{\partial \bar \psi}{\partial x} (x) dx \, , \\[3mm] \displaystyle c\left(\phi',\bar\psi \right) &:=\displaystyle \int_0^{L_z} \int_{0}^{L_x} A_{\perp}'(x,z) \frac{\partial \phi'}{\partial x}(x,z) \frac{\partial \bar \psi}{\partial x}(x) dx dz \, ,\\[3mm] \displaystyle d(\phi',\psi') &:=\displaystyle \frac{1}{{L_z}} \int_{0}^{L_x} \int_0^{L_z} \int_0^{L_z} A_{\perp}'(x,z) \frac{\partial \phi'}{\partial x}(x,z) \frac{\partial \psi'}{\partial x}(x,\zeta)\, dz d\zeta dx\, ,\\[3mm] b(\bar P,\psi') &:= \displaystyle \int_{0}^{L_x} \bar P(x) \int_0^{L_z} \psi'(x,z) dz dx\, ,\\[4mm] a(\phi',\psi')&:=\displaystyle a_0\left(\phi',\psi'\right) + \varepsilon a_1\left(\phi',\psi'\right) - \varepsilon d(\phi',\psi')\,, \end{array}$$ permits to rewrite the AP system (\[eq:compute:mean\])-(\[sys:phi:prime\]) under the weak form $$\label{eq:var:P} (AP)\,\, \left\{ \begin{array}{l} \displaystyle a_2 \left(\bar\phi,\bar\psi \right) = (\bar f, \bar\psi)- \frac{1}{L_z} c\left(\phi',\bar\psi \right)\, , \qquad \forall \bar \psi \in \mathcal{W} \, ,\\[3mm] \displaystyle a\left(\phi',\psi'\right) + b(\bar P,\psi')= \varepsilon (f',\psi')- \varepsilon c\left(\psi',\overline{\phi} \right)\, , \quad \forall \psi' \in \mathcal{V}\, , \\[3mm] \displaystyle b(\bar Q,\phi') = 0 \, , \qquad \forall \bar Q\in \mathcal{W} \, , \end{array} \right.$$ where $\phi'(x,z) \in \mathcal{V}$, $\bar \phi(x) \in \mathcal{W}$ as well as $\bar{P}(x) \in \mathcal{W}$ are the unknowns and $\psi' \in \mathcal{V}$, $\bar \psi \in \mathcal{W}$ and $ \bar{Q} \in \mathcal{W}$ the test functions. It can be observed that the constraint $\bar{\phi'}=0$ was introduced via the Lagrange multiplier $\bar{P}$. We will see in the next theorem that the weak formulation (\[eq:var:P\]) is equivalent for $\eps>0$ to the system a\_2 (|,|) = (|f, |)- c(’,| ) , | ,\[eq:compute:mean:var\]\ a (’,’) = (f’,’)- c(’, ) , ’ ,\[sys:phi:prime:var\] where the explicit constraint $\bar{\phi'}=0$ does not appear. Let us assume in the sequel\ [**Hypothesis A**]{} [*Let the diffusion functions $A_\perp \in L^{\infty} (\Omega)$ and $A_z \in L^{\infty} (\Omega)$ satisfy $$0<c_\perp\le A_{\perp}(x,z) \le M_\perp \,,\quad 0<c_z\le A_{z}(x,z) \le M_z\,, \quad \textrm{f.a.a.}\,\, (x,z) \in \Omega\,,$$ with some positive constants $c_\perp, c_z, M_\perp, M_z$. Let moreover $f \in L^2(\Omega)$.*]{}\ The next theorem analyzes the well-posedness of the AP-formulation. \[thm\_EX\] For every $\eps>0$ the problem (\[eq:compute:mean:var\])-(\[sys:phi:prime:var\]) admits under Hypothesis A a unique solution $(\phi'_\eps,\overline{\phi}_\eps) \in \mathcal{V} \times \mathcal{W}$, where $\phi_\eps:=\phi'_\eps+\overline{\phi}_\eps$ is the unique solution of the Singular Perturbation model (\[eq:ellipti:original\_bis\]). The function $\phi'_\eps$ has zero mean value along the $z$-coordinate, i.e. $\overline{\phi'}_\eps=0$ for every $\eps>0$.\ Consequently, $(\phi'_\eps,\overline{\phi}_\eps)\in \mathcal{V} \times \mathcal{W}$ is the unique solution of (\[eq:compute:mean:var\])-(\[sys:phi:prime:var\]) if and only if $(\phi'_\eps,\overline{\phi}_\eps,\overline{P}_\eps)\in \mathcal{V} \times \mathcal{W}\times \mathcal{W}$ is a solution of the AP-formulation (\[eq:var:P\]). In this last case, we have $\overline{P}_\eps =0$.\ Finally, these solutions satisfy the bounds $$||\phi_\eps||_{H^1(\Omega)} \le C ||f||_{L^2(\Omega)}\,, \quad ||\phi'_\eps||_{H^1(\Omega)} \le C||f||_{L^2(\Omega)}\,, \quad ||\overline{\phi}_\eps||_{H^1(\Omega_x)} \le C||f||_{L^2(\Omega)}\,,$$ with an $\eps$-independent constant $C>0$. In the limit $\eps \rightarrow 0$ there exist some functions $(\phi'_0,\overline{\phi}_0) \in \mathcal{V} \times \mathcal{W}$, such that [we have the following weak convergences in $H^1$]{} [$$\phi'_\eps \rightharpoonup_{\eps \rightarrow 0} \phi'_0 \quad \textrm{in} \quad H^1(\Omega)\,, \quad \overline{\phi}_\eps \rightharpoonup_{\eps \rightarrow 0} \overline{\phi}_0\quad \textrm{in} \quad H^1(\Omega_x)\,,$$]{} [and the strong $L^2$ convergences]{} [$$\phi'_\eps \rightarrow_{\eps \rightarrow 0} \phi'_0 \quad \textrm{in} \quad L^2(\Omega)\,, \quad \partial_z \phi'_\eps \rightarrow_{\eps \rightarrow 0} \partial_z \phi'_0 \quad \textrm{in} \quad L^2(\Omega)\,, \quad \overline{\phi}_\eps \rightarrow_{\eps \rightarrow 0} \overline{\phi}_0\quad \textrm{in} \quad L^2(\Omega_x)\,,$$]{} where $\phi'_0 \equiv 0$ and $\overline{\phi}_0$ is the unique solution of the Limit model (\[eq:ellipti:S\]). The Singular Perturbation model (\[eq:ellipti:original\_bis\]) and the Limit model (\[eq:ellipti:S\]) are standard elliptic problems and posses under Hypothesis A (and for every $\eps >0$) unique solutions $\phi_\eps \in \mathcal{V}$, respectively $\overline{\phi} \in \mathcal{W}$. It is then a simple consequence of the decomposition (\[eq:def:decomp:bis\]), that the problem (\[eq:compute:mean:var\])-(\[sys:phi:prime:var\]) admits a unique solution $(\phi'_\eps,\overline{\phi}_\eps) \in \mathcal{V} \times \mathcal{W}$, where $\overline{\phi}_\eps(x):= \frac{1}{L_z} \int_{0}^{L_z} \phi_\eps (x,z) dz$ is the mean and $\phi'_\eps:=\phi_\eps -\overline{\phi}_\eps$ the fluctuation part. Thus we have also $\overline{\phi'_\eps}=0$. This property can also be understood from the fact that the right-hand side of (\[sys:phi:prime\]), denoted in the sequel by $g$ $$g(x,z):=f'(x,z)+\frac{\partial}{\partial x} \left(A_\perp'(x,z) \frac{\partial \overline{\phi}}{\partial x}(x) \right)\,,$$ has zero mean value along the $z$-coordinate. Indeed, taking in (\[sys:phi:prime:var\]) test functions $\psi'(x) \in \mathcal{V}$ depending only on $x$, yields immediately that $\overline{\phi'_\eps}=0$ for all $\eps >0$.\ Standard stability results for elliptic problems yield now the $\eps$-independent estimate for the solution of the Singular Perturbation model (\[eq:ellipti:original\_bis\]) $$||\phi_\eps||^2_{H^1(\Omega)} \le ||\partial_x \phi_\eps||^2_{L^2(\Omega)} + {1 \over \eps} ||\partial_z \phi_\eps||^2_{L^2(\Omega)} \le C||f ||^2_{L^2(\Omega)} \,,$$ implying that $||\overline{\phi}_\eps||^2_{H^1(\Omega_x)} \le C||f ||^2_{L^2(\Omega)}$ and $||\phi'_\eps||^2_{H^1(\Omega)} \le C||f ||^2_{L^2(\Omega)}$, with a constant $C>0$ independent of $\eps>0$. Thus there exist some functions $(\phi'_0,\overline{\phi}_0) \in \mathcal{V} \times \mathcal{W}$, such that, up to a subsequence $\phi'_\eps \rightharpoonup_{\eps \rightarrow 0} \phi'_0$ in $H^1(\Omega)$ and $\overline{\phi}_\eps \rightharpoonup_{\eps \rightarrow 0} \overline{\phi}_0$ in $H^1(\Omega_x)$. Hence we have $$\int_0^{L_x}\int_0^{L_z} \phi'_\eps(x,z) \psi(x,z) dx \, dz \rightarrow_{\eps \rightarrow 0} \int_0^{L_x}\int_0^{L_z} \phi'_0(x,z) \psi(x,z) dx \, dz\,, \quad \forall \psi \in \mathcal{V}\,.$$ Taking here $\psi(x)\in \mathcal{V}$ depending only on the $x$-coordinate, we observe that the feature $\overline{\phi'}_\eps \equiv 0$ yields the crucial property of the limit solution $\overline{\phi'}_0 \equiv 0$. Passing now to the limit $\eps \rightarrow 0$ in (\[sys:phi:prime:var\]), we get that $\phi'_0$ is solution of $$\begin{array}{l} \displaystyle a_0(\phi_0',\psi')=0\,, \quad \forall \psi' \in \mathcal{V}\,,\quad \textrm{with} \quad \displaystyle \overline{ \phi'_0}= 0\quad \text{in} \quad \Omega_x\,, \end{array}$$ which is the weak form of (\[NRNR\]) and implies $\phi'_0 \equiv 0$. Finally, passing to the limit in (\[eq:compute:mean:var\]), yields that $\overline{\phi}_0$ is the unique solution of the Limit model (\[eq:ellipti:S\]). Because of the uniqueness of the limit $(\phi'_0,\overline{\phi}_0)$, we deduce that the whole sequence $(\phi'_\eps,\overline{\phi}_\eps)$ converges weakly towards this limit. To conclude the first part of the proof, we shall show [the strong $L^2$ convergences]{}. For this, taking in (\[sys:phi:prime:var\]) $\phi_\eps'$ as test function and passing to the limit $\eps \rightarrow 0$, yields $\partial_z \phi_\eps' \rightarrow 0$ in $L^2(\Omega)$. As $\phi_\eps' \in \mathcal{V}$ and $\bar{\phi_\eps'}=0$, the Poincaré inequality [$$||\phi_\eps'||_{L^2} \le C ||\partial_z \phi_\eps'||_{L^2}\,,$$ is valid and implies that $\phi_\eps' \rightarrow 0$ in $L^2(\Omega)$. The convergence $\bar{\phi_\eps}\rightarrow \bar{\phi_0}$ in $L^2(\Omega_x)$ is immediate by compacity.]{} It remains finally to prove the equivalence between (\[eq:var:P\]) and (\[eq:compute:mean:var\])-(\[sys:phi:prime:var\]). This is immediate. Indeed, if $(\phi'_\eps,\overline{\phi}_\eps) \in \mathcal{V} \times \mathcal{W}$ is solution of (\[eq:compute:mean:var\])-(\[sys:phi:prime:var\]), then $(\phi'_\eps,\overline{\phi}_\eps,0)$ is solution of (\[eq:var:P\]). And if $(\phi'_\eps,\overline{\phi}_\eps,\overline{P}_\eps)\in \mathcal{V} \times \mathcal{W} \times \mathcal{W}$ satisfies (\[eq:var:P\]), then $\overline{P}_\eps \equiv 0$ (obvious by taking as test function in (\[eq:var:P\]) $\psi'(x) \in \mathcal{V}$ depending only on $x$) and $(\phi'_\eps,\overline{\phi}_\eps)$ solves hence (\[eq:compute:mean:var\])-(\[sys:phi:prime:var\]). The subject of the next section will be the numerical resolution of the AP-formulation (\[eq:compute:mean\])-(\[sys:phi:prime\]) (or (\[eq:var:P\])) and this shall be done iteratively via a fixed-point application. Let us thus introduce here the fixed-point map, construct an iterative sequence and analyze its convergence. In the rest of this section, the parameter $\eps>0$ shall be considered as fixed. Due to the fact that the two systems (\[eq:var:P\]) and (\[eq:compute:mean:var\])-(\[sys:phi:prime:var\]) are equivalent, we shall concentrate on the simpler one, i.e. (\[eq:compute:mean:var\])-(\[sys:phi:prime:var\]). Let us define the Hilbert space $$\mathcal{U}:= \{ \psi(\cdot,\cdot) \in \mathcal{V}\,\, / \,\, \overline{\psi}=0\}\,,$$ associated with the [scalar product]{} [$$(\phi,\psi)_*:=\int_0^{L_x}\!\!\int_0^{L_z}\!\!{A}_z \partial_z \phi \, \partial_z \psi dz dx+ \eps \int_0^{L_x}\!\!\int_0^{L_z}\!\!{A}_\perp \partial_x \phi\, \partial_x \psi dz dx\,,$$]{} [which is equivalent to the scalar product $(\cdot,\cdot)_{\cal V}$ on $\cal V$, defined by (\[sc\_VW\]).]{}\ The fixed-point map $T: \mathcal{U} \rightarrow \mathcal{U}$ is defined as follows: With $\phi' \in \mathcal{U}$ we associate $\overline{\phi} \in \mathcal{W}$, solution of (\[eq:compute:mean:var\]). Then constructing the right-hand side of (\[sys:phi:prime:var\]) via this $\overline{\phi} \in \mathcal{W}$, we define $T(\phi')$ as the corresponding solution of (\[sys:phi:prime:var\]). Denoting by $(\phi'_*,\overline{\phi}_*) \in \mathcal{V} \times \mathcal{W}$ the unique solution of (\[eq:compute:mean:var\])-(\[sys:phi:prime:var\]), we remark by Theorem \[thm\_EX\] that $\phi'_* \in \mathcal{U}$ and that it is the unique fixed-point of the map $T$.\ \[FP\_thm\] Let $\eps>0$ be fixed and let $\phi'_* \in \mathcal{U}$ be the unique fixed-point of the application $T: \mathcal{U} \rightarrow \mathcal{U}$ constructed as follows $$\phi'\in \mathcal{U} \quad \xrightarrow{(\ref{eq:compute:mean:var})} \quad \overline{\phi}\in \mathcal{W} \quad \xrightarrow{(\ref{sys:phi:prime:var})} \quad T(\phi')\in \mathcal{U}\,.$$ Then for every starting point $\phi'_0 \in \mathcal{U}$, the sequence $\phi'_k:=T(\phi'_{k-1}) = T^k(\phi'_0)$ converges in [$(\mathcal{U},||\cdot||_*)$, and consequently also in $(\mathcal{U},||\cdot||_{\cal V})$,]{} towards the fixed-point $\phi'_* \in \mathcal{U}$ of $T$. The proof of this theorem is based on the following\ [@brezis] \[BR\] Let $(\mathcal{U},||\cdot||_{{*}})$ be a normed space and $T: \mathcal{U} \rightarrow \mathcal{U}$ a contractive application, i.e. $$||T(\phi)-T(\psi)||_{{*}} <||\phi-\psi||_{{*}}\,, \quad \forall \phi, \psi \in \mathcal{U}\quad \textrm{with} \quad \phi \neq \psi\,.$$ Then the set of fixed-points of T, denoted by $FP(T)$, is identical with the set of accumulation points of the sequences $\{ T^k(\phi)\}_{k \in \NN}$, with $\phi \in \mathcal{U}$, set which is denoted by $AP(T)$. Moreover, these two spaces contain at most one element. [**Proof of theorem \[FP\_thm\] :**]{}\ The linear application $T$ is well-defined. The first step $\phi'\in \mathcal{U} \quad \xrightarrow{(\ref{eq:compute:mean:var})} \quad \overline{\phi}\in \mathcal{W}$ is immediate by the Lax-Milgram theorem. For the second step, we remark that for given $\overline{\phi} \in \mathcal{W}$ the equation \[AP\_c\] a(,’)=(f’,’)-c(’,), ’ , has a unique solution $\theta \in \mathcal{U}$. Indeed, we notice first (by taking test functions only depending on the $x$-coordinate) that $\overline{\theta}=0$. This enables us to consider instead of (\[AP\_c\]), the variational formulation \[E\_APm\] m(,’)=(f’,’)-c(’,), ’ , where the bilinear form $a(\cdot,\cdot)$, which is not coercive, was replaced by the coercive bilinear form $m(\cdot,\cdot)$, given by \[E\_APmm\] m(,’):=a(,’)+[ M\_L\_z]{} \_0\^[L\_x]{} dx . Indeed, due to the property $\overline{\theta}=0$, the two equations (\[AP\_c\]) and (\[E\_APm\]) are equivalent and this time $m(\cdot,\cdot)$ is a continuous, coercive bilinear form, as for all $\psi' \in \mathcal{V}$ we have $$m(\psi',\psi')\ge \int_0^{L_x} \int_0^{L_z} A_z |\partial_z \psi'|^2\, dz dx + \varepsilon \int_0^{L_x} \int_0^{L_z} A_\perp |\partial_x \psi'|^2\, dz dx\ge C||\psi'||_{\mathcal{V}}^2\,.$$ Thus the Lax-Milgram theorem implies the existence and uniqueness of a solution $\theta \in \mathcal{U}$ of the continuous problem (\[E\_APm\]) and hence also of problem (\[AP\_c\]). We have shown by this that $T$ is a well-defined mapping.\ Furthermore we know that $T$ admits, for fixed $\eps>0$, a unique fixed-point, denoted by $\phi'_* \in \mathcal{U}$. Let us now suppose that we have shown that $T$ is contractive. Then lemma \[BR\] implies that $FP(T)=AP(T)=\{\phi'_*\}$. Thus choosing an arbitrary starting point $\phi'_0 \in \mathcal{U}$, and constructing the sequence $\phi'_k:=T(\phi'_{k-1}) = T^k(\phi'_0)$, we deduce that this sequence has a unique accumulation point $\phi'_*$ in $\mathcal{U}$. This means that the sequence $\{\phi'_{k}\}_{k\in \NN}$ converges in [$(\mathcal{U},||\cdot||_*)$ towards $\phi'_*$. Due to the fact that $||\cdot||_*$ and $||\cdot||_{\cal V}$ are equivalent norms, we have also the convergence in $(\mathcal{U},||\cdot||_\mathcal{V})$.]{}\ It remains to show that $T$ is contractive. For this let $\phi'_1, \phi'_2 \in \mathcal{U}$ be two given, distinct functions. Denoting by $\phi':=\phi'_1-\phi'_2$, $\overline{\phi}:=\overline{\phi}_1-\overline{\phi}_2$ (where $\overline{\phi_i} \in \mathcal{W}$ are the corresponding solutions of (\[eq:compute:mean:var\])) and $\theta':= T(\phi'_1)-T(\phi'_2)$, we have to show that $||\theta'||_{{*}} < ||\phi'||_{{*}}$. First we observe that $\overline{\phi}$ solves \[EQ\_T1\] a\_2(|,|)=-[1 L\_z]{} c(’,|), |, and $\theta'$ is solution of \[EQ\_T2\] a(’,’)=- c(’, ) , ’ . Taking in (\[EQ\_T1\]) $\overline{\phi}$ as test function, gives rise to $$\begin{array}{lll} \displaystyle \int_0^{L_x} \overline{A}_\perp |\partial_x \overline{\phi} (x)|^2 \, dx &=& \displaystyle - \int_0^{L_x} \left[ {1 \over L_z} \int_0^{L_z} {A}'_\perp \partial_x \phi' (x,z) dz \right] \partial_x \overline{\phi} (x)\, dx\\[5mm] &= &\displaystyle - \int_0^{L_x} \left[ {1 \over L_z} \int_0^{L_z} {A}_\perp \partial_x \phi' (x,z) dz \right] \partial_x \overline{\phi} (x)\, dx\\[5mm] &\le &\displaystyle {1 \over \sqrt{L_z}} \left[ \int_0^{L_x}\!\! \int_0^{L_z}\!\!{A}_\perp |\partial_x \phi'|^2 dz dx \right]^{1/2} \left[ \int_0^{L_x}\!\!\overline{A}_\perp |\partial_x \overline{\phi}|^2 dx \right]^{1/2} \,. \end{array}$$ Thus $$\left[ \int_0^{L_x} \overline{A}_\perp |\partial_x \overline{\phi} (x)|^2 dx \right]^{1/2} \le {1 \over \sqrt{L_z}} \left[ \int_0^{L_x}\int_0^{L_z}{A}_\perp |\partial_x \phi'|^2 dz dx \right]^{1/2}\,.$$ Equally, taking in (\[EQ\_T2\]) $\theta'$ as test function gives rise to $$\label{NR} \begin{split} \int_0^{L_x}\!\!\int_0^{L_z}\!\!&\!\!{A}_z |\partial_z \theta'|^2 dz dx + \eps \int_0^{L_x}\!\!\int_0^{L_z}\!\!\!\!{A}_\perp |\partial_x \theta'|^2 dz \le - \eps \int_0^{L_x}\!\!\int_0^{L_z}\!\!\!\!{A}_\perp \partial_x \overline{\phi}\, \partial_x \theta' dz dx \hspace*{-2cm}\\[5mm] & \le \eps \left[ \int_0^{L_x}\!\!\int_0^{L_z}\!\!{A}_\perp |\partial_x \overline{\phi}|^2 dz dx \right]^{1/2} \!\!\left[ \int_0^{L_x}\!\!\int_0^{L_z}\!\!{A}_\perp |\partial_x \theta'|^2 dz dx\right]^{1/2}\\[5mm] &\le \displaystyle \eps \sqrt{L_z} \left[ \int_0^{L_x}\!\!\overline{A}_\perp |\partial_x \overline{\phi}|^2 dx \right]^{1/2} \!\!\left[ \int_0^{L_x}\!\!\int_0^{L_z}\!\!{A}_\perp |\partial_x \theta'|^2 dz dx\right]^{1/2}\,. \end{split}$$ This last inequality yields $$\begin{array}{lll} \displaystyle \int_0^{L_x}\!\!\int_0^{L_z}\!\!{A}_z |\partial_z \theta'|^2 dz dx&+& \displaystyle \eps \int_0^{L_x}\!\!\int_0^{L_z}\!\!{A}_\perp |\partial_x \theta'|^2 dz dx \le \eps L_z \int_0^{L_x}\!\!\overline{A}_\perp |\partial_x \overline{\phi}|^2 dx\\[3mm] &\le & \displaystyle \eps \int_0^{L_x}\!\!\int_0^{L_z}\!\!{A}_\perp |\partial_x \phi'|^2 dz dx\\[3mm] &< & \displaystyle \int_0^{L_x}\!\!\int_0^{L_z}\!\!{A}_z |\partial_z \phi'|^2 dz dx+ \eps \int_0^{L_x}\!\!\int_0^{L_z}\!\!{A}_\perp |\partial_x \phi'|^2 dz dx\,. \end{array}$$ In this last step we would have the “equality” if and only if $\int_0^{L_x}\int_0^{L_z}{A}_z |\partial_z \phi'|^2 dz dx =0$. This is however only possible for functions depending exclusively on the $x$-coordinate, $\phi'(x)$, which is in contradiction with the fact that $\overline{\phi'}=0$ and $\phi'\neq 0$. Thus we have shown that $||T(\phi')||_{{*}} < ||\phi'||_{{*}}$ for $\phi' \neq 0$, $\phi' \in \mathcal{U}$, which means that $T$ is a contractive application on [$(\mathcal{U},||\cdot||_*)$.]{} Numerical discretization and simulation results {#SEC3} =============================================== This part of the paper is concerned with the numerical discretization of the AP-scheme (\[eq:compute:mean\])-(\[sys:phi:prime\]) and the comparison of the simulation results with those obtained via the Singular Perturbation model (\[eq:ellipti:original\_bis\]) and the Limit model (\[eq:ellipti:S\]). Discretization {#SEC31} -------------- The numerical resolution of the Asymptotic Preserving system (\[eq:compute:mean\])-(\[sys:phi:prime\]) is done by means of the standard finite element method.\ Let us recall the variational formulation of the AP-formulation $$\label{eq:var:P:bis} \left\{ \begin{array}{l} \displaystyle a_2 \left(\bar\phi,\bar\psi \right) = (\bar f, \bar\psi)- \frac{1}{L_z} c\left(\phi',\bar\psi \right)\, , \qquad \forall \bar \psi \in \mathcal{W} \, ,\\[3mm] \displaystyle a\left(\phi',\psi'\right) + b(\bar P,\psi')= \varepsilon (f',\psi')- \varepsilon c\left(\psi',\overline{\phi} \right)\, , \quad \forall \psi' \in \mathcal{V}\, , \\[3mm] \displaystyle b(\bar Q,\phi') = 0 \, , \qquad \forall \bar Q\in \mathcal{W} \, , \end{array} \right.$$ with the notation of section \[SEC2\]. Here $\phi'(x,z) \in \mathcal{V}$, $\bar \phi(x) \in \mathcal{W}$ as well as $\bar{P}(x) \in \mathcal{W}$ are the unknowns and $\psi' \in \mathcal{V}$, $\bar \psi \in \mathcal{W}$ and $ \bar{Q} \in \mathcal{W}$ the test functions.\ The introduction of the Lagrange multiplier $\overline{P}(x)$ was explained in a simplistic manner in the preceding sections and will be analyzed in more details in section \[SEC4\]. Due to the equivalence of (\[eq:var:P:bis\]) and (\[eq:compute:mean:var\])-(\[sys:phi:prime:var\]), one can comment that the introduction of $\overline{P}(x)$ is superfluous, but this is not the case for the discretized equations. The property $\overline{\phi'}=0$ is indeed automatically fulfilled since the right-hand side of equation has a zero mean value along the $z$-coordinate. However the discrete implementation of this quantity introduces round-off errors which probably will destroy the zero mean value property and justify the introduction of the Lagrange multiplier.\ For simplicity reasons we omitted here the $\eps$-index of the solution $(\phi'_\eps,\overline{\phi}_\eps)$, the parameter $\eps >0$ being considered as fixed.\ To discretize now the system (\[eq:var:P:bis\]) we introduce the grid $$0=x_0 \le \cdots \le x_n \le \cdots \le x_{N_x+1}=L_x\,, \quad 0=z_1 \le \cdots \le z_k \le \cdots \le z_{N_z}=L_z$$ and denote the cells by $I_n:=[x_{n},x_{n+1}]$ and $J_k:=[z_{k},z_{k+1}]$. The finite dimensional spaces $\mathcal{V}_h \subset \mathcal{V}$ and $\mathcal{W}_h \subset \mathcal{W}$ are constructed as usual, by means of the hat functions ($\mathcal{Q}_1$ finite elements) $$\chi_n(x):= \left\{ \begin{array}{ll} \displaystyle \frac{x-x_{n-1}}{x_{n}-x_{n-1}}\,,& x \in I_{n-1}\,,\\[2mm] \displaystyle \frac{x_{n+1}-x}{x_{n+1}-x_{n}}\,,& x \in I_n\,,\\[2mm] \displaystyle 0 \,,& \textrm{else} \end{array} \right. \,, \quad \kappa_k(x):= \left\{ \begin{array}{ll} \displaystyle \frac{z-z_{k-1}}{z_{k}-z_{k-1}}\,,& z \in J_{k-1}\,,\\[2mm] \displaystyle \frac{z_{k+1}-z}{z_{k+1}-z_{k}}\,,& z \in J_k\,,\\[2mm] \displaystyle 0 \,,& \textrm{else} \end{array} \right. \,.$$ Thus we are searching for approximations $\phi'_h \in \mathcal{V}_h$, $\bar \phi_h \in \mathcal{W}_h$ and $\bar P_h \in \mathcal{W}_h$, which can be written under the form $$\phi'_h(x,z)=\sum_{n=1}^{N_x} \sum_{k=1}^{N_z} \alpha_{nk} \chi_n(x) \kappa_k(z) \,, \quad \bar \phi_h(x)=\sum_{n=1}^{N_x} \beta_{n} \chi_n(x) \,, \quad \bar P_h(x)=\sum_{n=1}^{N_x} \gamma_{n} \chi_n(x)\,.$$ Inserting these decompositions in the variational formulation (\[eq:var:P:bis\]) and taking as test functions the hat-functions $\chi_n$ and $\kappa_k$ gives rise to the following linear system to be solved in order to get the unknown coefficients $\alpha_{nk}$, $\beta_n$ and $\gamma_n$ $$\begin{aligned} \label{eq:Dyn:moy} A_2 \beta &= {\bf w} \,,\\ \left( \begin{array}[c]{cc} A_0 + \varepsilon \left(A_1-D\right) &B \\ B^t & 0 \end{array}\right) \left( \begin{array}[c]{c} \alpha \\ \gamma \end{array} \right) &=\varepsilon \left( \begin{array}[c]{c} {\bf v}\\ 0 \end{array} \right)\,,\label{eq:Dyn:fluc}\end{aligned}$$ where the matrices $A_2 \in \RR^{N_x \times N_x}$, $A_0, A_1, D \in \RR^{N_x N_z \times N_x N_z}$ and $B \in \RR^{N_x N_z \times N_x}$ correspond to the bilinear forms (\[biVF\]) and the right-hand sides are defined by $${\bf w}_n:=(\bar f, \chi_n)-\frac{1}{L_z} c(\phi'_h,\chi_n)\,, \quad {\bf v}_{nk}:=(f', \chi_n \kappa_k)-c(\chi_n \kappa_k,\bar \phi_h) = (g, \chi_n \kappa_k)\,,$$ for all $\quad n=1,\cdots,N_x\,; \,\,\, k=1, \cdots N_z\,$ and $$\label{def:function:g} g(x,z):=f'(x,z)+\frac{\partial}{\partial x} \left(A_\perp'(x,z) \frac{\partial \overline{\phi}}{\partial x}(x) \right)\,.$$ Solving iteratively the linear systems (\[eq:Dyn:moy\])-(\[eq:Dyn:fluc\]) permits finally to get the unknown function $\phi_h(x,z)=\bar \phi_h (x) + \phi'_h(x,z)$. The convergence of the iterations was proved for the continuous case in theorem \[FP\_thm\] and can be identically adapted for the discrete case. Numerical results {#SEC33} ----------------- In this section we shall compare the numerical results obtained by the discretization of the Singular Perturbation model, the Limit model and the just presented Asymptotic Preserving reformulation. With this aim, we consider a test case where the exact solution is known. Let thus $$\begin{gathered} \label{SOL_EX} \phi_e(x,z) := \sin\left(\frac{2\pi}{L_x} x\right)+ \varepsilon \cos\left(\frac{2\pi}{L_z} z\right)\sin\left(\frac{2\pi}{L_x} x\right) \,,\end{gathered}$$ be the exact solution of problem (\[eq:ellipti:original\_bis\]), where we choose $ A_\perp(x,y) = c_1 + x z^2$ and $A_z(x,z) = c_2 + xz$, with two constants $c_1>0$, $c_2>0$. [The numerical experiments are performed with $L_x=L_z= 10$ and $c_1 = c_2 = L_z$.]{} The exact right-hand side $f$ is computed by inserting (\[SOL\_EX\]) in (\[eq:ellipti:original\_bis\]). We denote by $\phi_P$, $\phi_L$ and $\phi_A$, respectively, the numerical solutions of the Singular Perturbation model , the Limit model and the Asymptotic Preserving formulation (\[eq:compute:mean\])-(\[sys:phi:prime\]). The comparison will be done in the $l^{2}$-norm, that means $$|| \phi_e - \phi_{num}||_2 = {1 \over \sqrt{N}} \left( \sum_{i\in\mathcal{G}} \left| \phi_e(X_i) - \phi_{num,i} \right|^2 \right) ^{1/2} \,,$$ where $\phi_{num}$ stands for one of the numerical solutions and $\phi_e(X_i)$ is the exact solution evaluated in the grid point $X_i$. The index $i$ covers all possible grid indices, reassembled in the set $\mathcal{G}$, and $N$ is the total number of grid points. The linear systems obtained after the discretization of either the P-model, the L-model or the AP-formulation are solved thanks to the same numerical algorithm (MUMPS [@MUMPS]). The purpose here is not to design a specific preconditioner for the resolution of these linear systems, but to point out the efficiency of the presently introduced AP-method to deal with a large range of anisotropy ratios. \[l\]\[l\]\[.5\][(S)]{} \[l\]\[l\]\[.5\][(L)]{} \[l\]\[l\]\[.5\][(AP)]{} \[\]\[\]\[.8\][$\varepsilon$]{} \[\]\[\]\[1.\] As can be seen from Table \[tab:error\] and Figure \[fig:error:30\], the finite element resolution of the Singular Perturbation model is precise only for large $0<\eps<1$, whereas the Limit model is accurate for small $\eps \ll 1$. $\varepsilon$ $10$ $1$ $10^{-1}$ $10^{-4}$ $10^{-14}$ $10^{-16}$ --------------- ---------------------- --------------------- --------------------- --------------------- --------------------- --------------------- AP-scheme $3.4 \cdot 10^{-2}$ $7.8 \cdot 10^{-3}$ $3.8 \cdot 10^{-3}$ $2.7 \cdot 10^{-3}$ $2.7 \cdot 10^{-3}$ $2.7 \cdot 10^{-3}$ S-scheme $2.8\cdot10^{-2}$ $4.5 \cdot 10^{-3}$ $2.8 \cdot 10^{-3}$ $2.7 \cdot 10^{-3}$ $6.6 \cdot 10^{-2}$ $1.2 $ L-model $9.9 $ $1.0 \cdot 10^{1}$ $1.0 \cdot 10^{-1}$ $2.8 \cdot 10^{-3}$ $2.7 \cdot 10^{-3}$ $2.7 \cdot 10^{-3}$ : Absolute error in the $l^{\infty}$-norm for the approximation computed thanks to the AP-scheme, discretized Singular Perturbation and Limit models (S-scheme and L-model) as compared to the exact solution.[]{data-label="tab:error"} The range of $\eps$-values in which both the Singular Perturbation and the Limit models provide an accurate approximation of the solution shrinks as the mesh size is refined. For a coarse grid (with $50\times 50$ points see figure \[fig:error:coarse\]) this domain ranges from $10^{-12}$ to $10^{-3}$ while it is reduced to $10^{-9} - 10^{-5}$ for the refined $500 \times 500$ grid (figure \[fig:error:refined\]). This question is determinant for the development of a model coupling strategy. Indeed it requires an intermediate area where both discretized models furnish an accurate approximation and we observe that for refined meshes this area may not exist. This reduction of the validity domain can be explained for both the L-model and P-model but for quite different reasons. The numerical approximation computed via the Limit model is altered by both the discretization error of the numerical scheme and the approximation error introduced by the reduction of the initial Singular Perturbation problem to the Limit problem. For coarse grids, the global error is rapidly dominated by the scheme discretization error, but as the mesh is refined, the approximation error becomes preponderant, as the Limit model is precise only for small $\eps$-values. The schemes implemented here are of second order, thus when the mesh size is divided by ten, the discretization error is reduced by one hundred. The global error for the L-model displayed in figure \[fig:error:coarse\] does not depend on $\eps$ as soon as $\eps< 10^{-3}$. Below this limit the L-model is able to furnish a better approximation of the solution with vanishing $\eps$, however the numerical scheme is not precise enough and consequently the global error does not decrease. For the refined mesh, this discretization error is lowered by two order of magnitudes and the global error is a function of $\eps$ as long as its value is greater than $ 10^{-5}$ (Fig. \[fig:error:refined\]). The analysis for the Singular Perturbation model is quite complementary. The accuracy of the approximation provided by the P-model is good for large $\eps$-values and deteriorates rapidly for small ones. This can be explained by the conditioning of the linear system obtained by the P-model discretization. An estimate of the condition number for the matrix is displayed in figure \[fig:condition:number\] for two different grid sizes. This conditioning deteriorates with vanishing $\eps$-parameter, which is coherent with the fact that, working with a finite-precision arithmetic, the Singular Perturbation model degenerates into an ill-posed problem. This also explains the blow up of the error displayed in figure \[fig:error:30\] as soon as the conditioning of the matrix approaches the critical value of the double precision (materialized by the level $10^{15}$ in Fig. \[fig:condition:number\]). This limit is reached on more refined meshes for larger $\eps$-values ($\eps\approx10^{-12}$ on a $50\times 50$ grid and $\eps \approx 10^{-10}$ on a $200\times200$ grid). As expected, the P-model, though valid for all $\eps$-values, cannot be exploited numerically for small $\eps$. The $\eps$-region where both the P-model and the L-model are accurate all-together, shrinks dramatically with the size of the mesh, fact which motivates the development of the AP-method.\ The condition number estimate of the linear system providing the approximation of the solution for the AP-scheme is also plotted in Figure \[fig:condition:number\]. The conditioning of the system is rather $\eps$ independent and this is due to the introduction of the Lagrange multiplier, which forces the system in the limit to remain well-posed. The accuracy of the AP-scheme is totally comparable to the P-model for the large values of $\eps$ and to the L-model for the smallest ones. \[\]\[\]\[1.\][$\varepsilon$-values]{} \[\]\[\]\[1.\][Condition number estimate]{} \[l\]\[l\]\[.7\][S-scheme (50$\times$50)]{} \[l\]\[l\]\[.7\][S-scheme (200$\times$200)]{} \[l\]\[l\]\[.7\][AP-scheme (50$\times$50)]{} \[l\]\[l\]\[.7\][AP-scheme (200$\times$200)]{} ![The $l^2$ absolute error between the exact solution and the numerical approximation computed with the AP-scheme, as a function of the iteration number, with $\eps=10$ and a $200 \times 200$-mesh. Dashed line : mean part of the solution; Plain line : fluctuating part.[]{data-label="fig:relaxation"}](cond.eps "fig:"){width="\textwidth"} \[l\]\[l\]\[.8\][$\left\| \phi_A' -\phi_e' \right\|_2 $]{} \[l\]\[l\]\[.8\][$\left\| \bar \phi_A -\bar \phi_e \right\|_2 $]{} \[\]\[\]\[1.\][$l^2$-Error]{} \[\]\[\]\[1.\][Iterations]{} ![The $l^2$ absolute error between the exact solution and the numerical approximation computed with the AP-scheme, as a function of the iteration number, with $\eps=10$ and a $200 \times 200$-mesh. Dashed line : mean part of the solution; Plain line : fluctuating part.[]{data-label="fig:relaxation"}](Contractive.eps "fig:"){width="\textwidth"} The AP-formulation is a good tool for computing an approximation for the solution which is accurate uniformly in $0<\eps<1$ and is therefore of great practical interest. Note that this approximation is obtained thanks to an iterative sequence $\{ \phi'_k \}_{k \in \NN}$, constructed with the fixed-point mapping $T$ defined in theorem \[FP\_thm\]. The convergence of this iterative process is analysed in figure \[fig:relaxation\] on a $200\times 200$ grid for a large value of $\eps$. The $l^2$-absolute error between the mean respectively the fluctuating parts of the exact solution and the approximation provided by the AP-scheme are plotted as a function of the iteration number. The sequence is initiated with the zero function. [With the iterative process, both components converge towards the solution until the precision of the schemes is reached. At this point, after roughly 27 iterations, the approximation can not be improved and a plateau is observed.]{} The convergence of this sequence may be improved thanks to classical relaxation techniques. Finally we investigate the positivity of the AP-scheme. With this aim the anisotropic elliptic problem is solved with a positive source term, in this case an approximation of the Dirac $\delta$-function. This function denoted $\delta_a^h$ has a support included in a subset ($[-a,a]\times[-a,a]$, with $0<a<1$) of the simulation domainr $[-1,1]\times[-1,1]$. Two different parameters $a$ are chosen, $a=10^{-1}$ and $a=10^{-2}$. The simulation domain is discretized by a $500\times500$ mesh. For the smallest value of $a$ the support of the function is reduced to 5 cells in each direction. The source term $\delta_a^h$ is normalized, such that the maximal value of $\delta_a^h$ grows with vanishing a-parameter. In table \[table:positiv\] the maxima and minima of the numerical approximations computed by the AP-scheme ($\phi_A$) and the discretized Singular Perturbation model ($\phi_{P}$) are gathered for the two source functions $\delta_a^h$. Only large $\eps$-values are considered to verify the positivity of the numerical approximations. Indeed for very small $\varepsilon$ the solution is reduced to its mean part which is the solution of a classical elliptic problem preserving the maximum principle. This means that the relevant question is related to configurations where the fluctuating part $\phi'$ has a significant contribution to the elliptic problem solution. In this range of large and intermediate $\varepsilon$ values, both approximations are comparable. Only slight differences can be observed on the maxima for the smallest $\varepsilon$-parameters. The results of table \[table:positiv\] demonstrate the positivity of the approximations computed by either the AP-scheme or the Singular Perturbation model. [|p[0.4cm]{}|p[1.7cm]{}|p[1.3cm]{}|p[1.3cm]{}|p[1.3cm]{}|p[1.3cm]{}|p[1.3cm]{}|p[1.3cm]{}|]{} &$\varepsilon$ & $10^2$ & $10$ & $1$ & $10^{-1}$ & $10^{-2}$ & $10^{-3}$\ -- ----------------- ---------------- ---------------- ------------------ ------------------ ------------------ ------------------ -- -- max($\phi_{P}$) 77.58 3.82 1.63 8.93 7.22 6.93 max($\phi_A$) 77.58 3.82 1.63 8.93 6.89 6.89 min($\phi_{P}$) $1.9\,10^{-7}$ $2.5\,10^{-7}$ $2.4 \, 10^{-2}$ $2.4 \, 10^{-2}$ $2.8 \, 10^{-2}$ $2.8 \, 10^{-2}$ min($\phi_A$) $1.9\,10^{-7}$ $2.5\,10^{-7}$ $2.4 \, 10^{-2}$ $2.4 \, 10^{-2}$ $2.8 \, 10^{-2}$ $2.8 \, 10^{-2}$ -- ----------------- ---------------- ---------------- ------------------ ------------------ ------------------ ------------------ -- -- : Maxima and minima of the numerical solutions computed thanks to the AP-scheme ($\phi_A$) and the Singular Perturbation model ($\phi_{P}$). The elliptic problem is solved with the Dirac $\delta_a^h$ function as a source term on a $500\times500$ mesh.[]{data-label="table:positiv"} [|p[0.4cm]{}|p[1.7cm]{}|p[1.3cm]{}|p[1.3cm]{}|p[1.3cm]{}|p[1.3cm]{}|p[1.3cm]{}|p[1.3cm]{}|p[1.3cm]{}]{} &max($\phi_{P}$) & $1.8\, 10^{2}$ & $7.1\, 10^{1}$ & $2.6\, 10^{1}$ & $1.2\, 10^{1}$ & $8.29$& $7.34$\ &max($\phi_A$) & $1.8\, 10^{2}$ & $7.1\, 10^{1}$ & $2.6\, 10^{1}$ & $1.2\, 10^{1}$ & $7.14$& $7.11$\ &min($\phi_{P}$) &$1.6 \, 10^{-7}$ &$2.5 \, 10^{-3}$ &$2.4 \, 10^{-2}$ & $2.8 \, 10^{-2}$& $2.8 \, 10^{-2}$& $2.8 \, 10^{-2}$\ & min($\phi_A$) &$1.6 \, 10^{-7}$ &$2.5 \, 10^{-3}$ &$2.4 \, 10^{-2}$ & $2.8 \, 10^{-2}$& $2.8 \, 10^{-2}$& $2.8 \, 10^{-2}$\ Numerical analysis of the AP-scheme {#SEC4} =================================== In this last part of the paper we shall concentrate on the numerical analysis of the $\mathcal{Q}_1$ finite element scheme introduced in section \[SEC31\] for solving \[E\_APi\] { [l]{} - (A\_z ) - (A\_ ) + () = g, ,\ = 0 \_x \_z , = 0 \_x \_z, . where $g \in L^2(\Omega)$ is a given function, with mean value along the $z$-coordinate equal to zero, $\overline{g}=0$. Moreover we shall explain why we have to introduce the Lagrange multiplier in order to solve numerically this equation. We remark that in contrast to section \[SEC3\] we omitted for simplicity reasons the primes for $\phi$, which indicated the fluctuation functions with zero mean value.\ The weak form of (\[E\_APi\]) is \[E\_APw\] a(,)=(g,), , or equivalently \[E\_APwm\] m(,)=(g,), , where $m(\cdot,\cdot)$ is the coercive bilinear form defined in (\[E\_APmm\]). Let us now consider the corresponding discrete problem \[E\_APad\] a(\_h,\_h)=(g,\_h), \_h \_h, where the finite dimensional space $\mathcal{V}_h \subset \mathcal{V}$ was introduced in section \[SEC31\]. It can be seen that the property $\overline{g}=0$ induces also in the discrete case that $\overline{\phi_h} = 0$. Thus, following the same arguments as for the continuous case, we can show that equation (\[E\_APad\]) is equivalent to \[E\_APmd\] m(\_h,\_h)=(g,\_h), \_h \_h. The Lax-Milgram theorem implies then the existence and uniqueness of a discrete solution $\phi_h \in \mathcal{V}_h$. The next theorem gives an estimate of the discretization error $||\phi -\phi_h||_{\mathcal{V}}$.\ [ We shall suppose in the sequel, that the diffusion matrices $A_\perp$, $A_z$ and the function $f$ are regular enough, to be able to use standard regularity/interpolation results.]{}\ Let $\phi \in \mathcal{V}$ be the unique solution of the continuous problem (\[E\_APw\]) and $\phi_h \in \mathcal{V}_h$ the unique solution of the discrete problem (\[E\_APad\]). Both solutions are elements of the [normed]{} space $(\mathcal{U},||\cdot||_{\mathcal{U}})$, where $$\mathcal{U}:=\{ \psi(\cdot,\cdot) \in \mathcal{V}\,\, / \,\, \overline{\psi}=0 \} \quad \textrm{with}\quad ||\psi||_{\mathcal{U}}:=||\partial_z \psi||_{L^2(\Omega)}\,.$$ Then we have the following discretization error estimate \[E\_dis\] ||-\_h||\_\^2=||\_z -\_z \_h||\_[L\^2]{}\^2+ ||\_x -\_x \_h||\_[L\^2]{}\^2 Ch\^2, with a constant $C>0$ independent of $\varepsilon>0$. Moreover, as $\phi,\phi_h \in \mathcal{U}$, we have $$||\phi -\phi_h||_{\mathcal{U}}^2 \le Ch^2\,.$$ The fact that both solutions $\phi$ and $\phi_h$ belong to the space $\mathcal{U}$ is an immediate consequence of the fact that the right-hand side of the equation (\[E\_APw\]) (resp. (\[E\_APad\])) satisfies $\overline{g}=0$. The discretization error estimate is rather standard. Denoting by $\phi_I$ the interpolant of $\phi$ in the finite dimensional space $\mathcal{V}_h$, i.e. $$\phi_I(x,z):=\sum_{n=1}^{N_x} \sum_{k=1}^{N_z} \phi (x_n,z_k) \chi_n(x) \kappa_k(z)\,,$$ we have due to the coercivity of the bilinear form $m(\cdot,\cdot)$ $$c ||\phi -\phi_h||^2_{\mathcal{V}} \le m(\phi -\phi_h,\phi -\phi_h) = m(\phi -\phi_h,\phi -\phi_I) \le c||\phi -\phi_h||_{\mathcal{V}}||\phi -\phi_I||_{\mathcal{V}}\,.$$ Thus $$||\phi -\phi_h||_{\mathcal{V}} \le c||\phi -\phi_I||_{\mathcal{V}}\,.$$ Standard $\mathcal{Q}_1$ finite element interpolation results [[@raviart]]{} yield for the interpolation error $$||\partial_x \phi -\partial_x \phi_I||_{L^2}^2+ ||\partial_z \phi -\partial_z \phi_I||_{L^2}^2 \le ch^2 (||\partial_{xx} \phi||_{L^2}^2 +||\partial_{zz} \phi||_{L^2}^2 )\,,$$ and regularity results for the solution $\phi$ of (\[E\_APw\]), imply $\varepsilon^2 ||\partial_{xx} \phi||_{L^2}^2 +||\partial_{zz} \phi||_{L^2}^2 \le c \varepsilon^2$. [This last estimate can be found by applying standard $H^2$ regularity results on the solution $\phi_\eps$ of the initial Singular Perturbation problem (\[eq:ellipti:original\_bis\]) (after a change of variable $\xi:=\sqrt{\eps} x$) and then exploiting the decomposition $\phi_\eps=\phi'_\eps+\bar{\phi_\eps}$. Thus, we have altogether]{} with a constant $c>0$ independent of $\varepsilon >0$ $$\varepsilon ||\partial_x \phi -\partial_x \phi_h||_{L^2}^2+ ||\partial_z \phi -\partial_z \phi_h||_{L^2}^2 \le ch^2\,.$$\ What is important to observe from the error estimate (\[E\_dis\]) is that for $\varepsilon \rightarrow 0$ the error $||\phi -\phi_h||_{H^1}$ in the standard $\varepsilon$-independent $H^1$-norm blows up. This is one argument why the Singular Perturbation model is inaccurate for $\varepsilon \ll 1$. However, in the case where $\phi$ and $\phi_h$ are elements of the space $\mathcal{U}$, we have $||\phi -\phi_h||_{\mathcal{U}} \le Ch^2$ independently of $\varepsilon$, which means that we have convergence of the scheme [ in $({\cal U},||\cdot||_{\cal U})$]{}, uniformly in $\varepsilon>0$. [ The Poincaré inequality implies then the uniform convergence in the $||\cdot||_{L^2}$ norm.]{} The AP-scheme is thus equally accurate for every value of $0<\varepsilon<1$.\ The discretization error $\phi -\phi_h$ is not the only error we are introducing when solving numerically (\[E\_APad\]) instead of (\[E\_APw\]). Indeed, (\[E\_APad\]) is nothing but a linear system \[E\_ls\] M = v, to be solved to get the unknowns $\alpha_{nk}:=\phi_h(x_n,z_k)$, where $v_{nk}:=\varepsilon (g, \chi_n \kappa_k)$ and the discrete solution of (\[E\_APad\]) is then reconstructed as $$\phi_h(x,z)=\sum_{n=1}^{N_x} \sum_{k=1}^{N_z} \alpha_{nk} \chi_n(x) \kappa_k(z)\,.$$ Unfortunately the implementation of the system (\[E\_ls\]) introduces round-off as well as approximation errors due for example to the numerical computation of $a(\chi_n \kappa_k,\chi_r \kappa_p)$. Thus the numerical resolution of (\[E\_ls\]) does not yield the exact solution, but an approximation $(\tilde{\alpha}_{nk})_{nk}$, solution of the slightly perturbed system \[E\_lsd\] M = . We are now interested in the error estimate $||\phi_h -\tilde{\phi_h}||_{\mathcal{V}}$, as a function of the perturbation $||v -\tilde{v}||_2$, where $||\cdot||_2$ denotes the Euclidean norm in $\RR^{N_x N_z}$.\ Let $\alpha$ be the exact solution of (\[E\_ls\]) and $\tilde{\alpha}$ the exact solution of the perturbed system (\[E\_lsd\]). Let $\phi_h \in \mathcal{V}_h$ and $\tilde{\phi_h}\in \mathcal{V}_h$ denote the corresponding functions $$\phi_h(x,z)=\sum_{n=1}^{N_x} \sum_{k=1}^{N_z} \alpha_{nk} \chi_n(x) \kappa_k(z)\,, \quad \tilde{\phi_h}(x,z)=\sum_{n=1}^{N_x} \sum_{k=1}^{N_z} \tilde{\alpha}_{nk} \chi_n(x) \kappa_k(z)\,.$$ Then we have \[E\_round\] [ ||\_x \_h -\_x ||\_[L\^2]{}\^2 +||\_z \_h -\_z ||\_[L\^2]{}\^2 ]{} , with a constant $c>0$ independent of $\varepsilon>0$ and $h>0$. However, if both functions $\phi_h$ and $\tilde{\phi_h}$ belong to $\mathcal{U}$, then we have the $\varepsilon$-independent estimate $$||\phi_h - \tilde{\phi_h}||_{\mathcal{U}} \le c ||v-\tilde{v}||_2 \,.$$ Let us denote within this proof $E_{nk}:= \alpha_{nk}-\tilde{\alpha}_{nk}$ for $n=1,\cdots,N_x$, $k=1,\cdots,N_z$ and $e_h(x,z):= \phi_h(x,z) -\tilde{\phi_h}(x,z)$, such that $$e_h(x,z)=\sum_{n=1}^{N_x} \sum_{k=1}^{N_z} E_{nk} \chi_n(x) \kappa_k(z)\,.$$ Moreover let $N:=N_x N_z$ and $Y \in \RR^{N}$ be an arbitrary vector associated with the function $y_h(x,z)=\sum_{n=1}^{N_x} \sum_{k=1}^{N_z} Y_{nk} \chi_n(x) \kappa_k(z)$. Then we have with $(\cdot,\cdot)_2$ the euclidean scalar product in $\RR^N$ and $M$ the discretization matrix of (\[E\_ls\]) $$\begin{array}{lll} ||M E||_2 &=&\displaystyle \sup_{Y \in \RR^N\,, Y \neq 0} {(Y,ME)_2 \over ||Y||_2} = \sup_{Y \in \RR^N\,, Y \neq 0} {m(y_h,e_h) \over ||Y||_2}\,. \end{array}$$ Due to the fact that $$||Y||_2 \le c ||y_h||_{L^2} \le {c \over \sqrt{\varepsilon}} ||y_h||_{\mathcal{V}}\,,$$ we have $$\begin{array}{lll} ||M E||_2 &\=&\displaystyle \sup_{Y \in \RR^N\,, Y \neq 0} {m(y_h,e_h) \over ||Y||_2} \ge c \sqrt{\varepsilon} \sup_{y_h \in \mathcal{V}_h \,, y_h\neq 0} {m(y_h,e_h) \over ||y_h||_{\mathcal{V}}} \ge c \sqrt{\varepsilon} ||e_h||_{\mathcal{V}}\,. \end{array}$$ Thus we get with a constant $c>0$ independent of $\varepsilon$ $$||e_h||_{\mathcal{V}}\le {c \over \sqrt{\varepsilon}} ||M E||_2 = {c \over \sqrt{\varepsilon}} ||v-\tilde{v}||_2\,.$$ In the case the two functions $\phi_h$ and $\tilde{\phi_h}$ belong to $\mathcal{U}$, i.e. $e_h \in \mathcal{U}$, we can exploit the fact that in $\mathcal{U}$ [ the Poincaré inequality gives rise to]{} $||Y||_2 \le c ||y_h||_{L^2} \le c ||y_h||_{\mathcal{U}}$. This yields, as $m(\cdot,\cdot)$ is also coercive on $\mathcal{U}$, that $$\begin{array}{lll} ||M E||_2 &\=&\displaystyle \sup_{Y \in \RR^N\,, Y \neq 0} {m(y_h,e_h) \over ||Y||_2} \ge c \sup_{y_h \in \mathcal{U} \,, y_h\neq 0} {m(y_h,e_h) \over ||y_h||_{\mathcal{U}}} \ge c ||e_h||_{\mathcal{U}}\,. \end{array}$$ and thus the $\varepsilon$-independent estimate is proved.\ Similarly as for the discretization error, we can deduce from the round-off error estimate (\[E\_round\]) that, for $\varepsilon \rightarrow 0$, the standard $H^1$-norm $|| \phi_h -\tilde{\phi_h}||_{H^1}$ explodes. However if we impose that both solutions $\phi_h$ and $\tilde{\phi_h}$ are elements of the space $\mathcal{U}$, space of functions with mean value along the $z$-coordinate equal to zero, then we have the uniform estimate $|| \phi_h - \tilde{\phi_h}||_{\mathcal{U}} \le c ||v-\tilde{v}||_2 $ [, and by the Poincaré inequality $|| \phi_h - \tilde{\phi_h}||_{L^2} \le c ||v-\tilde{v}||_2$]{}. Unfortunately even if we know that $\phi_h \in \mathcal{U}$, this is not necessarily true for $\tilde{\phi_h}$, if we discretize (\[E\_APi\]). But it can be achieved by forcing the numerical solution $\tilde{\phi_h}$ to satisfy $\overline{\tilde{\phi_h}}=0$. Indeed, this can be done by introducing explicitly in the discrete problem (\[E\_APad\]) the constraint $\overline{\phi_h}=0$, such that it is much more ingenious to solve instead \[E\_APl\] { [l]{} a(\_h,\_h)+ b(P\_h,\_h)=(g,\_h), \_h \_h,\ b(Q\_h,\_h)=0, Q\_h \_h, . where $\mathcal{W}_h \subset \mathcal{W}$ was constructed in section \[SEC31\]. As mentioned in the continuous case this problem is equivalent for $\eps >0$ to the discrete problem (\[E\_APad\]). If $\phi_h \in \mathcal{V}_h$ is the unique solution of (\[E\_APad\]), then $(\phi_h ,0) \in \mathcal{V}_h \times \mathcal{W}_h$ is a solution of (\[E\_APl\]). And if $(\phi_h ,P_h) \in \mathcal{V}_h \times \mathcal{W}_h$ solves (\[E\_APl\]), then $P_h \equiv 0$ and $\phi_h \in \mathcal{V}_h$ is the unique solution of (\[E\_APad\]). This last statement is immediately proved by taking in the variational formulation (\[E\_APl\]) only $x$-dependent test functions $\psi_h(x) \in \mathcal{V}_h$. By doing this, we can be sure that the numerical solution $\tilde{\phi_h}$ of (\[E\_APl\]) satisfies $\overline{\tilde{\phi_h}}=0$, such that the error $||\phi_h -\tilde{\phi_h}||_{\mathcal{U}}$ is uniformly bounded. This proves that the introduction of the constraint $\overline{\phi_h}=0$ in the AP-formulation is crucial and avoids the numerical difficulties associated with the original P-model. Conclusion {#SEC5} ========== In this paper we have introduced an Asymptotic Preserving formulation for the resolution of a highly anisotropic elliptic equation. We have shown the advantages of the AP-formulation as compared to the initial Singular Perturbation model and to its limit model, when the asymptotic parameter goes to zero. It came out that the AP-scheme is a powerful tool for the resolution of elliptic problems presenting huge anisotropies along one coordinate, and gives access to the simulation in a very easy and precise manner. The Asymptotic-Preserving method developed here relies on the decomposition of the solution in its mean part along the anisotropy direction, and a fluctuation part. This integration along the anisotropy direction is easily performed in the context of Cartesian coordinate systems with one coordinate aligned with the direction of the anisotropy. In a forthcoming work [@brull] this procedure is extended to more general anisotropies. 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--- author: - 'G. Nandakumar' - 'M. Schultheis' - 'A. Feldmeier-Krause' - 'R. Schödel' - 'N. Neumayer' - 'F. Matteucci' - 'N. Ryde' - 'A. Rojas-Arriagada' - 'A. Tej' bibliography: - 'paper.bib' title: 'Near-infrared spectroscopic observations of massive young stellar object candidates in the Central Molecular Zone [^1]' --- [ The Central Molecular Zone (CMZ) is a $\sim$200 pc region around the Galactic Center. The study of star formation in the central part of the Milky Way is of highest interest as it provides a template for the closest galactic nuclei.]{} [We present a spectroscopic follow-up of photometrically-selected young stellar object (YSO) candidates in the CMZ of the Galactic center. Our goal is to quantify the contamination of this YSO sample by reddened giant stars with circumstellar envelopes and to determine the star formation rate in the CMZ. ]{} [We obtained KMOS low-resolution near-infrared spectra (R$\sim$4000) between 2.0 and 2.5$\mu$m of sources, many of them previously identified, by mid-infrared photometric criteria, as massive YSOs in the Galactic center. Our final sample consists of 91 stars with good signal-to-noise ratio. We separate YSOs from cool late-type stars based on spectral features of CO and $\rm Br{\gamma}$ at 2.3$\mu m$ and 2.16$\mu m$ respectively. We make use of SED model fits to the observed photometric data points from 1.25 to 24 $\mu m$ in order to estimate approximate masses for the YSOs. ]{} [ Using the spectroscopically identified YSOs in our sample, we confirm that existing colour-colour diagrams and colour-magnitude diagrams are unable to efficiently separate YSOs and cool late-type stars. In addition, we define a new colour-colour criterion that separates YSOs from cool late-type stars in the H-K$_{\rm S}$ vs H-\[8.0\] diagram. We use this new criterion to identify YSO candidates in the $|$l$|$ &lt; 15, $|$b$|$&lt;05 region and use model SED fits to estimate their approximate masses. By assuming an appropriate initial mass function (IMF) and extrapolating the stellar IMF down to lower masses, we determine a star formation rate (SFR) of $\sim$0.046$\pm$0.026M$_{\sun}$yr$^{-1}$ assuming an average age of 0.75 $\pm$ 0.25Myr for the YSOs. This value is lower than estimates found using the YSO counting method in the literature. ]{} [ Our SFR estimate in the CMZ agrees with the previous estimates from different methods and reaffirms that star formation in the CMZ is proceeding at a lower rate than predicted by various star forming models. ]{} Introduction ============ The CMZ is the innermost $\sim$200pc region of the Milky Way, covering about $\rm -0\fdg7 < l < 1\fdg8$ in longitude and $\rm -0\fdg3 < b < 0\fdg2$ in latitude. It is a giant molecular cloud complex with an asymmetric distribution of molecular clouds (see e.g. @morris1996 [@Martin2004; @oka2005]). The understanding of the physical processes occurring in the CMZ of our Galaxy is crucial for the insight in the formation/evolution of our own Milky Way. This prodigious reservoir of molecular gas is in an active region of star formation, with evidence of starburst activity in the last 100,000 years (@YZ2009, hereafter YHA09). The gas pressure and temperature are higher in the CMZ than in the Galactic disk, conditions that favor a larger Jeans mass for star formation and an IMF biased toward more massive stars (see @Serabyn1996 [@Fatuzzo2009]). Thus it is essential to understand the modes of star formation and star formation history in the unique environment of the CMZ, both to gain insight into our own Milky Way and to provide a template for circumnuclear star formation in the closest galactic nuclei. [@2013MNRAS.429..987L] carried out a detailed study of the variations in star formation across the Galactic plane using observational tracers of dense gas (NH$_{3}$(1,1)) as well as star formation activity (masers, HII regions). They showed that there is a deficiency in the star formation activity tracers in the CMZ given the large reservoir of dense gas available. On the other hand, they found that various star formation models predict much higher values of SFR. [@2017MNRAS.469.2263B] determined the average SFR across the CMZ using a variety of extra-galactic luminosity–SFR conversion and found it to be comparable to previous measurements made from YSO counting and the free-free emission. Thus they ruled out systematic uncertainties in the SFR measurements as the reason for low star formation in the CMZ. The central few parsecs of the Milky Way host a massive young population of stars. There is strong evidence that young stars in the central parsec formed *in situ* (see @2010RvMP...82.3121G and references therein). Recent spectroscopic observations have provided further strong evidence to support this. observed the central &gt;4 pc$^{2}$ of the Galactic centre and identified &gt;100 early-type young stars by spectral classification. They found that early-type stars are centrally concentrated favouring the *in situ* formation of the early-type stars. [@2015ApJ...808..106S] mapped a smaller area within 0.28-0.92 pc from SgrA$^{*}$ and found a break in the distribution of young stars at 0.52 pc. They concluded that this break possibly indicated an outer edge to the young stellar cluster in the Galactic center which is expected in the case of *in situ* star formation. Until recently, most studies of YSOs in the CMZ have been based on infrared photometry (@Felli2002 [@Schuller2006], YHA09). The YSO phase of a massive star is a relatively brief phase in which they are surrounded by dense envelopes of gas and dust . They are best identified by their point-source infrared radiation as well as excess flux values in the mid-infrared bands. YSOs are classified into three classes/stages depending on their spectral index [@1987IAUS..115....1L] as well as spectral energy distribution (SED) models [@2006ApJS..167..256R]. [@1987ApJ...312..788A] identified Class I YSOs as protostars with infalling envelopes, Class II YSOs as stars with disks and Class III YSOs as those stars having the SEDs of stellar photospheres. Analogous to this classification, [@2006ApJS..167..256R] defined three evolutionary stages based on their derived SED model properties : Stage I objects are young protostars embedded in an opaque infalling envelope, Stage II objects are stars surrounded by an opaque disk and dispersed envelope, and Stage III objects are stars with an optically thin disk. [@Felli2002] searched for YSO candidates using the mid-infrared excess derived from Infrared Space Observatory (ISO) photometry at 7 and 15 $\mu$m and found a strong concentration of YSO candidates in the inner Galaxy. [@Schuller2006] refined the ISO mid-infrared colour criteria, and argued that slightly extended mid-infrared sources were more likely to be YSOs than point-like mid-infrared sources. YHA09 identified YSO candidates with *Spitzer* photometry at 3.6 – 24 $\mu$m. Their SED fitting techniques associated most of their YSO candidates with Stage I objects; they concluded that a recent starburst took place in the CMZ. The CMZ, however, suffers from very large and spatially-variable interstellar extinction ($\rm A_{V}$ = 20 – 40 mag, see e.g. @Schultheis2009). The significant foreground extinction causes evolved stars with circumstellar envelopes, such as mass-losing asymptotic giant branch (AGB) stars, to have infrared colours similar to those of YSOs. [@Schultheis2003] demonstrated that near-infrared spectra are a powerful tool to distinguish YSOs from reddened AGB stars. They found that YSO samples in the CMZ selected by photometric colour criteria are heavily contaminated by AGB stars, red giants and even supergiants (see their Fig. 5). By contrast, they showed that moderate-resolution spectra in the H and K bands delineate YSOs from evolved stars by the absence or presence (respectively) of CO absorption at $\sim$ 2.3$\mu$m. Detectable YSOs at the distance of the Galactic center ($\sim$8 kpc; see eg: @2016ApJ...830...17B [@2017ApJ...837...30G]) are all massive, and thus never show 2.3 $\mu$m CO absorption; instead, they are featureless around 2.3 $\mu$m or show CO in emission (Geballe & Persson 1987; Carr 1989; Hanson et al. 1997; Bik et al. 2006). A recent improvement in YSO selection in the CMZ came from using *Spitzer*/*IRS* spectra to select YSOs. [@An2009; @An2011], hereafter An11, presented *Spitzer*/*IRS* 5–35 $\mu$m spectra of 107 YSO candidates selected from 3.6–8.0 $\mu$m *Spitzer* photometry [@Ramirez2008]. An11 identified massive YSOs in the CMZ by the presence of gas-phase absorption from $\rm C_{2}H_{2}$ (13.7 $\mu$m), HCN (14.0 $\mu$m), and $\rm CO_{2}$ (15.0 $\mu$m) as well as strong and broad 15.2 $\mu$m $\rm CO_{2}$ ice absorption. They found that mid-infrared spectra confirm only 33% of YSO candidates selected by their photometric criteria, and confirm 57% of YSO candidates selected photometrically by YHA09. [@Immer2012] analysed 5–40 $\mu$m *Spitzer*/*IRS* spectra of 57 YSO candidates selected from 7 and 15 $\mu$m ISO colours and spatial extent at 15 $\mu$m [@Schuller2006]. They identified 25% of their sources as YSOs, with an additional 37% identified as H II regions. There is disagreement in the YSO classification even among the common sources in An11 and [@Immer2012] samples, suggesting uncertainties in spectroscopic YSO classification schemes as well. [@2015ApJ...799...53K], using radiative transfer models and realistic synthetic observations, re-examined the YHA09 YSO sample and showed that embedded main sequence stars contaminate the YHA09 sample. These recent studies demonstrate significant contamination of photometrically-selected YSO candidate samples by non-YSOs, which has important implications for CMZ star formation rates derived from photometry (YHA09). In this paper, we present moderate-resolution spectroscopic follow-up observations of a sample of 91 photometrically-identified YSO candidates in the CMZ using K-band Multi Object Spectrograph (KMOS, @2013Msngr.151...21S) at VLT-UT1 (Antu). Our goal is to distinguish YSOs from evolved late-type stars by their near-infrared spectra. We discuss and show the contaminating evolved late-type stars in different colour-magnitude (CMD) and colour-colour (CCD) diagrams and define a new colour-colour criterion to distinguish them using our spectroscopically identified YSO sample. We estimate the SFR in the CMZ based on YSO counting and on SED fitting techniques. Sample selection, observations and data reduction ================================================= Sample selection ---------------- We select the sample for our observation from the photometric catalogue of SIRIUS [@2006ApJ...638..839N] and point-source catalogue of *Spitzer* IRAC survey of the Galactic center [@Ramirez2008]. The JHK$_{\rm S}$ photometry from the SIRIUS catalogue has average 10$\sigma$ limiting magnitudes of J=17.1, H=16.6 and K$_{\rm S}$=15.6 mag, while the 3.6, 4.5, 5.8 and 8.0 $\mu$m bands from the *Spitzer* IRAC catalogue has confusion limits of 12.4, 12.1, 11.7, and 11.2 mag respectively. We divide our sample into three categories of high, medium and low priorities, with the highest priority (priority 1) given to those sources in our sample that are photometrically identified YSO candidates in YHA09. We divide the rest of the sample into medium (priority 2) and low (priority 3) priorities using the following criteria that select sources exhibiting excess emission in mid-infrared regimes :\ \ Medium priority Low priority\ $[$3.6$]$-$[$4.5$]$ &gt; 0.5 \[3.6\]-\[8.0\] &gt; 2\ $[$4.5$]$-$[$5.8$]$ &gt; 0.5K$_{\rm S}$ &lt; 17\ $[$5.8$]$-$[$8.0$]$ &gt; 1\ K$_{\rm S}$ &lt; 17\ KMOS consists of 24 integral field units (IFU) that can be arranged in a 72 diameter field per configuration, and it is crucial to prevent the 24 IFUs on the 24 pick-off arms from blocking each other. We use the KMOS ARM Allocator (KARMA) which assigns the maximum number of highest priority targets to the 24 pick-off arms, followed by medium and low priority targets thereby leaving as few arms as possible unallocated. KMOS observations ----------------- Our spectroscopic observations were carried out with KMOS at VLT-UT1 (Antu) on June 23, 2016. Each of the 24 IFUs in KMOS has a field of view of 28$\times$28. The spectral resolution of KMOS is R$\sim$4300 with the wavelength range covering . We prepared 22 fields with unique IFU configuration covering a significant part of the CMZ. But due to bad weather conditions, only 8 fields could be observed with an integration time of 900s for each field in the *nod to sky* mode. The observations were carried out under photometric conditions with seeing $\sim$08. The observed field positions are shown in Figure \[planned\] and the field details with the number of different priority sources in each field are given in Table \[OBS\_fields\]. Field No. l() b() Priority 1 Priority 2 Priority 3 Sky Random source ----------- ---------- --------- ------------ ------------ ------------ ----- --------------- 1 359.2568 -0.0813 13 3 3 3 2 4 359.5854 -0.0351 13 5 4 1 1 5 359.5131 -0.0986 9 3 11 1 0 6 359.8000 -0.0500 15 1 7 1 0 8 0.0419 -0.0547 0 5 18 1 0 14 0.6674 -0.0527 12 9 3 1 0 15 0.0319 0.0506 3 9 12 2 0 18 359.1303 -0.0080 12 1 5 4 2 The fields are in regions with high stellar density as can be seen in the Figure \[planned\]. Hence, it was not possible to observe sky in each IFU by dithering or nodding to a new position within the respective field. So we observed a dark cloud G359.94+0.17 ($\alpha$ $\sim$ 2662, $\delta$ $\sim$ -289, ) in free dither mode with 28 dither in between two 900s exposures to carry out a proper sky subtraction. The sky observation was carried out after every two field observations. B and A type stars were observed for telluric corrections after every sky offset. Data reduction -------------- We used the ESO KMOS Recipe Flexible Execution Workbench (Reflex, ) for data reduction. It organises the science and associated sky and calibration data together based on the calibration source type as well as their proximity in time to science observations. This is followed by dark level correction of frames, flat fielding, wavelength calibration, spatial illumination correction, telluric correction, sky subtraction and cube reconstruction of the science data by dedicated pipeline recipes (or stages). We made use of an IDL routine to remove the Br$\gamma$ absorption line from each telluric spectrum by fitting the Br$\gamma$ line with a Lorentz profile and subtracting it from the telluric spectrum. This routine also removes the stellar continuum by dividing it by a blackbody spectrum. We also removed cosmic rays from the final reconstructed object cube with a 3D version of L.A.Cosmic [@2001PASP..113.1420V]. We extracted spectra from 173 data cubes using *kmos$\_$extract$\_$spec* recipe with the ESO Recipe Execution Tool (EsoRex). *kmos$\_$extract$\_$spec* extracts a spectrum from a data cube with the option of defining a mask manually or automatically by fitting a normalised profile to the image of the data cube. We identify multiple sources in 53 data cubes and extracted their spectra by defining the mask manually. Finally, we extracted nearly 250 spectra and used IRAF median filtering with 7 pixels as filter size to smoothen the spectra. After discarding spectra with a signal-to-noise ratio (S/N) below 20 or having negative flux values, there were 91 spectra left in our sample with good S/N. Among them, there are 15 spectra from data cubes with multiple sources, and we selected the spectrum of the brightest source in the data cube. Classification ============== In this section, we classify our targets as YSOs and late-type stars based on their spectra. Using this classification, we evaluate different photometric YSO classification criteria and suggest a new criterion to distinguish YSOs and late-type stars. Spectroscopic classification {#spectra_classify} ---------------------------- We classify our spectra mainly based on the presence or absence of the $\rm ^{12}CO$ (2,0)-band at 2.3 $\mu m$. CO absorptions bands are typically found in late-type G, K, M giants and AGB stars. In addition, we also use $\rm Br {\gamma}$, found in emission, absorption or with a P-Cygni-type profile in massive YSOs [@2013MNRAS.430.1125C]. We have carried out background subtraction during data reduction to make sure that the $\rm Br {\gamma}$ emission lines are intrinsic to the source. Still, we expect contamination from the OB main-sequence/post main-sequence/Wolf Rayet stars, the spectra of some of which also show $\rm Br {\gamma}$ in emission attributed to their stellar wind [@Mauerhan2010]. CO band emission at 2.3$\mu$m is also considered to be an indication for the presence of a dense circumstellar disk and hence a YSO signature . Some spectra show a featureless continuum at 2.0–2.5 $\mu$m; these could be either YSOs [@Greene1996] or dusty late-type carbon-type (WC) Wolf–Rayet stars [@Mauerhan2010]. We measure the equivalent width (EW) of the $\rm ^{12}CO$ (2,0) band at 2.3$\mu$m using the same CO band and continuum points as in [@Ramirez2000]. In addition, we measure the EW of $\rm Br{\gamma}$ line at 2.16$\mu$m. Figure \[EW\] shows the EW(CO) vs EW($\rm Br{\gamma}$) plot of our 91 sources. We find that there are two separate groups of stars with a very evident gap which we approximately mark by the dashed line at EW(CO) = 10 Å. Positive values of EW indicate that the line is in absorption while negative values indicate it is in emission. Thus we classify the stars to the right of the dashed line as cool, late-type stars and those to the left as YSOs. All cool, late-type stars (represented by filled black circles) lie very close to indicating the absence of this particular feature in their spectra. The majority of stars we classify as YSOs also show no $\rm Br{\gamma}$ feature, while approximately five stars show $\rm Br{\gamma}$ in emission (EW($\rm Br{\gamma}$)&lt;1Å) and only one star show $\rm Br{\gamma}$ in absorption (EW($\rm Br{\gamma}$)$>$1Å), which also show CO in emission (EW(CO) = -10 Å). Based on the above mentioned classification scheme, there are 23 spectroscopically identified YSOs in our sample. Figures \[Yso\_Spectra\] and \[M\_Spectra\] show the reduced normalised spectra of YSOs and cool late-type stars respectively. We searched for previously identified sources in the SIMBAD database with a search radius of 20 from each of the 23 YSOs to check their status in the literature. Table \[simbad\] lists all 23 sources with their SST (*Spitzer Space Telescope*) GC (Galactic center) No., equatorial coordinates in degrees, distance of the SIMBAD match from the source, source type along with corresponding references and JHK$_{\rm S}$ magnitudes from SIRIUS catalog. There are seven sources in common with An11, out of which only one is confirmed to be YSO and one is considered to be a possible YSO by An11. The remaining five sources have been classified as non-YSOs, with $\#$517724 classified also as a OB super giant star in [@2010ApJ...710..706M] based on absorption lines of $\rm Br{\gamma}$ at 2.1661$\mu$m, NIII at 2.115 $\mu$m and He I at 2.058, 2.113, and 2.1647$\mu$m. Two other sources ($\#$528828 and $\#$599826) have been classified as possible long period variable stars in [@matsunaga2009] and [@2001MNRAS.321...77G] based on periods estimated using near-infrared observations, though no clear periodicity was found for them. Two sources ($\#$531300 and $\#$584563) have been classified as blue super giant stars based on weak $\rm Br{\gamma}$ emission or absorption feature, NIII and CIV contributions as well as HeI absorption profile at 2.058$\mu$m in addition to the Paschen-$\alpha$ (P$\alpha$) excess detected in them [@Mauerhan2010]. The counterpart to the source $\#$238110 has been classified as X-ray source in [@2003ApJ...599..465M]. Three sources ($\#$358063, $\#$395315 and $\#$520760) are in common with the YHA09 sample, out of which $\#$520760 is classified also as a radio source in [@2015MNRAS.446..842D]. The remaining eight sources do not have any counterparts in the SIMBAD database within 20. In Figure \[EW\], we show the sources classified in the literature as X-ray source, OB/blue supergiants and non-YSOs using separate symbols. Regarding the classification of five sources as non-YSOs by An11, the large pixel sizes of *Spitzer/IRS* spectra can lead to mis-identification of sources in high stellar density regions like in the CMZ. Also one non-YSO ($\#$405235) shows $\rm Br{\gamma}$ in absorption and CO in emission, indicating the presence of a dense circumstellar disk and considered to be a massive YSO feature though rarely seen (. Thus we stick with our classification scheme for them. Since no clear periodicity was found for the two sources classified as long period variable stars, we assume them to be YSOs as well. Two blue supergiants ($\#$584563 and $\#$531300) show $\rm Br{\gamma}$ in emission, while no clear emission is seen for $\#$517724 classified as OB supergiant. We made an approximate EW measurement of NIII at 2.115 $\mu$m and found that all three sources mentioned above as well as the radio source $\#$238110 show clear emission feature, not seen in the rest of our YSOs. Thus we consider their classifcation as supergiants to be acceptable. Based on these measurements we conclude that none of the rest of our YSOs is a O/B supergiant. {width=".35\textwidth"} {width=".35\textwidth"} {width=".30\textwidth"} SST GC No. RA () DEC () Distance (in ) Source type J (mag) H (mag) K$_{\rm S}$ (mag) Field ------------ ----------- ----------- ---------------- --------------------------- --------- --------- ------------------- ------- -- 238110 265.93584 -29.67287 0.64 X-ray source 17.19 13.64 11.84 18 517724 266.40542 -28.89823 0.07 OB supergiant, non-YSO 15.75 12.80 11.23 15 520760 266.41002 -28.89086 1.54 radio source, YSO ... 15.59 13.17 15 524419 266.41569 -28.89559 0.50 non-YSO ... 15.86 14.02 15 525666 266.41759 -28.89117 0.04 non-YSO ... 14.49 12.74 15 531300 266.42633 -28.87979 0.17 Blue super giant 14.70 11.67 10.11 15 535007 266.43185 -28.87358 0.05 non-YSO ... 16.05 13.81 15 528828 266.42248 -28.90728 0.11 Long period variable star ... ... 13.69 8 541457 266.44147 -28.90716 ... ... ... 14.28 12.53 8 553700 266.45999 -28.91312 ... ... 15.11 13.09 12.27 8 563727 266.47539 -28.97628 ... ... ... 14.75 12.97 8 567598 266.48123 -28.93786 ... ... ... 14.71 12.92 8 609669 266.54470 -28.92073 ... ... ... 16.24 14.17 8 610642 266.54618 -28.92802 0.05 Maybe YSO ... 15.26 12.69 8 584563 266.50690 -28.92095 0.17 Blue super giant 13.45 10.69 9.12 8 585974 266.50900 -28.95653 ... ... 16.80 11.73 9.42 8 599826 266.52982 -28.93144 0.52 Long period variable star ... ... 13.87 8 373107 266.17890 -29.33702 ... ... 16.48 13.59 11.94 4 388790 266.20389 -29.39521 0.40 non-YSO ... 14.12 12.05 5 352034 266.14518 -29.39360 ... ... ... 12.52 10.95 5 395315 266.21438 -29.34097 0.02 YSO ... 15.95 13.65 4 405235 266.23023 -29.26057 0.28 non-YSO ... 14.77 12.97 4 358063 266.15475 -29.31017 0.14 YSO 15.59 13.42 11.79 4 @2003ApJ...599..465M @Mauerhan2010 @An2011 @2015MNRAS.446..842D @YZ2009 @matsunaga2009 [@2001MNRAS.321...77G] Classification using photometric criteria {#hkh8} ----------------------------------------- Several previous studies have made use of colour-colour diagrams (CCDs) to define criteria in order to classify YSOs. We are trying with our spectroscopic sample of YSOs and non-YSOs to establish new and more reliable photometric criteria in order to distinguish YSOs from non-YSOs. Initially, we make use of some YSO classification criteria using CCDs implemented in the literature to check if they are able to classify and separate our sample of spectroscopically identified YSOs from late-type stars. For this, we obtain the photometry of point sources at 3.6$\mu$m, 4.5$\mu$m, 5.8$\mu$m, 8.0$\mu$m [@Ramirez2008], and 24$\mu$m from the MIPSGAL catalog [@2015AJ....149...64G], and we merge these with our data by searching within radii of 20. We find 87 sources with valid 3.6 to 8.0$\mu$m photometry and 28 sources with valid 24$\mu$m photometry in our sample. Figure \[CC\_lit\] shows two CCDs and a colour-magnitude diagram (CMD) with respective criteria from the literature to classify YSOs. In Figure \[CC\_lit\](a), we plot \[5.8\]-\[8.0\] vs \[3.6\]-\[4.5\] with a small and big box representing the regions belonging to Class II and Class I YSOs respectively. Those areas come from the disk and envelope models of low mass YSOs as shown in [@2004ApJS..154..363A] and [@2004ApJS..154..367M]. Figure \[CC\_lit\](b) shows the same plot, but with a polygon used to define the region enclosing Stage I YSOs as defined in [@2006ApJS..167..256R]. YHA09 used the CMD (\[8.0\]-\[24\] vs \[24\]) in Figure \[CC\_lit\](c) to choose their sample of possible YSO candidates in the CMZ by considering all sources lying to the right of the dashed line as YSOs. We estimate and show the extinction vector (black arrow) for the two CCDs, assuming an A$_{K}$ of 2 mag (corresponding to A$_{V}$=30 mag; typical of the CMZ from [@Schultheis2009]) using the $\frac{A_{\lambda}}{A_{K}}$ relations from [@2009ApJ...696.1407N]. It is clear from each plot that, even after taking the extinction into account, there is severe contamination from the late-type stars in the regions defined to contain YSOs and that there is no clean colour-colour criterion visible. Hence, we test different combinations of colours and magnitudes in order to clearly separate YSOs from cool late-type stars in our sample. Only in the CCD, H-K$_{\rm S}$ vs H-\[8.0\], as shown in Figure \[CC\_our\], we see a clear linear trend followed by the late-type stars, while the YSOs exhibit redder H-\[8.0\] colours and are thus clearly separated. We define a rough criterion to separate them, taking into account the H-K$_{\rm S}$ cut at 1.5 estimated to remove foreground sources (assuming A$_{V}$=30mag and using extinction laws of [@2009ApJ...696.1407N]). A similar H-K$_{\rm S}$ cut was also suggested by to remove foreground sources based on their study toward the central parsec of the Galaxy. Here again, we estimate the extinction vector as mentioned before and show that it is almost parallel to the line separating YSOs from late-type stars. Thus the extinction does not greatly affect our criterion. We want to stress that our sample is small and this criterion needs to be confirmed by a larger sample: $$(H-[8.0]) = 2.75\times(H-K_{\rm S}) + 1.75 ; \,\,1.5 < (H-K_{\rm S}) \leq 5 \label{eq1}$$ Mass estimates using SED model fits ==================================== In the above section, we spectroscopically identified 23 YSOs from among 91 sources in the CMZ. In this section, we construct their near and mid-infrared SEDs and fit them using synthetic SED models for YSOs to constrain their stellar parameters (e.g. stellar radius R$_\star$, effective temperature T$_{\rm eff}$, stellar mass M$_{\star}$, total stellar luminosity L$_{\star}$, extinction in V-band A$_{V}$). Robitaille et al. models ------------------------ [@2006ApJS..167..256R] presented a set of approximately 20,000 radiative transfer models with corresponding SEDs assuming an accretion scenario with a central star surrounded by an accretion disk, infalling envelope and bipolar cavities, i.e., YSOs. [@2007ApJS..169..328R] presented a tool to fit these YSO model SEDs to observations, providing a range in the parameter (e.g. stellar mass, total luminosity, extinction in V-band, envelope accretion rate, age) space corresponding to a set of best fit models. These models are largely in use to estimate approximate values of stellar parameters for a photometrically or spectroscopically identified sample of YSOs. YHA09 and An11 used these models in order to classify YSOs into different evolutionary stages based on the envelope infall rate and disk accretion rate of each source. YHA09 in turn estimate the star formation rate (SFR) in the CMZ using masses they obtain from the SED fits. Recently, , hereafter R17, introduced an improved set of SED models for YSOs that covers a much wider range of parameter space and excluding most of model-dependent parameters in addition to several other improvements. Unlike previous models, there are 18 different sets of models with increasing complexity that varies from a single central star to a star in an ambient medium surrounded by accreting disk, infalling envelope and bipolar cavities as described in detail in R17. We use the latest R17 models to fit the SEDs of sources in our sample to estimate stellar parameter such as stellar radius, luminosity or effective temperature. We will use these parameters to determine the stellar masses of our YSO sample. SED fits -------- We construct the SEDs for our sample using wavelengths ranging from 1.25 - 24$\mu$m. As mentioned in section \[hkh8\], we use the JHK$_{\rm S}$ photometry from the SIRIUS catalogue, 3.6 to 8.0$\mu$m photometry from [@Ramirez2008] and 24$\mu$m photometry from [@2015AJ....149...64G]. In addition, we use the 15$\mu$m photometry from the ISOGAL point source catalog () so that we constrain the SEDs over a large wavelength range. Searching within 20 of YSOs in our sample, we find seven sources in the ISOGAL PSC out of which only two have valid 15$\mu$m magnitudes. Within the same search radius, we find six YSOs with a match in the 24 $\mu$m catalogue, all of which have valid photometry. Thus, among 23 spectroscopically identified YSOs in our sample, in addition to 1.25 - 8.0$\mu$m magnitudes, six sources have only 24$\mu$m magnitudes, two sources have only 15$\mu$m magnitudes and there are no sources with valid magnitude determined at both 15$\mu$m as well as 24$\mu$m. We find that the SED fitting by the model requires data points at $\lambda$&gt;12 $\mu$m to give reliable results. For that reason, we carried out SED fits using the SED fitting tool only for these eight sources using the above mentioned set of photometry. We assume the source distance to be in the range 7 kpc $<$ R $<$ 9 kpc from the Sun and interstellar extinction along the line of sight to the Galactic center to be in the range 20 mag $<$ A$_{V}$ $<$ 50 mag [@Schultheis2009], ensuring that these sources belong to the CMZ. These assumptions ensure that the conditions at the Galactic center are considered while fitting the model SEDs to our observed SEDs. We assume typical errors of 0.05mag for JHK$_{\rm S}$ photometry and 0.1 mag for 3.6 to 8.0$\mu$m photometry, while ISOGAL and MIPSGAL catalogues provide typical errors of $\sim$0.05 mag for 15$\mu$m and 0.1 mag for 24$\mu$m photometry respectively. To include reasonable fitting results, we select all SEDs that satisfy $\chi^{2}$ - $\chi_{\rm best}^{2}$ $<$ 5 per data points for each source in all 18 model sets. $\chi_{\rm best}^{2}$ represents $\chi^{2}$ value of the best fit for each model set. Thus each source SED is fitted with 18 different model sets, each of which gives a best fit SED with a $\chi_{\rm best}^{2}$ value. For each source, we select the model set corresponding to the best fit SED with the lowest $\chi_{\rm best}^{2}$ value as the one that best represents the evolutionary stage of the source. Figure \[sedfits\] displays typical examples of SED fitting results for eight YSOs in our sample. {width=".35\textwidth"} {width=".35\textwidth"} {width=".35\textwidth"} {width=".35\textwidth"} {width=".35\textwidth"} {width=".35\textwidth"} {width=".35\textwidth"} {width=".35\textwidth"} Fit parameters and mass {#mass} ----------------------- We choose the model set corresponding to the best fit SED with the lowest $\chi_{\rm best}^{2}$ value as mentioned above and estimate mean values of A$_{V}$, T$_{\rm eff}$ and stellar radius, R$_{\star}$ from all the fits satisfying the $\chi^{2}$ cut in the chosen model set. We estimate approximate values for the stellar luminosity, L$_{\star}$, using the Stefan-Boltzmann law from T$_{\rm eff}$ and R$_{\star}$, assuming solar T$_{\rm eff}$ to be 5772 K. To determine an approximate mass for each YSO, we use the pre-main sequence (PMS) tracks for stars with metallicity of Z = 0.02 and mass range of 0.8 M$_{\sun}$ to 60 M$_{\sun}$ from . The masses are sampled in a non-uniform manner with stellar tracks provided for 0.8, 1.0, 1.5, 2.0, 3.0, 5.0, 9.0, 15.0, 25.0 and 60.0 M$_{\sun}$. We calculate the separation of each source from the stellar track for each mass in the log$_{10}$(L$_{\star}$) - log$_{10}$(T$_{\rm eff}$) space, and assign them the mass corresponding to the track at the least separation. This exercise is carried out for all fits that satisfy the $\chi^{2}$ cut in the chosen model set and we estimate a mean mass from them. The standard deviation in mass from all SED fits can be used to make a rough estimate of the error. A zero error for mass is obtained when there is only one SED fit or when there is a single closest stellar track to source positions from all SED fits of the chosen model set. We assume their mass uncertainties to be limited by the mass sampling of PMS tracks. Figure \[LT\] shows the log$_{10}$(L$_{\star}$) vs log$_{10}$(T$_{\rm eff}$) plot with the stellar tracks and location of 8 spectroscopically identified YSOs. Mass estimates range from $\sim$ 8 to 20 M$_{\sun}$, as expected for high mass YSOs. We estimate the uncertainties for T$_{\rm eff}$, L$_{\star}$ and A$_{V}$ similarly from standard deviation in their values from all SED fits of the chosen model set. Table \[fit\_param\] lists the main model fit parameters and the estimated masses for the YSOs. The A$_{V}$ values estimated by the SED models are mostly close to the lower end of our constraints (20 mag$<$A$_{V}$$<$50 mag), which is not the expected case. So we used the extinction map of [@Schultheis2009] to estimate the foreground visual extinction close to the location of our sample, by searching within the radius corresponding to the pixel size of the extinction map (2). The estimated A$_{Vmap}$ values are listed in the last column of Table \[fit\_param\]. We find significant difference in A$_{V}$ from models and A$_{Vmap}$ from the extinction map (mean difference$\sim$9.3 mag), suggesting that the models need to be improved. A similar disagreement between A$_{V}$ from [@2007ApJS..169..328R] models and A$_{V}$ from [@Schultheis2009] extinction map was estimated by An11 for their spectroscopically identified YSOs. SST GC No. Model N$_{data}$ N$_{fits}$ $\chi^{2}_{\rm best}$ &lt;A$_{V}$&gt; (mag) &lt;Log$_{10}$(L$_{\star}$)&gt; (L$_{\sun}$) &lt;T$_{\rm eff}$&gt; (K) &lt;M$_{\star}$&gt; (M$_{\sun}$) A$_{Vmap}$ (mag) ------------ --------- ------------ ------------ ----------------------- ----------------------- ---------------------------------------------- --------------------------- ---------------------------------- ------------------ 238110 spubhmi 8 2 96.7 20.5 $\pm$ 0.4 4.4 $\pm$ 0.0 8329 $\pm$ 936 20.0 $\pm$ 5.0 25.5 $\pm$ 1.1 352034 sp–s-i 7 329 1.1 22.8 $\pm$ 2.4 3.7 $\pm$ 0.4 6668 $\pm$ 2213 12.4 $\pm$ 3.8 45.0 $\pm$ 11.3 358063 sp–s-i 8 45 30.1 20.3 $\pm$ 0.3 3.7 $\pm$ 0.1 9091 $\pm$ 979 9.4 $\pm$ 1.5 32.3 $\pm$ 5.8 373107 sp–h-i 8 9 104.5 24.8 $\pm$ 0.6 4.1 $\pm$ 0.0 10524 $\pm$ 499 15.0 31.7 $\pm$ 5.1 388790 spubsmi 7 13 0.9 22.2 $\pm$ 1.2 3.7 $\pm$ 0.2 6946 $\pm$ 1255 12.7 $\pm$ 2.9 43.0 $\pm$ 12.4 395315 spubsmi 5 91 1.9 24.2 $\pm$ 3.5 3.2 $\pm$ 0.2 9397 $\pm$ 3981 7.8 $\pm$ 2.0 34.3 $\pm$ 4.2 405235 spu-smi 7 1 32.7 20.0 3.9 6671 15.0 37.5 $\pm$ 2.5 609669 spubsmi 7 132 8.6 22.8 $\pm$ 2.7 3.3 $\pm$ 0.2 9932 $\pm$ 3436 8.2 $\pm$ 1.7 26.7 $\pm$ 2.6 A complex model in which the central star with a disk, a variable disk inner radius and bipolar cavities is enclosed in a rotationally flattened envelope structure surrounded by ambient interstellar medium. Disks around a central star, with non-variable inner radius. No surrounding envelope or ambient interstellar medium Same as b except that the disk inner radius is variable. Same as a except that the disk inner radius is set to the dust sublimation radius. Same as d except that there are no bipolar cavities Only one closest stellar track to source position (see Figure \[LT\]) from all SED fits of the chosen model set Only one SED fit to the observed photometry by the chosen model set Star formation rate in the CMZ ============================== One of the commonly used ways to estimate the star formation rate (SFR) in the CMZ is by YSO counting. In this method, masses of photometrically or spectroscopically confirmed YSOs in the region are estimated either from SED fits or from zero-age main sequence (ZAMS) luminosity-mass relation. By assuming an appropriate initial mass function (IMF) and extrapolating the stellar IMF down to lower masses, the total embedded stellar population mass of the region can then be estimated. Due to the low number of spectroscopically identified YSOs, it is not possible to apply this method to our spectroscopic sample. However it is possible to use our photometric selection criterion (see Section \[hkh8\]) based on the H-K$_{\rm S}$ vs H-\[8.0\] diagram to obtain a much more complete sample of YSOs in the CMZ. For this, we used the the photometric catalogue of SIRIUS towards the Galactic center from which we selected our observed sample. We combine this sample with 3.6 - 8.0$\mu$m photometry from [@Ramirez2008], 24$\mu$m photometry from [@2015AJ....149...64G] and 15$\mu$m photometry from ISOGAL PSC. Within $|$l$|$ $<$ 15 and $|$b$|$&lt;05 we find 16,180 sources with valid photometric magnitudes in H, K$_{\rm S}$, 8.0$\mu$m bands and in either of the two bands: 15$\mu$m or 24$\mu$m. We then apply our criterion (see Equation \[eq1\]), identifying 334 sources as YSOs. As seen in Section \[hkh8\], foreground sources are removed by default using this criterion. However, OB supergiants can still contaminate our YSO sample. Figure \[NGI\_YSOs\] shows the selected YSOs in H-K$_{\rm S}$ vs H-\[8.0\] diagram (left panel) and spatial distribution of YSOs in (l,b) plane colour coded with the number of YSOs in (l,b) bins of 005 each. The white patch close to $\sim$00 latitude and longitude is an observational artefact from the [@2015AJ....149...64G] and the ISOGAL PSC catalogues where data are lacking. As a result, the YSO count is higher at the negative longitudes than in positive longitudes, contrary to the fact that two-third of molecular gas is on positive longitudes (@1977ApJ...216..381B [@1988ApJ...324..223B; @morris1996; @oka2005] etc.). We perform the SED fitting using R17 models for these 334 sources, and we determine approximate masses for them as mentioned in section \[mass\]. We choose $\sim$190 sources with $\chi_{\rm best}^{2}$$<$35 (chosen based on the average value of $\chi_{\rm best}^{2}$ among the 8 spectroscopically confirmed YSOs) to plot the mass distribution, which ranges from 2.7M$_{\sun}$ to 35M$_{\sun}$. The distribution peaks at $\sim$8M$_{\sun}$, emphasizing that the majority of YSOs are in the high mass range. Thus we miss the low mass stars, and hence we adopt the Kroupa IMF [@2001MNRAS.322..231K] to fit it to the peak of our distribution extrapolate it to lower masses and estimate the total embedded stellar population in the CMZ. The Kroupa IMF for different mass ranges is given below : $$\label{A} \zeta (M) = A M^{-2.3} \ \text{for} \ 0.5 M_{\sun} \leq M \leq 120 M_{\sun}$$ $$\label{B} \zeta (M) = B M^{-1.3} \ \text{for} \ 0.08 M_{\sun} \leq M \leq 0.5 M_{\sun}$$ $$\label{C} \zeta (M) = C M^{-0.3} \ \text{for} \ 0.01 M_{\sun} \leq M \leq 0.08 M_{\sun}$$ \ where A, B and C are scaling factors. We follow the method described in [@Immer2012] and fit our mass distribution histogram with a curve of the form as in Equation \[A\] by non-linear least square fitting routine. The fitting results in a value of A = 7339, which we use to obtain $\zeta (M)$ at M = 0.5 M$_{\sun}$ assuming a continuous IMF and thus estimate B = 14677 from Equation \[B\]. We carry out the same exercise to estimate C = 183464. Figure \[IMF\] shows the mass distribution histogram and the fit we performed on the distribution. Finally we estimate the total mass of YSOs to be $\sim$ 35000 M$_{\sun}$ in the CMZ using $$M_{tot} = \int_{0.01}^{120} M \zeta (M) dM$$ [c c c c ]{} Method & Region covered & SFR (M$_{\sun}$yr$^{-1}$) & Reference\ YSO counting (photometric criterion) & $|l|$ $<$ 13, $|b|$ $<$ 017 & 0.14 & YHA09\ \ YSO counting (spectroscopic criterion) & ... & 0.07 & An11\ \ YSO counting (photometric criterion) & $|l|$ $<$ 15, $|b|$ $<$ 05 & 0.08 & [@Immer2012]\ \ & $|l|$ $<$ 10, $|b|$ $<$ 05 & 0.015 & [@2013MNRAS.429..987L]\ & $|l|$ $<$ 10, $|b|$ $<$ 10 & 0.06 & [@2013MNRAS.429..987L]\ \ Column density threshold & $|l|$ $<$ 10, $|b|$ $<$ 05 & 0.78 & [@2013MNRAS.429..987L]\ \ Volumetric SF relations & $|l|$ $<$ 10, $|b|$ $<$ 05 & 0.41 & [@2013MNRAS.429..987L]\ \ Infrared luminosity–SFR & $|l|$ $<$ 10, $|b|$ $<$ 05 & 0.09$\pm$0.02 & [@2017MNRAS.469.2263B]\ \ YSO counting & $|$l$|$ $<$ 15, $|$b$|$&lt;05 & 0.046$\pm$0.026 & This work\ Assumed age of YSOs $\sim$ 0.1 Myr Assumed age of YSOs $\sim$ 1 Myr Assuming all YSOs that constitute our sample have an average age of 0.75 $\pm$ 0.25Myr, we estimate the average SFR to be $\sim$0.046M$_{\sun}$yr$^{-1}$. If we assume a different IMF (e.g. Salpeter) as well as change the integration limits in the mass range in addition to including the mass uncertainties from the individual SED fitting (see Table \[fit\_param\]) and uncertainty in the assumed age, our estimated error in the derived SFR is in the order of $\pm$ 0.026M$_{\sun}$yr$^{-1}$. We also changed our colour criterion by reducing the H-K$_{\rm S}$ cut to 1.0 instead of 1.5 in order to account for the variability in extinction across CMZ and the estimated SFR is still within the uncertainty limit of +0.026M$_{\sun}$yr$^{-1}$. Our SFR estimate is lower than values from previous studies of YHA09, An11 and [@Immer2012]. YHA09 and [@Immer2012] applied YSO counting method of photometrically identified YSOs to calculate SFR of $\sim$0.14M$_{\sun}$yr$^{-1}$ (YSO lifetime $\sim$0.1 Myr) and $\sim$0.08M$_{\sun}$yr$^{-1}$ (YSO lifetime $\sim$1 Myr) respectively. An11 carried out a spectroscopic identification of YSOs among sources in common with YHA09 and derived a value of 0.07M$_{\sun}$yr$^{-1}$ based on the 50$\%$ contamination they found. Based on the re-examination of YHA09 sample using radiative transfer models and realistic synthetic observations, [@2015ApJ...799...53K] estimate the SFR to be lower by a factor of three or more. In addition to the YSO counting method, there have been studies that have employed the infrared luminosity-SFR relation, free-free emission from the ionised gas (i.e. bremsstrahlung radiation) at cm-wavelengths to estimate the mass of the underlying YSO population, column density threshold and volumetric star forming relations to estimate and predict the SFR in the CMZ (@2013MNRAS.429..987L [@2017MNRAS.469.2263B]). [@2013MNRAS.429..987L] estimated the SFR in the $|$l$|$&lt;10, $|$b$|$&lt;05 to be $\sim$0.015M$_{\sun}$yr$^{-1}$ based on the free–free emission contribution to the 33 GHz flux using Wilkinson Microwave Anisotropy Probe (WMAP) data. But the predictions from the column density threshold and volumetric star formation relations exceed the observed SFR with estimates of 0.78M$_{\sun}$yr$^{-1}$ and 0.41M$_{\sun}$yr$^{-1}$ respectively. Given that predictions from these star formation relations/models are largely dependent on the mass of dense gas, it is important to make sure that different tracers of the dense gas are reliable probes. For e.g., [@2017ApJ...835...76M] have shown that HNCO might be a better cloud mass probe than HCN 1-0 in the Galactic center environment. [@2017MNRAS.469.2263B] found an average global SFR of $\sim$ 0.09$\pm$0.02 M$_{\sun}$yr$^{-1}$ in the same l, b range from the luminosity-SFR relations using 24$\mu$m, 70$\mu$m and total infrared bolometric luminosity. Based on the observational evidence that the individual clouds and clusters are connected along a coherent velocity structure in position-position-velocity (PPV) space [@2016MNRAS.457.2675H], [@2017MNRAS.469.2263B] determined the SFR of individual clouds in the CMZ using the dynamical orbit model of [@2015MNRAS.447.1059K] assuming that star formation within these clouds is tidally triggered at the pericentre of the orbit [@2013MNRAS.433L..15L]. They find the total SFR within these clouds to be 0.03 to 0.071M$_{\sun}$yr$^{-1}$. Table \[compare\_lit\] lists the details of the SFR estimated in the CMZ using different methods based on past studies. Thus the SFR estimate in the CMZ from different methods (including our estimate) all point to a lower value than expected given the large reservoir of dense gas available. There are several physical explanations attributed to this dearth of star formation in the CMZ. [@2013MNRAS.429..987L] suggested that the additional turbulent energy in the gas, as indicated by the larger internal cloud velocity dispersion, could be providing support against gravitational collapse. The other explanations include episodic star formation in the CMZ due to spiral instabilities, high turbulent pressure in the CMZ, and the gas being not self-gravitating as discussed in detail in [@2014MNRAS.440.3370K]. Summary ======= With the aim of estimating the SFR in the CMZ using spectroscopic identification of YSOs, we prepared a detailed observation of 22 fields using KMOS. From the 8 fields we observed, we extracted clean spectra for 91 sources. Based on the CO absorption found in cool, late-type stars and Br$\gamma$ emission seen in YSOs, we were able to clearly separate YSOs from cool, late-type stars in the EW(CO) vs EW($\rm Br {\gamma}$) diagram. We plotted our spectroscopically classified YSOs and late-type stars in the colour-colour and colour-magnitude diagrams used in the literature to classify YSOs. We found that different criteria used to classify YSOs in such diagrams were not able to remove contaminants. We suggest a new criterion in the H-K$_{\rm S}$ vs H-\[8.0\] colour-colour diagram wherein we see a clear separation of YSOs and late-type stars. We used the new and improved version of SED models for YSOs in R17 to fit the observed photometry in the wavelength range of 1.25–24 $\mu$m for 8 YSOs. From the radii and temperatures we obtained from the SED fit, we estimated their masses to be greater than 8 M$_{\sun}$. Since we needed a bigger sample to estimate the SFR in the CMZ, we searched for sources within $|$l$|$ $<$ 15 and $|$b$|$ $<$ 05 with valid photometry in H, K$_{\rm S}$ (IRSF catalogue), 8.0$\mu$m [@Ramirez2008] and 15$\mu$m (ISOGAL PSC) or 24$\mu$m [@2015AJ....149...64G] bands. We identified 334 YSOs based on the criterion we defined in the H-K$_{\rm S}$ vs H-\[8.0\] diagram. We performed SED fits for these sources using R17 models resulting in  190 sources with a good fit, and their estimated masses range from 2.7 to 35M$_{\sun}$, peaking at $\sim$8M$_{\sun}$. The total mass of YSOs in the CMZ was then estimated to be $\sim$35000M$_{\sun}$ by extrapolating to lower masses using a Kroupa IMF between 0.01 and 120M$_{\sun}$. Assuming an average age of 0.75 $\pm$ 0.25Myr for YSOs, we estimate the SFR to be $\sim$0.046$\pm$0.026M$_{\sun}$yr$^{-1}$, that is slightly lower than found in previous studies. It is necessary to carry out follow up spectroscopic infrared observations to obtain a statistically significant YSO sample to further constrain our colour-colour criterion to identify YSOs. This will help in accurate determination of SFR, which is an important ingredient in the chemical evolution models of the Galaxy as well as to understand star formation of the Galactic center and as a template for circumnuclear star formation in the other galactic nuclei. We wish to thank the anonymous referee for the extremely useful comments that greatly improved the quality of this paper. G.N and M.S. acknowledges the Programme National de Cosmologie et Galaxies (PNCG) of CNRS/INSU, France, for financial support. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement n$^{o}$ \[614922\]. This research made use of the SIMBAD database (operated at CDS, Strasbourg, France). [^1]: Based on observations collected at the European Southern Observatory, Chile, program number 097.C-0208(A)

--- author: - Maria Grazia Izzo - Björn Wehinger - Giancarlo Ruocco - Aleksandar Matic - Claudio Masciovecchio - Alessandro Gessini - Stefano Cazzato bibliography: - 'pnas-sample.bib' title: 'Rayleigh anomalies and disorder-induced mixing of polarizations in amorphous solids' --- he reasearch on glasses in the last decades focused on the macroscopic anomalies which characterize as a whole these systems with respect to long-range ordered solids. In particular, a hump over the level predicted by the Debye theory is observed in the specific heat of glasses at about $10$ $K$ [@Phillips]. It is related to an excess over the Debye level of the Vibrational Density of States (VDOS) at energies of few $meV$, called Boson Peak (BP) [@Shintani; @Schirmacher_4th; @Schi_SCBorn; @Schirmacher_book; @Shir_gen1; @Schober1; @Marruzzo; @Chumakov1]. The physical origin of this feature has been largely debated in the literature without merging into a unified theory. It is noteworthy that all existing theories beg the question of the effect of elastic disorder on acoustic waves, whether it is the disorder treated as only a small perturbation with respect to the ordered structure of the crystals [@Chumakov1; @Giordano], modeled as defects embedded in an otherwise homogeneous medium [@Schober1; @Sheng] or represented as spatial fluctuations of elastic moduli - local [@Schirmacher_4th; @Schi_SCBorn; @Schirmacher_book; @Shir_gen1; @Schober1; @Marruzzo; @Ferrante] or with long-range correlations [@Tanaka]. In this context large emphasis was given to the so-called Rayleigh anomalies [@Schirmacher_4th; @Schi_SCBorn; @Schirmacher_book; @Shir_gen1; @Schober1; @Marruzzo; @Ferrante; @hydro_MonGio; @Mossa; @Ruffle2]. They consist in a strong increase of the acoustic wave attenuation and softening of the phase velocity with respect to the macroscopic value - the two quantities being related to each other by Kramers-Kroning relations [@Marruzzo] - observed for wavelengths of few nanometers. The experimental study of acoustic excitations at such length-scales with the experimental techniques so far available is quite difficult [@Ferrante; @hydro_MonGio], thus obfuscating the experimental verification of the different thoeries. An acoustic wave traveling in a three-dimensional material is characterized by its phase velocity, amplitude and polarization. It is known from elementary elasticity theory [@Landau] that a purely longitudinal (or transverse) wave impinging on an interface between two different elastic media is transformed in waves with mixed polarization. Acoustic waves with mixed polarization have been observed by both Inelastic X-ray and Neutron Scattering (IXS, INS) techniques [@Ruzi; @Scopigno1HFglasses_transverse; @zanatta_INS_GeO2; @Cunsolo] as well as by MD simulations [@Sampoli; @Bryk1; @Ribeiro3] in several amorphous solids and liquids in the first pseudo-Brillouin zone, typically at wavevectors equal to $4-5$ $nm^{-1}$. This latter phenomenon, however, has never been related to Rayleigh anomalies and quantitatively described as a phenomenon also originating from the disordered nature of the medium. The authors believe that a coherent, full-blown, experimentally verifiable, mathematical description of all the phenomena arising from the elastic heterogeneous structure of an amorphous solid and affecting its acoustic excitations finally could trace the way towards the understanding of how microscopic disorder can be exhaustively described and how it can affects macroscopic properties in glasses. A stochastic approach developed in the framework of the Random Media Theory (RMT) [@Sobczyk], or Heterogeneous-Elasticity Theory (HET) [@Schirmacher_4th; @Schi_SCBorn; @Schirmacher_book; @Shir_gen1] if one refers to the case of elastic constants inhomogeneity, would permit to reach this goal. However, the impossibility to solve of the Dyson equation, which describes the ensemble averaged elastodynamic response of the system to an impulsive force, leads to the introduction of scale-dependent approximations [@Sobczyk; @Schirmacher_book; @Schi_SCBorn; @B1; @Turner; @Kraichnan], thus hindering the unified description of these phenomena in turn occurring on different length scales. The Rayleigh anomalies appear at values of wavelength, $\lambda$, of elastic excitations lower than the average radius of inhomogeneity domains, $a$, whereas the coupling between longitudinal and transverse excitations (depolarization) is maximum when $\lambda$ becomes comparable to $a$ [@Calvet]. On the other hand, the parameters of the theory, such as the average size of heterogeneity domains or the strength of spatial fluctuations, only in rare cases can be *a priori* determined by experiments [@Sobczyk; @Schirmacher_4th; @Schi_SCBorn; @Schirmacher_book; @Shir_gen1]. They are usually chosen as *ad hoc* parameters to permit a correct description of the phenomena one aims to explain. This could give rise to tautologies. The goal of the present study is the building of a theoretical framework, able to make quantitative and verifiable predictions and allowing a unified description of the phenomena so far addressed. Within this aim we developed in the framework of the RMT, a simple enough, mathematically tractable approximation. It allows for a complete description of a real system, the ionic glass 1-Octyl-3-methylimidazolium chloride (\[C8MIM\]Cl), whose heterogeneous structure can be well assessed and experimentally characterized. Furthermore, the acoustic dynamics of this system has been characterized by IXS with a detail never offered so far in the whole pseudo-Brillouin zone of a glass. Because of the analytical approach, a certain level of generality of the proposed model is guaranteed. It can be thought as a starting point for describing acoustic dynamics in different kind of glasses, composites, ceramics, geophysical systems, or propagation of different kind of waves in disordered media. ![*Panels I.-IV.* IXS spectra of 1-Octyl-3-methylimidazolium chloride ionic glass at $T= 176$ $K$ for selected $Q$-values (black circles with error bars) together with best-fit curve obtained by using the single-DHO model (full red line). The inelastic component of the fitting model (DHO function) is shown as full black line. *Panels V.-VI.* Detail of a representative IXS spectrum at fixed Q in the high-Q region, $Q>5$ $nm^{-1}$, (black circles with error bars), best fit curve (full red line) and inelastic contribution (full black line) obtained by using the single-DHO model (Panel V.) and the two-DHO model (Panel VI.). Fit residuals (open black circles) are shown in the resepctive bottom panels.[]{data-label="esempio_spettri"}](Fig1_short.pdf){width="1\linewidth"} Results {#results .unnumbered} ======= Experimental characterization of longitudinal acoustic waves in 1-Octyl-3-methylimidazolium chloride ionic glass {#experimental-characterization-of-longitudinal-acoustic-waves-in-1-octyl-3-methylimidazolium-chloride-ionic-glass .unnumbered} ---------------------------------------------------------------------------------------------------------------- We report the experimental characterization of longitudinal dynamics and Vibrational Density of States (VDOS) of \[C8MIM\]Cl at $T=176$ $K$ ($T_g = 214$ $K$) obtained respectively by IXS and INS. Fig. \[esempio\_spettri\] *Panels I-IV* shows IXS spectra for selected wavevector (Q)-values [^1]. The measured signal reproduces the longitudinal dynamical structure factor, $S_L(Q,E)$. $E=\hbar \omega$ is the exchanged energy between the probe and the sample, $\hbar$ is the Planck constant, $\omega$ is the exchanged frequency. $S_L(Q,E)$ is modeled in turn with a sum of a delta function for the elastic component and a damped harmonic oscillator (DHO) function for the inelastic component. Such a protocol provides the characteristic energy and broadening (attenuation) of the inelastic excitation, respectively $\Omega$ and $\Gamma$. ![*Panel I.* Broadening ($\Gamma$) as a function of $Q$, obtained by fitting with the single-DHO model in the ’low-Q’ region (open circle), single-DHO function in the ’high-Q’ region (black circles) and two-DHO model in the ’high-Q’ region (blue and green circles for high- and low-frequency features respectively). Blue and dashed lines show $Q^2$ trend, black line shows $Q^4$ trend. *Panel II.* Reduced VDOS, $g(E)/E^2$, obtained by INS measurements. *Panel III.* Characteristic energy ($\Omega$) of the inelastic excitations. The lines describe the bending of the acoustic dispersion curve on approaching the edge of first pseudo-Brillouin zone, reproduced by using a sinusoidal function, $\Omega=(c\hbar) Q\frac{\pi}{Q_0}sin(\frac{Q\pi}{Q_0})$ [@Bosak]. The parameters $c$ and $Q_0$ are fixed to values that better reproduce the experimentally observed trend obtained by single-DHO model (dashed line) and two-DHO model fitting (blue and black line). The measured $S(Q)$ is shown.[]{data-label="dispersion1DHO"}](Fig2_short_color.pdf){width="1\linewidth"} In the high-wavevectors region ($Q>5$ $nm^{-1}$) an additional feature on the inelastic component is observed. Similar features were so far observed in several glasses, see e.g. Refs. [@Ruzi; @Scopigno1HFglasses_transverse; @zanatta_INS_GeO2], and commonly related to ’projection’ of transverse into longitudinal dynamics. To account for the presence of the extra feature, two DHO contributions can be inserted in the fitting model (see Fig. \[esempio\_spettri\] *Panel VI*). Within the aim to enforce the reliability of the comparison with theoretical outcomes, these latter will be compared with experimental results obtained both by using in the fitting model a single-DHO function in the whole explored Q-range or by using two DHO-functions in the high-Q region. Experimentally determined $\Omega$ and $\Gamma$ are displayed in Fig. \[dispersion1DHO\] *Panels I* and *III*. The figure, furthermore, shows the measured VDOS normalized to the square of the exchanged energy, $g(E)/E^2$, (*Panel II*) and the static structure factor, $S(Q)$, measured by X-ray scattering (*Panel III*). The reduced VDOS presents (i) a peak around $2$ $meV$, referred to be the Boson Peak (BP) [@Ribeiro_Raman2]; (ii) a broad feature at higher frequencies, in the region between $7$ and $12$ $meV$, related to librational modes of the imidazolium ring [@Ribeiro_Raman2] - referred hereafter as Intemolecular Vibrational Modes (IVMs). The static structure factor shows - as it is the case for most ionic liquids (ILs) - a First Sharp Diffraction Peak (FSDP) at $Q_{FSDP} = 2.8$ $nm^{-1}$. It is related to nanoscale segregation of the cations alkyl chains [@Fujii; @Ferde], which in turn results in a local structure formed by an alternation of polar and nonpolar domains [@Aoun]. The characteristic length scale defined by the FSDP, $2 \pi/Q_{FSDP}$, gives thus a rough estimation of the diameter of nonpolar domains [@Fujii], i.e. $2a \sim 2 \pi /Q_{FSDP}$ (see Fig. \[MODEL\_disp2\_ord2\]). Since, furthermore, intermolecular forces acting in polar and apolar regions are of different nature, it is possible to assume that (i) to the heterogeneous local structure corresponds an elastic heterogeneous structure [@Ribeiro3]; (ii) the elastic constants difference between the two kind of domains is quite large. The information that we can extract from inspection of Fig. \[dispersion1DHO\] are in summary: (i) the existence of a crossover in the $\Gamma$ trend at $Q_c=4.8 nm^{-1}$; (ii) a related kink at $Q_c$ in the $\Omega$ dispersion; (iii) for $Q>Q_c$ the existence of two inelastic features in $S_L(Q,E)$. The crossover in the $\Gamma$ trend has been observed in several other glasses, see e.g. Refs. [@hydro_MonGio; @Ruffle2]. It usually falls at frequency in the region of the BP, i.e. at slightly lower frequency than in the present case. A power law of $4$-th and $2$-th order can describe the experimental trend of $\Gamma$ for $Q$ respectively lower or highr than $Q_c$. We observe that $Q_{c} \sim 2Q_{FSDP} \sim 2 \pi /a$, i.e. the crossover appears at wavevectors related to the typical size of elastic heterogeneity domains. We notice, on the other hand, that $Q_c$ correspond as well to the energy cross point between longitudinal acoustic mode dispersion and the energy position of the barycenter of the broad high-frequency feature in VDOS at $\sim 9$ $meV$ related to IVMs (see Fig. \[dispersion1DHO\]). The possible coupling between acoustic waves and IVMs is discussed in Supporting Information. Random Media Theory in the Generalized Born Approximation {#random-media-theory-in-the-generalized-born-approximation .unnumbered} --------------------------------------------------------- The dynamic structure factor is related to the average Green propagator, $<G(\textbf{q},\omega)>$, via the relationship $S(\textbf{q},\omega) \propto \frac{q^2}{\omega} Im\{<G(\textbf{q},\omega)>\}$. The brackets $<\ \ >$ denotes ensemble average. In the RMT the disorder of the system is described by introducing in the Helmotz equation spatial fluctuations of density or elastic tensor. We here account for only fluctuations of the elastic tensor, $\textbf{C}(\textbf{r})$, where $\textbf{r}$ is the spatial coordinate. Purpose of the RMT is to obtain the average Green dyadic, $<\textbf{G}(\textbf{q},\omega)>$, solution of the Dyson equation [@Sobczyk], $$\begin{aligned} <\textbf{G}(\textbf{q},\omega)>= [\textbf{G}^0(\textbf{q},\omega)^{-1}-\Sigma (\textbf{q},\omega)]^{-1} \label{Dyson}.\end{aligned}$$ where $\textbf{G}^0(\textbf{q},\omega)$ is the bare Green’s dyadic describing the bare’ system in the absence of spatial fluctuations. All the information about the system’s inhomogeneity are embedded in the so-called self-energy or mass-operator, $\Sigma (\textbf{q},\omega)$. A formal expression of the self-energy can be given trough a Neumann-Liouville series [@Sobczyk]. Under the hypothesis of statistical homogeneity, its truncation to lowest non-zero order leads to the so-called Bourret [@Sobczyk; @B1; @Calvet; @Turner] or Born approximation [@Schirmacher_book; @Schi_SCBorn], $$\begin{aligned} \Sigma^B_{\beta j}(\textbf{q},\omega)=\hat{L}_{1 \beta \gamma j k}G_{\gamma k}^0(\textbf{q},\omega)=\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\ =\int \ d^3s q_{\alpha}q_{l} s_{\delta} s_{i} \tilde{R}_{\alpha \beta \gamma \delta ijkl}(\textbf{q}-\textbf{s})G_{\gamma k}^0(\textbf{s},\omega)\label{Sigma_Bo_F},\end{aligned}$$ where summation over repeated indeces is assumed, $\textbf{s}$ is a wavevector and the integral extends to $\Re^3$. The function $\tilde{R}_{\alpha \beta \gamma \delta ijkl}(\textbf{q})$ is the Fourier transform of the covariance of the elastic tensor fluctuations, $R_{\alpha \beta \gamma \delta ijkl}(\textbf{r}=\textbf{r}_1-\textbf{r}_2)=<\delta C_{\alpha \beta \gamma \delta}(\textbf{r}_1)\delta C_{ijkl}(\textbf{r}_2)>$. Thus, the self-energy in the Fourier space can be written as a convolution between the bare Green’s dyadic and the Fourier transform of the covariance of the elastic tensor fluctuations. It is possible to show that the Bourret approximation holds for small fluctuations intensity, wavevectors and frequencies [@B1]. The so-called Self-consistent Born Approximation (SCBA) [@Schirmacher_4th; @Schirmacher_book; @Shir_gen1] or Kraichnan model [@Kraichnan] is a generalization of the Bourret (Born) approximation, which states $$\Sigma_{\beta j}(\textbf{q},\omega)=\hat{L}_{1 \gamma j k l}<G_{\gamma k}(\textbf{q},\omega)>\label{Sigma_self}.$$ Eq. \[Sigma\_self\] together with Eq. \[Dyson\] corresponds to successive self-consistent approximations for $\Sigma(\textbf{q},\omega)$ and $<G(\textbf{q},\omega)>$. At the zero-th step it is $<G(\textbf{q},\omega)>=G^0(\textbf{q},\omega)$. The first-step of the iteration procedure thus corresponds to the Born approximation. The SCBA holds under the hypothesis of small fluctuations [@Schirmacher_book; @Schirmacher_4th] but it is not affected by the limitation in frequencies and wavevectors of the Born approximation. Analytical solutions of the self-consistent set of equations (Eqs. \[Dyson\] and \[Sigma\_self\]) while preserving their wavevector-dependence involves computational drawbacks, thus approximations or numerical approaches are required. Analytical expressions have been obtained in Rayleigh Region (RR) by assuming $q=0$ in the expression of $\Sigma(q,\omega)$ at each step of the self-consistent procedure ($SCBA^{RR}$) [@Schirmacher_4th; @Marruzzo; @Ferrante], allowing a description of the Rayleigh anomalies [@Marruzzo; @Ferrante]. Outcomes from this procedure are discussed in Supporting Information. We propose a novel approximate method for the calculation of $\Sigma(\textbf{q},\omega)$. It introduces corrective terms to the Born approximation in the framework of the SCBA. We will thus refer to it as to a Generalized Born Approximation. We report here the final expression for $\Sigma(\textbf{q},\omega)$, whereas we outline the passages which leads to it in Sec. Method. Under the hypothesis of local isotropy [^2], we introduce the orthonormal vector base defined by the direction of wave propagation and the two orthogonal directions [@Turner]. On this basis all the bare’ and average Green dyadics and the self-energy are diagonal. We introduce their longitudinal and transverse components, respectively $g_{L(T)}$, $g^0_{L(T)}$, $\Sigma_{L(T)}$. These latter, in turn, contain two terms accounting for the coupling with longitudinal and transverse dynamics respectively, i.e. $\Sigma_{L(T)} =\Sigma_{LL(TT)}+\Sigma_{LT(TL)}$. In the Generalized Born Approximation we find $$\begin{aligned} &\Sigma_{j}(\textbf{q},\omega) \simeq \cdot \hat{L}_{1 j k}\frac{1}{\tilde{c_{k}}^2}\big\{\frac{1}{\tilde{q}_{0 k}^2-q^2}+\frac{1}{(\tilde{q}_{0 k}^2-q^2)^2}\frac{\epsilon^2}{\tilde{c}_{k}^2}q^2\Delta\tilde{\Sigma}_{k}^{1}(0,\omega)\big\}. \label{Sigma_K_ap0}\end{aligned}$$ where $\Delta \tilde{\Sigma}_k^1(\textbf{q},\omega)=\tilde{\Sigma}_k^1(\textbf{q},\omega)-\tilde{\Sigma}_k^1(q=0,\omega=0)$, $\tilde{\Sigma}_k^1(q,\omega)=\hat{L}_{1k\alpha}'G_{\alpha}^0(\textbf{q},\omega)$, $\hat{\textbf{L}}_1=\epsilon^2q^2\hat{\textbf{L}}_1'$, $\tilde{q}_{0l}=\frac{\omega}{\tilde{c}_k}$, $\tilde{c}_k=[(c_k^0)^2+\epsilon^2\tilde{\Sigma}_k^1(q=0,\omega=0)]^{1/2}$ is the macroscopic velocity of the (first step) perturbed medium, $c_k^0$ is the phase velocity of the bare’ medium along a given direction (longitudinal or transverse), $\epsilon^2$ is the square of the intensity of spatial fluctuations per density. The suffix $1$ marks a quantity calculated to the first step of the self-consistent procedure. Generalizations of the Born approximation have attracted interest in several fields of physics [@Jin; @Li]. The validity of the present approximation can be demonstrated up to wavelengths of the order of the average size of heterogeneity domains [@izzo]. We consider only spatial fluctuations of shear modulus in agreement with previous literature studies [@Marruzzo; @Schirmacher_4th]. The introduction of fluctuations of the Lamé parameter [@Calvet; @Turner] is discussed in Supporting Information. We consider, furthermore, the simplest form of the shear modulus fluctuations correlation function, in real space an exponentially decaying function with correlation length $a$. The input parameters of the theory are $a$, whose value in the present case can be fixed by the *a priori* (experimental) knowledge of the local structure $a=\frac{\pi}{Q_{FSDP}}$, $\epsilon^2$ and $c^0_{L(T)}$. It has been shown that the short-range structure of glasses preserves residual order that characterizes the long-range structure of crystals [@Chumakov1; @Giordano] with the consequent occurrence of the pseudo-Brillouin zone and the related bending of the acoustic waves dispersion. We empirically superimpose it to acoustic waves dispersions calculated in the framework of the RMT (see Fig. \[MODEL\_FIT\_disp2\_ord2\]) by a suitable normalization of the frequency of $G^0(q,\omega)$ in Eq. \[Dyson\] as described in Sec. Method. ![*Panel I.* Projection on $(q,E)$ plane of the current $C_L(q,E) \propto \frac{E^2}{q^2} S_L(q,E)$ with its maximum value normalized to one, obtained by the Generalized Born Approximation and with frequency normalization to account for the bending related to the edge of the first pseudo-Brillouin zone. Black, blue and green triangles represent $\Omega$ obtained respectively by the single-DHO model and by the two-DHO model (high- and low-frequency excitations). *Panel II.* Corresponding $S_L(q,E)$. *Panel II.* Example of calculated $C_L(q, E)$ (black line) and $C_T(q, E)$ (red line) in the ’low-q’ region and best-fit curves (dashed lines) obtained by the single-DHO model. *Panel VII.* Example of calculated $C_L(q, E)$ (black line) and $C_T(q, E)$ (red line) in the ’high-q’ region and best-fit curves obtained with the single-DHO model (dashed curves).[]{data-label="MODEL_FIT_disp2_ord2"}](Fig_mod_short_giu2017_4.pdf){width="1.02\linewidth"} Discussion {#discussion .unnumbered} ========== The main insights which can be gained from our theoretical approach are enclosed in Figs. \[MODEL\_FIT\_disp2\_ord2\] and \[MODEL\_disp2\_ord2\] and outlined in the following. (1) In the Rayleigh region ($qa < 1$) the dynamic structure factor is characterized by one only inelastic excitation (Fig. \[MODEL\_FIT\_disp2\_ord2\] *Panel III*). Its characteristic phase velocity and attenuation show respectively the softening and the crossover in the $q$ trend, as it is possible to infer from Fig. \[MODEL\_disp2\_ord2\] *Panels I* and *III*, low $q$ points. (2) In the region $qa \approx 1$ the dynamic structure factor presents features, which cannot be accounted for by a scalar model or by $SCBA^{RR}$. It is observed in $S_L(q,\omega)$ a shoulder at a frequency close to the characteristic frequency of the transverse excitation (Fig. \[MODEL\_FIT\_disp2\_ord2\] *Panels II* and *IV*). This originates from the mixing of transverse and longitudinal dynamics. The endorsement comes from the fact that the shoulder disappears when the transverse contribution ($\Sigma_{LT}(q,\omega)$) is removed from the longitudinal self-consistent energy. Furthermore, depending on the disorder parameter and transverse to longitudinal phase velocity ratio, a hump in the wavevector trend (the stronger as the bigger it is the former and the smaller the latter) is observed. It can be seen as a prolongation of the $q^4$ behavior observed in the Rayleigh regime. The hump corresponds to a rapid increase of the phase velocity and has been reported in theoretical characterization of elastic waves in polycrystalline aggregates [@Calvet]. This latter fact emphasizes how in the case of longitudinal acoustic dynamics the coupling of polarizations contributes to the attenuation increase observed at the edge of the Rayleigh region ($qa \approx 1$). A scalar model thus can underestimate the attenuation observed in this wavevectors region in real systems. (3) In the high wavevectors region, $qa>1$, the co-existence of two excitations can be observed [@Calvet; @Sheng; @Baron]. ![*Panel I.* Phase velocity of inelastic excitations observed in experimental (circles with error bars) and calculated (stars) $S_L(q,E)$, obtained by the generalized Born approximation. *Panel II.* Ratio of intensities of the DHO functions modeling the two high-q excitations. *Panel III.* Broadening of inelastic excitations. textit[Panel IV.]{} Experimental (black lines) and calculated (red stars) $g(E)/E^2$. *Panel V.* Measured $S(Q)$ compared with the shear modulus fluctuations correlation function, $R_{\mu \mu}(Q)$. Thin black line shows Lorentzian function modeling the main peak of $S(Q)$, used to determine the first pseudo-Brillouin zone edge distribution function (see Sec. Method for details). The input parameters of the theory are $c^0_L=2.29$, $c^0_T/c^0_L=0.53$, which is a typical value for glasses [@Leonforte], $\tilde{\epsilon}^2=\frac{\epsilon^2}{\overline{\mu}^2}=0.4$ ($\overline{\mu}$ is the average shear modulus) and $a/2\pi=0.15$. The figure on the bottom is a pictorial representation of the spatial configuration of the 1-octyl-methylimadozolium chloride glass at the nanometric scale in non-polar domains.[]{data-label="MODEL_disp2_ord2"}](Fig6_2.pdf){width="1.0\linewidth"} This general picture is coherent with the experimental characterization of real systems found in the literature [@hydro_MonGio; @Ferrante; @Ruzi; @Scopigno1HFglasses_transverse]. The contrast between theoretical and experimental results obtained for the $[C8MIM]Cl$ glass shows an excellent agreement in the whole measured wavevectors range, as attested in Fig. \[MODEL\_disp2\_ord2\] *Panels I-IV*. In particular the following experimental features are quantitatively reproduced by the theory: (i) the Rayleigh anomalies; (ii) the increase of attenuation and phase velocity beyond the Rayleigh region (i.e. at frequencies higher than BP), which can be attributed to the strong coupling between transverse and longitudinal dynamics (see point (2)), in turn related to the strong intensity of elastic fluctuations; (ii) the presence of the low-frequency shoulder in $S_L(Q,E)$, related to the mixing of longitudinal and transverse polarization; (iii) the position of the BP. Such a quantitative agreement cannot be achieved by using the Born approximation. In Fig. \[MODEL\_Tdisp2ord2\] we report the features of transverse dynamics obtained for the same input parameters of the longitudinal dynamics. In the low-wavevectors region we observe a single excitation, showing: i) a crossover in the $q$-trend of the attenuation at frequency where the BP shows up; ii) a softening of the phase velocity in the Reileigh regime; iii) the Ioffe-Regel crossover at wavevector and frequency point where $\pi \Gamma$ becomes larger than $\Omega$, occurring near the BP-frequency. At higher wavevector a high-frequency shoulder appears in the calculated dynamic structure factors. This is not entirely related to the mixing of polarizations. It is indeed partially preserved when the term $\Sigma_{TL}$ is fixed to zero. These results are in qualitative agreement with results from simulations in glasses [@Mossa; @Marruzzo] or liquids at high-wavevectors [@Ribeiro3; @Bryk1]. ![Characteristic features of transverse dynamics obtained by using the generalized Born approximation. *Panel I.* $\pi\Gamma $ (open circles) and phase velocity (stars with line) of the low-frequency inelastic excitation. Red (dashed) line shows $q^4$ ($q^2$) trend. *Panel II.* Reduced VDOS, $g(E)/E^2$. *Panel III.* Phase velocities of the two inelastic excitations observed in the calculated $S_T(k,\omega)$. *Panel IV.* Projection on $(q,E)$ plane of $C_T(q,E)$ with frequency normalization to account for the bending related to the edge of first pseudo-Brillouin zone.[]{data-label="MODEL_Tdisp2ord2"}](Fig5_short_2_prova_2.pdf){width="1.01\linewidth"} Our results shows that the mixing of polarization can be generated by disorder. Beyond Rayleigh anomalies an approximate solution of the Dyson equation can quantitatively account also for this latter phenomenon. Appendix A {#appendix-a .unnumbered} ========== Expressions of the longitudinal and transverse self-energies in the Generalized Born Approximation {#expressions-of-the-longitudinal-and-transverse-self-energies-in-the-generalized-born-approximation .unnumbered} -------------------------------------------------------------------------------------------------- We derive the expressions of the longitudinal and transverse dynamic structure factors in the Generalized Born Approximation, by exploiting Eq. 4 in the main text. We re- write it here for sake of clarity, $$\begin{aligned} & \Sigma_j(\textbf{q},\omega) \simeq lim_{\eta \rightarrow 0^+}\hat{L}_{1jk}\big\{\frac{1}{(\tilde{q}_{0k}+i \tilde{\eta})^2-q^2}+\frac{1}{((\tilde{q}_{0k}+i \tilde{\eta})^2-q^2)^2} \cdot \nonumber \\& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot \frac{\epsilon^2}{\tilde{c}_k^2}q^2\Delta \tilde{\Sigma}_k^{1*}(0,\omega + i \eta)\big\}, \label{Sigma_K_ap1}\end{aligned}$$ where $\eta$ and $\tilde{\eta}$ are related to each other by the relationship $\tilde{\eta}=\frac{\eta}{\tilde{c}_k}$. We recall that $\hat{\textbf{L}}_1=\epsilon^2q^2\hat{\textbf{L}}_1'$, $\Delta \tilde{\Sigma}_k^1(\textbf{q},\omega)=\tilde{\Sigma}_k^1(\textbf{q},\omega)-\tilde{\Sigma}_k^1(q=0,\omega=0)$, $\tilde{\Sigma}_k^1(\textbf{q},\omega)=\hat{L}_1'G^0(\textbf{q},\omega)$, $\tilde{q}_{0k}=\frac{\omega}{\tilde{c}_k}$ and $\tilde{c}_k=(c_k^2+\epsilon^2\tilde{\Sigma}_k^1(q=0,\omega=0))^{1/2}$ is the macroscopic velocity of the (first step) perturbed medium. Under the hypothesis of local isotropy, the elastic tensor, $C_{ijkl}(\textbf{r})$, vary in the space by following the expressions $$\begin{aligned} &C_{ijkl}(\textbf{r})=\overline{C}_{ijkl}(1+\delta \tilde{C}_{ijkl}(\textbf{r}))= \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\ &=\overline{\lambda}(1+\delta \tilde{\lambda}(\textbf{r}))\delta_{ij}\delta_{kl}+\overline{\mu}(1+\delta \tilde{\mu}(\textbf{r}))(\delta_{ik}\delta_{jl}+\delta_{il}\delta{jk}), \label{elasticmoduli_r}\end{aligned}$$ where $\overline{\mu}$ and $\overline{\lambda}$ are the shear modulus and Lamé parameter of the bare medium and $\delta \tilde{\rho}=\frac{\delta \rho}{\overline{\rho}}$, $\delta \tilde{\lambda}=\frac{\delta \lambda}{\overline{\lambda}}$, $\delta \tilde{\mu}=\frac{\delta \mu}{\overline{\mu}}$. The operator $\hat{\textbf{L}}_1$ in the Fourier space is defined as $$\begin{aligned} \Sigma^B_{\beta j}(\textbf{q},\omega)=\hat{L}_{1 \beta \gamma j k}G_{\gamma k}^0(\textbf{q},\omega)=\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\ =\int \ d^3s q_{\alpha}q_{l} s_{\delta} s_{i} \tilde{R}_{\alpha \beta \gamma \delta ijkl}(\textbf{q}-\textbf{s})G_{\gamma k}^0(\textbf{s},\omega)\label{Sigma_Bo_F},\end{aligned}$$ $\textbf{s}$ is a wavevector and the integral extends to $\Re^3$. The function $\tilde{R}_{\alpha \beta \gamma \delta ijkl}(\textbf{q})$ is the Fourier transform of the covariance of the elastic tensor fluctuations, $R_{\alpha \beta \gamma \delta ijkl}(\textbf{r}=\textbf{r}_1-\textbf{r}_2)=<\delta C_{\alpha \beta \gamma \delta}(\textbf{r}_1)\delta C_{ijkl}(\textbf{r}_2)>$. We consider only fluctuations of the shear modulus. The introduction of spatial fluctuations of the Lamé parameter is discussed in the following. By assuming that in the real space the correlation function of the shear modulus fluctuations is an exponential decay function, in the Fourier space it is $$R_{\mu\mu}(\textbf{q})=\epsilon^2 \frac{1}{\pi^2}\frac{q^2a^{-1}}{(q^2+a^{-2})^2}, \label{R_mu_FT_exp}$$ where $q=|\textbf{q}|$ and $\int d^3q \tilde{R}_{\mu\mu}(\textbf{q})=1$. The bare Green’s dyadic, ensemble average Green’s dyadic and self-energy in the orthonormal basis defined by the direction of wave propagation, $\hat{q}$, and the two orthogonal ones, can be written as $$\begin{aligned} \textbf{G}^0 (\textbf{q},\omega)=g^0_L(\textbf{q},\omega) \hat{q}\hat{q}+g^0_T(\textbf{q},\omega)(I-\hat{q}\hat{q}); \nonumber \\ \Sigma(\textbf{q},\omega)=\Sigma_L(\textbf{q},\omega)\hat{q}\hat{q}+\Sigma_T(\textbf{q},\omega)(I-\hat{q}\hat{q}); \nonumber \\ <G(\textbf{q},\omega)>=<g_L(\textbf{q},\omega)>\hat{q}\hat{q}+<g_T(\textbf{q},\omega)>(I-\hat{q}\hat{q}).\nonumber \\ \label{G_base}\end{aligned}$$ The bare Green’s functions are $g^0_{L(T)}(\textbf{q},\omega)=lim_{\eta \rightarrow 0^+}\frac{1}{(\omega+i\eta)^2-c_{L(T)}^2q^2}$. The longitudinal and transverse average wave speeds are defined respectively by the relations: $\overline{\rho}(c_L^0)^2=\overline{\lambda}+2\overline{\mu}$, $\overline{\rho}c_T^2=\overline{\mu}$. As it follows from the properties of the inverse tensor, it is $$<g_{L(T)}(\textbf{q},\omega)>=\frac{1}{g_{L(T)}^0(\textbf{q},\omega)^{-1}-\Sigma_{L(T)}(\textbf{q},\omega)}. \label{gL}$$ The longitudinal and transverse self-energies are $$\begin{aligned} &&\Sigma_L(\textbf{q},\omega)= \Sigma_{LL}(\textbf{q},\omega)+\Sigma_{LT}(\textbf{q},\omega);\nonumber \\ &&\Sigma_L(\textbf{q},\omega)= \Sigma_{LL}(\textbf{q},\omega)+\Sigma_{LT}(\textbf{q},\omega).\end{aligned}$$ Each partial term of the self-energies, $\Sigma_{LL(LT)}$ and $\Sigma_{TT(TL)}$ in the Generalized Born Approximation is composed of two terms, $$\begin{aligned} &\Sigma_{LL(TT)}(\textbf{q},\omega)=\Sigma_{LL(TT)}^{(0)}(\textbf{q},\omega)+\Sigma_{LL(TT)}^{(1)}(\textbf{q},\omega), \nonumber \\ &\Sigma_{LT(TL)}(\textbf{q},\omega)=\Sigma_{LT(TL)}^{(0)}(\textbf{q},\omega)+\Sigma_{LT(TL)}^{(1)}(\textbf{q},\omega). \nonumber \end{aligned}$$ We outline in the following the respective expressions for each polarizations coupling, $$\begin{aligned} &\Sigma_{LL}^{(0)}(\textbf{q},\omega)=lim_{\eta\rightarrow 0^+}\int \hat{q}\hat{q} \hat{s}\hat{s}d^3s\tilde{R}_{\mu \mu}(\textbf{q}-\textbf{s}) \cdot \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot \frac{1}{\tilde{c}_L^2}\frac{1}{(\tilde{q}_{0L}+i\tilde{\eta})^2-q^2}; \label{ll_i} \\ &\Sigma_{LL}^{(1)}(\textbf{q},\omega)= lim_{\eta\rightarrow 0^+}\int \hat{q}\hat{q} \hat{s}\hat{s}d^3s\tilde{R}_{\mu \mu}(\textbf{q}-\textbf{s}) \cdot \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot \frac{1}{\tilde{c}_L^2} \frac{\frac{\epsilon^2}{\tilde{c}_L^2}s^2\Delta \tilde{\Sigma}_L^1(0,\omega+i\eta)}{[(\tilde{q}_{0L}+i\tilde{\eta})^2-q^2]^2} ; \label{ll_ii}\end{aligned}$$ $$\begin{aligned} &\Sigma_{LT}^{(0)}(\textbf{q},\omega)= lim_{\eta \rightarrow 0^+}\int \hat{q}\hat{q}(\textbf{I}-\hat{s}\hat{s})d^3s \tilde{R}_{\mu \mu}(\textbf{q}-\textbf{s}) \cdot \ \ \ \ \ \ \nonumber \\& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot \frac{1}{\tilde{c}_T^2} \frac{1}{(\tilde{q}_{0T}+i\tilde{\eta})^2-q^2}; \label{lt_i} \\&\Sigma_{LT}^{(1)}(\textbf{q},\omega)= lim_{\eta \rightarrow 0^+} \int \hat{q}\hat{q}(\textbf{I}-\hat{s}\hat{s})d^3s \tilde{R}_{\mu \mu}(\textbf{q}-\textbf{s}) \cdot \ \ \ \ \ \ \nonumber \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot \frac{1}{\tilde{c}_T^2} \frac{\frac{\epsilon^2}{\tilde{c}_T^2}s^2\Delta \tilde{\Sigma}_T^1(0,\omega+i\eta)}{[(\tilde{q}_{0T}+i\tilde{\eta})^2-q^2]^2} ; \label{lt_ii} \end{aligned}$$ $$\begin{aligned} &\Sigma_{TT}^{(0)}(\textbf{q},\omega)=lim_{\eta \rightarrow 0^+}\frac{1}{2} \int (\textbf{I}-\hat{q}\hat{q}) (\textbf{I}-\hat{s}\hat{s})d^3s\tilde{R}_{\mu \mu} (\textbf{q}-\textbf{s}) \cdot \nonumber \\& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot \frac{1}{\tilde{c}_T^2}\frac{1}{(\tilde{q}_{0T}+i\tilde{\eta})^2-q^2};\label{tt_i} \\&\Sigma_{TT}^{(1)}(\textbf{q},\omega)= lim_{\eta \rightarrow 0^+}\frac{1}{2} \int (\textbf{I}-\hat{q}\hat{q}) (\textbf{I}-\hat{s}\hat{s})d^3s\tilde{R}_{\mu \mu} (\textbf{q}-\textbf{s})\cdot \nonumber \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot \frac{1}{\tilde{c}_T^2} \frac{\frac{\epsilon^2}{\tilde{c}_T^2}s^2\Delta \tilde{\Sigma}_T^1(0,\omega+i\eta)}{[(\tilde{q}_{0T}+i\tilde{\eta})^2-q^2]^2} ; \label{tt_ii}\end{aligned}$$ $$\begin{aligned} &\Sigma_{TL}^{(0)}(\textbf{q},\omega)= lim_{\eta \rightarrow0^+}\frac{1}{2}\int (\textbf{I}-\hat{q}\hat{q}) \hat{s}\hat{s}d^3s\tilde{R}(\textbf{q}-\textbf{s}) \cdot \ \ \ \ \ \ \ \ \ \ \nonumber \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot \frac{1}{\tilde{c}_L^2}\frac{1}{(\tilde{q}_{0L}+i\tilde{\eta})^2-q^2}; \label{tl_i} \\& \Sigma_{TL}^{(1)}(\textbf{q},\omega)= lim_{\eta \rightarrow0^+}\frac{1}{2} \int (\textbf{I}-\hat{q}\hat{q})\hat{s}\hat{s}d^3s \tilde{R}(\textbf{q}-\textbf{s}) \cdot \ \ \ \ \ \ \ \ \ \ \nonumber \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot \ \frac{1}{\tilde{c}_L^2} \frac{\frac{\epsilon^2}{\tilde{c}_L^2}s^2\Delta \tilde{\Sigma}_L^1(0,\omega+i\eta)}{[(\tilde{q}_{0L}+i\tilde{\eta})^2-q^2]^2}. \label{tl_ii}\end{aligned}$$ By using spherical coordinates in Eqs. \[ll\_i\]-\[tl\_ii\] , it is obtained $$\begin{aligned} &\Sigma^{(0)[(1)]}_{LL}(\textbf{q},\omega)=\epsilon^2 q^2 \int_{-1}^{+1} dx 4x^4 \frac{1}{\tilde{c}_L^2} \frac{2}{\pi} a^{-1} I^{(0)[(1)]}_{L}(\textbf{q},\omega,x), \ \ \ \label{I_Sigma_LT_2}\end{aligned}$$ $$\begin{aligned} &\Sigma^{(0)[(1)]}_{LT}(\textbf{q},\omega)=\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&=\epsilon^2 q^2 \int_{-1}^{+1} dx 4(1-x^2)x^2 \frac{1}{\tilde{c}_T^2} \frac{2}{\pi} a^{-1} I^{(0)[(1)]}_{T}(\textbf{q},\omega,x), \label{I_Sigma_LT_2}\end{aligned}$$ $$\begin{aligned} &\Sigma^{(0)[(1)]}_{TT}(\textbf{q},\omega)=\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&=\epsilon^2 q^2 \int_{-1}^{+1} dx (1-3x^2+4x^4) \frac{1}{\tilde{c}_T^2} \frac{1}{\pi} a^{-1}I^{(0)[(1)]}_{T}(\textbf{q},\omega,x), \ \ \ \label{I_Sigma_TT}\end{aligned}$$ $$\begin{aligned} &\Sigma^{(0)[(1)]}_{TL}(\textbf{q},\omega)=\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&=\epsilon^2 q^2 \cdot \int_{-1}^{+1} dx 4(1-x^2)x^2 \frac{1}{\tilde{c}_L^2} \frac{1}{\pi} a^{-1} I^{(0)[(1)]}_{L}(\textbf{q},\omega,x), \label{I_Sigma_TL}\end{aligned}$$ where $x=cos(\theta)$ and $\theta$ is the $\hat{\textbf{q}\textbf{s}}$ angle [^3].$I_{k}^{(n)}(\textbf{q},\omega,x)$, where $n=0,1,...$, is an integral which remains implicitely defined in the previous equation. By exploiting the Sokhotski-Plemelj-Fox theorem [@Galapon], integration by part and the Cauchy’s Residue Theorem the integral $I_k^{(n)}(\textbf{q},\omega,x)$ for a generic polarization $k$, is given by [@izzo] $$\begin{aligned} &&I_k^{(n)}(\textbf{q},\omega,x)=lim_{\eta \rightarrow 0^+}\int_{0}^{\infty}ds \ \ s^2\tilde{R}_{\mu \mu}(q,s,x) \frac{[s^2\frac{\epsilon^2}{\tilde{c}_k^2}\Delta\tilde{\Sigma}_k^{1}(0,\omega+i\tilde{c}\eta)]^n}{((\tilde{q}_{0k}+i\tilde{\eta})^2-s^2)^{n+1}}=\# \int_{0}^{\infty}ds \ s^2 \tilde{R}_{\mu \mu}(q,s,x) \frac{[s^2\frac{\epsilon^2}{\tilde{c}_k^2}\Delta\tilde{\Sigma}_k^{1}(0,\omega)]^n}{(\tilde{q}_{0k}^2-s^2)^{n+1}}+ \nonumber \\ &&+(-1)^{n+1}\frac{i\pi}{n!} \frac{d^n}{dz^n}\Big\{z^2 \tilde{R}_{\mu \mu}(q,z)\frac{[z^2\frac{\epsilon^2}{\tilde{c}_k^2}\Delta\tilde{\Sigma}_k^{1}(0,\omega)]^n}{(\tilde{q}_{0k}+z)^{n+1}}\Big\}|_{z=\tilde{q}_{0k}}. \label{In}\end{aligned}$$ The symbol $\#$ states for the Hadamard finite part integral (equal to the Cauchy principal value when $n=0$). The quantities $\Delta \tilde{\Sigma}_{L(T)}^1(\textbf{q},\omega)=\tilde{\Sigma}_{L(T)}^1(\textbf{q},\omega)-\tilde{\Sigma}_{L(T)}^1(0,0)$, which appear in the expressions of $\Sigma^{(1)}_{LL(LT)}$ and $\Sigma^{(1)}_{TT(TL)}$ are related to the first step self-energies, coincident with the self-energies obtained trough the Born approximation, as reported in the following, $$\begin{aligned} &\Sigma_L^1(\textbf{q},\omega)=\epsilon^2q^2\tilde{\Sigma}_L^1(\textbf{q},\omega)= \epsilon^2q^2[\tilde{\Sigma}_{LL}^1(\textbf{q},\omega)+\tilde{\Sigma}_{LT}^1(\textbf{q},\omega)] =\nonumber \\&=\int \hat{q}\hat{q} \tilde{R}_{\mu \mu}(\textbf{q}-\textbf{s})[g_L^0(s,\omega)\hat{s}\hat{s}+g_T^0(s,\omega)(\textbf{I}-\hat{s}\hat{s})]d^3s; \ \ \ \ \ \ \ \label{Sigma_L_app} \\& \Sigma_T^1(\textbf{q},\omega)=\epsilon^2q^2\tilde{\Sigma}_T^1(\textbf{q},\omega)= \epsilon^2q^2[\tilde{\Sigma}_{TT}^1(\textbf{q},\omega)+\tilde{\Sigma}_{TL}^1(\textbf{q},\omega)] =\nonumber \\&=\int (\textbf{I}-\hat{q}\hat{q}) \tilde{R}_{\mu \mu}(\textbf{q}-\textbf{s})[g_L^0(s,\omega)\hat{s}\hat{s}+g_T^0(s,\omega) (I-\hat{s}\hat{s})]d^3s. \ \ \label{Sigma_T_app}\end{aligned}$$ It can be calculated by exploiting the Sokhotski-Plamelj theorem and the Cauchy’s residue theorem. We report in the following as an example the expression for $\tilde{\Sigma}_{LL}^1(\textbf{q},\omega)$, $$\begin{aligned} &\tilde{\Sigma}_{LL}^{1}(\textbf{q},\omega)=lim_{\eta\rightarrow 0^+}\int \hat{q}\hat{q} \hat{s}\hat{s}d^3s\tilde{R}_{\mu \mu}(\textbf{q}-\textbf{s}) \frac{1}{(c^0_L)^2}\frac{1}{{q}_{0L}+i\eta)^2-q^2}=\int_{-1}^{1} dx \ 4x^4 \frac{1}{c_L^2} 2 a^{-1} \int_{0}^{\infty} dq \frac{(aq)^4}{(1+(aq)^2+(as)^2-2(as)(aq)x)^2} \frac{1}{q_{0L}^2-q^2}= \nonumber \\& \ \ \ = \int_{-1}^{1} dx 4x^4\frac{1}{c_L^2}2 a^{-1} i \{ \frac{q_{0L}^3}{(a^{-2}+q^2+q_{0L}^2+2qq_{0L}x)^2}-\frac{1}{a^2(x,q)} \frac{(qx+ia(x,q))^3}{q^2_{0L}-(qx+ia(x,q))^2} \cdot [3/2+i\frac{qx}{a(x,q)}+\frac{(qx+ia(x,q))^2}{q_{0L}^2-(qx+ia(x,q))^2}]\}. \label{Sigma_LL_ii}\end{aligned}$$ By comparing Eqs. \[Sigma\_L\_app\] and \[Sigma\_T\_app\] with Eqs. \[ll\_i\], \[lt\_i\], \[tt\_i\], \[tl\_i\] it is immediate to verify that $\Sigma^1$ is equivalent to $\Sigma^{(0)}$ under the transformation $\tilde{c}_{L(T)} \rightarrow c_{L(T)}$. By using integration by parts, it follows that $$\begin{aligned} \# \int_{0}^{\infty}ds s^2 \tilde{R}_{\mu \mu}(q,s,x) \frac{[s^2\frac{\epsilon^2}{\tilde{c}^2}\Delta \tilde{\Sigma}^{1}(0,\omega)]^n}{(\tilde{q}_0^2-s^2)^{n+1}}=p.v. \int_{0}^{\infty}ds \frac{1}{n!} (-1)^{n+1}\frac{1}{(\tilde{q}_0-s)}\frac{d^n}{dq^n}\big\{s^2 \tilde{R}_{\mu \mu}(q,s,x) \frac{[s^2\frac{\epsilon^2}{\tilde{c}^2}\Delta \tilde{\Sigma}^{1}(0,\omega)]^n}{(\tilde{q}_0+s)^{n+1}}\big\} . \label{pv_B}\end{aligned}$$ The Hadamard finite part integral exists because it exists the Cauchy principal value of the integral on the right side of Eq. \[pv\_B\], since it is possible to demonstrate [@izzo] that the integrand satisfy the Lipschitz property. The integral in Eq. \[pv\_B\] can be calculated by exploiting the Residue Theorem because the function $z^2\tilde{R}_{\mu \mu}(z,q,x)\frac{[z^2\frac{\epsilon^2}{\tilde{c}^2}\Delta \tilde{\Sigma}^{1}(0,\omega)]^n}{\tilde{q}_0+z}$ and its $n$-th order derivatives have only non-essential singularities [@izzo]. Since we truncated the Taylor series of $<G(\textbf{q},\omega)>^1$ to the first order (see Sec. Method), we only need to calculate $I_k^{(0)}(\textbf{q},\omega,x)$ and $I_k^{(1)}(\textbf{q},\omega,x)$. It is $$\begin{aligned} &I_{k}^{(0)}(\textbf{q},\omega,x)=i\pi Res_{I^{(0)}}^{(2)}\{qx+i a(q,x)\}-i\pi Res_{I^{(0)}}^{(1)}\{q_{0k}(\omega)\}= i \pi \{ - \frac{1}{2}\frac{(qx+ia(q,x))^3}{q_{0k}^2(\omega)-(qx+ia(q,x))^2} \frac{1}{(a(q,x))^2} [2-\frac{qx+ia(q,x)}{2ia(q,x)}+\ \ \ \ \nonumber \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\frac{(qx+ia(q,x))^2}{q_{0k}^2(\omega)-(qx+ia(q,x))^2} ] +\frac{q_{0k}(\omega)^3}{2} \frac{1}{(a^{-2}+q^2+q_{0k}^2(\omega)+2qq_{0k}(\omega)x)^2}\}; \label{I_Sigma_LL} \\ &I_{k}^{(1)}(\textbf{q},\omega,x)=i \pi Res_{I^{(1)}}^{(2)}\{qx+i a(q,x)\}-i\pi Res_{I^{(1)}}^{(2)}\{q_{0L}(\omega)\}=i \pi \{ - \frac{1}{4} \frac{(qx+ia(q,x))^5}{[q_{0k}^2(\omega)-(qx+ia(q,x))^2]^2} \frac{1}{(a(q,x))^2} [5-\frac{qx}{ia(q,x)}+\ \ \ \ \nonumber \\&\ \ \ \ \ \ \ \ \ +\frac{4(qx+i a(q,x))^2}{q_{0k}^2(\omega)-(qx+i a(q,x))^2}]+ \frac{1}{4}\frac{q_{0k}^3(\omega)}{(a^{-2}+q_{0k}^2(\omega)+q^2+2qq_{0k}(\omega)x)^2}[4q_{0k}(\omega)(q_{0k}(\omega)+qx) \frac{1}{a^{-2}+q_{0k}^2(\omega)+q^2+2q_{0k}(\omega)qx}-5]\}, \nonumber \\&\end{aligned}$$ with $a(q,x)=\sqrt{q^2(1-x^2)+a^{-2}}$, $Res_{I^{(n)}}^{(m)}\{ p\}$ is the residue of the integrand of $I^{(n)}$ around the pole $p$ of order $m$. The first term in Eq. \[I\_Sigma\_LL\] is related to the Hadamard finite part integral in Eq. \[In\] [^4], whereas the second term to the second term in Eq. \[In\]. Appendix B {#appendix-b .unnumbered} ========== #### **Bending of acoustic wave dispersion and definition of the edge of the pseudo-Brillouin zone** {#bending} It has been shown that the short-range structure of glasses preserves residual order that characterizes the long-range structure of crystals [@Chumakov1]. The most evident consequence is the occurrence in glasses of a pseudo-Brillouin zone whose extension depends on nearest-neighbour average distance. In the present model, the occurrence of the pseudo-Brillouin zone with the consequent bending of the acoustic waves dispersion is empirically accounted by superimposing it to longitudinal and transverse acoustic dispersions calculated in the framework of HET. The value of the edge of the first pseudo-Brillouin zone, $Q_0=17.9$ $nm^{-1}$, is determined by the experimental longitudinal acoustic dynamics dispersion. The frequency of the bare Green dyadic $G^0(\textbf{q},\omega)$ in the Dyson equation is thus normalized, as specified in the main text, i.e. $$\tilde{\omega}=\omega[\frac{Q_0}{q\pi} \cdot sin(\frac{\pi}{Q_0})]^{-1} \label{bending}.$$ It is possible to introduce a distribution function for the values of the pseudo-Brillouin edge related to the nearest-neighbor values distribution function. This latter can be empirically extrapolated from the atomic-form-factors-weighted static structure factor measured by X-ray Scattering, similarly to what is done in Ref. [@Giordano]. The main peak in $S(Q)$ is first shaped by a Lorentz function centered in $Q=17.9$ $nm^{-1}$, i.e. at the edge of the first pseudo-Brillouin zone experimentally determined. The FWHM of the Lorentz function is fixed such that the area of the main peak, i.e. the area of the static structure factor in between $13$ and $24$ $nm^{-1}$, is equal to the total area of the Lorentz function, each function being normalized to their respective maxima. By using the Lorentzian shaped nearest-neighbor values distribution function and the dispersion relation given in Eq. \[bending\] the distribution function of the values of the pseudo-Brillouin zone edge can be given in turn by a Lorentz function. At each wavevector the calculated dynamic static structure factor is convoluted with this latter Lorentz function, which accounts for the spectral frequency broadening generated by the distribution function of the values of the edge of the first pseudo-Brillouin zone. Examples of the acoustic waves dispersion obtained by RMT and including the bending related to the edge of the first pseudo-Brillouin zone are shown in Fig. 3, *Panels* I and II of the main text.  Appendix C {#appendix-c .unnumbered} ========== Coupling of acoustic waves with intermolecular vibrational modes {#coupling-of-acoustic-waves-with-intermolecular-vibrational-modes .unnumbered} ---------------------------------------------------------------- It is possible to introduce intermolecular vibrational modes (IVMs) coupled to acoustic waves. The self-energy operator in this case becomes $\Sigma=\Sigma_{RMT}+\Sigma_{IVM}$, where it is added to the self-energy calculated by using the Random Media Theory in the Generalized Born Approximation, $\Sigma_{RMT}$, the term $\Sigma_{IVM}=\frac{q^2A^s_{L(T)}}{\omega^2-\omega_0^2+i\omega \Gamma_s}$ (). The IVM’s self-energy is characterized by the characteristic frequency ($\omega_0$), broadening ($\Gamma_s$) and the polarization dependent coupling factor, $A^s_{L(T)}$. The coupling of IVM’s with acoustic waves is treated to lowest order [@Shir_gen1]. The dynamic structure factor of IVM (taking into account also its coupling with the acoustic modes) and the relative contribution to reduced VDOS are respectively given by $$\begin{aligned} &S_{IVM}(\textbf{q},\omega)=\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&=\frac{q^2}{\omega} \textit{Im}{\frac{1}{\omega^2-\omega_0^2+i \omega \Gamma_s+\sum_{\{P\}} q^2A_{P}^s<G(\textbf{q},\omega)>_P^{(RMT)}}} \label{S0_res};\label{Sqw_loc} \end{aligned}$$ $$\begin{aligned} \frac{g_{IVM}(E)}{E^2}=3 \frac{2}{\pi q_D^3}\int_0^{q_D} dq S_{IVM}(\textbf{q},\omega), \label{VDOS_loc}\end{aligned}$$ where $P=L,T_1,T_2$ is the polarization (longitudinal and transverse degenerate). The high-frequency region of the measured VDOS can be described only after the introduction of IVMs. Their occurrence has been observed by Raman scattering [@Ribeiro_Raman2]. To cope with literature data we introduce two IVMs with respectively characteristic frequency and attenuation $\omega_{0}^{1}(\omega_{0}^{2})=7.2(10)$ $meV$ and $\Gamma_s^{1}(\Gamma_s^{2})=4.0(3.8)$ $meV$. In Fig. \[long\_HET\_LO\] the results are compared with experimental findings. By taking into account also for the presence of IVMs it is possible to quantitatively reproduce the feature related to measured $S_L(Q,E)$ in the first-pseudo Brillouin zone and, simultaneously, the VDOS in the whole measured energy range. The presence of IVMs also influences the broadening of the longitudinal and transverse current at frequencies higher than BP frequency, as well as the position of BP in VDOS. It cannot, however, account for the mixing of polarization, which can be instead described by the RMT. We furthermore stress that taking under consideration only the effect of the coupling of acoustic modes with IVMs (completely neglecting the effect of the heterogeneous elastic structure on the acoustic dynamics described in the framework of RMT) permits to achieve a reliable description of the experimentally observed feature in $S_L(Q,E)$ only at those frequencies of acoustic excitations near the characteristic frequency of IVMs. By looking at Fig. 5 in the main text, one can think that the broadening of the transverse current could be underestimated in the present approximation at high wavevectors. This can depend on the fact that we neglected there a possible coupling with IVMs. Actually this latter effect, as it is possible to deduce by the observation of Fig. \[long\_HET\_LO\], leads to larger values of the transverse broadenings. [^1]: In the present text capital $Q$ states for wavevectors experimentally determined (thus affected by undeterminancy), non-capital $q$ states for a wavevevector introduced inside the theory. If experimental and theoretical outputs are put in contrast, this latter notation will be preferred. [^2]: Pure transverse modes do not contribute to the measured IXS signal in the first Brillouin zone. In solids with local anisotropy quasi-longitudinal or quasi-transverse modes (i.e. modes composed by different polarizations with a dominant polarization component) exist in the mesoscopic region. They can contribute to the measured IXS signal giving rise to the shoulder at a frequency near the one characteristic of the transverse excitation. It is possible, in the frame of RMT, to account both for local and statistical anisotropy (A. J. Turner, ’Elastic wave propagation and scattering in heterogeneous, anisotropic media: Textured polycrystalline materials’, J. Acoust. Soc. Am. **106**, 541 (1999)). We neglected such effects because our aim is to point out that, even in a isotropic medium, the disorder can generate the mixing of polarizations. [^3]: The x-integration is performed numerically. [^4]: We also exploit the fact that the integrand function is even with respect to $s$, thus $\# \int_{0}^{\infty} ds \ ... =\frac{1}{2} \# \int_{-\infty}^{\infty}ds \ ... \ .$

--- abstract: | In this paper, the performance of a binary phase shift keyed random time-hopping impulse radio system with pulse-based polarity randomization is analyzed. Transmission over frequency-selective channels is considered and the effects of inter-frame interference and multiple access interference on the performance of a generic Rake receiver are investigated for both synchronous and asynchronous systems. Closed form (approximate) expressions for the probability of error that are valid for various Rake combining schemes are derived. The asynchronous system is modelled as a chip-synchronous system with uniformly distributed timing jitter for the transmitted pulses of interfering users. This model allows the analytical technique developed for the synchronous case to be extended to the asynchronous case. An approximate closed-form expression for the probability of bit error, expressed in terms of the autocorrelation function of the transmitted pulse, is derived for the asynchronous case. Then, transmission over an additive white Gaussian noise channel is studied as a special case, and the effects of multiple-access interference is investigated for both synchronous and asynchronous systems. The analysis shows that the chip-synchronous assumption can result in over-estimating the error probability, and the degree of over-estimation mainly depends on the autocorrelation function of the ultra-wideband pulse and the signal-to-interference-plus-noise-ratio of the system. Simulations studies support the approximate analysis. *Index Terms—*$\,$Ultra-wideband (UWB), impulse radio (IR), Rake receivers, multiple access interference (MAI), inter-frame interference (IFI). author: - | Sinan Gezici$^{2,4}$, *Student Member, IEEE*, Hisashi Kobayashi$^{2}$, *Life Fellow, IEEE*,\ H. Vincent Poor$^{2}$, *Fellow, IEEE*, and Andreas F. Molisch$^{3}$, *Senior Member, IEEE*\ To appear in *IEEE Transactions on Signal Processing* title: 'Performance Evaluation of Impulse Radio UWB Systems with Pulse-Based Polarity Randomization$^\textrm{\small{1}}$' --- Introduction ============ Since the US Federal Communications Commission (FCC) approved the limited use of ultra-wideband (UWB) technology [@FCC], communications systems that employ UWB signals have drawn considerable attention. A UWB signal is defined to possess an absolute bandwidth larger than $500$MHz or a relative bandwidth larger than 20% and can coexist with incumbent systems in the same frequency range due to its large spreading factor and low power spectral density. UWB technology holds great promise for a variety of applications such as short-range high-speed data transmission and precise location estimation. Commonly, impulse radio (IR) systems, which transmit very short pulses with a low duty cycle, are employed to implement UWB systems ([@scholtz]-[@andy]). In an IR system, a train of pulses is sent and information is usually conveyed by the position or the polarity of the pulses, which correspond to Pulse Position Modulation (PPM) and Binary Phase Shift Keying (BPSK)$^{5}$, respectively. In order to prevent catastrophic collisions among different users and thus provide robustness against multiple-access interference, each information symbol is represented by a sequence of pulses; the positions of the pulses within that sequence are determined by a pseudo-random time-hopping (TH) sequence specific to each user [@scholtz]. The number $N_{f}$ of pulses representing one information symbol can also be interpreted as pulse combining gain. In “classical" impulse radio, the polarity of those $N_{f}$ pulses representing an information symbol is always the same, whether PPM or BPSK is employed ([@scholtz], [@gian]). Recently, pulse-based polarity randomization was proposed, where each pulse has a random polarity code ($\pm1$) in addition to the modulation scheme ([@eran1], [@sadler]). The use of polarity codes can provide additional robustness against multiple-access interference [@eran1] and help optimize the spectral shape according to FCC specifications by eliminating the spectral lines that are inherent in IR systems without polarity randomization [@paul]. A TH-IR system with pulse-based polarity randomization can be considered as a random CDMA (RCDMA) system with a generalized signature sequence, where the elements of the sequence take values from $\{-1,0,+1\}$ and are not necessarily independent and identically distributed (i.i.d.) [@eran1]. The performance of RCDMA systems with i.i.d. binary spreading codes has been investigated thoroughly in the past (see e.g. [@pursley_1987]-[@zang]). Recently, [@ternaryUWB_ciss03] and [@ternaryUWB_ciss04] have considered the problem of designing ternary codes for TH-IR systems. Moreover, in [@eran1], the performance of random TH-IR systems with pulse-based polarity randomization has been investigated over additive white Gaussian noise (AWGN) channels, assuming symbol-synchronized users. To the best of our knowledge, no study concerning the bit error probability (BEP) performance of Rake receivers (with various combining schemes) for a random TH-IR system with pulse-based polarity randomization in a multiuser, frequency-selective environment has been reported in the literature. In this paper, we investigate such a system and provide (approximate) closed-form expressions for its performance. We consider an important case in practice, where the different users are completely asynchronous. We begin by considering the chip-synchronous case where the symbols of different users are misaligned but this misalignment is an integer multiple of the chip interval. Subsequently, we treat a more general asynchronous case, where we show that the system can be represented as a chip-synchronous system with uniform timing jitter between zero and the chip interval for each interfering *user*. We consider frequency-selective channels and analyze the performance of Rake receivers with various combining schemes. The remainder of the paper is organized as follows. Section describes the transmitted signal model for a TH-IR system with pulse-based polarity randomization. In Section , both chip-synchronous and asynchronous systems over frequency-selective channels are considered, and the performance of Rake receivers is analyzed for various combining schemes. Simulation studies are presented in Section , followed by some concluding remarks in Section . Signal Model ============ We consider a BPSK random TH-IR system with $N_u$ users, where the transmitted signal from user $k$ is represented by $$\begin{gathered} \label{eq:tran1} s^{(k)}_{tx}(t)=\sqrt{\frac{E_k}{N_f}}\sum_{j=-\infty}^{\infty}d^{(k)}_j\, b^{(k)}_{\lfloor j/N_f\rfloor}w_{tx}(t-jT_f-c^{(k)}_jT_c),\end{gathered}$$ where $w_{tx}(t)$ is the transmitted UWB pulse with duration $T_c$, $E_k$ is the bit energy of user $k$, $T_f$ is the “frame" time, $N_f$ is the number of pulses representing one information symbol, and $b^{(k)}_{\lfloor j/N_f \rfloor}\in \{+1,-1\}$ is the information symbol transmitted by user $k$. In order to allow the channel to be shared by many users without causing catastrophic collisions, a time-hopping sequence $\{c^{(k)}_j\}$ is assigned to each user, where $c^{(k)}_j \in \{0,1,...,N_c-1\}$ with equal probability, with $N_c$ denoting the number of possible pulse positions in a frame ($N_c=T_f/T_c$), and $c^{(k)}_j$ and $c^{(l)}_i$ are independent for $(k,j)\ne (l,i)$. This TH sequence provides an additional time shift of $c^{(k)}_jT_c$ seconds to the $j$th pulse of the $k$th user where the pulse width $T_c$ is also considered as the chip interval. $N=N_fN_c$ represents the total processing gain of the system. Due to the regulations by the FCC [@FCC], each user can transmit a certain amount of energy in a given time interval. Since the symbol (bit) energy of the signal defined in (\[eq:tran1\]) is constant (denoted by $E_k$), we consider a fixed symbol interval; hence, a constant total processing gain $N$ throughout the paper. The random polarity codes $d^{(k)}_j$’s are binary random variables taking $\pm1$ with equal probability, and such that $d^{(k)}_j$ and $d^{(l)}_i$ are independent for $(k,j)\ne (l,i)$ [@eran1]. Use of random polarity codes helps reduce the spectral lines in the power spectral density of the transmitted signal [@paul] and mitigate the effects of MAI [@eran1]. The receiver for user $k$ is assumed to know its polarity code. Defining a sequence $\{s^{(k)}_j\}$ as $$\begin{gathered} \label{eq:spread} s^{(k)}_j=\begin{cases}d^{(k)}_{\lfloor j/N_c\rfloor} \,\,\,\, j-N_f\lfloor j/N_c\rfloor=c^{(k)}_{\lfloor j/N_c\rfloor} \\ \quad 0 \quad\quad\quad\quad \textrm{Otherwise}. \end{cases},\end{gathered}$$ we can express (\[eq:tran1\]) as $$\label{tran_CDMA} s^{(k)}_{tx}(t)=\sqrt{\frac{E_k}{N_f}}\sum_{j=-\infty}^{\infty}s^{(k)}_j\, b^{(k)}_{\lfloor j/N_fN_c\rfloor}w_{tx}(t-jT_c),$$ which indicates that a TH-IR system with polarity randomization can be regarded as an RCDMA system with a generalized spreading sequence $\{s^k_j\}$ ([@gian2], [@eran1]). Note that the main difference of the signal model in (\[eq:tran1\]) from the “classical" RCDMA model ([@pursley_1987]-[@zang]) is the use of $\{-1,0,+1\}$ as the spreading sequence, instead of $\{-1,+1\}$. The system model given by equation (\[eq:tran1\]) can represent an RCDMA system with a processing gain of $N_f$, by considering the special case when $T_f=T_c$. An example TH-IR signal is shown in Figure \[fig:codedIR\], where six pulses are transmitted for each information symbol ($N_f=6$) with the TH sequence $\{2,1,2,3,1,0\}$. Performance Analysis ==================== We consider transmission over frequency selective channels, where the channel for user $k$ is modelled as $$\begin{gathered} \label{eq:chan_k} h^{(k)}(t)=\sum_{l=1}^{L}\alpha^{(k)}_l\delta(t-(l-1)T_c-\tau_k),\end{gathered}$$ where $\alpha_l^{(k)}$ and $\tau_k$ are the fading coefficient of the $l$th path and the delay of user $k$, respectively; $T_c$ is the minimum resolvable path interval. We set $\tau_1=0$ without loss of generality. We assume that the channel characteristics remain unchanged over a number of symbol intervals, which can be justified by considering that the symbol duration in a typical application is on the order of tens or hundreds of nanoseconds, and the coherence time of an indoor wireless channel is on the order of tens of milliseconds. Note that the channel model in (\[eq:chan\_k\]) is quite general in that it can model any channel of the form $\sum_{l=1}^{\hat{L}}\hat{\alpha}_l^{(k)}\delta(t-\hat{\tau}_l^{(k)})$ if the channel is bandlimited to $1/T_c$. Thus, each realization of an arbitrary (and nonuniformly sampled) channel model, e.g., the 802.15.3a UWB channel model [@Molisch; @et; @al.; @2003], can be represented in the form of equation (4). Only the statistics of the tap amplitude are changed when the tap spacing is changed to a uniform spacing. Using (\[eq:tran1\]) and (\[eq:chan\_k\]), the received signal can be expressed as follows: $$\begin{aligned} \label{eq:rec_MP} r(t)=\sum_{k=1}^{N_u}\sqrt{\frac{E_k}{N_f}}\sum_{j=-\infty}^{\infty}d^{(k)}_j\, b^{(k)}_{\lfloor j/N_f\rfloor}u^{(k)}(t-jT_f-c^{(k)}_jT_c-\tau_k)+\sigma_nn(t),\end{aligned}$$ where $n(t)$ is a white Gaussian noise with zero mean and unit spectral density, and $$\begin{gathered} \label{eq:u_k} u^{(k)}(t)=\sum_{l=1}^{L}\alpha_l^{(k)}w_{rx}\left(t-(l-1)T_c\right),\end{gathered}$$ with $w_{rx}(t)$ being the received UWB pulse with unit energy. We consider a Rake receiver for the user of interest, say user $1$, and express the template signal at the Rake receiver as follows: $$\begin{gathered} \label{eq:temp_RAKE} s^{(1)}_{temp}(t)=\sum_{j=iN_f}^{(i+1)N_f-1}d_j^{(1)}v(t-jT_f-c_j^{(1)}T_c),\end{gathered}$$ where $$\begin{gathered} \label{eq:v} v(t)=\sum_{l=1}^{L}\beta_lw_{rx}\left(t-(l-1)T_c\right),\end{gathered}$$ with $\boldsymbol{\beta}=[\beta_1,...,\beta_L]$ being the Rake combining weights. The template signal given by (\[eq:temp\_RAKE\]) and (\[eq:v\]) can represent different multipath diversity combining schemes by choosing an appropriate weighting vector $\boldsymbol{\beta}$: In an $M$-finger Rake the weights for $(L-M)$ multipath components not used in the Rake receiver are set to zero while the remaining $M$ weights are determined according to the combining scheme, such as “Equal Gain Combining (EGC)" or “Maximum Ratio Combining (MRC)". The output of the Rake receiver can be obtained from (\[eq:rec\_MP\])-(\[eq:v\]) as follows: $$\label{eq:RAKE_out} y_1=b_i^{(1)}\sqrt{E_1N_f}\sum_{l=1}^{L}\alpha^{(1)}_l\beta_l+\hat{a}+a+n,$$ where the first term is due to the desired signal, $\hat{a}$ is the self interference of the received signal from user $1$ itself, which we call inter-frame interference (IFI), $a$ is the MAI from other users and $n$ is the output noise, which is approximately distributed as $n\sim\mathcal{N}\left(0\,,\,N_f\sigma_n^2\sum_{l=1}^{L}\beta_l^2\right)$ for large $N_f$ (Appendix \[app:noise\]). Inter-frame interference (IFI) occurs when a pulse of user $1$ in a frame spills over to an adjacent frame due to the multipath effect and consequently interferes with the pulse in that frame (Figure \[fig:IFI\]). The IFI in (\[eq:RAKE\_out\]) can be expressed, from (\[eq:rec\_MP\]) and (\[eq:temp\_RAKE\]), as $$\label{eq:IFI} \hat{a}=\sqrt{\frac{E_1}{N_f}}\sum_{m=iN_f}^{(i+1)N_f-1}\hat{a}_m,$$ where $$\begin{gathered} \label{eq:IFI_m} \hat{a}_m=d_m^{(1)}\underset{j\ne m}{\sum_{j=-\infty}^{\infty}}d^{(1)}_jb^{(1)}_{\lfloor j/N_f\rfloor}\phi_{uv}^{(1)}\left((j-m)T_f+(c_j^{(1)}-c_m^{(1)})T_c\right),\end{gathered}$$ with $\phi_{uv}^{(k)}(x)$ denoting the cross-correlation between $u^{(k)}(t)$ of (\[eq:u\_k\]) and $v(t)$ of (\[eq:v\]): $$\begin{gathered} \label{eq:phi} \phi_{uv}^{(k)}(x)=\int_{-\infty}^{\infty} u^{(k)}(t-x)v(t)dt.\end{gathered}$$ Note that $\hat{a}_m$ in (\[eq:IFI\_m\]) denotes the IFI due to the transmitted pulse in the $m$th frame of user $1$, and the sum of such IFI terms over $N_f$ frames is equal to $\hat{a}$, as seen in (\[eq:IFI\]). In Appendix \[app:IFI\], we show that these $N_f$ terms form a 1-dependent sequence$^{6}$ when $L\leq N_c+1$ and their sum converges to a Gaussian random variable for a large $N_f$. This result is summarized in the following lemma: \[lem:IFI\] As $N_f\longrightarrow \infty$, the IFI $\hat{a}$ in (\[eq:IFI\]) is asymptotically normally distributed as $$\label{eq:lemma_IFI} \hat{a}\sim\mathcal{N}\left(0\,,\,\frac{E_1}{N_c^2}\sum_{j=1}^{L-1}j \left[\sum_{l=1}^{L-j}\left(\beta_l\alpha^{(1)}_{l+j}+\alpha^{(1)}_l\beta_{l+j}\right)\right]^2\right),$$ for $L\leq N_c+1$. *Proof:* See Appendix \[app:IFI\]. Note that for a Rake receiver with one finger such that $\beta_1=1$ and $\beta_l=0$ for $l=2,...,L$, the expression reduces to $\hat{a}\sim\mathcal{N}\left(0\,,\,\frac{E_1}{N_c^2}\sum_{l=1}^{L-1}l\,(\alpha^{(1)}_{l+1})^2\right)$. Due to the FCC’s regulation on peak to average ratio (PAR), $N_f$ cannot be chosen very small in practice. Since we transmit a certain amount of energy in a constant symbol interval, as $N_f$ gets smaller, the signal becomes more peaky as shown in Figure \[fig:tradeOff\]. Therefore, the approximation for large $N_f$ values can be quite accurate for real systems depending on the system parameters. When $L>N_c+1$, the pulses in a frame always spill over to the adjacent frame(s). In this case, the $N_f$ terms in (\[eq:IFI\]) form a $\lceil (L-1)/N_c\rceil$-dependent sequence and the asymptotic distribution of the IFI is given by the following lemma: \[lem:IFI2\] As $N_f\longrightarrow \infty$, the IFI $\hat{a}$ in (\[eq:IFI\]) is asymptotically normally distributed as $$\label{eq:lemma_IFI2} \hat{a}\sim\mathcal{N}\left(0\,,\,\frac{E_1}{N_c}\sum_{j=1}^{L-N_c} \left[\sum_{l=1}^{j}\left(\beta_l\alpha^{(1)}_{l+L-j}+\alpha^{(1)}_l\beta_{l+L-j}\right)\right]^2 +\frac{E_1}{N_c^2}\sum_{j=1}^{N_c-1}j \left[\sum_{l=1}^{L-j}\left(\beta_l\alpha^{(1)}_{l+j}+\alpha^{(1)}_l\beta_{l+j}\right)\right]^2\right),$$ for $L>N_c+1$. *Proof:* See Appendix \[app:IFI2\]. The MAI term in (\[eq:RAKE\_out\]) can be considered as the sum of MAI terms from each user, that is, $a=\sum_{k=2}^{N_u}a^{(k)}$, where each $a^{(k)}$ is in turn the sum of interference due to the signals in the frames of the template: $$\label{eq_MAI_MP} a^{(k)}=\sqrt{\frac{E_k}{N_f}}\sum_{m=iN_f}^{(i+1)N_f-1}a_m^{(k)},$$ with $$\begin{gathered} \label{eq:MAI_m_MP} a_m^{(k)}=d_m^{(1)}\sum_{j=-\infty}^{\infty}d^{(k)}_jb^{(k)}_{\lfloor j/N_f\rfloor}\phi_{uv}^{(k)}\left((j-m)T_f+(c_j^{(k)}-c_m^{(1)})T_c+\tau_k\right),\end{gathered}$$ where $\phi_{uv}^{(k)}(x)$ is as in (\[eq:phi\]) and $\tau_k$ is the delay of the $k$th user. The effects of MAI will be different for synchronous and asynchronous systems, as investigated in the following subsections. Symbol-Synchronous and Chip-Synchronous Cases --------------------------------------------- In the symbol-synchronous case, the symbols from different users are exactly aligned. In other words, $\tau_k=0$ for $k=2,...,N_u$. On the other hand, for a chip-synchronous scenario, the symbols are misaligned but the amount of misalignment is an integer multiple of the chip interval $T_c$. That is, $\tau_k=\Delta_kT_c$, for $k=2,...,N_u$, where $\Delta_k$ is uniformly distributed in $\{0,1,...,N-1\}$ with $N=N_cN_f$. Note that the assumption of synchronism may not be very realistic for a UWB system due to its high time resolution. However, the aim of this subsection is two-fold. First, we will show that the BEP performance of the UWB system is the same whether the users are symbol-synchronized or chip-synchronized. Second, we will extend the result for the chip-synchronous case to a more practical asynchronous case by modelling asynchronous interfering users as chip-synchronous users with uniform timing jitter, as will be shown in the next subsection. The following lemma gives the asymptotic distribution of MAI from a user for a large number of pulses per symbol. \[lem:syc\_MAI\_MP\] As $N_f\longrightarrow\infty$, the MAI from user $k$, which is chip-synchronized to user $1$, is asymptotically normally distributed as $$\begin{gathered} \label{eq:syc_MAI_MP_lemma} a^{(k)}\sim\mathcal{N}\left(0\,,\,\frac{E_k}{N_c}\left[\sum_{j=1}^{L}\left(\sum_{l=1}^{j}\beta_l\alpha^{(k)}_{l+L-j}\right)^2 +\sum_{j=1}^{L-1}\left(\sum_{l=1}^{j}\alpha^{(k)}_l\beta_{l+L-j}\right)^2\right]\right).\end{gathered}$$ The result is also valid for a symbol-synchronous scenario. *Proof:* See Appendix \[app:syc\_MAI\_MP\]. Note that when $\beta_1=1$ and $\beta_l=0$, for $l=2,...,L$, we have $a^{(k)}\sim\mathcal{N}\left(0\,,\,\frac{E_k}{N_c}\sum_{l=1}^{L}\alpha_l^2\right)$, which represents the result for a Rake receiver with a single finger that picks up the first path signal component only. Note that the Gaussian approximation in Lemma \[lem:syc\_MAI\_MP\] is different from the standard Gaussian approximation (SGA) used in analyzing a system with many users ([@pursley_1977]-[@gardner]). Lemma \[lem:syc\_MAI\_MP\] states that when the number of *pulses* per information symbol is large, the MAI from an interfering user is approximately distributed as a Gaussian random variable. We also note from Lemma \[lem:syc\_MAI\_MP\] that the effect of the MAI is the same for symbol-synchronized and chip-synchronized cases. This is due mainly to the pulse-based polarity randomization, which makes the probability distribution of the MAI independent of the information bits of the interfering user, as can be observed from (\[eq:MAI\_m\_MP\]). Since the probability that a pulse of the template signal overlaps with any of the pulses of an interfering user is the same whether the users are symbol-synchronous or chip-synchronous, the probability distributions turn out to be the same for both cases. An approximate expression for BEP can be derived from (\[eq:RAKE\_out\]), using Lemma \[lem:IFI\], Lemma \[lem:IFI2\] and Lemma \[lem:syc\_MAI\_MP\] as follows: $$\label{eq:BER_syc_MP} P_{e}\approx Q\left({\frac{\sqrt{E_1}\sum_{l=1}^{L}\alpha^{(1)}_l\beta_l}{\sqrt{\frac{E_1}{N_cN}\sigma_{IFI,1}^2+\frac{E_1}{N}\sigma_{IFI,2}^2+ \frac{1}{N}\sum_{k=2}^{N_u}E_k\sigma_{MAI,k}^2+\sigma_n^2\sum_{l=1}^{L}\beta_l^2}}}\,\,\right),$$ where $$\begin{aligned} \label{eq:sig2_IFI} \sigma_{IFI,1}^2&=\sum_{j=1}^{\min\{N_c,L\}-1}j \left[\sum_{l=1}^{L-j}\left(\beta_l\alpha^{(1)}_{l+j}+\alpha^{(1)}_l\beta_{l+j}\right)\right]^2,\\\label{eq:sig2_IFI2} \sigma_{IFI,2}^2&=\begin{cases}\sum_{j=1}^{L-N_c} \left[\sum_{l=1}^{j}\left(\beta_l\alpha^{(1)}_{l+L-j}+\alpha^{(1)}_l\beta_{l+L-j}\right)\right]^2, \quad\quad L>N_c\\0\,,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\,L\leq N_c \end{cases},\end{aligned}$$ and $$\begin{gathered} \label{eq:sig2_MAI_k} \sigma_{MAI,k}^2=\sum_{j=1}^{L}\left(\sum_{i=1}^{j}\beta_i\alpha^{(k)}_{i+L-j}\right)^2 +\sum_{j=1}^{L-1}\left(\sum_{i=1}^{j}\alpha^{(k)}_i\beta_{i+L-j}\right)^2.\end{gathered}$$ Equation (\[eq:BER\_syc\_MP\]) implies that, for a fixed total processing gain $N$, increasing $N_c$, the number of chips per frame, will decrease the effects of IFI, while the dependency of the expressions on the MAI remains unchanged. Hence, an RCDMA system, where $N_f=N$, can suffer from IFI more than any other TH-IR system with pulse-based polarity randomization, where $N_f<N$, if the amount of IFI is comparable to the MAI and thermal noise. Asynchronous Case ----------------- Now consider a completely asynchronous scenario. In this case, it is assumed that $\tau_k$ in (\[eq:MAI\_m\_MP\]) is uniformly distributed according to $\mathcal{U}[0,NT_c)$ for $k=2,...,N_u$. In order to calculate the statistics of the MAI term in (\[eq:RAKE\_out\]), the following simple result will be used. \[prop1\] The MAI in the asynchronous case has the same distribution as the MAI in the chip-synchronous case with interfering user $k$ having a jitter $\epsilon_k$, for $k=2,...,N_u$, which is the same for all pulses of that user and is drawn from the uniform distribution $\mathcal{U}[0,T_c)$. *Proof:* Consider (\[eq:MAI\_m\_MP\]). For $k=2,\ldots,N_u$, $\tau_k$ is uniformly distributed in the discrete set $\{0,T_c,...,(N-1)T_c\}$ in the chip-synchronous case. In the asynchronous case, $\tau_k$ is a continuous random variable with distribution $\mathcal{U}[0,NT_c)$. If the jitter $\epsilon_k$ in the chip-synchronous case is uniformly distributed with $\mathcal{U}[0,T_c)$, then $\tau_k+\epsilon_k$ is uniformly distributed as $\mathcal{U}[0,NT_c)$ hence is equivalent to the distribution of $\tau_k$ in the asynchronous case.$\,\square$ Proposition \[prop1\] reduces the performance analysis of asynchronous systems to the calculation of the statistical properties of $$\begin{gathered} \label{eq:MAI_m_MP_asy} a_m^{(k)}=d_m^{(1)}\sum_{j=-\infty}^{\infty}d^{(k)}_jb^{(k)}_{\lfloor j/N_f\rfloor}\phi_{uv}^{(k)}\left((j-m)T_f+(c_j^{(k)}-c_m^{(1)})T_c+\tau_k+\epsilon_k\right),\end{gathered}$$ where $\tau_k=\Delta_kT_c$ takes on the values $\{0,T_c,...,(N-1)T_c\}$ with equal probabilities and $\epsilon_k\sim\mathcal{U}[0,T_c)$. This problem is similar to the analysis of TH-IR systems in the presence of timing jitter, which is studied in [@sinan2]. However, in the present case, the timing jitter of all pulses of an interfering user is the same instead of being i.i.d. The following lemma approximates the distribution of the MAI from an asynchronous user, conditioned on the timing jitter of that user when the number of pulses per symbol, $N_f$, is large. \[lem:asyc1\_MAI\_MP\] As $N_f\longrightarrow\infty$, the MAI from user $k$ given $\epsilon_k$ has the following asymptotic distribution: $$\begin{gathered} \label{eq:asyc1_MAI_MP_lemma} a^{(k)}|\epsilon_k\sim\mathcal{N}\left(0\,,\,\frac{E_k}{N_c}\sigma^2_{MAI,k}(\epsilon_k)\right),\end{gathered}$$ where $$\begin{aligned} \label{eq:MAI_MP_jit}\nonumber \sigma^2_{MAI,k}(\epsilon_k)&=\sum_{j=0}^{L-1}\left(\sum_{l=1}^{j}\beta_l[\alpha^{(k)}_{l+L-j-1}R(T_c-\epsilon_k)+\alpha^{(k)}_{l+L-j}R(\epsilon_k)]+\beta_{j+1}\alpha^{(k)}_{L}R(T_c-\epsilon_k)\right)^2\\ &+\sum_{j=0}^{L-1}\left(\sum_{l=1}^{j}\alpha^{(k)}_l[\beta_{l+L-j-1}R(\epsilon_k)+\beta_{l+L-j}R(T_c-\epsilon_k)]+\alpha_{j+1}^{(k)}\beta_LR(\epsilon_k)\right)^2,\end{aligned}$$ with $R(x)=\int_{-\infty}^{\infty}w_{rx}(t-x)w_{rx}(t)dt$. *Proof:* See Appendix \[app:asyc1\_MAI\_MP\]. Note that when $\epsilon_k=0$, which corresponds to the chip-synchronized case, (\[eq:MAI\_MP\_jit\]) reduces to (\[eq:sig2\_MAI\_k\]). From Lemma \[lem:asyc1\_MAI\_MP\], we can calculate, for large $N_f$, an approximate conditional BEP given $\boldsymbol{\epsilon}=[\epsilon_2\ldots\epsilon_{N_u}]$ as $$\label{eq:BER_MP_jit} P_{e}|\boldsymbol{\epsilon}\approx Q\left({\frac{\sqrt{E_1}\sum_{l=1}^{L}\alpha^{(1)}_l\beta_l}{\sqrt{\frac{E_1}{N_cN}\sigma_{IFI,1}^2+\frac{E_1}{N}\sigma_{IFI,2}^2+ \frac{1}{N}\sum_{k=2}^{N_u}E_k\sigma_{MAI,k}^2(\epsilon_k)+\sigma_n^2\sum_{l=1}^{L}\beta_l^2}}}\,\,\right),$$ where $\sigma_{MAI,k}^2(\epsilon_k)$ is as in (\[eq:MAI\_MP\_jit\]) and $\sigma_{IFI,1}^2$ and $\sigma_{IFI,2}^2$ are as in (\[eq:sig2\_IFI\]) and (\[eq:sig2\_IFI2\]), respectively. By taking the expectation of (\[eq:BER\_MP\_jit\]) with respect to $\boldsymbol{\epsilon}=[\epsilon_2,\ldots,\epsilon_{N_u}]$, where $\epsilon_k\sim\mathcal{U}[0,T_c)$ for $k=2,...,N_u$, we find the BEP: $$\begin{gathered} \label{eq:accurate_ber} P_{e}\approx\frac{1}{T_c^{N_u-1}}\int_{0}^{T_c}\ldots\int_{0}^{T_c} P_{e}|\boldsymbol{\epsilon}\,\,d\epsilon_2\ldots d\epsilon_{N_u}.\end{gathered}$$ However, when the number of users is large, calculation of (\[eq:accurate\_ber\]) becomes cumbersome since it requires integration of $P_{e}|\boldsymbol{\epsilon}$ over $(N_u-1)$ variables. In this case, the SGA [@pursley_1977]-[@gardner] can be employed in order to approximate the BEP in the case of large number of equal energy interferers: \[lem:asyc2\_MAI\_MP\] Assume that all the interfering users have the same bit energy $E$. Then, as $N_u\longrightarrow\infty$, $a/\sqrt{N_u-1}$, where $a$ is the MAI term in (\[eq:RAKE\_out\]), is asymptotically normally distributed as $$\begin{gathered} \label{eq:asyc2_MAI_MP_lemma} a\sim\mathcal{N}\left(0\,,\,\frac{E}{N_c}\textrm{E}\{\sigma^2_{MAI,k}(\epsilon_k)\}\right),\end{gathered}$$ where $$\begin{aligned} \label{eq:exp_value}\nonumber \textrm{E}\{\sigma^2_{MAI,k}(\epsilon_k)\}&=\frac{1}{T_c}\sum_{j=0}^{L-1}\int_{0}^{T_c}\left(\sum_{l=1}^{j}\beta_l[\alpha^{(k)}_{l+L-j-1}R(T_c-\epsilon_k)+\alpha^{(k)}_{l+L-j}R(\epsilon_k)]+\beta_{j+1}\alpha^{(k)}_{L}R(T_c-\epsilon_k)\right)^2d\epsilon_k\\ &+\frac{1}{T_c}\sum_{j=0}^{L-1}\int_{0}^{T_c}\left(\sum_{l=1}^{j}\alpha^{(k)}_l[\beta_{l+L-j-1}R(\epsilon_k)+\beta_{l+L-j}R(T_c-\epsilon_k)]+\alpha_{j+1}^{(k)}\beta_LR(\epsilon_k)\right)^2d\epsilon_k.\end{aligned}$$ *Proof:* See Appendix \[app:asyc2\_MAI\_MP\]. The BEP can be approximated from Lemma \[lem:asyc2\_MAI\_MP\] as $$\label{eq:PER_MP_asy} P_{e}\approx Q\left({\frac{\sqrt{E_1}\sum_{l=1}^{L}\alpha^{(1)}_l\beta_l}{\sqrt{\frac{E_1}{N_cN}\sigma_{IFI,1}^2+\frac{E_1}{N}\sigma_{IFI,2}^2+ \frac{E}{N}(N_u-1)\textrm{E}\{\sigma^2_{MAI,k}(\epsilon_k)\}+\sigma_n^2\sum_{l=1}^{L}\beta_l^2}}}\,\,\right),$$ for large $N_f$ and $N_u$, and for equal energy interferers. From (\[eq:PER\_MP\_asy\]) we make the same observations as in the synchronous case. Namely, for a given value of the total processing gain $N=N_cN_f$, the effect of the MAI on the BEP remains unchanged while the effect of the IFI increases as the number of chips per frame, $N_c$, decreases. Hence, the IFI could be more effective for an RCDMA system, where $N_c=1$. Different Rake Receiver Structures ---------------------------------- In the previous derivations, we have considered a Rake receiver with $L$ fingers, one at each resolvable multipath component (see (\[eq:temp\_RAKE\]) and (\[eq:v\])). A Rake receiver combining all the paths of the incoming signal is called an *all-Rake* (*ARake*) receiver. Since a UWB signal has a very large bandwidth, the number of resolvable multipath components is usually very large. Hence, an ARake receiver is not implemented in practice due to its complexity. However, it serves as a benchmark for the performance of more practical Rake receivers. A feasible implementation of diversity combining can be obtained by a *selective-Rake* (*SRake*) receiver, which combines the $M$ best, out of $L$, multipath components. Although an SRake receiver is less complex than an ARake receiver, it needs to keep track of all the multipath components and choose the best subset of multipath components before feeding it to the combining stage. A simpler Rake receiver, which combines the first $M$ paths of the incoming signal, is called a *partial-Rake* (*PRake*) receiver [@cassiICC02]. The BEP expressions derived in the previous subsections for synchronous and asynchronous cases are general since one can express different combining schemes by choosing appropriate combining weight vector, $\boldsymbol{\beta}$. For example, if we consider the maximum ratio combining (MRC) scheme, the weights can be expressed as follows for ARake, SRake and PRake receivers: ### ARake In this case, the combining weights are chosen as $\boldsymbol{\beta}=\boldsymbol{\alpha}^{(1)}$, where $\boldsymbol{\beta}=[\beta_1\ldots\beta_L]$ are the Rake combining weights in (\[eq:v\]) and $\boldsymbol{\alpha}^{(1)}=[\alpha^{(1)}_1\ldots\alpha^{(1)}_L]$ are the fading coefficients of the channel for user $1$. ### SRake An SRake receiver combines the best $M$ paths of the received signal. Let $\mathcal{S}$ be the set of indices of these best fading coefficients with largest amplitudes. Then, the combining weights $\boldsymbol{\beta}$ in (\[eq:v\]) are chosen as follows: $$\begin{gathered} \beta_l=\begin{cases}\alpha_{l}^{(1)},\quad l\in\mathcal{S}\\ 0,\quad\quad\,l\notin\mathcal{S} \end{cases}.\end{gathered}$$ ### PRake A PRake receiver combines the first $M$ paths of the received signal. Therefore, the weights of an SRake receiver with MRC scheme are given by the following: $$\begin{gathered} \beta_l=\begin{cases}\alpha_{l}^{(1)},\quad l=1,\ldots,M\,\\ 0,\quad\quad\,\, l=M+1,\ldots,L \end{cases},\end{gathered}$$ where $M<L$. Special Case: Transmission over AWGN Channels --------------------------------------------- From the analysis of frequency-selective channels, we can obtain the expressions for AWGN channels as a special case, which might be useful for intuitive explanations. Considering the expressions in (\[eq:rec\_MP\])-(\[eq:v\]), and setting $\alpha_1=\beta_1=1$ and $\alpha_l=\beta_l=0$ for $l=2,...,L$, the output of the matched filter (MF) receiver can be expressed as $$\begin{gathered} \label{eq:MF_out} y_1=\sqrt{E_1N_f}\,b_i^{(1)}+a+n,\end{gathered}$$ where the first term is the signal part of the output, $a$ is the multiple-access interference (MAI) due to other users and $n$ is the output noise, distributed as $n\sim \mathcal{N}(0,\,N_f\sigma_n^2)$. Note that there is no IFI in this case since a single path channel is assumed. The MAI is expressed as $a=\sum_{k=2}^{N_u}a^{(k)}$, where the distribution of $a^{(k)}$ in the symbol-synchronous and chip-synchronous cases can be obtained from Lemma \[lem:syc\_MAI\_MP\] as $$\begin{gathered} \label{eq:MAI_k_syc} a^{(k)}\sim\mathcal{N}\left(0\,,\,\frac{E_k}{N_c}\right).\end{gathered}$$ Then, the BEP can be obtained as follows: $$\begin{gathered} \label{eq:BER_syc} P_{e}\approx Q\left(\sqrt{\frac{E_1}{\frac{1}{N}\sum_{k=2}^{N_u}E_k+\sigma_n^2}}\,\,\right),\end{gathered}$$ where $N=N_cN_f$, which is the total processing gain of the system. Note from (\[eq:BER\_syc\]) that the BEP depends on $N_c$ and $N_f$ only through their product. Hence, the system performance does not change by changing the number of symbols per information symbol $N_f$ and the number of chips per frame $N_c$ as long as $N_cN_f$ is held constant. This is different from the general case of (\[eq:BER\_syc\_MP\]), where the IFI is reduced for larger $N_c$. Therefore, for AWGN channels, the BEP performance of a TH-IR system with pulse-based polarity randomization is the same as the special case of an RCDMA system. Considering [@eran1], the BEP for TH-IR systems *without* pulse-based polarity randomization is given by the following expression for the case of a synchronous environment with a large number of equal energy interferers: $$\begin{gathered} \label{eq:uncoded} P_e\approx Q\left(\sqrt{\frac{E_1}{(N_u-1)\frac{E}{N}\left(1+\frac{N_f-1}{N_c}\right)+\sigma_n^2}}\right),\end{gathered}$$ where $E$ is the energy of an interferer. Comparing (\[eq:BER\_syc\]) and (\[eq:uncoded\]), we observe that, for $N_f>1$, the MAI affects a TH-IR system without polarity randomization more than it affects a TH-IR system with pulse-based polarity randomization and that the gain obtained by polarity randomization increases as $N_f$ increases (in an interference-limited scenario). The main reason behind this is that random polarity codes make each interference term to a pulse of the template signal (see (\[eq:MAI\_m\_MP\])) a random variable with zero mean since it can be plus or minus interference with equal probability. On the other hand, without random polarity codes, the interference terms to the pulses of the template signal have the same sign, hence add coherently, which increases the effects of the MAI. Note that the effects of the MAI reduce if the UWB system without pulse-based polarity randomization is in an asynchronous environment. Because, in such a case, the MAI terms from some of the pulses add up among themselves while the remaining ones add up among themselves and the polarities of these two groups are independent from each other. Hence, the average MAI is smaller than the symbol-synchronous case but it is still larger than or equal to the MAI for the UWB system *with* pulse-based polarity randomization, where the sign of each interference term is independent (see [@sinan3] for the trade-off between processing gains in TH-IR systems with and without polarity randomization). For TH-IR systems with polarity randomization, we can approximate, using Lemma \[lem:asyc2\_MAI\_MP\], the total MAI in the asynchronous case for a large number of equal energy interferers as $$\begin{gathered} \label{eq:MAI_AWGN_asyc} a \sim {\mathcal{N}}\left(0\,,\,(N_u-1)\frac{2E}{N_cT_c}\int_{0}^{T_c}R^2(\epsilon)d\epsilon\right).\end{gathered}$$ Let $\gamma=\frac{2}{T_c}\int_{0}^{T_c}R^2(\epsilon)d\epsilon=\frac{1}{T_c}\int_{-T_c}^{T_c}R^2(\epsilon)d\epsilon$. Then, from (\[eq:MAI\_AWGN\_asyc\]), $a \sim {\mathcal{N}}\left(0\,,\,\gamma(N_u-1)E/N_c\right)$. Note from (\[eq:MAI\_k\_syc\]) that for equal energy interfering users, the MAI in the symbol/chip-synchronous case is distributed as $a \sim {\mathcal{N}}\left(0\,,\,(N_u-1)E/N_c\right)$. Hence we see that the difference between the powers of the MAI terms depends on the autocorrelation function of the UWB pulse. For example, for the autocorrelation function of (\[eq:R1\]) below, $\gamma\approx0.2$ and symbol/chip-synchronization assumption could possibly result in an over-estimate of the BEP depending on the signal-to-interference-pulse-noise ratio (SINR) of the system. From (\[eq:MF\_out\]) and (\[eq:MAI\_AWGN\_asyc\]), the BEP of an asynchronous system can be approximately expressed as follows: $$\label{eq:BER_asycAWGN} P_{e}\approx Q\left(\frac{\sqrt{E_1}}{\sqrt{{(N_u-1)\frac{2E}{NT_c}\int_{0}^{T_c}R^2(\epsilon)d\epsilon+\sigma_n^2}}}\right),$$ for large values of $N_u$. Similar to the synchronous case, the performance is independent of the distribution of $N$ between $N_c$ and $N_f$. Therefore, the TH-IR system performs the same as an RCDMA system in this case. Average Bit Error Probability ----------------------------- In order to calculate the average BEP, the previous expressions for probability of bit error need to be averaged over all fading coefficients. That is, $P_{avg}=\textrm{E}\{P_e(\boldsymbol{\alpha}^{(1)},\ldots,\boldsymbol{\alpha}^{(k)})\}$, which does not lend itself to simple analytical solutions. However, this average can be evaluated numerically, or by Monte-Carlo simulations. Simulation Results ================== In this section, the BEP performance of a TH-IR system with pulse-based polarity randomization is evaluated by conducting simulations in MATLAB. The following two types of (unit energy) UWB pulses and their autocorrelation functions are employed as the received UWB pulse $w_{rx}(t)$ in the simulations (Figure \[fig:pulse\_auto\]): $$\begin{aligned} \label{eq:w_1} w_1(t)&=\left(1-\frac{4\pi t^2}{\tau^2}\right)e^{-2\pi t^2/\tau^2}/\sqrt{E_p},\\\label{eq:R1} R_1(\Delta t)&=\left[1-4\pi(\frac{\Delta t}{\tau})^2+\frac{4\pi^2}{3}(\frac{\Delta t}{\tau})^4\right]e^{-\pi (\frac{\Delta t}{\tau})^2},\\\label{eq:w_2} w_2(t)&=\frac{1}{\sqrt{T_c}},\quad -0.5T_c\leq t \leq 0.5T_c,\\\label{eq:R2} R_2(\Delta t)&=\begin{cases}-\Delta t/T_c+1,\quad\,\,\, 0\leq \Delta t\leq T_c\\\Delta t/T_c+1,\quad -T_c\leq \Delta t <0 \end{cases},\end{aligned}$$ where $E_p$ of $w_1(t)$ is the normalization constant, $\tau=T_c/2.5$ is used in the simulations, and the rectangular pulse $w_2(t)$ is chosen as an approximate pulse shape in order to compare the performance of the system with different pulse shapes. Figure \[fig:SIR\_vs\_BER\] shows the BEP performance of a $10$-user system ($N_{u}=10$) over an AWGN channel, where $N_{f}=15$ and $N_{c}=5$. The bit energy of the user of interest, user $1$, is $E_{1}=0.5$, whereas the interfering users transmit bits with unit energy ($E_{k}=1$ for $k=2,...,10$), and the attenuation due to the channel is set equal to unity. The SINR is defined by $\textrm{SINR}=10\log_{10}\left({E_1}/{(\frac{1}{N}\sum_{k=2}^{N_u}E_k+\sigma_n^2})\right)$. In Figure \[fig:SIR\_vs\_BER\], the SINR is varied by changing the noise power $\sigma_{n}^{2}$ and the BEP is obtained for different SINR values in the cases of symbol-synchronous, chip-synchronous and asynchronous TH-IR systems with pulse-based polarity randomization and a synchronous TH-IR system without pulse-based polarity randomization$^7$. For the asynchronous case, performance is simulated for different pulse shapes $w_{1}(t)$ and $w_{2}(t)$, given by (\[eq:w\_1\]) and (\[eq:w\_2\]), respectively. From Figure \[fig:SIR\_vs\_BER\], we see that the simulation results match closely with the theoretical results. Also note that for small SINR, all the systems perform quite similarly since the main source of error is the thermal noise in that case. As the SINR increases, i.e., as the MAI becomes the limiting factor, the systems start to perform differently. The asynchronous systems perform better than the chip-synchronous and symbol-synchronous cases since $\frac{2}{T_{c}}\int_{0}^{T_{c}}R^{2}(\epsilon)d\epsilon$ in (\[eq:BER\_asycAWGN\]) is about $0.2$ for $w_{1}(t)$ and $2/3$ for $w_{2}(t)$, which also explains the reason for the lowest bit error rate of the asynchronous system with UWB pulse $w_{1}(t)$. Also it is observed that for an IR-UWB system with pulse-based polarity randomization, the chip-synchronous and the symbol-synchronous systems perform the same as expected. Moreover, we observe that without pulse-based polarity randomization, the MAI is more effective, which results in larger BEP values. In order to compare the approximate analytical expressions and the simulation results for multipath channels, we consider the following channel coefficients for all users: $\boldsymbol{\alpha}=[0.4653\,\,\,0.5817\,\,\,0.2327\,-0.4536 \,\,\,0.3490\,\,\,0.2217\,-0.1163\,\,\,0.0233\,-0.0116\,-0.0023]$. Then, the Rake combining fingers are $\boldsymbol{\beta}=\boldsymbol{\alpha}$ for an ARake receiver, $\boldsymbol{\beta}=[0.4653\,\,\,0.5817\,\,\,0\,-0.4536 \,\,\,0\,\,\,0\,\,\,0\,\,\,0\,\,\,0\,\,\,0]$ for an SRake receiver with $3$ fingers, and $\boldsymbol{\beta}=[0.4653\,\,\,0.5817\,\,\,0.2327\,\,\,0$ $\,\,\,0\,\,\,0\,\,\,0\,\,\,0\,\,\,0\,\,\,0]$ for a PRake receiver with $3$ fingers. The system parameters are chosen as $N_u=10$, $N_c=5$, $N_f=15$, $E_1=0.5$ and $E_k=1$ for $k=2,...,10$. Figure \[fig:MP1\] plots BEPs of different Rake receivers for synchronous and asynchronous systems with pulse-based polarity randomization. From the figure, we have the same conclusions as in the AWGN channel case about synchronous and asynchronous cases. Namely, chip-synchronous and symbol-synchronous systems perform the same and asynchronous systems with received pulses $w_1(t)$ and $w_2(t)$ perform better. The asynchronous system with $w_1(t)$ performs the best due to the properties of its correlation function. Note that the performance is poor when there is synchronism (chip or symbol level) among the users. However, the asynchronous system performs reasonably well even in this harsh multiuser environment. Hence, when computing the BEP of a system, the assumption of synchronism can result in over-estimating the BEP. Apart from those, it is also observed from the figure that the ARake receiver performs the best as expected. Also the SRake performs better than the PRake since the former collects more energy because the fourth path is stronger than the third path. For the next simulations, we model the channel coefficients as $\alpha_l=\textrm{sign}(\alpha_l)|\alpha_l|$ for $l=1,\ldots,L$, where $\textrm{sign}(\alpha_l)$ is $\pm1$ with equal probability and $|\alpha_l|$ is distributed lognormally as $\mathcal{LN}(\mu_l,\sigma^2)$. Also the energy of the taps is exponentially decaying as $\textrm{E}\{|\alpha_l|^2\}=\Omega_0e^{-\lambda(l-1)}$, where $\lambda$ is the decay factor and $\sum_{l=1}^{L}\textrm{E}\{|\alpha_l|^2\}=1$ (so $\Omega_0=(1-e^{-\lambda})/(1-e^{-\lambda L})$). All the system parameters are the same as the previous case, except we have $E_1=1$ in this case. For the channel parameters, we have $L=20$, $\lambda=0.25$, $\sigma^2=1$ and $\mu_l$ can be calculated from $\mu_l=0.5\left[\textrm{ln}(\frac{1-e^{-\lambda}}{1-e^{-\lambda L}})-\lambda(l-1)-2\sigma^2\right]$, for $l=1,\ldots,L$. Figure \[fig:MP2\] plots the BEP versus $E_b/N_0$ for different Rake receivers in an asynchronous environment where $w_1(t)$ models the received UWB pulse. We consider ARake, SRake and PRake receivers for the TH-IR system with pulse-based polarity randomization and an ARake receiver for the one without pulse-based polarity randomization. The SRake and PRake receivers have $5$ fingers each. As can be seen from the figure, the theoretical results are quite close to the simulation results. More accurate results can be obtained when the number of users is larger. It is also observed that the performance of the SRake receiver with $5$ fingers is close to that of the ARake receiver in this setting. Moreover, the ARake receiver for the system without polarity randomization performs almost as worst as the PRake receiver for the UWB system with polarity randomization, which indicates the benefit of polarity randomization in reducing the effects of MAI. In Figure \[fig:MP3\], we set $E=2$ and keep all the other parameters the same as in the previous case. Here we consider a UWB system with polarity randomization and observe the performances of the SRake and the PRake receivers for different number of fingers $M$, using (\[eq:PER\_MP\_asy\]). It is observed from the figure that the performance of the SRake receiver with $10$ fingers is very close to that of the ARake receiver whereas the PRake receiver needs around $15$ fingers for a similar performance. Conclusion ========== In this paper, the performance of random TH-IR systems with pulse-based polarity randomization has been analyzed and approximate BEP expressions for various combining schemes of Rake receivers have been derived. Starting from the chip-synchronous case, we have analyzed the completely asynchronous case by modelling the latter by an equivalent chip-synchronous system with uniform timing jitter at interfering users. 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COM-27, pp. 143–149, Jan. 1979. D. Cassioli, M. Z. Win, F. Vatalaro and A. F. Molisch, “Performance of low-complexity RAKE reception in a realistic UWB channel," *Proceedings of the IEEE International Conference on Communications, 2002 (ICC 2002),* vol. 2, pp. 763-767, New York, NY, April 28-May 2, 2002. S. Gezici, A. F. Molisch, H. V. Poor, and H. Kobayashi, “The trade-off between processing gains of an impulse radio UWB system in the presence of timing jitter," *in preparation*, 2004. P. Billingsley, *Probability and Measure,* John Wiley & Sons, New York, 2nd edition, 1986. Asymptotic Distribution of $n$ in (\[eq:RAKE\_out\]) {#app:noise} ---------------------------------------------------- The noise term $n$ in (\[eq:RAKE\_out\]) can be obtained from (\[eq:rec\_MP\]) and (\[eq:temp\_RAKE\]) as $n=\sigma_n\int s^{(1)}_{temp}(t)n(t)dt$, where $n(t)$ is a zero mean white Gaussian process with unit spectral density. Hence, $n$ is a Gaussian random variable for a given template signal. Since the process has zero mean, $n$ has zero mean for any template signal. The variance of $n$ can be calculated as $\textrm{E}\{n^2\}=\sigma_n^2\int(s_{temp}^{(1)}(t))^2dt$ using the fact that $n(t)$ is white. Using the expressions in (\[eq:temp\_RAKE\]) and (\[eq:v\]), we get $$\begin{aligned} \label{eq:noise_var1} \textrm{E}\{n^2\}=\sigma_n^2\sum_{j=iN_f}^{(i+1)N_f-1}\int f_j^2(t)dt+2\sigma_n^2\underset{j\ne k}{\sum_{j,k=iN_f}^{(i+1)N_f-1}}d^{(1)}_jd^{(1)}_k\int f_j(t)f_k(t)dt,\end{aligned}$$ where $f_j(t)=\sum_{l=1}^{L}\beta_l\,w_{rx}(t-jT_f-c^{(1)}_jT_c-(l-1)T_c-\tau_1)$. It can be shown that $\int f_j^2(t)dt=\sum_{l=1}^{L}\beta_l^2$ for all $j$ since $w_{rx}(t)$ is assumed to be a unit energy pulse. Now consider $\int f_j(t)f_k(t)dt$. By definition, $f_j(t)f_k(t)$ is zero when there is no overlap between the pulses from the $j$th and the $k$th frames. Assume that $L\leq N_c$. Then, $f_j(t)f_k(t)=0$ for $|j-k|>1$. In other words, there can be spill-over from one frame only to a neighboring frame. In this case, (\[eq:noise\_var1\]) becomes $$\begin{aligned} \label{eq:noise_var2} \textrm{E}\{n^2\}=\sigma_n^2N_f\sum_{l=1}^{L}\beta_l^2 +2\sigma_n^2{\sum_{j=iN_f}^{(i+1)N_f-2}}d^{(1)}_jd^{(1)}_{j+1}\int f_j(t)f_{j+1}(t)dt.\end{aligned}$$ Note that $f_j(t)$ is a random variable at a given time instant $t$ due to the presence of the random time-hopping sequence $\{c_j^{(1)}\}$, and $\{f_j(t)f_{j+1}(t)\}$ are identically distributed for $j=iN_f,...,(i+1)N_f-2$. Since $\{d^{(1)}_jd^{(1)}_{j+1}\}$ has zero mean and forms an i.i.d. sequence for $j=iN_f,...,(i+1)N_f-2$, $\{d^{(1)}_jd^{(1)}_{j+1}\int f_j(t)f_{j+1}(t)dt\}$ forms a zero mean i.i.d. sequence. Hence, the second summation in (\[eq:noise\_var2\]) converge to zero as $N_f\longrightarrow\infty$, by the Strong Law of Large Numbers. When the $L\leq N_c$ assumption is removed, we can still use the same approach to prove the result for finite values of $L$. In that case, we can write a more general version of (\[eq:noise\_var2\]) as $$\begin{aligned} \label{eq:noise_var3} \textrm{E}\{n^2\}=\sigma_n^2N_f\sum_{l=1}^{L}\beta_l^2 +2\sigma_n^2\sum_{k=1}^{D}{\sum_{j=iN_f}^{(i+1)N_f-1-k}}d^{(1)}_jd^{(1)}_{j+k}\int f_j(t)f_{j+k}(t)dt,\end{aligned}$$ where $f_j(t)f_k(t)=0$ for $|j-k|>D$. Since $L$ is assumed to be finite, $D$ is also finite. Hence, the second term in (\[eq:noise\_var3\]) still converges to zero as $N_f\longrightarrow\infty$. Thus for large $N_f$, $\textrm{E}\{n^2\}\approx\sigma_n^2N_f\sum_{l=1}^{L}\beta_l^2$, and so $n$ is approximately distributed as $n\sim\mathcal{N}\left(0\,,\,\sigma_n^2N_f\sum_{l=1}^{L}\beta_l^2\right)$. Proof of Lemma \[lem:IFI\] {#app:IFI} -------------------------- The aim is to approximate the distribution of $\hat{a}=\sqrt{\frac{E_1}{N_f}}\sum_{m=iN_f}^{(i+1)N_f-1}\hat{a}_m$, where $\hat{a}_m$ is given by (\[eq:IFI\_m\]). Note that $\hat{a}_m$ denotes the interference to the $m$th frame coming from the other frames. Assuming that $L\leq N_c+1$, there can be interference to the $m$th frame only from the $(m-1)$th or $(m+1)$th frames. Hence, $\hat{a}_m$ can be expressed as: $$\begin{gathered} \label{eq:IFI_m_ap1} \hat{a}_m=d_m^{(1)}\sum_{j\in\{m-1,m+1\}}d^{(1)}_jb^{(1)}_{\lfloor j/N_f\rfloor}\phi_{uv}^{(1)}\left((j-m)T_f+(c_j^{(1)}-c_m^{(1)})T_c\right).\end{gathered}$$ Note that $\hat{a}_{iN_f},\ldots,\hat{a}_{(i+1)N_f-1}$ are identically distributed but not independent. However, they form a 1-dependent sequence [@meas] since $\hat{a}_{m}$ and $\hat{a}_{n}$ are independent whenever $|m-n|>1$. The expected value of $\hat{a}_m$ is equal to zero due to the random polarity code. That is, $\textrm{E}\{\hat{a}_m\}=0$. The variance of $\hat{a}_m$ can be calculated from (\[eq:IFI\_m\_ap1\]) as $$\begin{gathered} \label{eq:var_a_m_ap1} \textrm{E}\{\hat{a}_m^2\}=\sum_{j\in\{m-1,m+1\}}\textrm{E}\left\{\left[\phi_{uv}^{(1)}\left((j-m)T_f+(c_j^{(1)}-c_m^{(1)})T_c\right)\right]^2\right\},\end{gathered}$$ where the fact that the random polarity codes are zero mean and independent for different indices is employed. Since the TH sequence can take any value in $\{0,1,\ldots,N_c-1\}$ with equal probability, the variance can be calculated as $$\begin{gathered} \label{eq:var_a_m_ap2} \textrm{E}\{\hat{a}_m^2\}=\frac{1}{N_c^2}\sum_{j=1}^{L-1}j\left\{[\phi_{uv}^{(1)}(jT_c)]^2+[\phi_{uv}^{(1)}(-jT_c)]^2\right\},\end{gathered}$$ which can be expressed as $$\begin{gathered} \label{eq:var_a_m_ap3} \textrm{E}\{\hat{a}_m^2\}=\frac{1}{N_c^2}\sum_{j=1}^{L-1}j \left[\left(\sum_{l=1}^{L-j}\beta_l\alpha^{(1)}_{l+j}\right)^2+ \left(\sum_{l=1}^{L-j}\alpha^{(1)}_l\beta_{l+j}\right)^2\right],\end{gathered}$$ using (\[eq:phi\]), (\[eq:u\_k\]) and (\[eq:v\]). Now consider the correlation terms. Since $L\leq N_c$, $\textrm{E}\{\hat{a}_m\hat{a}_n\}=0$ when $|m-n|>1$. Hence, we need to consider $\textrm{E}\{\hat{a}_m\hat{a}_{m+1}\}$ only. Similar to the derivation of the variance, $\textrm{E}\{\hat{a}_m\hat{a}_{m+1}\}$ can be obtained, from (\[eq:IFI\_m\_ap1\]), as follows: $$\begin{gathered} \label{eq:corr_a_m_ap} \textrm{E}\{\hat{a}_m\hat{a}_{m+1}\}=\frac{1}{N_c^2}\sum_{j=1}^{L-1}j \left(\sum_{l=1}^{L-j}\beta_l\alpha^{(1)}_{l+j}\right)\left(\sum_{l=1}^{L-j}\beta_l\alpha^{(1)}_{l+j}\right).\end{gathered}$$ Since $\{\hat{a}_m\}_{m=iN_f}^{(i+1)N_f-1}$ is a zero mean 1-dependent sequence, $\sqrt{\frac{E_1}{N_f}}\sum_{m=iN_f}^{(i+1)N_f-1}\hat{a}_m$ converges to $$\begin{gathered} \mathcal{N}\left(0\,,\,E_1\left[\textrm{E}\{(\hat{a}_m)^2\}+2\textrm{E}\{\hat{a}_m\hat{a}_{m+1}\}\right]\right)\end{gathered}$$ as $N_f\longrightarrow\infty$ [@meas]. Hence, (\[eq:lemma\_IFI\]) follows from (\[eq:var\_a\_m\_ap3\]) and (\[eq:corr\_a\_m\_ap\]). Proof of Lemma \[lem:IFI2\] {#app:IFI2} --------------------------- In this section we derive the distribution of IFI for $L>N_c+1$. Consider the case where $(D-1)N_c+1<L\leq DN_c+1$, with $D$ being a positive integer. Hence, $\{\hat{a}_m\}_{m=iN_f}^{(i+1)N_f-1}$ forms a $D$-dependent sequence in this case. Similar to Appendix \[app:IFI\], we need to calculate the mean, the variance and the correlation terms for $\hat{a}_m$ in (\[eq:IFI\_m\]). Due to the polarity codes, it is clear that $\textrm{E}\{\hat{a}_m\}=0$. The variance can be expressed as follows, using (\[eq:IFI\_m\]) and the fact that the polarity codes are zero mean and independent for different indices: $$\begin{gathered} \label{eq:appIFI2_var1} \textrm{E}\{\hat{a}_m^2\}=\underset{j\ne m}{\sum_{j=-\infty}^{\infty}}\textrm{E}\left\{\left[\phi_{uv}^{(1)}\left((j-m)T_f+(c_j^{(1)}-c_m^{(1)})T_c\right)\right]^2\right\},\end{gathered}$$ which can be calculated as $$\begin{gathered} \label{eq:appIFI2_var2} \textrm{E}\{\hat{a}_m^2\}=\frac{1}{N_c^2}\sum_{i=0}^{N_c-1}\sum_{l=0}^{N_c-1}\underset{j\ne m}{\sum_{j=-\infty}^{\infty}}\left[\phi_{uv}^{(1)}\left((j-m)T_f+(c_j^{(1)}-c_m^{(1)})T_c\right)\right]^2,\end{gathered}$$ using that fact that the TH sequence is uniformly distributed in $\{0,1,\ldots,N_c-1\}$. Then, the variance term can be expressed as $$\begin{gathered} \label{eq:appIFI2_var3} \textrm{E}\{\hat{a}_m^2\}=\frac{1}{N_c^2}\sum_{j=1}^{N_c-1}j\left\{ \left[\phi_{uv}^{(1)}(jT_c)\right]^2+\left[\phi_{uv}^{(1)}(-jT_c)\right]^2\right\} +\frac{1}{N_c}\sum_{j=N_c}^{L-1}\left\{ \left[\phi_{uv}^{(1)}(jT_c)\right]^2+\left[\phi_{uv}^{(1)}(-jT_c)\right]^2\right\},\end{gathered}$$ which can be obtained, using (\[eq:phi\]), (\[eq:u\_k\]) and (\[eq:v\]), as follows: $$\begin{aligned} \label{eq:exp2_IFI}\nonumber \textrm{E}\{(\hat{a}_m)^2\}&=\frac{1}{N_c}\sum_{j=1}^{L-N_c}\left[\left(\sum_{i=1}^{j}\beta_i\alpha^{(1)}_{L+i-j}\right)^2+\left(\sum_{i=1}^{j}\alpha^{(1)}_i\beta_{L+i-j}\right)^2\right]\\ &+\frac{1}{N_c^2}\sum_{j=1}^{N_c-1}j\left[\left(\sum_{i=1}^{L-j}\beta_i\alpha^{(1)}_{i+j}\right)^2+\left(\sum_{i=1}^{L-j}\alpha^{(1)}_i\beta_{i+j}\right)^2\right].\end{aligned}$$ Since $\hat{a}_{iN_f},\ldots,\hat{a}_{(i+1)N_f-1}$ form a $D$-dependent sequence, we need to calculate $\textrm{E}\{\hat{a}_m\hat{a}_{m+n}\}$ for $n=1,\ldots,D$. Then, the IFI term $\hat{a}$ in (\[eq:IFI\]) can be approximated by $$\begin{gathered} \label{eq:gaus_IFI2} {\mathcal{N}}\left(0\,,\,E_1\left[\textrm{E}\{(\hat{a}_{iN_f})^2\}+2\sum_{n=1}^{D}\textrm{E}\{\hat{a}_{iN_f}\hat{a}_{iN_f+n}\}\right]\right),\end{gathered}$$ as $N_f\longrightarrow\infty$ [@meas]. Using (\[eq:IFI\_m\]), (\[eq:phi\]), (\[eq:u\_k\]) and (\[eq:v\]), the correlation term in (\[eq:gaus\_IFI2\]) can be calculated, after some manipulation, as $$\begin{aligned} \label{eq:totalCrossIFI}\nonumber \sum_{n=1}^{D}\textrm{E}\{\hat{a}_{iN_f}\hat{a}_{iN_f+n}\}&=\frac{1}{N_c}\sum_{j=1}^{L-N_c}\left(\sum_{i=1}^{j}\beta_i\alpha^{(1)}_{L+i-j}\right)\left(\sum_{i=1}^{j}\alpha^{(1)}_i\beta_{L+i-j}\right)\\ &+\frac{1}{N_c^2}\sum_{j=1}^{N_c-1}j\left(\sum_{i=1}^{L-j}\beta_i\alpha^{(1)}_{i+j}\right)\left(\sum_{i=1}^{L-j}\alpha^{(1)}_i\beta_{i+j}\right).\end{aligned}$$ Hence, (\[eq:lemma\_IFI2\]) can be obtained by inserting (\[eq:exp2\_IFI\]) and (\[eq:totalCrossIFI\]) into (\[eq:gaus\_IFI2\]). Proof of Lemma \[lem:syc\_MAI\_MP\] {#app:syc_MAI_MP} ----------------------------------- In order to calculate the distribution of the MAI from user $k$, $a^{(k)}=\sqrt{\frac{E_k}{N_f}}\sum_{m=iN_f}^{(i+1)N_f-1}a_m^{(k)}$, we first calculate the mean and variance of $a_m^{(k)}$ given by (\[eq:MAI\_m\_MP\]), where the delay of the user, $\tau_k$, is an integer multiple of the chip interval: $\tau_k=\Delta_kT_c$. Due to the polarity codes, the mean is equal to zero for any delay value $\tau_k$; that is, $\textrm{E}\{a_m^{(k)}|\Delta_k\}=0$. In order to calculate the variance, we make use of the facts that the polarity codes are independent for different user and frame indices, and that the TH sequence is uniformly distributed in $\{0,1,\ldots,N_c-1\}$. Then, we obtain the following expression: $$\begin{gathered} \label{eq:mai_syc_ap1} \textrm{E}\{(a_m^{(k)})^2|\Delta_k\}=\frac{1}{N_c^2}\sum_{i=0}^{N_c-1}\sum_{l=0}^{N_c-1}\sum_{j=-\infty}^{\infty} \left\{\phi_{uv}^{(k)}\left[(i-l+(j-m)N_c+\Delta_k)T_c\right]\right\}^2,\end{gathered}$$ which is equal to $$\begin{gathered} \label{eq:mai_syc_ap2} \textrm{E}\{(a_m^{(k)})^2|\Delta_k\}=\frac{1}{N_c}\sum_{j=-(L-1)}^{L-1} \left[\phi_{uv}^{(k)}(jT_c)\right]^2.\end{gathered}$$ Using (\[eq:phi\]), (\[eq:u\_k\]) and (\[eq:v\]), (\[eq:mai\_syc\_ap2\]) can be expressed as $$\begin{gathered} \label{eq:mai_syc_ap3} \textrm{E}\{(a_m^{(k)})^2|\Delta_k\}=\frac{1}{N_c}\left[\sum_{j=1}^{L}\left(\sum_{i=1}^{j}\beta_i\alpha^{(k)}_{i+L-j}\right)^2 +\sum_{j=1}^{L-1}\left(\sum_{i=1}^{j}\alpha^{(k)}_i\beta_{i+L-j}\right)^2\right].\end{gathered}$$ Moreover, we note that $\textrm{E}\{a_m^{(k)}a_n^{(k)}|\Delta_k\}=0$ for $m\ne n$ due to the polarity codes. Similar to the proofs in Appendices \[app:IFI\] and \[app:IFI2\], $\{a_m^{(k)}\}_{m=iN_f}^{(i+1)N_f-1}$ forms a dependent sequence and the MAI from user $k$, $a^{(k)}=\sqrt{\frac{E_k}{N_f}}\sum_{m=iN_f}^{(i+1)N_f-1}a_m^{(k)}$, converge to ${\mathcal{N}}\left(0\,,\,E_k\textrm{E}\{(a_m^{(k)})^2\}\right)$ since the correlation terms are zero. Hence, (\[eq:syc\_MAI\_MP\_lemma\]) can be obtained from (\[eq:mai\_syc\_ap3\]). Note that the result is true for any value of $\Delta_k$ since $\textrm{E}\{(a_m^{(k)})^2|\Delta_k\}$ in (\[eq:mai\_syc\_ap3\]) is independent of $\Delta_k$. Hence, the result is valid for both symbol and chip synchronous cases. Proof of Lemma \[lem:asyc1\_MAI\_MP\] {#app:asyc1_MAI_MP} ------------------------------------- The proof of Lemma \[lem:asyc1\_MAI\_MP\] is an extension of that of Lemma \[lem:syc\_MAI\_MP\]. Considering (\[eq:MAI\_m\_MP\_asy\]), we have an additional offset $\epsilon_k$, which causes a partial overlap between pulses from the template signal and those from the interfering signal. Due to the presence of random polarity codes, the mean of $a_m^{(k)}$ in (\[eq:MAI\_m\_MP\_asy\]) is equal to zero. Using the fact that the polarity codes are zero mean and independent for different frame indices and that the TH codes are uniformly distributed in $\{0,1,\ldots,N_c-1\}$, we can calculate the variance of $a_m^{(k)}$ conditioned on $\Delta_k$ and $\epsilon_k$ as $$\begin{gathered} \label{eq:mai_asyc_ap1} \textrm{E}\{(a_m^{(k)})^2|\Delta_k,\epsilon_k\}=\frac{1}{N_c^2}\sum_{i=0}^{N_c-1}\sum_{l=0}^{N_c-1}\sum_{j=-\infty}^{\infty} \left\{\phi_{uv}^{(k)}\left[(i-l+(j-m)N_c+\Delta_k)T_c+\epsilon_k\right]\right\}^2,\end{gathered}$$ which can be shown to be equal to $$\begin{gathered} \label{eq:mai_asyc_ap2} \textrm{E}\{(a_m^{(k)})^2|\Delta_k,\epsilon_k\}=\frac{1}{N_c}\sum_{j=-L}^{L-1} \left[\phi_{uv}^{(k)}(jT_c+\epsilon_k)\right]^2.\end{gathered}$$ Note that since the expression in (\[eq:mai\_asyc\_ap2\]) is independent of $\Delta_k$, $\textrm{E}\{(a_m^{(k)})^2|\Delta_k,\epsilon_k\}=\textrm{E}\{(a_m^{(k)})^2|\epsilon_k\}$. From (\[eq:phi\]), (\[eq:u\_k\]) and (\[eq:v\]), we can obtain an expression for $\phi_{uv}^{(k)}(jT_c+\epsilon_k)$ when $j\geq0$ as $$\begin{gathered} \label{eq:phi1} \phi_{uv}(jT_c+\epsilon_k)=\sum_{l=1}^{L-j-1}\alpha_l^{(k)}\left[\beta_{l+j}R(\epsilon_k)+\beta_{l+j+1}R(T_c-\epsilon_k)\right]+\alpha_{L-j}^{(k)}\beta_LR(\epsilon_k),\end{gathered}$$ where $R(x)=\int_{-\infty}^{\infty}w_{rx}(t-x)w_{rx}(t)dt$. Similarly, the expression for $\phi_{uv}^{(k)}(-jT_c+\epsilon_k)$ can be expressed as follows for $j>0$: $$\begin{gathered} \label{eq:phi2} \phi_{uv}(-jT_c+\epsilon_k)=\sum_{l=1}^{L-j}\beta_l\left[\alpha^{(k)}_{l+j}R(\epsilon_k)+\alpha^{(k)}_{l+j-1}R(T_c-\epsilon_k)\right]+\beta_{L-j+1}\alpha^{(k)}_LR(T_c-\epsilon_k).\end{gathered}$$ Using (\[eq:phi1\]) and (\[eq:phi2\]), $\textrm{E}\{(a_m^{(k)})^2|\epsilon_k\}$ can be expressed from (\[eq:mai\_asyc\_ap2\]) as $$\begin{aligned} \label{eq:son_bu}\nonumber \textrm{E}\{(a_m^{(k)})^2|\epsilon_k\}&=\frac{1}{N_c}\sum_{j=0}^{L-1}\left(\sum_{i=1}^{j}\beta_i[\alpha^{(k)}_{i+L-j-1}R(T_c-\epsilon_k)+\alpha^{(k)}_{i+L-j}R(\epsilon_k)]+\beta_{j+1}\alpha^{(k)}_{L}R(T_c-\epsilon_k)\right)^2\\ &+\frac{1}{N_c}\sum_{j=0}^{L-1}\left(\sum_{i=1}^{j}\alpha^{(k)}_i[\beta_{i+L-j-1}R(\epsilon_k)+\beta_{i+L-j}R(T_c-\epsilon_k)]+\alpha_{j+1}^{(k)}\beta_LR(\epsilon_k)\right)^2.\end{aligned}$$ Also, due to the polarity codes, the correlation terms are zero. That is, $\textrm{E}\{a_m^{(k)}a_n^{(k)}\}=0$ for $m\ne n$. Then, from the central limit argument in [@meas], we see that $a^{(k)}$ in (\[eq\_MAI\_MP\]), conditioned on $\epsilon_k$, converge to the distribution given in Lemma \[lem:asyc1\_MAI\_MP\]. Proof of Lemma \[lem:asyc2\_MAI\_MP\] {#app:asyc2_MAI_MP} ------------------------------------- Consider $(N_u-1)$ interfering users, each with bit energy $E$. Then, the total MAI $a=\sum_{k=2}^{N_u}a^{(k)}$ is the sum of $(N_u-1)$ i.i.d. random variables, where $a^{(k)}=\sqrt{\frac{E}{N_f}}\sum_{m=iN_f}^{(i+1)N_f-1}a_m^{(k)}$. Using the results in Appendix \[app:asyc1\_MAI\_MP\], namely, $\textrm{E}\{a_m^{(k)}\}=0$, $\textrm{E}\{a_m^{(k)}a_n^{(k)}\}=0$ for $m\ne n$ and (\[eq:son\_bu\]), we obtain $$\begin{gathered} \frac{1}{\sqrt{N_u-1}}\sum_{k=2}^{N_u}a^{(k)}\sim\mathcal{N}\left(0\,,\,\frac{E}{N_c}\textrm{E}\{\sigma^2_{MAI,k}(\epsilon_k)\}\right),\end{gathered}$$ as $N_u\longrightarrow\infty$, where $\textrm{E}\{\sigma^2_{MAI,k}(\epsilon_k)\}$ can be obtained as in (\[eq:exp\_value\]) from (\[eq:son\_bu\]) using the fact that $\epsilon_k\sim\mathcal{U}[0,T_c)$.

--- author: - 'T. Ruiz-Lara, C. G. Few, E. Florido, B. K. Gibson, I. Pérez,' - 'P. Sánchez-Blázquez' bibliography: - 'bibliography.bib' date: 'Received —; accepted —' title: The Role of Stellar Radial Motions in Shaping Galaxy Surface Brightness Profiles --- [The physics driving features such as breaks observed in galaxy surface brightness (SB) profiles remains contentious. Here, we assess the importance of stellar radial motions in shaping their characteristics.]{} [We use the simulated Milky Way-mass, cosmological discs, from the Ramses Disc Environment Study ([RaDES]{}) to characterise the radial redistribution of stars in galaxies displaying type I (pure exponentials), II (downbending), and III (upbending) SB profiles. We compare radial profiles of the mass fractions and the velocity dispersions of different sub-populations of stars according to their birth and current locations.]{} [Radial redistribution of stars is important in all galaxies regardless of their light profiles. Type II breaks seem to be a consequence of the combined effects of outward-moving and accreted stars. The former produces shallower inner profiles (lack of stars in the inner disc) and accumulate material around the break radius and beyond, strengthening the break; the latter can weaken or even convert the break into a pure exponential. Further accretion from satellites can concentrate material in the outermost parts, leading to type III breaks that can coexist with type II breaks, but situated further out. Type III galaxies would be the result of an important radial redistribution of material throughout the entire disc, as well as a concentration of accreted material in the outskirts. In addition, type III galaxies display the most efficient radial redistribution and the largest number of accreted stars, followed by type I and II systems, suggesting that type I galaxies may be an intermediate case between types II and III. In general, the velocity dispersion profiles of all galaxies tend to flatten or even increase around the locations where the breaks are found. The age and metallicity profiles are also affected, exhibiting different inner gradients depending on their SB profile, being steeper in the case of type II systems (as found observationally). The steep type II profiles might be inherent to their formation rather than acquired via radial redistribution.]{} Introduction ============ The observed properties of spiral galaxies are the outcome of complex, non-linear, and inter-related, formation and evolution processes. According to the current paradigm, after the assembly of large galaxies via a number of high-redshift minor mergers, a period of secular evolution follows. In parallel with the latter, non-axisymmetric structures such as bars or spiral arms may be formed [e.g. @1978MNRAS.183..341W; @2003ApJ...591..499A] with the ability to redistribute material [@2002MNRAS.336..785S; @2008ApJ...675L..65R; @2008ApJ...684L..79R; @2010ApJ...722..112M] across the entire galaxy. Simultaneous to this secular evolution, satellite accretion may continue to influence the characteristics of the host , particularly those found in the outermost regions. A growing number of theoretical and observational works are focused on analysing the characteristics of these outer parts of spiral galaxies . The lower surface densities (and, hence, gravitational effects) and the longer dynamical times of stars populating those regions make the study of the outskirts of disc galaxies a unique place to test current galaxy formation and evolution models. One of the defining metrics of these outer regions is the presence, or lack thereof, of deviations from a pure exponential surface brightness (SB) profile . While some galaxies possess an essentially pure exponential SB profile into the outer parts (type I, continuation of the inner behaviour), others show a deficiency (type II) or excess of light (type III) that can be characterised by two exponentials . The drivers behind these different profiles remain unclear, although changes in the ages of stellar populations, the effect of radial migration, and satellite accretion have all been proposed . In the last decade a clear link between an outer upturn (“U-shape”) in the age profiles and type II galaxies was found via observations and theory, with the age upturn causing the lack of light in the outer regions [e.g. @2008ApJ...683L.103B; @2008ApJ...675L..65R; @2009MNRAS.398..591S; @2009ApJ...705L.133M; @2012ApJ...752...97Y]. Conversely, in @2016MNRAS.456L..35R, analysing spectroscopic information from the CALIFA[^1] survey, we found that “U-shape” age profiles are found for both type I and II galaxies [similar to that found by @2012ApJ...758...41R from photometric data]. As a consequence, we suggested that the mechanisms causing the light distributions and the stellar age profiles might not be coupled. In the same vein, showed that all galaxies in the [RaDES]{} simulation suite show “U-shape” stellar age profiles, regardless of their light distribution. Other observational studies have also found the presence of old stellar populations in the outer parts of nearby spiral galaxies [e.g. @2010ApJ...712..858G; @2012MNRAS.420.2625B; @2015MNRAS.446.2789B]. All these works seem to indicate that a change in the stellar population age is not necessarily the main agent shaping the observed light profiles. Other studies have concentrated on the role of satellite bombardment in shaping light profiles [e.g. @2007ApJ...670..269Y; @2009MNRAS.397.1599Q; @2012MNRAS.420..913B]. For instance, @2007ApJ...670..269Y show that minor mergers can cause type III SB profiles by concentrating mass in the inner regions or expanding the outer discs. However, most of the effort to understand the occurrence of the different SB profiles using simulations has been focused on analysing the effect of stellar radial motions/redistribution . @2009MNRAS.398..591S found that downbending light profiles (type II) arise (together with “U-shape” age profiles) from the combined effect of i) an abrupt change in the radial star formation profile due to a change in the gas volume density profile linked to a warped disc (causing the truncation in the light profile) and ii) radial migration of stars formed in the inner parts towards positions located beyond the break radius. The authors claim that it is the first effect (flaring of the disc) which drives the “U-shape” age and downbending light profiles, while the second effect modifies the final shape of such profiles. However, they also speculate that different SB profiles might arise as a consequence of differences in the efficiency of both processes. In this way, if the distribution of gas volume density changes smoothly with galactocentric distance and the outwards radial redistribution of material is more efficient than in a type II galaxy, one might obtain a type III profile (upbending light profile). The same conditions might be applicable to pure exponential discs if the radial redistribution of material is less efficient than in type III systems. Indeed, U-shaped age profiles can be a natural outcome in analytical models of pure exponentials with radially varying gas infall prescriptions, even in the complete absence of radial stellar (or gas) motions [@BuenosAires]. In a similar line of reasoning, recently found different stellar age and metallicity inner gradients for galaxies displaying type I, II, and III SB profiles. They interpret those results as the outcome of a gradual increase in the radial redistribution efficiency from type II to type I and III galaxies.[^2] Recently, several studies have tried to shed further light onto the effect of radial redistribution of material in shaping SB profiles. @2015MNRAS.448L..99H [@2017MNRAS.470.4941H] investigated the role of the halo spin parameter ($\lambda$) in shaping the outer SB profiles by analysing a set of controlled simulations of isolated galaxies. They found a clear transition from type III systems displaying low spin parameters to type II galaxies showing higher values with type I discs having intermediate values ($\lambda$ $\sim$ 0.035). In particular, they suggested that orbital resonances with a strong central bar, coupled with the low initial halo spin, can produce stellar migration which leads to upbending SB profiles. According to this work, stars populating these outer regions present high radial velocity dispersions and a lower degree of rotation than expected in disc-like systems. found that after 2 Gyr of smooth, in-plane gas accretion, galaxies displaying typical type II profiles acquire light upbendings in the outermost regions. This results in galaxies displaying a combination of an inner type II and an outer type III profiles with the latter being populated by stars that also present high velocity dispersions. revisit the idea that flares in galaxies can give rise to discs following a type II SB distribution. In @2016ApJ...830..115E, the authors present numerical experiments on the effect of stochastic scattering of random particles shaping single exponentials discs. They also report that type II and III profiles could be found if a difference in the scattering bias for the inner and outer regions exists. In this work, we make use of the [RaDES]{} suite of simulated galaxies to investigate two of the processes that can drive the different types of SB profiles: the effect of radial motions of stars and satellite accretion. In §\[simulations\], we outline the simulations employed and characterise the associated galaxies. The main results and discussion are given in §\[result1\] and §\[result2\]; our conclusions are presented in §\[conclusions\]. Simulations and sample of galaxies {#simulations} ================================== The [RaDES]{} galaxies are simulated using the adaptive mesh refinement code [<span style="font-variant:small-caps;">ramses</span>]{} tracking dark matter, stars and gas, on cosmological scales. The hydrodynamical evolution of gas uses a refining grid such that the resolution of the grid evolves to follow overdensities, reaching a peak resolution of 436 pc (16 levels of refinement). In order to prevent numerical collapse, a polytropic equation of state is used for dense gas. If gas is more dense than 0.1 cm$^{-3}$ star formation occurs at a rate given by $\dot{\rho} = -\rho/t_{\star}$, where $t_{\star} = t_0(\rho/\rho_0$)$^{-1/2}$ with $t_0=8$ Gyr. Stellar feedback is delayed to occur $10^7$ years after star formation whereupon it distributes kinetic energy, mass and metals to the gas within a two grid cell radius sphere. The mass fraction of stellar particles that explode as supernovae (SNe) is 10%, with each SN providing 10$^{51}$ erg of energy, and the 10% of non-metals (H and He) are converted to metals. The cosmological parameters employed in generating these realisations were: H$_0$=70 km s$^{-1}$Mpc$^{-1}$, $\Omega_{\mathrm{m}}$=0.28, $\Omega_{\mathrm{\Lambda}}$=0.72, $\Omega_{\mathrm{b}}$=0.045, and $\sigma_8$=0.8.[^3] Two different box volumes sizes of 20 h$^{-1}$ Mpc and 24 h$^{-1}$ Mpc were used. The mass resolution of dark matter particles was either 5.5$\times$10$^{6}$ M$_\odot$ or 9.5$\times$10$^{6}$ M$_\odot$, respectively, for each of aforementioned two volumes. Further details of the halo selection process and the simulation parameters may be found in . The RaDES suite are a powerful tool in assessing the effect of radial motions (including satellite accretion) in shaping SB profiles. In , we showed that these Milky-Way-mass, disc-dominated galaxies, possess characteristics resembling those of observed systems, such as metallicity gradients, matter content (total, dark, stellar, baryonic, and gaseous mass), and rotation curves . Unlike other comparable simulated cosmological samples, the [RaDES]{} galaxies are somewhat unique in possessing the full range of SB profiles . Characterisation of the [RaDES]{} galaxies light distribution {#SB_prof_char} ------------------------------------------------------------- The light profiles that we analyse in this work have been previously shown in . The radial profiles were computed by azimuthally-averaging the light distribution from the mock images presented in . These images were produced using the [SUNRISE]{} [@2006MNRAS.372....2J] code mimicking the SDSS bandpasses. [SUNRISE]{} uses the stellar and gaseous distributions, as well as Spectral Energy Distributions (SEDs) for each composite stellar particle, drawn from the [Starburst99]{} stellar population models [@1999ApJS..123....3L], in order to generate the bandpass-dependent mock images. The SB profiles computed in this way were then fitted with the function presented in [@2008AJ....135...20E equations 5 and 6] with a broken exponential profile implemented. This allows us to characterise the light distribution in these simulated galaxies in a similar fashion as usually done with observed photometric data (see Table \[tab:galaxies\]). In the following analysis we will concentrate on the results using the SDSS $r$-band light profiles, although identical results are found with the other two filters analysed in (SDSS $g$ and $i$-bands). ----------- --------- ------------------ ---------------------- ------------------------ --------- Galaxy SB type $\rm h_{\rm in}$ $\rm h_\mathrm{out}$ $\rm R_\mathrm{break}$ $\beta$ (kpc) (kpc) (kpc) (kpc) Apollo II 2.34 1.39 4.96 0.95 Artemis III 0.79 5.87 5.30 -5.08 Atlas II 4.39 2.47 7.43 1.92 Ben II 5.28 3.36 12.18 1.93 Castor II 5.70 0.96 5.35 4.74 Daphne III 1.26 3.84 10.08 -2.58 Eos III 2.95 8.55 18.32 -5.60 Helios III 1.93 7.76 11.49 -5.83 Hyperion II 4.31 2.77 14.97 1.54 Krios III 2.62 8.45 16.11 -5.83 Leia III 3.81 7.48 21.21 -3.67 Leto III 0.99 4.07 6.59 -3.08 Luke I 5.78 - - 0.00 Oceanus II 8.08 4.18 23.17 3.90 Pollux III 1.25 4.03 8.61 -2.78 Selene II 5.68 2.01 11.98 3.66 Tethys II 4.43 2.12 9.82 2.31 Tyndareus III 1.33 3.39 7.12 -2.06 Zeus I 0.92 - - 0.00 ----------- --------- ------------------ ---------------------- ------------------------ --------- : Main disc properties for the [RaDES]{} galaxies from the analysis of their r-SDSS band light distribution. First column: Galaxy name. Second column: Surface brightness type according to the classification. Third column: Inner disc scale-length in kpc. Fourth column: Outer disc scale-length in kpc. Fifth column: Break radius in kpc. Sixth column: strength of the break ($\beta$) defined as h$_{\rm in}$ - h$_{\rm out}$ in kpc.[]{data-label="tab:galaxies"} A visual morphological classification from the mock images of these simulated galaxies suggests that the [RaDES]{} systems are mainly late-type disc galaxies. Thus, we have decided to compare these results with those found in analysing 1-dimensional SB profiles of a sample of 98 late-type spiral galaxies in a similar way than in this work to ensure that these simulated galaxies display realistic light distributions. Table \[tab:galaxies\] shows the values of the main parameters describing the light profiles for all the [RaDES]{} galaxies. The average values of the inner disc scale-length (h$_{\rm in}$) that we obtain for the [RaDES]{} galaxies for type I, II, and III galaxies are $\sim$ 3.3 $\pm$ 1.2, 5.0 $\pm$ 0.8, and 1.9 $\pm$ 0.5 kpc, respectively. These values are in agreement (within errors) with those found in (2.8 $\pm$ 0.8, 3.8 $\pm$ 1.2, and 1.9 $\pm$ 0.6 kpc for types I, II, and III disc galaxies). Similar claims can be outlined regarding the position of the break. The [ RaDES]{} type II and III galaxies present values of R$_{\rm break}$/h$_{\rm in}$ of 2.2 $\pm$ 0.4 and 6.4 $\pm$ 0.4, respectively while the values from the work are 1.9 $\pm$ 0.6 and 4.9 $\pm$ 0.6. Finally, we also find a good agreement if we consider the outer disc scale-length (h$_{\rm out}$). The values of h$_{\rm out}$ for the [RaDES]{} type II and III galaxies are 2.4 $\pm$ 0.5 and 5.9 $\pm$ 1.0 kpc, respectively, while for the sample those values are 2.1 $\pm$ 0.9 and 3.6 $\pm$ 1.2 kpc, respectively. Greater differences are found for the h$_{\rm out}$ and R$_{\rm break}$/h$_{\rm in}$ values for the type III galaxies. These discrepancies might arise as a consequence of the well known ‘angular momentum problem’ [e.g. @2002NewA....7..155S] causing the over-production of the spheroid component due to enhanced star formation at early epochs in cosmological simulations. Thus, we can claim that, although there are some small discrepancies with the typical observed SB profiles (especially for type III galaxies), the light distributions in the [ RaDES]{} set of galaxies are quite realistic and consistent with observations. In order to establish a continuous parameter to characterise these SB profiles, from downbending to upbending (with pure exponentials as intermediate cases), we define the strength of the break ($\beta$) as h$_{\rm in}$ - h$_{\rm out}$. These $\beta$ values are shown in Table \[tab:galaxies\] as well. We must highlight the continuity of this parameter, with the [RaDES]{} set of galaxies being comprised by galaxies presenting weak and strong type II and III breaks as well as pure exponentials. We should note that all [RaDES]{} type II galaxies also show an outer type III break whose origin will be also analysed in the next section . These type-II+type-III combined profiles have been also found in as a consequence of in-plane gas accretion. Despite the existence of this secondary break, we will focus our SB characterisation on the $\beta$ parameter of the break located closer to the centre (with $\beta$ = 0 for type I galaxies, negative for type III galaxies, and positive for type II galaxies). In the following we will name each analysed system according to the name given in . For further information in each particular galaxy we encourage the reader to check , , and Table \[tab:galaxies\]. Role of radial redistribution in shaping SB profiles {#result1} ==================================================== {width="95.00000%"}\ {width="95.00000%"}\ {width="95.00000%"}\ {width="95.00000%"}\ {width="95.00000%"}\ {width="95.00000%"}\ Several works seem to suggest that the role of radial redistribution of stars could be a decisive agent in shaping the SB profiles of spiral galaxies [@2008ApJ...675L..65R; @2009ApJ...705L.133M; @2009MNRAS.398..591S]. In this study we take advantage of the fact that the [RaDES]{} sample presents realistic type I, II, and III galaxies to assess if different radial redistribution patterns might give rise to the different observed SB profiles. To this aim, we characterise the radial redistribution of stars in the whole [RaDES]{} sample with SB profiles ranging from strong type II breaks ($\beta \sim$ 5, e.g. Castor) to strong type III breaks ($\beta \sim$ -6, e.g. Helios). Such characterisation is based on the comparison of radial profiles of the mass fraction of stars with some specific characteristics according to their birth and current locations (see Figs. \[fractions\_1\] and \[fractions\_2\]). On the left-hand panel ([*a*]{}) of Figs. \[fractions\_1\] and \[fractions\_2\] we show the SB profiles of six representative examples of galaxies displaying different SB profile types and break strengths. The second ([*b*]{}) panel of the same figures shows the radial redistribution of stars in those galaxies; to that aim, this panel show the radial profiles of the mass fraction of different stellar sub-populations according to their birth and current locations. The different sub-populations under analysis are: i) stars that are currently located at given galactocentric distances (denoted by $\rm R$) coming from the inner parts (outward-moving stars, red line); ii) stars that are currently located at different galactocentric distances ($\rm R$) coming from the outer parts (inward-moving stars, blue line); iii) “in situ” stars (stars that have not move from their birth place, green line); iv) stars born there and now located at a larger radius (magenta line); and v) accreted stars (black line). Although some of the above sub-populations definitions are self-explanatory, “in situ” and accreted star require precise definition that could potentially affect our results. We define accreted stars as those whose R$_\mathrm{birth} > $20 kpc and $|\rm z_{\rm birth}| > $ 3 kpc . We consider “in situ” stars as those whose absolute value of R$_\mathrm{birth} - $R$_\mathrm{current}$ is less than 0.2 disc scale-lengths (see Sect. \[SB\_prof\_char\]). In the subsequent analysis we will distinguish, not only among galaxies with different SB profiles (I, II, or III), but we will also consider the fact that [RaDES]{} type II galaxies have a more external upbending profile (type II breaks denoted by the gray dotted-dashed vertical lines while type III breaks by the gray dotted vertical lines). According to Figs. \[fractions\_1\] and \[fractions\_2\] we can claim that all galaxies have similar characteristics (regardless of their SB profile) in the bulge region[^4], delimited by the black vertical dashed line. Roughly speaking (and valid for most of the systems), the very centre is characterised by stars coming from regions at larger radius (blue line) but it is particularly noticeable the amount of stars born in the very centre that have moved outward (magenta line). As we move outward, the amount of “in situ” stars (green line) rises to a peak around the middle of the bulge-dominated region (closer to the centre). From the location of this peak until the boundary between the bulge and the disc, the fraction of stars coming from the inner regions (red line) increases. Accreted stars in this central region do not show a common behaviour in all the galaxies, but display behaviours ranging from domination of accreted stars (e.g. Ben), to those where the amount of accreted stars is negligible (Helios or Selene). It is in the disc-dominated region (beyond the bulge-dominated part of the galaxy) where discrepancies are larger between type I, II, and III galaxies. The region of the disc prior to the break radius for type II galaxies is mainly dominated by stars coming from the inner regions (red line) with the peak located at the break radius or beyond. It is worth noticing that the region where the disc starts to dominate (inner disc, before the break) presents an exodus of stars towards the outer regions (magenta line) that is slightly more evident for galaxies with strong rather than weaker type II breaks. This exodus of stars plays an important role in shaping type II breaks (as we demonstrate in Sect. \[facts\]). Beyond the type II break, the more important stellar sub-populations that we find are accreted stars (black line) and stars from the inner regions (red line), with the former gaining in importance as we move towards weaker breaks (Castor to Ben) and as we move outwards. All type II galaxies from [RaDES]{} are characterised by an excess of light (type III secondary break) in the outer parts. The excess of light in these outermost regions is the result of outward motions (red line) and satellite accretion (black line), i.e. the accumulation of outward and accreted stars. “In situ” stars (green line) and stars from the outer parts (blue line) are found throughout the disc region prior to the type III secondary break (more important for strong type II breaks), with a decrement in the mass fraction of both sub-populations from the position of the type II break outwards. However, there is a change in the behaviour of the “in situ” stars (green line) and stars from outer parts (blue line) for type I and III galaxies with respect to the type II systems. The fraction of stars in these 2 sub-populations is lower across the entire disc (after the bulge-dominated region) for type I and III galaxies than for type II systems (see Sect. \[facts\] where we further quantify this statement). Regarding these two types of galaxies (I and III), the main difference displayed between them is found in the radial distribution of accreted stars (black line) and stars coming from the inner parts (red line). At intermediate radii outward-moving stars (red line) dominate, while at larger galactocentric distances accreted stars are the main component for type I galaxies. The importance of accreted stars is higher for type III galaxies with respect to type I systems, especially if they display strong type III breaks (Helios). While type I and weak type III galaxies display quite extended distributions (across tens of kpcs), Helios (a type III galaxy presenting a strong break) presents a concentration of outward-moving stars right after the bulge dominated region, probably as a consequence of the huge amount of accreted stars in those parts. The middle-right and right-hand panels ([*c*]{} and [*d*]{}) of Figs. \[fractions\_1\] and \[fractions\_2\] show the radial ([*c*]{}) and vertical ([*d*]{}) velocity dispersion profiles for the different subpopulations analysed. In this case, we have decided to add the profiles considering all the stellar particles in the simulation (black dashed lines, current locations). It is clear that the inner and outer velocity dispersion profiles display distinct behaviours. All galaxies (regardless of their SB profiles) exhibit the highest values of the velocity dispersion (radial and vertical) in the centre, followed by a radial decrement along the inner regions, i.e. from the bulge until a location around the main break. In contrast, the outer parts may display either a flattening in the dispersion profiles at larger radii or an upturn depending on the galaxy SB profile. Type II galaxies exhibit outer velocity dispersion profiles for “in situ” stars (green) that are “U-shaped”, with the highest values found in the type III part of the profile and the minimum generally located beyond the type II break but before the type III one. The same behaviour is displayed by outward-moving stars (red) and considering all stellar particles together (dashed black line). Similar “U-shaped” profiles are also displayed by the “in situ” subpopulation in the case of type I galaxies. However, an outer flattening (with a slight increase) is displayed by the outward-moving stars and this shape is also found if we compute the velocity dispersion profiles considering all stellar particles together. On the other hand, outer flattenings or oscillating outer profiles are found for all subpopulations in the case of type III galaxies starting slightly before the location of the type III break. We must note that the reported inner decrement in the velocity dispersion profiles of all subpopulations is steeper for stronger type III breaks. On the other hand, the velocity dispersion profiles displayed by accreted stars is very similar for all galaxies (including types I, II, and III), namely a gradual radial decrement from the highest values (at the centre) to the lowest ones (at around the break location) followed by a flattening or a smoother decrement (in the outskirts). The extension of the region dominated by the gradual decrement for the velocity dispersion of the accreted stars decreases as we move from type II galaxies to type I and III. The fact that the boundary between both regimes (inner and outer behaviour) seems to be located near the break location may suggest a common origin between the light and the velocity dispersion profiles as well as several processes working at once. Interpretation {#scenario} -------------- These findings can be interpreted in such way that radial redistribution of material (regardless of what physically causes it and including accretion) can explain the different SB profiles. In such scenario, the presence and strength ($\beta$) of type II breaks are a direct consequence of the combination of the radial redistribution of stars and accretion. Stars moving outwards produce a deficit of stars in the disc region (right after the bulge-dominated zone) before the break radius (see magenta line) and, as a consequence, large inner disc scale-lengths are found in contrast to the outer part (type II break). As this deficit becomes less important and more stars are found in the region inmediately located after the bulge region, shorter disc scale-lengths are displayed, i.e. weaker breaks (see trend from Castor to Ben). However, the formation of a type II break does not only depends on the distribution of outward-moving stars; it is the complex combination of this distribution and the distribution of accreted stars what finally shapes type II galaxies and determine the position of the break. The region where type II breaks are found and beyond is again dominated by outward-moving stars (solid red line) and accreted stars (black line) that are accumulated in that region. As the fraction of accreted stars is more important at these intermediate galactocentric distances (and also before the type II break) we detect weaker type II breaks. Accreted stars smoothen the light profiles giving rise to weak type II breaks. The secondary type III breaks that are found in type II galaxies are basically formed because of the accumulation of accreted and outward-moving (especially accreted) stars in these outer parts which produces an outer excess of light. Despite the tendency of outward-moving and accreted stars to dominate the mass fraction for all galaxies, it is for type I and III galaxies where this superiority is even more important. The fact that the fraction of inward-moving and “in situ” stars is lower in the case of type I and III galaxies make radial redistribution a more important agent in shaping SB profiles in these galaxies than in type II galaxies. Therefore, it is the mixing induced by outward-moving and, especially, accreted stars what causes type II breaks to form pure exponential profiles. The final profile (I or III) will depend on the relative fraction of outward-moving and accreted stars. As accreted stars start to dominate, especially in the outer parts, type III breaks appear and become stronger. The overpopulation of outward-moving (red line) stars right after the bulge-dominated region found in type III galaxies with strong breaks (Helios) make the inner disc scale-length shorter (i.e. a steeper inner light profile). We must warn the reader here that this interpretation does not necessary establish a time evolution from type II galaxies to type III systems. We are only trying to obtain patterns in the radial redistribution of material that can link the different observed SB profiles. Those patterns suggest that radial redistribution of material can change the SB profile of a galaxy from type II, to I, and finally to type III in favourable conditions. However, this interpretation is qualitative and needs some quantification to be more robust. In Sect. \[facts\] we quantify some of the previous analysis. Facts favouring this interpretation {#facts} ----------------------------------- If the previous qualitative reasoning is correct and radial motions do play a role in shaping SB profiles, then Luke (a galaxy displaying a pure exponential light profile) should display a type III break in the region located between 25 and 35 kpc attending to Fig. \[fractions\_2\] (upper panel), where accreted stars begin to dominate. The lack of such a break in Luke suggest that, although the radial redistribution of material accumulating in the outer parts in this galaxy is also important, there must be some peculiarities swamping out any possible type III breaks. In the following we consider all the systems in [ RaDES]{} and quantify radial redistribution as a function of the break strength ($\beta$), a continuous quantity, to distinguish between SB types, not focusing on individual galaxies as previously done. This will allow us to further investigate the reasons for the appearance of type I galaxies (and other SB profiles) as well as further support our previous interpretation. This analysis of the radial redistribution as a function of $\beta$ allows us to assess the effect of radial redistribution, not only in shaping type I, II, or III SB profiles, but also in shaping the strength of the different breaks. For simplicity, we will focus in two regions that will be named [*inner*]{} region and [*outer*]{} region. The [*inner* ]{} region corresponds to the disc-dominated region before the break (type II or III) or before 3 disc scale-lengths for type I galaxies. The [*outer*]{} region will be the region between the type II break and the type III break for type II galaxies, the region beyond the break for type III galaxies, and the region beyond 3 disc scale-lengths for type I galaxies. We have decided to use 3 disc scale-lengths as the separation radius as it is there where the breaks are usually found in real galaxies . These two regions are key locations in which we can focus the analysis in order to check the validity of the previous interpretation. In the left-hand, top panel of Fig. \[beta\] we represent the mass fraction of accreted stars currently located in the [*outer*]{} region with respect to the total stellar mass in the same region ($\rm \gamma_{acc/tot, out}$) as a function of $\beta$. We find a clear correlation with strong type II galaxies having the lowest values of $\rm \gamma_{acc/tot, out}$ and strong type III galaxies displaying the highest ones. In the right-hand, top panel of Fig. \[beta\] we represent the mass fraction of stars coming from inner radius (outward-moving stars) with respect to the stellar mass of accreted stars in the [*outer*]{} region ($\rm \gamma_{out/acc, out}$) for the [RaDES]{} galaxies. In this case, type II galaxies with strong breaks display the highest values of $\rm \gamma_{out/acc, out}$ while type III galaxies with strong breaks displays the lowest ones. The mass fraction of “in situ” stars (along the entire galaxy) with respect to the total galaxy mass ($\rm \gamma_{insitu/tot}$) is represented in the left-hand, bottom panel. In this case we find a “U-shaped” relation between both magnitudes with type II galaxies displaying the highest values and weak type III systems showing the lowest. Galaxies with strong type III breaks also display a good amount of “in situ” stars (comparable with systems with weak type II breaks). Finally, in the right-hand, bottom panel we show the variation of the mass fraction of stars currently located in the [*inner*]{} region with respect to the mass of stars stars born in the same [*inner*]{} region ($\rm \gamma_{current/birth, in}$) as a function of $\beta$. Type II galaxies exhibit an interesting correlation with some scatter in which strong type II breaks display low values of $\rm \gamma_{current/birth, in}$ and weak type II breaks or even type I systems present higher values. {width="80.00000%"} The four correlations support the proposed scenario in which different degrees of radial redistribution of material give rise to the different observed SB profiles. The correlation found for $\rm \gamma_{acc/tot, out}$ reflects that accretion is more important in type I and III galaxies than in type II galaxies, confirming that accretion can convert type II breaks into pure exponentials or even type III galaxies. The behaviour of $\rm \gamma_{out/acc, out}$ with type II galaxies displaying larger values reflects again that accreted stars are less abundant in the [*outer*]{} region for these galaxies (causing outer disc scale-length to be shorter) and that outward-moving stars accumulate around the type II break and beyond. The shape of the $\rm \gamma_{insitu/tot}$ vs. $\beta$ relation allows us to confirm that the amount of “in situ” stars for type II galaxies is higher than for type I or III galaxies, verifying our interpretation that radial redistribution being less important in type II galaxies. However, the three type III galaxies with the strongest breaks invert this trend and display similar values of weak type II discs. It is the vast amount of accreted stars in these systems (see the upper left-hand panel of Fig. \[beta\]) that causes them to have such strong type III breaks despite the quantity of “in situ” stars. It is the correlation of $\rm \gamma_{current/birth, in}$ with $\beta$ for type II galaxies that leads us to the conclusion that one of the main factors shaping type II breaks is the exodus of stars from [*inner*]{} regions. The lack of stars in the [*inner*]{} region produces a shallower inner light profiles which strengthens the break. The fact that this behaviour is not found for type III galaxies reinforces the interpretation provided in this work. It is worth noticing that the importance of all the processes described is always intermediate for type I galaxies with respect to the other two types. This allows us to conclude that type I galaxies are “boundary” galaxies with properties in between those of type II and type III galaxies and that they exhibit small peculiarities (that need to be further analysed) swamping out any possible breaks. As the reader may have noticed, in the four correlations there is always one type I galaxy slightly off the correlation, Zeus (represented in Fig. \[beta\] with green diamonds). Zeus still has a satellite companion and thus it is not the best example for this analysis. The presence of the companion in Zeus explains its position out of the correlation in all the relations . Coming back to Luke, the lack of a type III break can be explained based on the radial redistribution of material in these outer regions. Although accreted stars dominate with respect to the other subpopulations (black line), it is not as important in absolute terms as for the rest of type III galaxies (see top-right panel of Fig. \[beta\]). In addition, there is a clear negative radial trend of the stars coming from the inner regions that end up at such galactocentric regions (red line). Both facts, along with the exodus of stars located at those external parts and that have moved outwards (magenta line), make that stars do not accumulate in the outskirts preventing any type III break to be observed. On the behaviour of the velocity dispersion radial profiles {#vel_disp} ----------------------------------------------------------- Previous works have found that type III profiles come from a kinematically hot component. @2017MNRAS.470.4941H suggested that a combination of a strong bar with a low initial halo spin is able to make stars migrate outward to form type III breaks that are populated by stars that acquire high velocity dispersions in the migration process. In addition, another way to inhabit the outer parts of type II galaxies with a hot component was found by via smooth accretion of gas. In that work, this accretion leads to the formation of outer type III breaks coexisting with the former downbending profile. The resemblance between their type-II+type-III combined profiles with the ones shown in this work is remarkable. However, while found a clear correlation between the location of this secondary type III break and an upturn in the stellar radial velocity dispersion (see their figure 17), we do not find such clear relation. We obtain that the stellar velocity dispersion profiles of the type II [RaDES]{} galaxies display upturns starting at intermediate locations between the inner type II break and the outer type III one (e.g. Castor) or related with the location of the primary type II break (e.g. Selene and Ben). The discrepancies among both works are understandable considering the differences in the simulation recipes as well as the fact that the analysis do not include stellar accretion. In fact, the location of the upturn in velocity dispersion seems to be related (in all cases) with the position where the mass fraction of accreted stars (see panel b) starts to increase, suggesting that accreted stars have a greater impact on the global velocity dispersion than on the light profile. However, not only the “global” radial and vertical velocity dispersion profiles (considering all particles, including accretion) show this upturn, but also other subpopulations of non-accreted stars (see Figs. \[fractions\_1\] and \[fractions\_2\]). Regarding pure type III galaxies, they do not display this velocity dispersion upturns at all. In contrast, they mainly show flattenings that start generally before the type III breaks, again related with the radial position where the accreted stars begin to dominate. To sum up, we can claim that all the [RaDES]{} galaxies display stellar velocity dispersions in the outer parts that are higher than that expected from the extrapolation of the behaviour in the inner parts (before the regions where the breaks appear), although only type I and II galaxies exhibit clear upturns that might be related with the accretion. Effect of radial redistribution in shaping stellar population age and metallicity profiles {#result2} ========================================================================================== According to the previous section, radial redistribution of stars seems to play an important role in shaping the different observed SB profiles. However, from an observational point of view, this statement is difficult to confirm and might fall into the box of the speculative theories. To solve this issue, other observables have to be analysed in order to provide this hypothesis with predictions that can be tested. Some of the observables that can be computed using modern observing techniques are stellar age and metallicity profiles, especially in the inner regions where the signal is higher. (hereafter TRL17) recently studied such profiles using the spectroscopic data provided by the CALIFA survey . The authors, analysing a sample of 214 spiral galaxies, found that type II systems tend to display steeper stellar age and metallicity profiles than type I and III galaxies, with the latter showing the shallowest ones. They interpret those findings as a consequence of radial redistribution of material being more effective in type III galaxies than in type I or II systems, in agreement with the scenario proposed in this work. In order to compare with those observational results, we compute mass-weighted age and metallicity profiles for all the [RaDES]{} galaxies by averaging all disc particles using 0.5kpc-wide radial bins . Afterwards, we perform error weighted linear fits to those profiles in the inner and the outer regions, separately. We define these inner and outer regions following TRL17 prescription and matching the inner and outer regions defined previously (see Sect. \[facts\]). We decided to perform this analysis only in the inner parts of the profiles as in the analysis presented in TRL17. In Fig. \[rades\_ste\_pop\_grads\] we show the distributions of the inner gradients for the stellar age (left-hand panel) and metallicity (right-hand panel) profiles distinguishing between type I (green), II (blue), and III (red) galaxies over-plotting the results coming from the CALIFA observations (TRL17, symbols with transparency). Table \[tab:av\_gradients\] shows the error-weighted average and dispersion values of the inner gradients of the stellar age and metallicity for the type I, II, and III galaxies from [RaDES]{}. The values of the age and metallicity gradients for individual galaxies from the simulations are consistent with the values presented in TRL17 for the CALIFA sample (see Fig. \[rades\_ste\_pop\_grads\]). The low number of simulated galaxies with respect to the observed ones hampers a complete comparison of both distributions, however, some similarities can be highlighted. Type II galaxies display a larger dispersion in age gradient for the simulations and the observations while in the case of metallicity gradients the dispersions are compatible for the three types of galaxies. Type II galaxies present the steepest average gradients for the age (-0.09 dex/h$_{\rm in}$) and the metallicity (-0.07 dex/h$_{\rm in}^{-1}$). On the other hand, type I systems show the shallowest profiles (0.004 and -0.02 dex/h$_{\rm in}$ for age and metallicity, respectively) with type III systems displaying intermediate values (or even positive values in the case of the age: 0.02 and -0.05 dex/h$_{\rm in}$, for the age and metallicity gradients, respectively). In TRL17, type III systems are found to have the shallowest profiles with a clear trend from type II (steepest) to type I (intermediate) and type III. The differences between the simulated values presented in this work and the observational results from CALIFA are reasonable and easily explained considering the low number of simulated systems analysed. However, the similarities and the consistency between the gradients for individual galaxies are reassuring. {width="80.00000%"}\ -------------------------------------------------------------- -------------- -------------- -------------- Type III Type I Type II [$ \rm \triangledown$]{} log(Age\[yr\]) \[dex/h$_{\rm in}$\] 0.02 (0.03) 0.004 (0.01) -0.09 (0.06) [$ \rm \triangledown$]{} \[M/H\] \[dex/h$_{\rm in}$\] -0.05 (0.03) -0.02 (0.02) -0.07 (0.02) -------------------------------------------------------------- -------------- -------------- -------------- In Fig. \[rades\_ste\_pop\_grads\_beta\] we study the dependency of the stellar age and metallicity gradients displayed by the [RaDES]{} galaxies as a function of the strength of their breaks ($\beta$). If radial redistribution of material does play an important role in shaping SB profiles (see Sect. \[scenario\]) and it also affects age or metallicity gradients of the inner disc (see Fig. \[rades\_ste\_pop\_grads\]), then there should be some kind of correlation between these gradients and the detailed shape of the break. In fact, this is what we show in Fig. \[rades\_ste\_pop\_grads\_beta\]. Type II galaxies with strong breaks (less influenced by radial redistribution, especially accretion) display steeper stellar age and metallicity profiles than type II systems with weak breaks or even type I galaxies. A consistent behaviour is also found in the case of the age profiles for type III galaxies in which systems with weak breaks (less affected by accretion) display steeper profiles than galaxies with strong breaks, in which case, even positive gradients are displayed. However, in the case of the metallicity gradients this trend is inverted. Type III galaxies with weak breaks have shallower profiles than systems with strong breaks (which are more affected by accretion). This inverted trend can be explained by the characteristics of the accreted stars. In we showed that accreted stars populating the [RaDES]{} galaxies are mainly old, metal-poor stars. As a consequence, the accumulation of accreted stars in the outer parts of type III galaxies giving rise to these kind of systems causes the outer parts to be old (positive age gradients) and relatively metal-poor (producing steeper negative profiles). Based on Fig. \[rades\_ste\_pop\_grads\_beta\] we can conclude that, not only does radial redistribution of material affect the shape of observed light distributions, but also that accretion has to play a key role in shaping observed stellar age and metallicity profiles. {width="80.00000%"}\ All these results and the similarities between the observational results (TRL17) and the theoretical work presented in this paper seem to indicate that radial redistribution of material and accretion shape the SB profiles and generally flatten the age and the metallicity profiles with the exception of the metallicity profiles for type III galaxies due to the accumulation of metal-poor, accreted stars in the outer disc. The greater the efficiency of this redistribution, the larger the flattening effect would be. As a consequence, we would expect steeper age and metallicity gradients at birth that what we observe presently due to the effect of the radial redistribution of material. To shed light onto this aspect and the true role of this redistribution in shaping the age and metallicity profiles we have also analysed the inner gradients of these stellar parameters in the absence of radial motions, i.e. considering birth locations (see Fig. \[new\_test\_patri\]). Figure \[new\_test\_patri\] shows the same information as Fig. \[rades\_ste\_pop\_grads\] but focusing on the effect of radial motion of stars on the stellar age and metallicity gradients. In this case we show mean values for type I, II and III systems, not the values for all the individual galaxies. Grey circles represent the mean values of the stellar age and metallicity gradients (averaging among SB types) considering stars located at their current locations. Grey squares represent the same mean quantities but this time computed using stars located at their birth locations (thus avoiding the effect of radial motions). Strikingly, the inner stellar parameter gradients for type I and III galaxies do not seem to be specially affected by radial redistribution of material despite steeper gradients being expected when considering birth locations and shallower ones at the current locations. The case of type II galaxies however shows that the migration of stars flattens the stellar profiles considerably. The fact that, even at birth locations type I and III galaxies seem to display shallow age and metallicity profiles (and definitely shallower than type II galaxies), suggests that the differences found in Fig. \[rades\_ste\_pop\_grads\] must have been imprinted at birth. This points toward the existence of formation mechanisms producing steep stellar age and metallicity profiles for type II galaxies and shallow profiles for type I and III systems. These findings deserve further investigation focusing on the early stages of the formation of these systems that is beyond the scope of this paper, focused on the relation between radial redistribution and the shape of the SB profiles. {width="95.00000%"}\ Conclusions =========== In this work we propose a scenario in which radial redistribution of stars can significantly affect the light distributions shaping observed SB profiles. Redistribution of material, especially outward-moving and accreted stars, is important in all [RaDES]{} galaxies regardless of their SB profile. The accretion and migration of stars particularly prevails in type I and III galaxies while the fraction of “in situ” stars is slightly higher in type II systems. Both aspects suggest that radial redistribution is less significant in systems with downbending light profiles. Outward-moving stars from the region just beyond the bulge produce extended inner light profiles and the accumulation of these and accreted stars determine the presence and strength of type II breaks. An increase in the fraction of accreted stars causes the fading of the type II breaks into pure exponentials and even the shaping of type III breaks whose strength depends on the amount of accreted stars settled in the outer parts. We have proven that type I galaxies present properties that lie between those of type II and type III galaxies. In addition, the stars populating the locations where breaks are found and beyond (where accreted stars begin to dominate) present higher velocity dispersions than that expected by extrapolating the behaviour in the inner parts in agreement with previous works . This scenario seems to leave signatures in the radial distribution of stellar properties such as age and metallicity, especially due to the effect of accreted stars. Shallower inner profiles are found in galaxies where radial redistribution is more important (type I and III) while type II galaxies present steeper inner gradients. These signatures have been observed recently in a sample of nearby galaxies from the CALIFA survey . However, these differences in the inner gradients of stellar population properties can be also found in the absence of stellar radial motion suggesting that they may be imprinted at birth. Further work merging observational analysis with realistic cosmological simulations[^5] in a larger number of systems to improve the statistics is needed to properly understand how the different SB profiles come into shape. We thank the referee for very useful suggestions and comments that have helped improve the current version of this manuscript. This research has been partly supported by the Spanish Ministry of Science and Innovation (MICINN) under grants AYA2014-53506-P and AYA2014-56795-P; and by the Junta de Andalucía (FQM-108). We acknowledge the generous allocation of resources from the Partnership for Advanced Computing in Europe (PRACE) via the DEISA Extreme Computing Initiative (PRACE-3IP Project RI-312763 and PRACE-4IP Project 653838), STFC’s DiRAC Facility (COSMOS: Galactic Archaeology - ST/J005673/1, ST/H008586/1, ST/K00333X/1), and the University of Hull’s High Performance Computing Facility (<span style="font-variant:small-caps;">viper</span>). TRL thanks the support of the Spanish Ministerio de Educación, Cultura y Deporte by means of the FPU fellowship. CGF acknowledges support from the FP7 European Research Council Starting Grant LOCALSTAR. [^1]: <http://califa.caha.es/> [^2]: An interpretation, admittedly, lacking a theoretical framework. [^3]: Being H$_0$ the Hubble constant, $\Omega_{\mathrm{m}}$ the fraction of total matter, $\Omega_{\mathrm{\Lambda}}$ the fraction of the dark energy, and $\sigma_8$ the strength of the primordial density fluctuations [^4]: The bulge region is defined along this work as the inner part where the difference between the observed light profile and the best fit of the disc light is higher than 0.1 mag arcsec$^{-1}$ [^5]: Future analyses will aim to identify star particles more rigosourly accordingly to their observational characteristics, as proposed in @2017arXiv170901523T.

8.75in 6.2in -0.45in -0.8in =cmr10 scaled0 v\#1\#2[V\_[\#1\#2]{}]{} \#1\#2[V\^\*\_[\#1\#2]{}]{} \#1 \#1[ [O]{}(\^\#1)]{} **ELECTROWEAK PENGUINS AND TWO-BODY $B$ DECAYS** *Michael Gronau* *Department of Physics* *Technion – Israel Institute of Technology, Haifa 32000, Israel* and *Oscar F. Hernández[^1] [and]{} David London[^2]* *Laboratoire de Physique Nucléaire* *Université de Montréal, Montréal, PQ, Canada H3C 3J7* and *Jonathan L. Rosner* *Enrico Fermi Institute and Department of Physics* *University of Chicago, Chicago, IL 60637* **ABSTRACT** > We discuss the role of electroweak penguins in $B$ decays to two light pseudoscalar mesons. We confirm that the extraction of the weak phase $\alpha$ through the isospin analysis involving $B\to\pi\pi$ decays is largely unaffected by such operators. However, the methods proposed to obtain weak and strong phases by relating $B\to\pi\pi$, $B\to\pi K$ and $B\to KK$ decays through flavor SU(3) will be invalidated if electroweak penguins are large. We show that, although the introduction of electroweak penguin contributions introduces no new amplitudes of flavor SU(3), there are a number of ways to experimentally measure the size of such effects. Finally, using SU(3) amplitude relations we present a new way of measuring the weak angle $\gamma$ which holds even in the presence of electroweak penguins. **I. INTRODUCTION** The $B$ system is the ideal place to measure the phases of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. The weak phases $\alpha$, $\beta$ and $\gamma$ can be measured in numerous ways through asymmetries and rate measurements of various $B$ decays [@CPreview]. Ultimately it will be possible to verify the relation $\alpha=\pi-\beta-\gamma$, predicted within the Standard Model. The conventional method for obtaining the angle $\alpha$ is through the measurement of the time-dependent rate asymmetry between the process $B^0 \to \pi^+\pi^-$ and its CP-conjugate. This assumes that the decay is dominated by one weak amplitude – the tree diagram. However, there is also a penguin contribution to the decay, which has a different weak phase than the tree diagram. This introduces a theoretical uncertainty into the extraction of $\alpha$. Fortunately, this uncertainty can be removed by the use of isospin [@GL]. The two final-state pions can be in a state with $I=2$ or $I=0$. But the penguin diagram, which is mediated by gluon exchange, contributes only to the $I=0$ final state. Thus, by isolating the $I=2$ component, one can isolate the tree contribution, thereby removing the uncertainty due to the penguin diagrams. This can be done through the use of an isospin triangle relation among the amplitudes for $B^+\to\pi^+\pi^0$, $B^0\to\pi^+\pi^-$ and $B^0\to\pi^0\pi^0$. By measuring the rates for these processes, as well as their CP-conjugate counterparts, it is possible to isolate the $I=2$ component and obtain $\alpha$ with no theoretical uncertainty. The crucial factor in this method is that the $I=2$ amplitude is pure tree and hence has a well-defined CKM phase. Recently, it was proposed that the phases of the CKM matrix could be determined through the measurement of various decay rates of $B$ mesons to pairs of light pseudoscalars [@BPP; @PRL; @PLB]. This was based on two assumptions: (i) a flavor SU(3) symmetry [@DZ; @SW; @Chau] relating $B\to\pi\pi$, $B\to\pi K$ and $B\to KK$ decays, and (ii) the neglect of exchange- and annihilation-type diagrams, which are expected to be small for dynamical reasons. For example, it was suggested that the weak phase $\gamma$ (equal to Arg$(V_{ub}^*)$ in the Wolfenstein parametrization [@LW]), could be found by measuring rates for the decays $B^+ \to \pi^0 K^+$, $B^+ \to \pi^+ K^0$, $B^+ \to \pi^+ \pi^0$, and their charge-conjugate processes [@PRL]. The $\pi K$ final states have both $I=1/2$ and $I=3/2$ components. The key observation is that the gluon-mediated penguin diagram contributes only to the $I=1/2$ final state. Thus, a linear combination of the $B^+ \to \pi^0 K^+$ and $B^+ \to \pi^+ K^0$ amplitudes, corresponding to $I = 3/2$ in the $\pi K$ system, could be related via flavor SU(3) to the purely $I = 2$ amplitude in $B^+ \to \pi^+ \pi^0$, permitting the construction of an amplitude triangle. The difference in the phase of the $B^+ \to \pi^+ \pi^0$ side and that of the corresponding triangle for $B^-$ decays was found to be $2 \gamma$. Taking SU(3) breaking into account, the analysis is unchanged, except that one must include a factor $f_K/f_\pi$ in relating $B\to\pi\pi$ decays to the $B\to\pi K$ decays [@GHLR]. The weak phase $\gamma$ can also be extracted in an [*independent*]{} way, along with the CKM phase $\alpha$ and all the strong final-state phases, by measuring the rates for another set of 7 decays, along with the rates for the charge-conjugate decays [@PLB]. (SU(3)-breaking effects are discussed in [@GHLR].) This method also relies on the SU(3) relation between the $I=3/2$ $\pi K$ amplitude and the $I=2$ $\pi\pi$ amplitude. The crucial ingredient in the above analyses is that the penguin is mediated by gluon exchange. However, there are also electroweak contributions to the processes $b\to sq{\bar q}$ and $b\to dq{\bar q}$, consisting of $\gamma$ and $Z$ penguins and box diagrams. (From here on, we generically refer to all of these as “electroweak penguins.”) Since none of the electroweak gauge bosons is an isosinglet, these diagrams can affect the above arguments. For the $B\to\pi\pi$ isospin analysis, the result is that the $I=2$ state will no longer have a well-defined weak CKM phase. For the $B\to\pi\pi/\pi K$ analyses, in the presence of electroweak penguins there are no longer triangle relations among the $B\to \pi K$ and $B\to\pi\pi$ amplitudes. Theoretical estimates [@DH] have indicated that electroweak penguins are expected to be relatively unimportant for $\pi\pi$. However, they are expected to play a significant role in the $\pi K$ case, introducing considerable uncertainties in the extraction of $\gamma$ as described above. The purpose of the present paper is to examine the role of electroweak penguins in all $B\to PP$ decays, where $P$ denotes a light pseudoscalar meson. We wish to address the following questions: \(1) To what $B$ decays do electroweak penguins contribute? \(2) Can one obtain information on their magnitude directly from the data? \(3) Can one extract weak CKM phases in the presence of electroweak penguins? We answer the first question by including the electroweak penguin contributions in a general graphical description of all $B\to PP$ amplitudes, which was shown to be a useful representation of flavor SU(3) amplitudes [@BPP]. The second question is answered in the affirmative. An explicit calculation of electroweak penguins [@DHT] suggests that they could dominate in decays of the form $B_s \to (\phi~{\rm or}~\eta) + (\pi~{\rm or}~\rho)$. We find that there are additional measurements which are indirectly sensitive to such contributions. As to the third question, we find that it is indeed possible to obtain information about the CKM angle $\gamma$, even in the presence of electroweak penguins. While the method proposed makes use of a considerably larger number of measurements than the original simple set proposed in [@BPP; @PRL; @PLB], there is no difficulty [*in principle*]{} in obtaining the necessary information from experiment alone. Whether these measurements are feasible [*in practice*]{} in the near term is another story, which we shall address as well. The four amplitudes for different charge states in $B \to \pi K$ decays satisfy a quadrangle relation dictated entirely by isospin. When sides are chosen in an appropriate order, we find that one diagonal of the quadrangle is related to the rate for $B_s \to \pi^0 \eta$, so that (up to discrete ambiguities) the quadrangle is of well-defined shape. The difference between the other diagonal and the corresponding quantity for charge-conjugate processes, when combined with the rate for $B^+ \to \pi^+ \pi^0$, provides information on $\sin \gamma$. We discuss general aspects of electroweak penguins in Sec. II, with particular emphasis on estimates of the size of such effects. In Sec. III we examine the electroweak penguin contributions to $B\to PP$ decays. The quadrangle for $B \to \pi K$ decays is treated in Sec. IV. Experimental prospects are noted in Sec. V, while Sec. VI summarizes. **II. ELECTROWEAK PENGUINS: GENERAL CONSIDERATIONS** The standard penguin diagram involves a charge-preserving, flavor-changing transition of a heavy quark to a lighter one by means of a loop diagram involving a virtual $W$ and quarks, and emission of one or more gluons. The penguin diagrams involving $\bar b \to \bar d$ transitions change isospin by 1/2 unit, while $\bar b \to \bar s$ transitions leave isospin invariant. Penguin diagrams in which the $\bar b \to \bar q$ system is coupled to other quarks through the photon or $Z$ (or through box diagrams involving $W$’s) instead of through gluons have more complicated isospin properties. There will be contributions in which the additional quark pair is isoscalar (as in the conventional penguin graphs), but others in which it is isovector. The importance of electroweak penguin (EWP) diagrams was realized in the calculation of the parameter $\epsilon'/\epsilon$ describing direct CP violation in $K_L \to \pi \pi$ [@KEWP]. That parameter requires an imaginary part of the ratio $A_2/A_0$, where the subscript denotes the isospin $I_{\pi \pi}$ of the $\pi \pi$ system. The EWP can provide an $I_{\pi \pi} = 2$ contribution, whereas the conventional penguin cannot. The numerical importance of the EWP diagram involving $Z$ exchange is enhanced by a factor of $m_t^2/M_Z^2$ [@GSW]. A similar circumstance was realized by Deshpande and He [@DH] to apply to two cases: (a) An isospin triangle for $B \to \pi \pi$ decays, while continuing to hold, receives small contributions from electroweak penguins. This can in principle affect the analysis proposed in [@GL] for extracting the weak phase $\alpha$. (b) The validity of the SU(3) triangle proposed in [@BPP; @PRL; @PLB], involving the comparison of $B \to \pi \pi$ and $B \to \pi K$ decays, is also affected. The dominant electroweak penguin contribution arises from $Z$ exchange. There are two such diagrams, shown in Fig. \[Zpenguin\]. The distinction between the two is that the diagram of Fig. \[Zpenguin\](a) is color-allowed, while that of Fig. \[Zpenguin\](b) is color-suppressed. We refer to these as $\pew$ and $\pewc$, respectively. Thus, EWP effects will be most important when the $\pew$ diagram is involved, that is, when there is a nonstrange neutral particle in the final state, such as $\pi^0$, $\eta$, $\rho^0$ or $\phi$. All-charged final states will be less affected by the presence of electroweak penguins, since in this case only the $\pewc$ diagram can arise. EWP diagrams which involve the annihilation of the quarks in the initial $B$ meson are suppressed by a factor of $f_B/m_B \approx 5\%$. As we will see from the hierarchy of diagrams discussed in the next section, this means that we will always be able to ignore annihilation-type EWP diagrams. The ratio of a $\pew$ electroweak penguin to a gluonic penguin contribution $P$ in $b$ quark decays contains a factor of $\alpha_2/\alpha_s \approx (1/30)/0.2 \approx 1/6$, where we have evaluated both couplings at $m_b$. The electroweak penguin for $Z$ exchange contains a factor of $(m_t/M_Z)^2 \approx 4$ in contrast to a logarithm $\ln(m_t^2/m_c^2) \approx 9$ in the gluonic penguin. Thus, the overall electroweak penguin’s amplitude should be ${\cal O}(10\%)$ that of the gluonic penguin, modulo group-theoretic factors. This is in qualitative accord with the result of [@DH]. A more quantitative calculation of the ratio $\pew/P$ will necessarily involve hadronic physics. In particular, the matrix elements for $\pew$ and $P$ are almost certainly different, since the two diagrams clearly have different dynamical structures. Such model-dependent calculations are fraught with uncertainties [@uncertainties]. (For example, although it might be argued that factorization applies to the $\pew$ diagram, it is considerably more doubtful for $P$.) Thus, theoretical calculations of $\pew/P$ [@DH] should be viewed with a certain amount of skepticism. Still, the magnitude of this ratio is very important. As we will see in the following sections, the methods presented in [@BPP; @PRL; @PLB] for the extraction of weak and strong phases will be invalidated if EWP’s are too large, say $\pew/P \gsim 20\%$. For these reasons it is important to try to obtain information about electroweak penguins from the data. There are, of course, other diagrams which contribute to $B\to PP$ decays, and it is equally important to estimate the size of electroweak penguins relative to these other contributions. Excluding electroweak penguins, there are six distinct diagrams which contribute to $B$ decays: (1) a (color-favored) “tree” amplitude $T$, $T'$; (2) a “color-suppressed” amplitude $C$, $C'$; (3) a “penguin” amplitude $P$, $P'$; (4) an “exchange” amplitude $E$, $E'$; (5) an “annihilation” amplitude $A$, $A'$; (6) a “penguin annihilation” amplitude $PA$, $PA'$. (We refer the reader to Ref. [@BPP] or [@GHLR] for a more complete discussion of the diagrams.) For $T$, $C$, $E$ and $A$, the unprimed and primed amplitudes contribute to the decays $\bar b \to \bar u u \bar d$ and $\bar b \to \bar u u \bar s$, respectively, and the primed amplitudes are related to their unprimed counterparts by a factor of $|V_{us}/V_{ud}| \simeq \lambda = 0.22$. For $P$ and $PA$ the unprimed and primed amplitudes contribute to the decays $\bar b \to \bar d$ and $\bar b \to \bar s$, respectively. In this case, the primed amplitudes are actually [*larger*]{} than the unprimed amplitudes by a factor of $|V_{ts}/V_{td}|$, which is of order $1/\lambda$. In Ref. [@GHLR] we estimated the relative sizes of these diagrams in $B\to PP$ decays. Here we include electroweak penguins, justifying our estimates of their magnitudes after presenting the expected hierarchies. [*1. $\bar b \to \bar u u \bar d$ and $\bar b \to \bar d$ transitions:*]{} The dominant diagram is $T$. Relative to the dominant contribution, we expect $$\begin{aligned} \label{buudhierarchy} 1 & : & |T|, \nonumber \\ {\cal O}(\lambda) & : & |C|,~|P|, \nonumber \\ {\cal O}(\lambda^2) & : & |E|,~|A|,~|\pew| \nonumber \\ {\cal O}(\lambda^3) & : & |PA|,~|P_{EW}^{\prime C}|.\end{aligned}$$ [*2. $\bar b \to \bar u u \bar s$ and $\bar b \to \bar s$ transitions:*]{} Here the dominant diagram is $P'$. Relative to this, we estimate $$\begin{aligned} \label{buushierarchy} 1 & : & |P'|, \nonumber \\ {\cal O}(\lambda) & : & |T'|,~|\pew'|\nonumber \\ {\cal O}(\lambda^2) & : & |C'|,~|PA'|,~|P_{EW}^{\prime C}| \nonumber \\ {\cal O}(\lambda^3) & : & |E'|,~|A'|.\end{aligned}$$ The use of the parameter $\lambda = 0.22$ here is unrelated to CKM matrix elements – it is simply used as a measure of the approximate relative sizes of the various contributions. For instance, $|C/T| \sim \lambda$ is due to color suppression, while $E$ and $A$ are suppressed relative to $T$ by the factor $f_B/m_B \approx 0.05 \sim \lambda^2$. Similarly, $PA/P \sim f_B/m_B$. Although it is fairly certain that $P'$ dominates the second class of decays, the value of the ratio $|T'/P'|$ is less clear. Our value of $\lambda$ for this ratio is probably a reasonable estimate. Finally as discussed in Ref. [@GHLR], we expect the SU(3) corrections to a diagram to be roughly 20% ($\sim \lambda$) of that particular diagram. We shall discuss SU(3)-breaking effects in the cases of several specific processes of interest in Sections III and IV. Note that both of the above hierarchies are educated guesses – it is important not to take them too literally. Since $\lambda$ is not that small a number, a modest enhancement or suppression (due to hadronic matrix elements, for example) can turn an effect of ${\cal O}(\lambda^n)$ into an effect of ${\cal O}(\lambda^{n\pm 1})$. Ultimately experiment will tell us exactly how large the various diagrams are. Some combination of the decays $B^0 \to \pi^+\pi^-$ and $B^0 \to K^+\pi^-$ has been observed [@Kpisep]. The most likely branching ratios for these two modes are both about $10^{-5}$ (though all that can be conclusively said is that their sum is about $2 \times 10^{-5}$). One then concludes that the $T$ and $P'$ amplitudes are about the same size. In this case, the estimated hierarchies in Eqs. (\[buudhierarchy\]) and (\[buushierarchy\]) can be combined. The above estimated hierarchies can be used to judge how large electroweak penguin effects should be. Our naive estimate of $\pew/P$ was ${\cal O}(10\%)$. Allowing for some variation in either direction, we have $\pew/P \sim {\cal O}(\lambda)$ – ${\cal O}(\lambda^2)$. Thus, for $\bar b \to \bar u u \bar d$/$\bar b \to \bar d$ decays, EWP’s are at most ${\cal O}(\lambda^2)$ of the dominant $T$ contribution. For this reason it is unlikely that electroweak penguins will significantly affect $B\to\pi\pi$ decays. On the other hand, for $\bar b \to \bar u u \bar s$/$\bar b \to \bar s$ decays, EWP contributions can be as much as ${\cal O}(\lambda)$ of the dominant $P'$ diagram, which is why they may be important in $B\to\pi K$ decays. As discussed in the previous section the color-suppressed electroweak penguin $\pewc$ should be smaller than its color-allowed counterpart $\pew$ by approximately a factor of $\lambda$. Thus this contribution is probably completely negligible in $\bar b \to \bar u u \bar d$/$\bar b \to \bar d$ decays, and is at most a 5% effect in $\bar b \to \bar u u \bar s$/$\bar b \to \bar s$ decays relative to the dominant $P'$ contribution. **III. $B\to PP$ DECAYS** We review briefly the SU(3) discussion of [@BPP]. The weak Hamiltonian operators associated with the transitions $\bar b \to \bar q u \bar u$ and $\bar b \to \bar q$ ($q = d$ or $s$) transform as a ${\bf 3^*}$, ${\bf 6}$, or ${\bf 15^*}$ of SU(3). These combine with the triplet light quark in the $B$ meson and couple to a symmetric product of two octets (the pseudoscalar mesons) in the final state, leading to decays characterized by one singlet, three octets, and one ${\bf 27}$-plet amplitude. Separate amplitudes apply to the cases of strangeness-preserving and strangeness-changing transitions. The diagrams $T$–$PA$ are a useful representation of flavor SU(3) amplitudes. Although there are 6 types of diagram (excluding electroweak penguins), they only appear in 5 linear combinations in $B\to PP$ decays, in accord with the group theory result. The inclusion of electroweak penguins does not affect this picture. The ratio of transitions $\bar b \to \bar q u \bar u$, $\bar b \to \bar q d \bar d$, and $\bar b \to \bar q s \bar s$ is altered, but the $\bar b \to \bar q d \bar d$ and $\bar b \to \bar q s \bar s$ terms remain equal. (This is obvious for the $\gamma$- and $Z$-penguins. For the box diagrams, this equality is ensured by the GIM mechanism. There are contributions from the boxes which break this equality, but they are much suppressed relative to the dominant term.) The weak Hamiltonian thus continues to contain terms transforming as a ${\bf 3^*}$, ${\bf 6}$, or ${\bf 15^*}$ of SU(3), but in different proportions. Thus, even if one includes electroweak penguin graphs, there must continue to be five independent amplitudes describing $\Delta S = 0$ decays and five other amplitudes describing $|\Delta S| = 1$ decays. However, some of the correspondence between $\Delta S = 0$ and $|\Delta S| = 1$ decays present in the previous description will be altered. In this section we extend the decomposition of $B\to PP$ decays in terms of the diagrams $T$–$PA$ to include the electroweak penguin diagrams of Fig. \[Zpenguin\]. In this way we see explicitly how $B\to\pi\pi$ and $B\to\pi K$ decays are affected by electroweak penguins. In [@BPP] it was argued that the diagrams $E$, $A$ and $PA$ (and their primed counterparts) are negligible since they are suppressed by a factor of $f_B/m_B = {\cal O} (\lambda^2)$ and hence are unlikely to be important in many cases. However, there are processes such as $B^0\to \pi^0\pi^0$, $B^+\to K^+\bar K^0$ and $B_s\to \pi^0\bar K^0$ which are dominated by the $\olambdai$ terms $C$ and/or $P$. In these cases diagrams suppressed by $\olambda 2$ with respect to the dominant $T$ contributions, such as $E,A$ and $\pew$, can cause a significant change in the rate. There are situations, which we will soon discuss, when one cannot neglect such seemingly small diagrams. These are precisely the cases where EWP’s are important. We continue to use the approximation of ignoring $E$, $A$ and $PA$-type diagrams when considering electroweak penguin effects as long as their effects are $\olambda2$ with respect to the dominant contribution to a process. Annihilation-type electroweak penguin amplitudes will always be subdominant by at least $\olambda2$ in all the processes we will consider and hence we can ignore them. In $|\Delta S = 1|$ decays, the $C'$ contribution should really be dropped, since it is expected to be of the same order as the $PA'$ diagram, which has been neglected. Nevertheless, we continue to keep track of the $C'$ contribution in such decays, since it is related to the non-negligible $C$ diagram in $\Delta S = 0$ decays. (Obviously our results should not, and do not, depend on keeping or ignoring the $C'$ contribution.) The distinction between the gluonic penguin $P$ and the electroweak penguin $\pew$ is the coupling to the light quarks. In $P$, the quarks $u$, $d$ and $s$ have equal couplings to the gluon. In $\pew$, however, the $u$ and $d$/$s$ quarks are treated differently. Schematically, we can represent the couplings of the strong and electroweak penguins as follows: $$\begin{aligned} \label{cucd} P & : & u{\bar u} + d{\bar d} + s{\bar s} ~~~, \nonumber \\ \pew,~\pewc & : & c_u \, u{\bar u} + c_d (d{\bar d} + s{\bar s})~~~.\end{aligned}$$ Although the precise values of $c_u$ and $c_d$ depend on the detailed structure of the electroweak penguin, they are taken to be numbers of order 1. For example, if the electroweak penguin coupled to the charge of the quarks (as it would if it arose purely from photon exchange), we would have $c_u=2/3$ and $c_d=-1/3$. In Tables \[BPPtablei\] and \[BPPtableii\] we present the decomposition of the 13 $B\to PP$ decays in terms of the various diagrams, for $P = \pi$ or $K$. We warn the reader that non-negligible SU(3)-breaking corrections can lead to differences in certain decays that appear equal in the above Tables. For example, according to Table 2, $B^+\to\pi^+K^0$ and $B_s \to K^0 \bar K^0$ will have the same rate. However, SU(3)-breaking effects introduce a rate difference here. We refer the reader to Ref. [@GHLR] for more details. We shall, however, correctly include SU(3)-breaking effects when discussing specific examples in the following sections. There are several interesting aspects of Tables \[BPPtablei\] and \[BPPtableii\] worth mentioning. ---------- ------------------ ------------------ ---------------------------------------- -- -- -- -- Final $T$,$C$,$P$ Electroweak state contributions Penguins $B^+\to$ $\pi^+\pi^0$ ${ -(T+C)/\s}$ $-[ (c_u-c_d)\pew +(c_u-c_d)\pewc]/\s$ $K^+ \bar K^0$ $P+A$ $c_d\pewc$ $B^0\to$ $\pi^+\pi^-$ ${ -(T+P)} $ $- c_u\pewc $ $\pi^0\pi^0$ ${ -(C-P-E)/\s}$ $-[{ (c_u-c_d)\pew} + c_d\pewc]/\s$ $K^0 \bar K^0$ ${ P}$ $ c_d\pewc$ $B_s\to$ $\pi^+K^-$ ${ -(T+P)}$ $-c_u\pewc$ $\pi^0 \bar K^0$ ${ -(C-P)/\s}$ $-[{ (c_u-c_d)\pew} - c_d\pewc]/\s$ ---------- ------------------ ------------------ ---------------------------------------- -- -- -- -- : Decomposition of $B\to PP$ amplitudes for $\Delta C= \Delta S=0$ transitions in terms of graphical contributions of Refs. [@BPP], [@GHLR] and Fig. 1. For completeness we include color-suppressed $\pewc$ contributions even when they are estimated to be negligible. \[BPPtablei\] ---------- ---------------- --------------------- --------------------------------------------------- -- -- -- -- Final $P'$,$T'$,$C'$ Electroweak state contributions Penguins $B^+\to$ $\pi^+K^0$ ${ P'}$ $ c_d P_{EW}^{\prime C}$ $\pi^0K^+$ $-({ P'+T'}+C')/\s$ $-[{ (c_u-c_d)\pew'}+c_u P_{EW}^{\prime C}]/\s$ $B^0\to$ $\pi^-K^+$ ${ -(P'+T')}$ $- c_u P_{EW}^{\prime C}$ $\pi^0K^0$ $-({ P'}-C')/\s$ $-[{ (c_u-c_d)\pew'} - c_d P_{EW}^{\prime C}]/\s$ $B_s\to$ $K^+K^-$ ${ -(P'+T')}$ $-c_u P_{EW}^{\prime C}$ $K^0 \bar K^0$ ${ P'}$ $c_d P_{EW}^{\prime C}$ ---------- ---------------- --------------------- --------------------------------------------------- -- -- -- -- : Decomposition of $B \to PP$ amplitudes for $\Delta C = 0,~ |\Delta S| = 1$ transitions in terms of graphical contributions of Ref. [@BPP], [@GHLR] and Fig. 1. For completeness we include $C'$ and the color-suppressed $P_{EW}^{\prime C}$ contributions even though they are estimated to be negligible. \[BPPtableii\] [*1. $B\to \pi\pi$ decays:*]{} Consider the $B\to\pi\pi$ decays in Table \[BPPtablei\]. The decay $B^+\to\pi^+\pi^0$, which is purely $I=2$, has an electroweak penguin component. If our estimated hierarchy is accurate, this component should be between ${\cal O}(\lambda^2)$ and ${\cal O}(\lambda^3)$ of the dominant $T$ contribution. This is in agreement with Deshpande and He [@DH], who find that $|A_{EWP}/A_T| \approx 1.6\%\, |V_{td}/V_{ub}|$ for this decay. In other words, the EWP contribution to $B^+\to\pi^+\pi^0$ is very small. It is even smaller in the decay $B^0\to\pi^+\pi^-$, since only the color-suppressed EWP can contribute here. On the other hand, electroweak penguins can be more significant in $B^0\to\pi^0\pi^0$ decays, since this decay suffers color suppression. The size of EWP’s is relevant to the extraction of $\alpha$ via the analysis proposed in [@GL]. Let us study this effect in detail. This analysis requires measuring the (time-integrated) rates of $B^+\to\pi^+\pi^0$, $B^0\to\pi^+\pi^-$, $B^0\to\pi^0\pi^0$ and their CP-conjugate counterparts, and observing the time-dependence of $B^0(t)\to\pi^+\pi^-$. The amplitudes of these six processes form two triangles, as shown in Fig. \[isoanalysis\], in which the CP-conjugate amplitudes have been rotated by a common phase $\tilde{A}(\bar B\to\pi\pi)\equiv {\rm exp}(2i\gamma)A(\bar B\to\pi\pi)$ (and similarly for $\tilde{P}_{EW}$ and $\tilde{P}_{EW}^C$). The CKM phase $\alpha$ is measured from the time-dependent rate of $B^0(t)\to\pi^+\pi^-$, which involves a term (2+)(mt) , where $\Delta m$ is the neutral $B$ mass difference. The angle $\theta$ is measured as shown in Fig. \[isoanalysis\]. The effect of the EWP amplitudes on determining $\theta$ and correspondingly fixing $\alpha$ is rather clearly represented by the small vectors at the right bottom corner of the Fig. \[isoanalysis\]. These terms, given by $(c_u-c_d)(P_{EW}+P_{EW}^C)$ and its CP-conjugate, have unknown phases relative to the $T+C$ term which dominates $A(B^+\to\pi^+\pi^0)$ and its charge-conjugate. This leads to a very small uncertainty in the relative orientation of the two triangles. \[In the limit of neglecting EWP amplitudes, one would have $\tilde{A}(B^-\to\pi^-\pi^0)=A(B^+\to\pi^+\pi^0)$\]. The uncertainty in measuring $\theta$, and consequently in determining $\alpha$, is given by  . We therefore conclude that the effects of EWP amplitudes on the measurement of $\alpha$ are at most of order $\lambda^2$ and are negligible. Since a different conclusion has been claimed in [@DH; @Desh], let us clarify the apparent disagreement. The authors of [@DH; @Desh] have only shown that the error in determining $\alpha$ from the rate of $B^0(t)\to\pi^+\pi^-$ is large. This is dominantly the effect of the gluonic penguin, as already noted in [@MG]. They have not separated the effect of EWP amplitudes. Fig. \[isoanalysis\] shows clearly how small this effect is. [*2. $B\to \pi K$ decays:*]{} We now turn to the $B\to\pi K$ decays in Table \[BPPtableii\]. In the absence of electroweak penguins, one can write two triangle relations involving amplitudes in both the $\Delta S = 0$ and $|\Delta S| = 1$ sectors: $$\begin{aligned} \label{tri-rel} \s A(B^0 \to \pi^0 K^0) + A(B^0 \to \pi^- K^+) &=& \lambda \s A(B^+ \to \pi^+ \pi^0) \nonumber \\ - (C'-P') - (P'+T') & = & - \lambda (T + C)\end{aligned}$$ $$\begin{aligned} \label{PRLtriangle} \s A(B^+\to\pi^0 K^+) + A(B^+\to\pi^+ K^0) & = & \lambda \s A(B^+\to\pi^+\pi^0) \nonumber \\ - (T'+C'+P') + (P') & = & - \lambda (T + C)\end{aligned}$$ SU(3) breaking can be taken into account by including a factor of $f_K/f_\pi$ on the right-hand side [@GHLR]. In Eq. (\[PRLtriangle\]) above, SU(3) relates the $I=3/2$ $\pi K$ amplitude to the $I=2$ $\pi\pi$ amplitude. By measuring the three rates involved in the triangle relation, as well as their CP-conjugates, the weak CKM angle $\gamma= {\rm Arg}(V_{ub}^*)$, which is the weak phase of $A(B^+\to\pi^+\pi^0)$, can be extracted [@PRL]. By using both Eqs. (\[tri-rel\]) and (\[PRLtriangle\]), strong final-state phases and the sizes of the different diagrams can also be extracted [@PLB]. When electroweak penguins are included, however, these two triangle relations no longer hold. For example, the left-hand side of Eq. (\[PRLtriangle\]) is now equal to -\[T’+C’+(c\_u-c\_d)(’+P\_[EW]{}\^[C]{})\], while the right-hand side is - . Despite their similarity, these two expressions are not equal since the relation between non-penguin contributions ($T'/T = C'/C = \lambda$) does not hold for the electroweak penguins: $|\pew'/\pew| = |V_{ts}/V_{td}| \sim 1/\lambda$. This relation would only hold if $c_u$ were equal to $c_d$, which cannot happen since EWP’s are not isosinglets. From our previous discussion, we estimate that $|\pew'/T'|$ may be as much as $\sim 1$. Eventually, it will be up to experiment to determine the size of electroweak penguins. However, in a realistic scenario, with hierarchies such as those discussed Sec. II B, EWP’s lead to large uncertainties in the extraction of weak CKM angles and strong phases through the analyses of Refs. [@PRL; @PLB]. In Sec. IV we extend the SU(3) triangle analysis of Ref. [@PRL] to a quadrangle relation, using more decay rate measurements to exhibit a new way of measuring the weak angle $\gamma$ which holds even in the presence of electroweak penguins. As discussed above, the fate of the analyses of Refs. [@PRL; @PLB] for extracting weak CKM phase information depends crucially on the size of electroweak penguins. Rather than relying on theoretical calculations, which inevitably have uncertainties due to hadronic matrix elements, it would be preferable to obtain this information from experiment. Electroweak penguins are expected to dominate decays of the form $B_s \to (\phi~{\rm or}~\eta) + (\pi~{\rm or}~\rho)$ [@DHT]. This is easy to understand in terms of diagrams: \[ewpdominant\] A\[B\_s ( [or]{} ) + ( [or]{} )\] \~-C’ + E’ - (c\_u-c\_d) ’  . We have already argued that the $E'$ diagram is small, so, from Eq. (\[buushierarchy\]) and the discussion following it, we see that the dominant contribution is $\pew'$. Unfortunately, even though these decays are dominated by electroweak penguins, their branching ratios are all small, less than $O(10^{-6})$. Furthermore, they all involve the decays of $B_s$ mesons, which are not as accessible experimentally. This leads to the obvious question: are there signals for electroweak penguins which involve decays of $B^\pm$ or $B^0$ mesons, and which have large branching ratios? Indeed there are. Consider the decays $B^+ \to \pi^0 K^+$ and $B^0\to\pi^- K^+$. From Table \[BPPtableii\], we have A(B\^+ \^0 K\^+) -\[T’+P’+(c\_u-c\_d) ’ \]   ,    A(B\^0\^- K\^+) -\[T’+P’\]    , where we have dropped the (much smaller) terms $C'$ and $P_{EW}^{\prime C}$. Both of these decays should have branching ratios of ${\cal O}(10^{-5})$ as a result of the dominant $P'$ contribution. A difference in the branching ratios of these decays can only be due to the presence of electroweak penguins. Though indirect, this is very likely to be the first experimental test of such effects. Similarly, the most likely source of a difference in the branching ratios of $B^0\to\pi^0 K^0$ and $B^+ \to \pi^+ K^0$ will be the contribution of electroweak penguins. **IV. AMPLITUDE QUADRANGLES** The decays $B \to \pi K$ involve a weak Hamiltonian with both $I = 0$ and $I = 1$ terms. The $I=0$ piece can lead only to a $\pi K$ final state with $I = 1/2$, while the $I = 1$ piece can lead to both $I = 1/2$ and $I = 3/2$ final states. Thus, there are two decay amplitudes leading to $I_{\pi K} = 1/2$ and one leading to $I_{\pi K} = 3/2$. Since there are four amplitudes for $B \to \pi K$ decays, they satisfy a quadrangle, which we may write as [@NQ; @GQ] A(B\^+ \^+ K\^0) + A(B\^+ \^0 K\^+) = A(B\^0 \^0 K\^0) + A(B\^0 \^- K\^+) = A\_[3/2]{}   . \[eqn:quad\] With the phase conventions adopted in [@BPP], the quadrangle has the shape shown in Fig. \[piKquad\], with two short diagonals. These diagonals are: D\_1 & = & -\[T’+C’+(c\_u-c\_d)(’+P\_[EW]{}\^[C]{})\]    ,\ D\_2 & = & -C’ -(c\_u-c\_d) ’ - A’    . The first of these diagonals, $D_1$, is just the amplitude $A_{3/2}$. The key point is that $A(B_s \to \pi^0 \eta) = -[C' +(c_u-c_d) \pew' - E']/\sqrt{3}$, for an octet $\eta$. Thus, ignoring the very small $E'$ and $A'$ diagrams, the second diagonal, $D_2$, is in fact equal to $\sqrt{3} A(B_s \to \pi^0 \eta)$. Therefore the shape of the quadrangle is uniquely determined, up to possible discrete ambiguities. The case of octet-singlet mixtures in the $\eta$ simply requires us to replace the $\sqrt{3}$ by the appropriate coefficient [@GK], since one can show that the singlet piece of $\eta$ does not contribute appreciably here. The quadrangle has been written in such a way as to illustrate the fact, noted in Refs. [@BPP; @PRL; @PLB], that the $B^+ \to \pi^+ K^0$ amplitude receives only penguin contributions in the absence of ${\cal O}(f_B/m_B)$ corrections. The weak phases of both gluonic and electroweak $\bar b \to \bar s$ penguins, which are dominated by a top quark in the loop, are expected to be $\pi$. We have oriented the quadrangle to subtract out the corresponding strong phase. The $I = 3/2$ amplitude is composed of two parts: A\_[3/2]{} = |A\^T\_[K]{}| e\^[i ]{} e\^[i \_T]{} - |A\^[EWP]{}\_[K]{}| e\^[i \_[EWP]{}]{}   , where we have explicitly exhibited electroweak and final-state phases, and the tildes denote differences with respect to the strong phase shift in the $B^+ \to \pi^+ K^0$ amplitude. The corresponding charge-conjugate quadrangle has one diagonal equal to |A\_[3/2]{} = |A\^T\_[K]{}| e\^[- i ]{} e\^[i \_T]{} - |A\^[EWP]{}\_[K]{}| e\^[i \_[EWP]{}]{}   , so that one can take the difference to eliminate the electroweak penguin contribution: \[eqn:diff\] A\_[3/2]{} - |A\_[3/2]{} = |A\^T\_[K]{}| 2 i e\^[i \_T]{}    . In diagrammatic language, the quantity $|A^T_{\pi K}|$ is just $|T'+C'|$. But this can be related to the $I = 2$ $\pi \pi$ amplitude in order to obtain $\sin \gamma$. Specifically, if we neglect electroweak penguin effects in $B^+ \to \pi^+ \pi^0$ (a good approximation, as noted in Sec. IIIB), we find that \[eqn:pipi\] |A\^T\_[K]{}| = (f\_K/f\_) |A(B\^+ \^+ \^0)|   . Thus, we can extract not only $\sin \gamma$, but also a strong phase shift difference $\tilde{\delta}_T$, by comparing (\[eqn:diff\]) and (\[eqn:pipi\]). Of course, if such a strong phase shift difference exists, the $B$ and $\bar B$ quadrangles will necessarily have different shapes, and CP violation in the $B$ system will already have been demonstrated. We should remark that the quadrangle construction for $B \to \pi K$ decays introduced in [@NQ] and refined in [@GQ] assumed the presence of a single weak phase in the amplitude $A_{3/2}$, and no longer is valid in the presence of electroweak penguins. The analysis presented above relies on the equality of two small amplitudes – the diagonal $D_2$ of the $\pi K$ quadrangle and the decay amplitude $\sqrt{3} A(B_s \to \pi^0 \eta)$. Thus one might worry that small effects, which we have ignored up to now, might break this equality. We address this question here. First, we have ignored $E'$ and $A'$ diagrams in equating these two amplitudes. This should not cause any problems. We expect that $\pew'$ is roughly of the same size as $T'$. But $E'$ and $A'$ are suppressed by $f_B/m_B \approx 5\%$ relative to $T'$. Thus their neglect introduces at most a small error into our analysis. The second possibility involves SU(3) breaking. The effects of SU(3) breaking in two-body decays of $B$ mesons have been analyzed by us in more detail in a longer paper [@GHLR]. The largest terms in the present case involve the effect of SU(3) breaking on the dominant gluonic penguin term ($P'$) in $B \to K \pi$. These terms are of the same strength in all the $B \to K \pi$ amplitudes illustrated in Fig. \[piKquad\], and hence cancel in the construction of the two diagonals. The next most important term involves SU(3) breaking in the ratio of the $|\Delta S| = 1$ and $\Delta S = 0$ non-penguin amplitudes. However, this is expected to be well-approximated by the ratio $f_K/f_{\pi}$ [@GHLR] (see also [@BPP; @SilWo]), as in Eq. (\[eqn:pipi\]). The critical term turns out to be the effect of SU(3) breaking on the electroweak penguin. Specifically, the $B_s \to \pi^0 \eta$ decay involves a spectator $s$ quark, whereas the spectator quark in the $B\to\pi K$ decays is $u$ or $d$. Thus, the SU(3) breaking corresponds here to a difference in the form factors for the two types of decays. Although we expect SU(3)-breaking effects to be typically of order 25% (i.e. the difference between $f_\pi$ and $f_K$), here they are expected to be smaller, since the mass ratio $m_\eta/m_K$ is much closer to unity than is $m_K/m_\pi$. Still, this SU(3) breaking does introduce some theoretical uncertainty into this method for obtaining $\gamma$. We have carried out a similar analysis for the decays $B \to \pi K^*$. Clearly it is still possible to write an amplitude quadrangle for these processes; the question is simply the interpretation of the diagonals. There are more SU(3) amplitudes in $B\to PV$ decays since the final-state particles do not belong to the same octet. Nevertheless, one can still use a graphical analysis in the spirit of Ref. [@BPP] – there are just more diagrams. For example, instead of one $T$ diagram, there are two ($T_P$ and $T_V$), corresponding to the cases where the spectator quark hadronizes into the $P$- or $V$-meson in the final state. Carrying out such a graphical analysis, we find that the diagonals of the $\pi K^*$ quadrangle are D\_1\^\* & = & -\[T\_P’+C\_P’+(c\_u-c\_d) (P\_[EW,V]{}’ + P\_[EW,V]{}\^[C]{})\]    ,\ D\_2\^\* & = & -C\_P’ -(c\_u-c\_d) P\_[EW,V]{}’    , where the subscripts $P$ and $V$ represent the spectator quark hadronizing into the $\pi$ and $K^*$, respectively. (In the above we have ignored annihilation-type contributions.) Remarkably, the diagonal $D_2^*$ (labeled by (e) in Fig. \[piKquad\]) corresponds to $\sqrt{2} A(B_s \to \pi^0 \phi)$. Again, the shape of the quadrangle can be specified by experimental measurements! The other diagonal $D_1^*$ contains both an electroweak penguin piece (which we can eliminate in the manner noted in Sec. IV A above), and a non-penguin piece $-(T_P' + C_P')$. This latter piece is closely related to the amplitude for the decay $B^+ \to \pi^0 \rho^+$: A(B\^+ \^0 \^+) = -\[T\_P + C\_P - P\_P + P\_V + (c\_u-c\_d) P\_[EW,V]{}\] . \[pirhoamp\] If the penguin diagrams are unimportant in this decay, or if the two types of penguin contributions $P_P$ and $P_V$ cancel (the EWP is expected to be quite small here), the analysis can be carried through exactly as in Sec. IV A. In this case, the precision on the measurement of $\gamma$ is roughly of order $|(P_P-P_V)/(T_P+C_P)|$. Another quadrangle relation holds for the amplitudes of $B\to \rho K$. They are obtained from the amplitudes of $B \to \pi K^*$ by replacing $T_P', C_P', P_{EW,V}'$, etc., by $T_V', C_V', P_{EW,P}'$, etc. Here one of the diagonals of the quadrangle is given by $\sqrt{3}A(B_s \to \eta \rho^0)$. The other diagonal (obtained from $D_1^*$ by substituting $P \leftrightarrow V$ in (18)) contains $-(T_V'+C_V')$ and an electroweak penguin term. When the latter is eliminated as in Sec. IV A, the remaining $-(T_V'+C_V')$ term is approximately equal to $\sqrt{2}A(B^+ \to \pi^+ \rho^0)$. **V. DATA: STATUS AND PROSPECTS** The measurements proposed here are not all easy. The $B \to \pi K$ decays should be characterized by branching ratios of order $10^{-5}$ for charged pions and about half that for neutral pions if the $B \to \pi^- K^+$ decay really has been observed at the $10^{-5}$ level [@Kpisep] and if the gluonic penguin amplitude is dominant. The amplitudes in Fig. \[piKquad\] are drawn to scale using the calculations of Ref. [@DH], neglecting strong final-state phase differences, and assuming $\gamma = \pi/2$. The effects of electroweak penguins can be seen not only in the rotation of the phase of $A_{3/2}$ from its non-penguin value, but in substantial differences in the lengths of the sides of the quadrangle. It may well be that electroweak penguin effects make their first appearance in such rate differences, as mentioned at the end of Sec. III. The $B_s \to \pi^0 \eta$ decay will be very difficult to measure. The calculations of Ref. [@DH] indicate a branching ratio of a couple of parts in $10^7$. One has to distinguish a $B_s$ from a $\bar B_s$. In order to observe the $\pi^0 \eta$ decay at a hadron machine, where the displaced vertex of the $B_s$ would seem to be a prerequisite, one would have to observe the $\eta$ in a mode involving charged particles. Somewhat more hope is offered in the corresponding $B \to \pi K^*$ case, if we can trust the very small branching ratio for $B_s \to \pi^0 \phi$ of a couple of parts in $10^8$ predicted in Ref. [@DHT]. (See also [@Du].) The corresponding electroweak penguin effects (characterizing the diagonal (e) in Fig. \[piKquad\]) are expected to be smaller here, whereas it is quite likely that the basic $B \to \pi K^*$ decays can be observed soon. The possibility of degeneracies in lengths of the sides of the quadrangles can lead to a large amplification of errors in the amplitudes (e) when used to predict the length of side (f). For example, imagine that (e) were really zero and (a) = (c), (b) = (d). The length of (f) then would be indeterminate. On the other hand, if the diagonal (e) of the quadrangle is sufficiently small, the quadrangle reduces to two nearly degenerate triangles in which the effects of electroweak penguins are negligible. In this case, the second diagonal is given to a good approximation by $\sqrt{2}A(B^+\to\pi^0\rho^+)$ \[assuming some cancellation between the $P_P$ and $P_V$ terms of Eq. (\[pirhoamp\])\], and the relative phase between this amplitude and its charge-conjugate measures $2\gamma$. Indeed, the very small value of ${\rm BR}(B_s\to\pi^0 \phi)$ calculated in Ref. [@DHT] suggests that this may be happening for the decays $B \to \pi K^*$. **VI. CONCLUSIONS** We have found the following results. \(a) Electroweak penguins (EWP’s) are not expected to substantially affect the discussion in Ref. [@GL] regarding $B \to \pi \pi$ decays. \(b) EWP’s [*are*]{} more likely to be important in the comparisons [@BPP; @PRL; @PLB] of $B \to \pi K$ and $B \to \pi \pi$ decays, though such conclusions are dependent on the evaluation of hadronic matrix elements of operators. \(c) EWP’s do not introduce new amplitudes of flavor SU(3), so that one cannot detect their presence merely by modification of flavor-SU(3) amplitude relations. \(d) A deviation of the rate ratio $2 \Gamma(B^+\to\pi^0 K^+)/\Gamma(B^0\to \pi^- K^+)$ from unity indicates the presence of EWP’s, and similarly for $2 \Gamma(B^0\to\pi^0 K^0)/\Gamma(B^+\to \pi^+ K^0)$. Since all of these branching ratios are expected to be ${\cal O}(10^{-5})$, these are likely to be the first (indirect) experimental signals of EWP’s. Electroweak penguins are expected to dominate decays of the form $B_s \to (\phi~{\rm or}~\eta) + (\pi~{\rm or}~\rho)$ [@DHT], but the branching ratios for these processes are expected to be significantly smaller. \(e) A quadrangle analysis has been presented for such decays as $B \to \pi K$, $B \to \pi K^*$, and $B \to \rho K$. One diagonal of the quadrangle is related to the amplitude for a physical process such as $B_s \to \pi^0 \eta$ or $B_s \to \pi^0 \phi$, so that one can perform a construction to obtain the other diagonal. From the magnitude and phase of this amplitude, one can obtain $\sin \gamma$, where $\gamma \equiv {\rm Arg} (V^*_{ub})$. \(f) The $B \to \pi K^*$ processes hold out hope for a small electroweak penguin contribution, if the $B_s \to \pi^0 \phi$ branching ratio is as small as cited in Ref. [@DHT]. In such a case, the quadrangle will degenerate into two nearly identical triangles, so that the original analysis of Ref. [@PRL], suitably modified to take account of the presence of one vector and one pseudoscalar meson, may be more trustworthy. We have presented the ingredients of such an analysis in Sec. IV C. **ACKNOWLEDGMENTS** We thank J. Cline, A. Dighe, I. Dunietz, G. Eilam, A. Grant, K. Lingel, H. Lipkin, R. Mendel, S. Stone, L. Wolfenstein, and M. Worah for fruitful discussions. J. Rosner wishes to acknowledge the hospitality of the Fermilab theory group and the Cornell Laboratory for Nuclear Studies during parts of this investigation. M. Gronau, O. Hernández and D. London are grateful for the hospitality of the University of Chicago, where part of this work was done. 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[^1]: e-mail: oscarh@lps.umontreal.ca [^2]: e-mail: london@lps.umontreal.ca

--- address: 'Department of Physics, Brown University, 182 Hope St, Providence, RI 02912, USA' author: - GREG LANDSBERG title: MOVING BEYOND EFFECTIVE FIELD THEORY IN DARK MATTER SEARCHES AT COLLIDERS --- Introduction ============ Effective field theory (EFT) has been an important tool to study various processes where a detailed description of the interaction and its carrier is either unknown or model-dependent. The EFT is used to parameterize our ignorance of the fine details of the process and has been successfully applied to a number of cases, including Fermi’s model of muon decay and searches for compositeness. It is therefore logical that the original theoretical papers [@DM1; @DM2; @DM3] that proposed the initial-state radiation (ISR) tagging to detect dark matter production (DM) at colliders, relied on the EFT description of the scattering process in order to allow for a comparison of the sensitivity of these searches with that for direct detection (DD) experiments. A classical example of such a collider process is production of a single jet recoiling against a pair of DM particles that escape the detection, resulting in a spectacular “monojet" signature. Similar, “monophoton" signature is also possible in the case of a photon ISR. Unfortunately, as has been realized recently, the use of EFT in this particular case is subject of a number of explicit and implicit assumptions, and important constraints, which severely limit the applicability of the EFT approach, sometimes to the point when it becomes all but useless. In this particular application, the EFT often fails in all three possible ways: - As an “E” — not being effective in probing certain regions of parameter space; - As an “F” — sometimes not even dealing with realistic fields; and - As a “T” — not even holding as a viable theory. The goal of these proceedings is to illustrate the limitations of the EFT approach and discuss more constructive ways of comparing the DM reach of collider experiments with that of the DD experiments, and potentially also with the reach of indirect detection experiments. Such a proper comparison would become particularly important if a significant excess in any of these experiments is seen. EFT formalism and assumptions ============================= Collider experiments are capable of setting limits on production cross section of DM particles in ISR-triggered processes, e.g. production of monojets [@CMS-monojet; @ATLAS-monojet]. These limits only require theoretical calculations, which properly describe the ISR process. While next-to-leading-order calculations are available for many such processes, often leading-order precision with an extra jet emission included in the matrix elements, suffices, making it relatively easy to calculate collider cross sections. The real issue comes when collider limits are being translated into limits on DM-nucleon scattering cross section, which is the variable used by DD experiments to represent their results. Note that fundamentally the process responsible for pair production of DM particles at colliders is the same as for the DM-nucleon scattering, or annihilation of a pair of DM particles used in indirect detection experiments. Assuming that the former process is mediated via an s-channel exchange of a certain particle, which we will refer to as the “mediator", the process is completely described by four parameters: the masses of the DM particle ($m$) and the mediator ($M$), and the two couplings of the mediator to quarks ($g_q$) and DM particles ($g_\chi$), see Fig. \[fig:Feynman\] (left). (A similar diagram can be drawn to describe collider DM pair production via a t-channel exchange of a mediator, with the caveat that in this case the mediator must be a colored particle.) In order to compare the s-channel collider process with the t-channel DM-nucleon scattering, we “contract" the s-channel exchange in the EFT four-point interaction vertex, as shown in Fig. \[fig:Feynman\] (right), which then can be used to describe both. In order to perform this contraction we move from three fundamental parameters $M$, $g_q$, and $g_\chi$ to a single parameter $\Lambda$, the EFT cutoff, thus losing the full information about the underlying process, which is an inherent feature of the EFT approach. ![Left: Feynman diagram of dark matter interaction with quarks via exchange of a mediator. Right: “contraction" of the s-channel mediator exchange diagram for the monojet or monophoton production into an EFT four-point interaction.[]{data-label="fig:Feynman"}](Feynman){width="0.9\linewidth"} ![Left: Feynman diagram of dark matter interaction with quarks via exchange of a mediator. Right: “contraction" of the s-channel mediator exchange diagram for the monojet or monophoton production into an EFT four-point interaction.[]{data-label="fig:Feynman"}](EFT){width="0.7\linewidth"} One can now directly equate the amplitude squared of the s-channel exchange in the limit of a heavy mediator ($M^2 \gg q^2$ in the event) with the one from the effective four-point interaction, which for, e.g. a mediator with scalar couplings, yields: $$\left|\frac{ig_q g_\chi}{q^2 - M^2}(\bar q q)(\bar\chi\chi)\right|^2 \approx \left|\frac{-ig_q g_\chi}{M^2}(\bar q q)(\bar\chi\chi)\right|^2 = \left|\frac{1}{\Lambda^2}(\bar q q)(\bar\chi\chi)\right|^2,$$ leading to a crucial expression: $\frac{1}{\Lambda^2} = \frac{g_q g_\chi}{M^2}$. The EFT approach is strictly valid for $q^2 \ll M^2$, which implies (from the kinematics of the s-channel exchange) $M^2 > (2m)^2$. Furthermore, in order for theory to be calculable, each of the two mediator couplings has to be less than $\sqrt{4\pi}$. Combining these two inequalities with the expression for $\Lambda$, we obtain: $2m < M < \Lambda\sqrt{g_q g_\chi} < 4\pi\Lambda$, or $\Lambda > \frac{m}{2\pi}$, which leads to an important conclusion that the validity region of the EFT grows when one deals with light DM. Similar validity regions in case of non-scalar couplings can be found, e.g. in Ref. [@Whiteson] The case of light DM is particularly important for colliders as the sensitivity of DD experiments to light DM is reduced due to low-momentum recoil, and since for very light DM ($m < 10$ GeV), the DD experiments will soon reach the solar neutrino floor. Nevertheless, it’s important to keep in mind that the above inequality really corresponds to the case when all the EFT assumptions break down spectacularly, and actual validity region really corresponds to $\Lambda \gg \frac{m}{2\pi}$. The most tricky scenario is the case of a light mediator, for which EFT certainly fails. This case was explicitly studied in one of the early phenomenological papers on collider searches [@DM4], with an explicit use of the s-channel exchange diagram instead of the EFT approach. In this case, collider searches offer an increased sensitivity to the DM production as they can produce light mediator on-shell, and hence the production cross section receives a resonant enhancement. However, the problem with the approach taken in Ref. [@DM4] is that it treats the mediator width as a free parameter, whereas one can’t do this, as the width of the mediator depends on the $\sqrt{g_q^2 + g_\chi^2}$, and if even one of the couplings approaches the $\sqrt{4\pi}$ limit, the width becomes comparable to the mass of the mediator, independent on how small the other coupling is. Since a single-resonance exchange description stops being physically reasonable for mediators that broad, this seemingly correct approach can still give incorrect comparison with the DD experiments [@Oliver]. Beyond the EFT ============== Given this situation, it is clear that EFT, while a convenient way to simplify the problem, has too many hidden caveats and simply does not allow for a fair comparison between the collider and DD experiments. The key to the proper comparison is to treat the problem as fundamentally four-dimensional and represent the reach of both the DD and collider experiments in various planes given by a pair of these parameters (e.g., $M$ and $m$), with the other two (in this case $g_q$ and $g_\chi$) being fixed to certain values, which can be scanned. In order to do this, one could use simplified models of DM, which assume certain type of couplings of the mediator to quarks and DM particles, e.g., vector or axial vector. Given that the number of such models is quite limited, one could rather easily span the relevant DM model space with just a handful of simplified models with s-channel or t-channel mediator exchange. Similar simplified model approach is successfully and broadly used in supersymmetry searches at the LHC. This is the approach advocated in the recent work [@MSDM; @MSDMA] coming from the two groups of experimentalists and theorists (the first one generally affiliated with the CMS experiment, whereas the second one – with ATLAS). Both ATLAS and CMS are now transitioning to this approach to be used in the LHC Run 2. ![Left: comparison of the EFT-based and simplified model limits on the DM-neutron scattering. Right: Comparison of the projected reach of the LHC and next generation of DD experiments. From Ref. [@MSDM][]{data-label="fig:EFT"}](EFT-MSDM){width="0.9\linewidth"} ![Left: comparison of the EFT-based and simplified model limits on the DM-neutron scattering. Right: Comparison of the projected reach of the LHC and next generation of DD experiments. From Ref. [@MSDM][]{data-label="fig:EFT"}](MSDMF){width="0.91\linewidth"} Figure \[fig:EFT\] (left) shows how the limits set using a simplified model with axial-vector couplings of the mediator to both DM particles and quarks compare with the limits from the EFT approach based on the CMS monojet analysis [@CMS-monojet], as well as with the limit from the LUX experiment [@LUX] in the canonical plane of DM-nucleon scattering cross section vs. the DM particle mass. While for relatively large couplings $g_q = g_\chi = 1.45$ the EFT results are close to those from the simplified model calculations up to DM particle mass of about 300 GeV, for smaller values of couplings the EFT grossly underestimates the LHC reach for light DM and grossly overestimates it for relatively heavy DM. Figure \[fig:EFT\] (right) shows the projection of the CMS monojet analysis for LHC Run 2 and High-Luminosity LHC, as well as projected sensitivity of the next generation of DD experiments, in the more relevant plane of $M$ vs. $m$, for the case of axial-vector mediator couplings. One can see a nice complementarity between the reach of the two types of experiments, with LHC winning over DD experiments for the case of small couplings and relatively heavy mediators and in the case of very light DM particles (with the mass less than about 5 GeV), while DD experiments offering higher reach for rather heavy DM with the mass above 200-400 GeV. Similar comparison is possible with indirect detection experiments. To conclude, the simplified model approach allows for a fair comparison of the DM reach of different types of experiments and provides a more clear and advantageous way to present the results of future collider searches. Acknowledgments {#acknowledgments .unnumbered} =============== This work is partially supported by the DOE Award No. DE-SC0010010-003376. The author is grateful to the Imperial College, London CMS group for organizing and supporting the brainstorming workshop on dark matter at colliders, which sparkled this work and Ref. [@MSDM] References {#references .unnumbered} ========== [99]{} M. Beltran [*et al.*]{}, JHEP [**09**]{} (2010) 037; arXiv:1002.4137. J. Goodman [*et al.*]{}, Phys. Lett. B [**695**]{} (2011) 185, arXiv:1005.1286. Y. Bai, P.J. Fox, and R. Harnik, JHEP [**12**]{} (2010) 048, arXiv:1005.3797. CMS Collaboration, arXiv:1408.3583. ATLAS Collaboration, arXiv:1502.01518. S. Malik [*et al.*]{}, Phys. Dark Universe [**03**]{} (2015) 003, arXiv:1409.4075. N. Zhou, D. Berge, and D. Whiteson, Phys. Rev. D [**87**]{} (2013) 095013, arXiv:1302.3619. P.J. Fox [*et al.*]{}, Phys. Rev. D [**85**]{} (2012) 056011, arXiv:1109.4398. O. Buchmueller, M. J. Dolan and C. McCabe, JHEP [**01**]{} (2014) 025, arXiv:1308.6799. J. Abdallah [*et al.*]{}, arXiv:1409.2893. LUX Collaboration, Phys. Rev. Lett. [**112**]{} (2014) 091303, arXiv:1310.8214.

--- abstract: | Kempf \[1976\] studied proper, $G$-equivariant maps from equivariant vector bundles over flag manifolds to $G$-representations $V$, which he called [*collapsings*]{}. We give a simple formula for the $G$-equivariant cohomology class on $V$, or [*multidegree*]{}, associated to the image of a collapsing: apply a certain sequence of divided difference operators to a certain product of linear polynomials, then divide by the number of components in a general fiber. When that number of components is $1$, we construct a desingularization of the image of the collapsing. If in addition the image has rational singularities, we can use the desingularization to give also a formula for the $G$-equivariant $K$-class of the image, whose leading term is the multidegree. Our application is to quiver loci and quiver polynomials. Let $Q$ be a quiver of finite type (A, D, or E, in arbitrary orientation), and assign a vector space to each vertex. Let ${\mathrm{Hom}}$ denote the (linear) space of representations of $Q$ with these vector spaces. This carries an action of $GL$, the product of the general linear groups of the individual vector spaces. A [*quiver locus*]{} $\Omega$ is the closure in ${\mathrm{Hom}}$ of a $GL$-orbit, and its multidegree is the corresponding [*quiver polynomial*]{}. Reineke \[2004\] proved that every ADE quiver locus is the image of a birational Kempf collapsing (giving a desingularization directly). Using Reineke’s collapsings, we give formulae for ADE quiver polynomials, previously only computed in type A (though in this case, our formulae are new). In the $A$ and $D$ cases quiver loci are known to have rational singularities \[Bobiński-Zwara 2002\], so we also get formulae for their $K$-classes, which had previously only been computed in equioriented type $A$ (and again our formulae are new). address: - 'Department of Mathematics, University of California, San Diego' - | Mathematics Department\ Virginia Tech\ Blacksburg, Virginia author: - Allen Knutson - Mark Shimozono title: Kempf collapsing and quiver loci --- Introduction ============ Kempf collapsings ----------------- Let $G$ be a reductive algebraic group, and $P$ a parabolic subgroup. Let $Y$ be a linear representation of $G$, and $Z\leq Y$ a $P$-invariant subspace (or more generally, a closed subvariety with at worst rational singularities). In [@K], Kempf considers the map $$\begin{aligned} G \times^P Z &\overset{\kappa}{\to} Y & \qquad \hbox{here $G\times^P Z := (G\times Z)\big/ \left \{ [g,\vec v] \sim [gp^{-1}, p\vec v] \right\}$} \\ [g, \vec z] &\mapsto g\cdot \vec z &\end{aligned}$$ which he calls a [[****]{}collapsing]{} of $G\times^P Z$. This space is the associated fiber bundle with fiber $Z$ over the homogeneous projective variety $G/P$. The map $\kappa$ is proper (it factors as $G\times^P Z {\operatorname*{\hookrightarrow}}G/P \times Y {\mathop{\twoheadrightarrow}}Y$), hence its image $G\cdot Z$ is closed. When $\kappa$ is birational, $\kappa$ serves as a resolution of singularities of the variety $G\cdot Z$. Since $G\cdot Z$ is a $G$-invariant subvariety of ${\mathrm{Hom}}$, it has a [*multidegree*]{} $[G\cdot Z]$, which can be defined as the associated $G$-equivariant cohomology class (or Chow class) in the ring $H^*_G(Y)$. Our first result, Theorem \[thm:multidegree\] below, is a formula for $[G\cdot Z]$. In Section \[sec:mdegs\] we recall the properties we need of multidegrees. If $T\leq P$ is a maximal torus of $G$, then $H^*_G(Y)$ naturally includes into $H^*_T(Y) = {\mathrm{Sym}}^\bullet(T^*)$, so it is enough to compute $[G\cdot Z]$ as a $T$-equivariant cohomology class. Since $Z$ is $P$-invariant and hence $T$-invariant, it too has a $T$-multidegree $[Z] \in {\mathrm{Sym}}^\bullet(T^*)$. This $[Z]$ turns out to be particularly simple for $Z$ a linear subspace: $[Z]$ is the formal product of the $T$-weights in $Y/Z$, each of which lives in the weight lattice $T^*$ of $T$. If none of the $T$-weights on $Y$ are $0$, then this product cannot be zero. For nonlinear $Z\subseteq Y$, e.g. a union of linear subspaces, there might still be some cancelation giving $[Z]=0$. This can’t happen (as follows from Theorem D in [@KM]) if all the weights of $T$ acting on $Y$ live in an open half-space in $T^*$; in this very common case[^1] [*any*]{} closed $T$-invariant scheme $Z\subseteq Y$ has a nonzero multidegree $[Z]$. \[thm:multidegree\] Let $\kappa : G\times^P Z\to Y$ be a Kempf collapsing, where $Lie(P)$ contains all the [*negative*]{} root spaces. Let $d$ be the number of components in a general fiber of $\kappa$. Assume that all the weights of $T$ acting on $Y$ live in an open half-space in $T^*$. Let $m_0 = [Z]$, and construct a sequence of polynomials $m_1, m_2,\ldots$ by applying divided difference operators $\partial_\alpha := \frac{1}{\alpha} (1 - r_\alpha)$ to $m_0$, where $\alpha$ varies over the set of simple roots of $G$, and $r_\alpha$ acts on ${\mathrm{Sym}}^\bullet(T^*)$ from the reflection action on $T^*$. Don’t apply a divided difference operator if the result is $0$, and only stop when all $\partial_\alpha$ give the result $0$. This process always terminates after the same number of steps (namely, $\dim G\times^P Z - \dim G\cdot Z$), and the last polynomial in this sequence is $d$ times $[G\cdot Z]$. In the case that $\kappa$ is generically finite, the sequence of simple roots can be taken to give a reduced expression for $w_0 w_0^P$, the product of the long elements of the Weyl groups $W$ of $G$ and $W_P$ of $P$ respectively. In this case there is an alternate formula $$d\,[G\cdot Z] = \sum_{w\in W^P} w\cdot \frac{[Z]}{\prod_{\beta \in \Delta\setminus \Delta_P} \beta}$$ where $W^P$ is the set of minimal coset representatives in $W/W_P$, and $\Delta$ and $\Delta_P$ are the sets of roots of $G$ and $P$ respectively. In Section \[sec:mdegs\] we give an example in which $\kappa$ does not have connected general fiber. When $\kappa$ does have connected fibers and $Z$ and $G\cdot Z$ have rational singularities, we can use the collapsing to compute a more precise invariant than the multidegree, which is the [*$K$-polynomial*]{} $[G\cdot Z]^K_Y$. Essentially, this is the numerator of the multigraded Hilbert series of the sheaf ${\mathcal{O}}_{G\cdot Z}$ on $Y$; we recall the precise definition in Section \[sec:mdegs\]. \[thm:Kpoly\] Let $\kappa : G\times^P Z\to Y$ be a Kempf collapsing whose general fiber is connected (so $d=1$ in the notation of Theorem \[thm:multidegree\]), and assume $Z$ and $G\cdot Z$ have rational singularities. Let $m_0 = [Z]^K_Y$, and construct a sequence of Laurent polynomials $m_1, m_2,\ldots$ by applying Demazure operators $d_\alpha := (1 - \exp(-\alpha))^{-1} (1 - \exp(-\alpha) r_\alpha)$ to $m_0$, where $\alpha$ varies over the set of simple roots of $G$. Don’t bother applying any $d_\alpha$ that acts as the identity, and only stop when all $d_\alpha$ act as the identity. The sequence of simple roots can be taken to give a reduced expression for $w_0 w_0^P$. This process terminates after finitely many steps (namely, $\dim G\times^P Z - \dim G\cdot Z$). The last Laurent polynomial in this sequence is the $K$-polynomial $[G\cdot Z]^K_Y$. Moreover $$[G\cdot Z]^K_Y = \sum_{w\in W^P} w\cdot \frac{[Z]^K_Y}{\prod_{\beta\in \Delta\setminus\Delta_P} (1-\exp(-\beta))}.$$ In the cases where Theorem \[thm:Kpoly\] applies, it implies Theorem \[thm:multidegree\], by viewing the multidegree as the lowest-order homogeneous component of the $K$-polynomial. Kempf worked only with the case that $Z \subseteq Y$ is a linear subspace (in which case its $K$-polynomial $[Z]^K_Y$ is again a very simple product over the weights in $Y/Z$), which will also suffice for our main application. It is frequently the case that the weights of $T$ on $Y$ are distinct, which implies that there are only finitely many $P$-invariant linear subspaces $Z$ on which to apply Kempf’s construction. Quiver loci ----------- A [[****]{}quiver]{} $Q=(Q_0,Q_1)$ is a finite directed graph, which consists of a set $Q_0$ of vertices and a set $Q_1$ of directed edges or arrows, such that each arrow $a\in Q_1$ has a [[****]{}tail]{} $ta\in Q_0$ and a [[****]{}head]{} $ha\in Q_0$. A [[****]{}representation]{} $V$ of $Q$ is a choice of a vector space $V(i)$ for each vertex $i\in Q_0$ and a linear map $V(a)\in{\mathrm{Hom}}(V(ta),V(ha))$ for each arrow $a\in Q_1$. There are obvious notions of isomorphism, direct sum, and indecomposable, for representations of $Q$. The [[****]{}dimension vector]{} of $V$ is the map $Q_0\rightarrow {{\mathbb N}}$ defined by $i\mapsto \dim V(i)$. Fix a dimension vector $d:Q_0\rightarrow{{\mathbb N}}$ and define $$GL := GL(Q,d) = \prod_{i\in Q_0} GL({\mathbb{C}}^{d(i)}), \qquad {\mathrm{Hom}}:= {\mathrm{Hom}}(Q,d) = \prod_{a\in Q_1} {\mathrm{Hom}}({\mathbb{C}}^{d(ta)},{\mathbb{C}}^{d(ha)}).$$ A typical element of ${\mathrm{Hom}}$ is denoted $V$, and for $a\in Q_1$ the $a$ component is denoted $V(a)$. The notation comes from thinking of $V$ as a functor from the free category on $Q$ to the category [**Vec**]{}. The group $GL$ acts linearly on ${\mathrm{Hom}}$ by change of basis: $(g\cdot V)(a) = g(ta) V(a) g(ha)^{-1}$ for all $g\in GL$, $V\in {\mathrm{Hom}}$, and $a\in Q_1$. Two points in ${\mathrm{Hom}}$ are in the same $GL$-orbit if and only if they define isomorphic representations of $Q$. The closures of the $GL$-orbits are called [[****]{}quiver loci]{}.[^2] - [@G] The action of $GL(Q,d)$ on ${\mathrm{Hom}}(Q,d)$ has finitely many orbits for all dimension vectors $d:Q_0\rightarrow{{\mathbb N}}$, if and only if $Q$ is a [[****]{}Dynkin quiver]{}, i.e. if the undirected graph underlying $Q$ is a Dynkin diagram of type $A_{n\geq 1}$, $D_{n\geq 4}$, $E_6$, $E_7$, or $E_8$. - [@LM; @BZ] For $Q$ of type $A,D$, the quiver loci have rational singularities. (To our knowledge the $E$ cases are still open.) - [@R] If $Q$ is a Dynkin quiver, each quiver locus $\Omega \subseteq {\mathrm{Hom}}$ is the image of a birational linear Kempf collapsing, i.e. there exists a parabolic subgroup $P \leq GL$ and a $P$-invariant linear subspace $Z\leq {\mathrm{Hom}}$ such that $GL \times^P Z {\mathop{\twoheadrightarrow}}GL\cdot Z = \Omega$ is birational. In [@R], Reineke constructs each $Z$ explicitly using the [*Auslander-Reiten quiver*]{} of $Q$. We recapitulate this construction precisely in Section \[sec:finitequivers\]. Modulo the construction of a certain ordering, we can state the resulting quiver formulae here. It is well-known [@G] that the indecomposable representations of a Dynkin quiver $Q$ are in bijection with the set of positive roots $R^+$ of the root system corresponding to the underlying Dynkin diagram. Fix an ordering $R^+=\{\beta_1,\beta_2,\dotsc,\beta_N\}$ of the set of positive roots and write $I_j$ for the (isomorphism class of an) indecomposable representation of $Q$ corresponding to $\beta_j$. The correspondence $I_j \leftrightarrow \beta_j$ is determined as follows: the dimension vector of the indecomposable $I_j$ is given by the expansion $\beta_j = \sum_{i\in Q_0} \dim I_j(i) \, \alpha_i$ of the corresponding positive root $\beta_j$ in the basis $\{\alpha_i\mid i\in Q_0\}$ of simple roots. Thus there is a bijection between the $GL$-orbits in ${\mathrm{Hom}}={\mathrm{Hom}}(Q,d)$ and the direct sums $\bigoplus_{j=1}^N I_j^{\,\oplus m_j}$ where $(m_j\mid j = 1,\ldots,N)$ satisfies the obvious dimension condition $$\begin{aligned} \label{E:dim} \text{for all $i\in Q_0$},\quad d(i) = \sum_{j=1}^N d_j(i), \qquad\text{where } \ d_j(i) := m_j \dim(I_j(i)).\end{aligned}$$ Fix such a tuple of multiplicities $m = (m_j)$ and let $\Omega_m \subset {\mathrm{Hom}}$ be the closure of the corresponding $GL$-orbit. Based on $m$ we define a parabolic $P_m\subset GL$ and a linear subspace $Z_m\subset{\mathrm{Hom}}$ as follows. For each vertex $i\in Q_0$ we divide the sets of row and column indices of $GL({\mathbb{C}}^{d(i)})$ into contiguous subsets of sizes $d_j(i)$ as $j$ runs from $1$ to $N$. This defines a standard parabolic subgroup $P_m\subset GL$ whose $i$th component (for $i\in Q_0$) is the block lower triangular subgroup of $GL({\mathbb{C}}^{d(i)})$ with the given diagonal block sizes. The decompositions $d(i) = \sum_{j=1}^N d_j(i)$ also induce a block structure on each component ${\mathrm{Hom}}({\mathbb{C}}^{d(ta)},{\mathbb{C}}^{d(ha)})$ of ${\mathrm{Hom}}$, whose $(j,j')$ block is a $d_j(ta) \times d_{j'}(ha)$ rectangle. Define the linear subspace $Z_m\subset{\mathrm{Hom}}$ to be those elements with zeroes in all blocks strictly above the “block diagonal". This $Z_m$ is easily seen to be $P_m$-invariant. Reineke proves that [*for certain choices of ordering*]{} (built using reduced words for $w_0$ adapted to the quiver, as spelled out in Section \[sec:finitequivers\]) on $R_+$, the Kempf collapsing $GL\times^{P_m} Z_m \to {\mathrm{Hom}}$ is birational to $\Omega_m$. Let $\{x_k^{(i)}\mid i\in Q_0,\, k \in \{1,\ldots,d(i)\} \}$ be a basis for the weight lattice $T^*$ of the standard maximal torus $T$ given by the tuples of diagonal matrices in $GL$. Then the $(k,k')$th matrix entry in the $a$th component of ${\mathrm{Hom}}$ has weight $x_k^{(ta)}-x_{k'}^{(ha)}$. \[thm:application\] Let $Q$ be a Dynkin quiver, and $\{\beta_1,\beta_2,\dotsc,\beta_N\}$ a certain order on the set $R^+$ of positive roots (constructed explicitly in Section \[sec:finitequivers\]). Let $\Omega_m$ be a quiver locus, with associated multiplicities $m$, parabolic $P_m \leq GL$, and subspace $Z_m \leq {\mathrm{Hom}}$. Then $[\Omega_m]$ may be computed by Theorem \[thm:multidegree\] where $[Z_m]$ is the product of the weights of all blocks in ${\mathrm{Hom}}$ that are strictly above the “block diagonal". For types $A$ and $D$ the $K$-polynomial $[\Omega_m]^K_{{\mathrm{Hom}}}$ may be computed by Theorem \[thm:Kpoly\], in which $[Z_m]^K_{\mathrm{Hom}}$ is the product of terms of the form $1-e^{-\gamma}$ where $\gamma$ runs over those same weights as in $[Z_m]$. Let $Q$ be the equioriented $A_n$ quiver: $$\begin{aligned} \pspicture(-.5,-.5)(4.5,.5) \psset{arrows=->,arrowsize=3pt 3} \psdots(0,0)(1,0)(2,0)(3,0) \psline[arrows=->](0,0)(1,0) \psline[arrows=->](1,0)(2,0) \psline[arrows=->,linestyle=dashed](2,0)(3,0) \rput(0,-.4){$1$} \rput(1,-.4){$2$} \rput(2,-.4){$3$} \rput(3,-.4){$n$} \endpspicture \end{aligned}$$ The simple roots of $A_n$ are given by $\alpha_{ij}=\alpha_i+\alpha_{i+1}+\dotsm+\alpha_j$ for $1\le i\le j\le n$. Write $I_{ij}$ for the indecomposable representation of $A_n$ corresponding to $\alpha_{ij}$. A suitable ordering on the indecomposables is given by $I_{11},I_{12},I_{22},I_{13},I_{23},I_{33},\dotsc,I_{nn}$. Let us consider the specific example $n=3$, $d=(2,3,2)$, and $\Omega$ given by the $GL$-orbit closure of $I_{12}^{\oplus 2} \oplus I_{23}\oplus I_{33}$. Geometrically, $\Omega$ is defined by requiring the map $V(2) \to V(3)$ to have rank $\leq 1$, and the composite map $V(1)\to V(3)$ to vanish. The decompositions $d(i) = \sum_{j=1}^N d_j(i)$ are $$d(1) = 2+0+0, \quad d(2) = 2+1+0, \quad d(3) = 0+1+1,$$ so the parabolic $P_m\subset GL$ and the linear subspace $Z_m\subset{\mathrm{Hom}}$ take the following form: $$P_m = \left\{\left(\begin{pmatrix} *&* \\ *&* \end{pmatrix},\begin{pmatrix} *&*&0 \\ *&*&0 \\ *&*&* \end{pmatrix}, \begin{pmatrix} *&0 \\ *&* \end{pmatrix}\right)\right\}, \qquad Z_m = \left\{\left( \begin{pmatrix} *&*&0\\ *&*&0 \end{pmatrix} , \begin{pmatrix} 0&0 \\ 0&0 \\ *&0 \end{pmatrix} \right)\right\}.$$ By Theorem \[thm:multidegree\] $$\begin{aligned} [Z_m] &= (x^{(1)}_1-x^{(2)}_3)(x^{(1)}_2-x^{(2)}_3)\ (x^{(2)}_1-x^{(3)}_1)(x^{(2)}_1-x^{(3)}_2) (x^{(2)}_2-x^{(3)}_1)(x^{(2)}_2-x^{(3)}_2) (x^{(2)}_3-x^{(3)}_2) \\ [\Omega] &= \partial_{x^{(2)}_1-x^{(2)}_2} \,\partial_{x^{(2)}_2-x^{(2)}_3} \,\partial_{x^{(3)}_1-x^{(3)}_2} \, [Z_m] \end{aligned}$$ By Theorem \[thm:Kpoly\] $$\begin{aligned} [Z_m]^K_{\mathrm{Hom}}&= (1-e^{-x^{(1)}_1+x^{(2)}_3})(1-e^{-x^{(1)}_2+x^{(2)}_3}) (1-e^{-x^{(2)}_1+x^{(3)}_1})(1-e^{-x^{(2)}_1+x^{(3)}_2}) \\ &\,\,\,\,\,\,\,(1-e^{-x^{(2)}_2+x^{(3)}_1})(1-e^{-x^{(2)}_2+x^{(3)}_2}) (1-e^{-x^{(2)}_3+x^{(3)}_2}) \\ [\Omega]^K_{\mathrm{Hom}}&= d_{x^{(2)}_1-x^{(2)}_2}\, d_{x^{(2)}_2-x^{(2)}_3}\, d_{x^{(3)}_1-x^{(3)}_2}\, [Z_m]^K_{\mathrm{Hom}}. \end{aligned}$$ We shall work out the multidegree calculation explicitly. Let $a_i=x^{(1)}_i$, $b_i=x^{(2)}_i$, and $c_i=x^{(3)}_i$. We shall use the following properties of $\partial_\alpha$: $\partial_\alpha(f)=0$ if $r_\alpha(f)=f$, and $\partial_\alpha(fg)=\partial_\alpha(f)g+r_\alpha(f) \partial_\alpha(g)$. In particular if $r_\alpha(f)=f$ then $\partial_\alpha(fg)=f\partial_\alpha(g)$. Using the notation $\partial^a_i = \partial_{a_i-a_{i+1}}$ (and similarly for $b,c$) we have $$\begin{aligned} [\Omega] &= \partial^b_1 \partial^b_2 \partial^c_1 (a_1-b_3)(a_2-b_3)(b_1-c_1)(b_1-c_2)(b_2-c_1)(b_2-c_2)(b_3-c_2) \\ &= \partial^b_1 \partial^b_2 (a_1-b_3)(a_2-b_3)(b_1-c_1)(b_1-c_2)(b_2-c_1)(b_2-c_2) \\ &= \partial^b_1 [ (a_2-b_3)(b_1-c_1)(b_1-c_2)(b_2-c_1)(b_2-c_2) \\ &+ (a_1-b_2)(b_1-c_1)(b_1-c_2)(b_2-c_1)(b_2-c_2) \\ &+ (a_1-b_2)(a_2-b_2)(b_1-c_1)(b_1-c_2)(b_2-c_2) \\ &+ (a_1-b_2)(a_2-b_2)(b_1-c_1)(b_1-c_2)(b_3-c_1) ] \\ &= [0 ]+ [(b_1-c_1)(b_1-c_2)(b_2-c_1)(b_2-c_2)] \\ &+ [(a_2-b_2)(b_1-c_1)(b_1-c_2)(b_2-c_2) \\ &+ (a_1-b_1)(b_1-c_1)(b_1-c_2)(b_2-c_2) \\ &+ (a_1-b_1)(a_2-b_1)(b_1-c_2)(b_2-c_2)] \\ &+ [ (a_2-b_2)(b_1-c_1)(b_1-c_2)(b_3-c_1) \\ &+ (a_1-b_1)(b_1-c_1)(b_1-c_2)(b_3-c_1) \\ &+ (a_1-b_1)(a_2-b_1)(b_1-c_2)(b_3-c_1) \\ &+ (a_1-b_1)(a_2-b_1)(b_2-c_1)(b_3-c_1) ].\end{aligned}$$ We check this against the component formula [@KMS Cor. 6.17], which is a sum over three minimal length lacing diagrams $$\begin{aligned} \psset{xunit=.5cm,yunit=.5cm} \pspicture(0,1)(2,3) \psdots(0,1)(0,2)(1,1)(1,2)(1,3)(2,1)(2,2) \psline(0,1)(1,1) \psline(0,2)(1,2) \psline(1,3)(2,1) \endpspicture \qquad \pspicture(0,1)(2,3) \psdots(0,1)(0,2)(1,1)(1,2)(1,3)(2,1)(2,2) \psline(0,1)(1,1) \psline(0,2)(1,3) \psline(1,2)(2,1) \endpspicture \qquad \pspicture(0,1)(2,3) \psdots(0,1)(0,2)(1,1)(1,2)(1,3)(2,1)(2,2) \psline(0,1)(1,2) \psline(0,2)(1,3) \psline(1,1)(2,1) \endpspicture\end{aligned}$$ which give the three tuples of partial permutation matrices $$\begin{aligned} \left(\begin{pmatrix} 1&0&0\\ 0&1&0 \end{pmatrix},\begin{pmatrix} 0&0 \\ 0&0 \\ 1&0 \end{pmatrix}\right)\qquad \left(\begin{pmatrix} 1&0&0\\ 0&0&1 \end{pmatrix},\begin{pmatrix} 0&0 \\ 1&0 \\ 0&0 \end{pmatrix}\right)\qquad \left(\begin{pmatrix} 0&1&0\\ 0&0&1 \end{pmatrix},\begin{pmatrix} 1&0 \\ 0&0 \\ 0&0 \end{pmatrix}\right).\end{aligned}$$ The formula is then $$\begin{aligned} [\Omega] &= {\mathfrak{S}}_{123}(a;b){\mathfrak{S}}_{3412}(b;c)+{\mathfrak{S}}_{132}(a;b){\mathfrak{S}}_{3142}(b;c)+{\mathfrak{S}}_{231}(a;b){\mathfrak{S}}_{1342}(b;c) \\ &= [(b_1-c_1)(b_1-c_2)(b_2-c_1)(b_2-c_2)] \\ &+ [(a_1+a_2-b_1-b_2)((b_1-c_1)(b_1-c_2)(b_2+b_3-c_1-c_2))] \\ &+ [(a_1-b_1)(a_2-b_1))((b_2-c_1)(b_3-c_1)+(b_1-c_2)(b_3-c_1)+(b_1-c_2)(b_2-c_2))]\end{aligned}$$ using, say, the pipe dream formula [@FK] [@KMS Thm. 5.3] to evaluate the double Schubert polynomials ${\mathfrak{S}}_w(x;y)$. The multidegrees of quiver loci are particularly important for studying the singularities of composites of differential mappings (see [@BF; @FR; @BFR] and the references therein). Until now, the only formulae for these multidegrees were in type $A$. The first such formula was in [@BF], and applied only to the case that the directed arrows are all oriented the same direction. This type $A$ formula has been improved in three ways: it has been made manifestly positive in an appropriate sense, the $K$-polynomial has been computed [@KMS], and the orientation has been generalized [@BR]. Some of these have been combined: the $K$-polynomial has been computed positively [@Bu; @Mi], and the multidegree has been computed positively for arbitrary orientations [@BR]. Using Theorems \[thm:multidegree\] and \[thm:Kpoly\], and the rationality of the singularities (from [@BZ]), we give the first formulae for - the multidegrees of type $D$ and $E$ quiver loci, - the $K$-polynomials for type $A$ quiver loci in non-equioriented cases, and - the $K$-polynomials for type $D$ quiver loci. Unfortunately, our formulae are not positive in the senses of [@KMS; @Bu; @Mi]. Some positivity of the answers is expected on very general grounds (e.g. Theorem D in [@KM]). The Bott-Samelson crank {#sec:geom} ======================= The inductive processes in Theorems \[thm:multidegree\] and \[thm:Kpoly\] have their geometric origin in the Bott-Samelson crank [@BS]. Fix a Borel subgroup $B$ with $P\geq B\geq T$. For each simple root $\alpha$ of $G$, let $P_\alpha$ be the corresponding minimal parabolic. Then if $f: C\to Y$ is a $B$-equivariant map, the space $P_\alpha \times^B C$ has also a natural $B$-equivariant map to $Y$, which we will call $P_\alpha \times^B f$. We call this functor (on the category of $B$-equivariant maps $f:C\to Y$ to a fixed $G$-space) [[****]{}one turn of the Bott-Samelson crank]{}. By projecting onto the first factor, we see that the resulting space is a $C$-bundle over $P_\alpha/B {\mathop{\cong}}{{\mathbb P}^1}$, and in particular $\dim P_\alpha\times^B C = \dim C + 1$. This $C$-bundle is trivial if $f$ is not just $B$- but $P_\alpha$-equivariant, with the projection onto the $C$ factor given by the $B$-quotient of the action map $P_\alpha \times C \to C$; we study this further in Lemma \[lem:GeqvtBS\] below. Since we generally turn the crank many times in succession, using a sequence $\vec \alpha = (\alpha_1,\ldots,\alpha_k)$, we will denote products of these functors $P_{\alpha_i} \times^B$ by $BS_{\vec\alpha} := P_{\alpha_k} \times^B \ldots \times^B P_{\alpha_1} \times^B$. A space $BS_{\vec\alpha}\cdot pt$ is a [[****]{} Bott-Samelson manifold]{}. The natural $G$-space for a point to map to $B$-equivariantly is $G/B$, so each Bott-Samelson manifold comes with a [[****]{}Bott-Samelson map]{} to $G/B$. Seeing a Bott-Samelson manifold as a free quotient by $B$ on the right of $P_{\alpha_k} \times^B \ldots \times^B P_{\alpha_1}$, any Bott-Samelson manifold tautologically carries a principal $B$-bundle. It is sometimes useful to see the space $BS_{\vec\alpha} Z$ as the associated $Z$-bundle over the Bott-Samelson manifold $BS_{\vec\alpha}\cdot pt$. \[lem:GeqvtBS\] Let $G$ act on two varieties $C,Y$ (which need not be linear), and let $f: C\to Y$ be a $G$-equivariant map. Let $\alpha_1,\ldots,\alpha_j$ be a sequence of simple roots. Then the general fibers of $BS_{\vec\alpha} f : BS_{\vec\alpha} C\to Y$ have the same number of components as the general fibers of $f$. Consider the diagram $$\begin{array}{cccc} (BS_{\vec\alpha} \cdot pt) \times C &\widetilde{\longrightarrow}& BS_{\vec\alpha} C& \\ \downarrow & & \downarrow& \\ C & \stackrel{f}{{\longrightarrow}} & Y& \end{array}$$ The left vertical map is projection onto the second factor, and the right vertical map is $BS_{\vec\alpha} f$. If the top map is $([p_k,\ldots,p_1],c) \mapsto [p_k,\ldots,p_1,p_1^{-1}\cdots p_k^{-1} c]$, which is easily seen to be well-defined and an isomorphism, then the diagram commutes. We can now study the right-hand map $BS_{\vec\alpha} f$ by reversing the isomorphism on the top of the diagram. The fibers of the map from the northwest corner to the southeast are just products of the fibers of $f$ with Bott-Samelson manifolds, which are connected. \[prop:BSshrinkdim\] Let $G$ act on a scheme $Y$, and $\iota:Z{\operatorname*{\hookrightarrow}}Y$ be the inclusion of a $B$-invariant subvariety. (In fact we may as well replace $Y$ by the subvariety $G\cdot Z$.) Let $\mu : G\times^B Z \to G\cdot Z$ be the projective map $[g,z] \mapsto g\cdot z$. Then there exists a sequence of simple roots $(\alpha_1,\ldots,\alpha_k)$, such that $BS_{\vec\alpha}\iota$ is surjective and generically finite, and its degree is the number of components in a general fiber of the map $\mu$. We will show there exist [*two*]{} sequences of simple roots $(\alpha_1,\ldots,\alpha_k)$, $(\beta_1,\ldots,\beta_j)$ and a natural commutative diagram $$\begin{array}{ccccc} BS_{\vec\alpha} Z &\stackrel{BS_{\vec\alpha} \iota} {\longrightarrow}& G\cdot Z &\stackrel{\mu}{\longleftarrow}& G\times^B Z \\ &&\uparrow&&\uparrow \\ && BS_{\vec\beta} (G\cdot Z) &{\longleftarrow}& BS_{\vec\beta} BS_{\vec\alpha} Z \end{array}$$ in which all maps are onto, the map $BS_{\vec\beta} BS_{\vec\alpha} Z \to G\times^B Z$ is generically $1$:$1$, and the map $BS_{\vec\alpha}\iota$ is generically finite to one. From this diagram we will derive the conclusions of the proposition. Let $Z_0 = Z$. Since the subgroups $\{P_\alpha\}$ generate $G$, for each $i$ either $Z_i$ is $G$-invariant or we may pick a simple root $\alpha_i$ such that $Z_i$ is not $P_{\alpha_i}$-invariant. Define $Z_i := P_{\alpha_i} \cdot Z_{i-1}$. Since $Z_i$ is the image of the proper map $P_{\alpha_i}\times^B Z_{i-1} {\mathop{\twoheadrightarrow}}Z_i$, and by inductive assumption $Z_{i-1}$ is closed, reduced, and irreducible, we find $Z_i$ is too. Since $Z_{i-1}$ was not $P_{\alpha_i}$-invariant, $Z_i \supset Z_{i-1}$ and $\dim Z_i = \dim Z_{i-1} + 1 = \dim Z + i$. Obviously $G\cdot Z_i = G\cdot Z$, so $Z_i \subseteq G\cdot Z$ is only $G$-invariant if $Z_i = G\cdot Z$. Hence this process stops when $\dim G\cdot Z = \dim Z_k = \dim Z + k$, i.e. $k = \dim G\cdot Z-\dim Z$. The map $BS_{\vec\alpha}\iota: BS_{\vec\alpha} Z \to Z_k = G\cdot Z$ is surjective. By dimension count it is generically finite-to-one. To construct the sequence $(\beta_j)$, consider the $B$-equivariant map $\{pt\} \to G/B$ taking a point to the identity coset, and apply $BS_{\vec\alpha}$ to that. The result $BS_{\vec\alpha} \cdot pt \to G/B$ is a [[****]{}Demazure-Hansen resolution]{} [@De; @Ha] of a Schubert variety in $G/B$, where the source is a Bott-Samelson manifold. Now select $(\beta_i)$ following the same procedure as was used above, to construct a finite-to-one map $BS_{\vec\beta} BS_{\vec\alpha}\cdot pt \to G/B$. In fact the resulting map is generically $1$:$1$ [@BS]. This obviously extends to a map of $B$-bundles, and our map $BS_{\vec\beta} BS_{\vec\alpha} Z \to G \times^B Z$ is the corresponding map of associated $Z$-bundles. Consequently it too is generically $1$:$1$. To finish setting up the diagram, define $$\begin{array}{rclrcl} \mu : G\times^B Z &{\mathop{\twoheadrightarrow}}& G\cdot Z &\qquad [g,z] &\mapsto& g\cdot z \\ BS_{\vec\beta} BS_{\vec\alpha} Z &{\mathop{\twoheadrightarrow}}& G\times^B Z &\qquad [g_1,\ldots,g_{j+k},z] &\mapsto& [g_1 g_2\cdots g_{k+j},z] \\ BS_{\vec\beta} BS_{\vec\alpha} Z &{\mathop{\twoheadrightarrow}}& BS_{\vec\beta} Z_k = BS_{\vec\beta} (G\cdot Z) &\qquad [g_1,\ldots,g_{j+k},z] &\mapsto& [g_1 g_2\cdots g_j,g_{j+1}\cdots g_{j+k} \cdot z] \end{array}$$ which are all visibly onto and define the commuting square above. It remains to prove our claims about these maps. Applying Lemma \[lem:GeqvtBS\] to the maps $BS_{\vec\alpha} \iota : BS_{\vec\alpha} Z\to G\cdot Z$, $BS_{\vec\beta} BS_{\vec\alpha} \iota : BS_{\vec\beta} BS_{\vec\alpha} Z\to G\cdot Z$, we see that the general fiber of $BS_{\vec\beta} BS_{\vec\alpha} \iota : BS_{\vec\beta} BS_{\vec\alpha} Z\to G\cdot Z$ has the same number of connected components as the general fiber of $BS_{\vec\alpha} \iota : BS_{\vec\alpha} Z\to G\cdot Z$, which (since it is generically finite-to-one) is just its degree. In the case $Z=pt$, the following is a standard result about Bott-Samelson manifolds for partial flag manifolds. \[lem:BSforGmodP\] Let $Z$ be a $B$-space, and $\vec \alpha$ a list of simple roots whose corresponding reflections $(r_{\alpha_i})$ give a reduced word for the Weyl group element $w_0 w_0^P$ where $w_0$ is the long element of $G$’s Weyl group and $w_0^P$ the long element of $P$’s. Then the map $BS_{\vec\alpha} \cdot Z \to G\times^P Z$ (constructed by applying $BS_{\vec\alpha}$ to the inclusion $Z {\mathop{\cong}}P\times^P Z {\operatorname*{\hookrightarrow}}G\times^P Z$ of the fiber over the basepoint) is a birational isomorphism. These two spaces are $Z$-bundles, and the map takes fibers to fibers; as such it is equivalent to check that $BS_{\vec\alpha}\cdot pt\to G/P$ is a birational isomorphism. Writing this as a composite $$BS_{\vec\alpha}\cdot pt\to G/B {\mathop{\twoheadrightarrow}}G/P,$$ the first map is birational, by the assumption of reducedness, to the (opposite) Schubert variety $\overline{B w_0 w_0^P B}/B$. The fiber over $gP \in G/P$ of the second map is $gP/B = g\overline{B w_0^P B}/B$. Hence the fiber over $gP$ of the composite is the intersection $$g\overline{B w_0^P B}/B \cap \overline{B w_0 w_0^P B}/B$$ which for generic $g$ is a point, since the $w_0$ makes these opposed Schubert varieties. Multidegrees, $K$-polynomials, and the proofs of Theorems \[thm:multidegree\] and \[thm:Kpoly\] {#sec:mdegs} =============================================================================================== Multidegrees and the proof of Theorem \[thm:multidegree\]. ---------------------------------------------------------- Let a torus $T$ act on a vector space $Y$. To each $T$-invariant subscheme $Z\subseteq Y$, we can associate a [[****]{}multidegree]{} $[Z]_Y$ living in the symmetric algebra on the weight lattice $T^*$ of $T$, satisfying the following properties: 1. If $Z=Y=\{0\}$, then $[Z]_Y=1$. 2. If as a cycle $Z = \sum_i m_i Z_i$, where the $\{Z_i\}$ are varieties occurring with multiplicities $\{m_i\}$, then $[Z]_Y = \sum_i m_i [Z_i]_Y$. 3. If $H \leq Y$ is a $T$-invariant hyperplane, and $Z$ is a variety, then 1. if $Z \not\subseteq H$, then $[Z]_Y = [Z\cap H]_H$, but 2. if $Z \subseteq H$, then $[Z]_Y = [Z]_H \cdot wt(Y/H)$, where $wt(Y/H) \in T^*$ is the $T$-weight on the line $Y/H$. The multidegree generalizes the notion of degree of a projective variety ${\mathbb P}Z \subseteq {\mathbb P}Y$. If $T$ is just a circle acting on $Y$ by rescaling, and $Z$ is the affine cone (hence $T$-invariant) over a projective variety ${\mathbb P}Z$, then $[Z]_Y = (\deg {\mathbb P}Z) a^{{\mathrm{codim}\,}_Y Z}$ where $a$ is the generator of $T^*$. Multidegrees (in ${\mathrm{Sym}}(T^*)$) are a special case of equivariant Chow classes (in $A_T(Y)$); since $Y$ is equivariantly contractible we have $A_T(Y) {\mathop{\cong}}A_T(pt) {\mathop{\cong}}{\mathrm{Sym}}(T^*)$. It is easy to see that properties (1)-(3) characterize multidegrees. One can show existence in several ways, one being through multigraded Hilbert series, as in the next section. Multidegrees were introduced by [@Jo]. Our reference for them is [@MS]. We only use three results about them. One that follows immediately from the properties above is that for $Z\leq Y$ a linear subspace, $[Z]_Y$ is the product of the weights in $Y/Z$. The second is that if all the $T$-weights in $Y$ lie in an open half-space, then $[Z]_Y \neq 0$ for $Z$ nonempty. (This follows from Theorem D in [@KM], and is also easily derived from the above properties.) The third is a technical result in equivariant Chow theory: \[lem:Hlocalization\] Let $Z$ be a $P$-variety and let $A_T(pt)_{frac}$ denote the field of fractions of the polynomial ring $A_T(pt)$. Then we have a formula in the localization $A_T(G\times^P Z){{\otimes}}_{A_T(pt)} A_T(pt)_{frac}$ of the equivariant Chow ring $A_T(G\times^P Z)$: $$1 = \sum_{w\in W^P} w\cdot \frac{[Z]_{G\times^P Z}}{\prod_{\beta\in\Delta\setminus\Delta_P} \beta}$$ where $[Z]_{G\times^P Z} \in A_T(G\times^P Z)$ is the class induced by the regularly embedded subvariety $Z$. As the map $G\times^P Z {\mathop{\twoheadrightarrow}}G/P$ is $T$-equivariant (indeed, $G$-equivariant), all the $T$-fixed points in $G\times^P Z$ lie over the $T$-fixed points $\{wP : w\in W^P\}$ in $G/P$. So we get inclusions $(G\times^P Z)^T {\operatorname*{\hookrightarrow}}\bigcup_{w\in W^P} wZ {\operatorname*{\hookrightarrow}}G\times^P Z$. Then we use the fact, proven in [@Br section 3.2], that the inclusion of fixed points (the composite of the two above) induces an injective pullback $A_T(G\times^P Z) {\operatorname*{\hookrightarrow}}A_T((G\times^P Z)^T)$. Hence the map $A_T(G\times^P Z) {\operatorname*{\hookrightarrow}}A_T(\cup_{w\in W^P} wZ) {\mathop{\cong}}\bigoplus_{w\in W^P} A_T(wZ)$ is injective, and to prove the two sides of the formula agree it will suffice to check their images. Let $i:Z{\operatorname*{\hookrightarrow}}G\times^P Z$ take $z \mapsto [1,z]$. Then $i^* [Z]_{G\times^P Z} = i^* i_* 1 = $ the equivariant Euler class of the normal bundle of $Z$ inside $G\times^P Z$. This normal bundle is the pullback of the normal bundle to the basepoint $P/P \in G/P$, hence its equivariant Euler class is the product $\prod_{\beta\in\Delta\setminus\Delta_P} \beta$ of the weights in the tangent space. Applying $i^*$ to both sides of the formula, we therefore get $1 = \prod_{\beta\in\Delta\setminus\Delta_P} \beta/ \prod_{\beta\in\Delta\setminus\Delta_P} \beta$. By the Weyl-invariance of both sides, the same confirmation holds for the pullback to each $wZ$. Now apply the injectivity above to conclude the formula on $G\times^P Z$ itself. This has a well-known corollary due to Joseph: [@Jo look in BBM]\[cor:Joseph\] Let $P_\alpha$ act on $Y$, and $Z$ be a $B$-invariant subscheme. Let $d$ be the degree of the map $P_\alpha \times^B Z \to P_\alpha\cdot Z$ unless $Z$ is $P_\alpha$-invariant, in which case let $d=0$. Let $\partial_\alpha$ denote the divided difference operator $\frac{1}{\alpha}(1 - r_\alpha)$, acting on ${\mathrm{Sym}}^\bullet(T^*)$. Then $$\partial_\alpha [Z]_Y = d\, [P_\alpha \cdot Z]_Y.$$ Let $L$ denote the Levi factor of $P_\alpha$ containing $T$, with semisimple part $L' {\mathop{\cong}}SL_2$. Then $P_\alpha = L B$, so $P_\alpha \cdot Z = L \cdot Z$. Applying Lemma \[lem:Hlocalization\], we learn $$\frac{[Z]_{L\times^B Z}}{\alpha} + \frac{r_\alpha\cdot [Z]_{L\times^B Z}}{-\alpha} = 1$$ as elements of $A_T(L\times^{B\cap L} Z)$. Let $\kappa$ denote the action map $L\times^{B\cap L} Z \to Y$, and apply $\kappa_*$ to both sides: $$\frac{[Z]_Y - r_\alpha\cdot [Z]_Y}{\alpha} = \kappa_*(1).$$ If $\kappa$ is generically finite of degree $d$, the right-hand side is $d\, [L\cdot Z]$, and otherwise $0$. (In this corollary we see the reason for $Lie(P)$ to contain all the [*negative*]{} root spaces rather than the positive ones; divided difference operators are usually defined for application to Schubert polynomials, which come from Schubert varieties that are $B_-$-invariant not $B$-invariant.) Use Proposition \[prop:BSshrinkdim\] to create a sequence $(\alpha_i)$. The condition in Proposition \[prop:BSshrinkdim\] on $(\alpha_i)$ is that $Z_i$ should grow in dimension at each step, which is the condition that the $d$ from Corollary \[cor:Joseph\] is nonzero. By the assumption that all the weights of $Y$ lie in a half-space, $[P_\alpha \cdot Z]_Y \neq 0$. Hence the dimension grows if and only if $\partial_\alpha$ does not act as zero. So the conditions on $(\alpha_i)$ in the theorem’s statement match those used in Proposition \[prop:BSshrinkdim\]. By Proposition \[prop:BSshrinkdim\], the map $BS_{\vec\alpha} Z \to Y$ has image $G\cdot Z$. The number of components in a general fiber of $G\times^B Z {\mathop{\twoheadrightarrow}}G\cdot Z$ is the degree of the map $BS_{\vec\alpha} Z {\mathop{\twoheadrightarrow}}G\cdot Z$. That degree is in turn the product of the degrees $d_i$ of the maps $P_{\alpha_i} \times^B Z_{i-1} \to Z_i$, since $BS_{\vec\alpha} Z {\mathop{\twoheadrightarrow}}G\cdot Z$ factors as $$\left(\prod_{i=1}^k P_\alpha \times^B\right) Z {\mathop{\twoheadrightarrow}}\left(\prod_{i=1}^{k-1} P_\alpha \times^B\right) Z_1 {\mathop{\twoheadrightarrow}}\left(\prod_{i=1}^{k-2} P_\alpha \times^B\right) Z_2 {\mathop{\twoheadrightarrow}}\cdots {\mathop{\twoheadrightarrow}}Z_k = G\cdot Z$$ where the $\{Z_i\}$ are as in the proof of Proposition \[prop:BSshrinkdim\], and the $j$th map is the associated map of bundles over $\left(\prod_{i=1}^j P_\alpha \times^B\right) \cdot pt$ to the $B$-equivariant map $P_{\alpha_i} \times^B Z_{i-1} \to Z_i$. Hence by $k$ applications of Corollary \[cor:Joseph\], $$d\, [G\cdot Z]_Y = \left(\prod_{i=1}^k d_i\right) [G\cdot Z]_Y = \left(\prod_{i=1}^k \partial_{\alpha_i}\right) [Z]_Y.$$ In the case $\kappa:G\times^P Z \to G\cdot Z$ is generically a finite map, we can use Lemma \[lem:BSforGmodP\] to know that for $\vec\alpha$ giving a reduced word for $w_0 w_0^P$, the map $BS_{\vec\alpha}\cdot Z \to G\cdot Z$ is also generically finite (with the same degree). To see the alternate formula, we apply (as in the proof of Joseph’s Lemma) the pushforward $\kappa_*$ to the equation from Lemma \[lem:Hlocalization\]: $$\kappa_*(1) = \sum_{w\in W^P} w\cdot \frac{[Z]_Y}{\prod_{\beta\in\Delta\setminus\Delta_P} \beta}.$$ Since $\kappa$ is generically finite of degree $d$, the left-hand side is $\kappa_*(1) = d\, [{\mathop{\mathrm{Im}}}\kappa]_Y$. The first part of this theorem only used Joseph’s Lemma (our Corollary \[cor:Joseph\]), rather than Lemma \[lem:Hlocalization\] directly. This will not be possible in the proof of Theorem \[thm:Kpoly\], where we will use a slightly different approach. Kempf assumed a condition on $Z$ that, among other things, forced the general fiber of a collapsing to be connected. While his extremely restrictive condition does not hold in our main application, we will at least have this connectedness, which is not shared by the following example. Let $G=SL_2({{\mathbb C}})$ act on $Y = {{\mathfrak sl}}_2({{\mathbb C}})$ via the adjoint action, and let $Z = {{\mathfrak b}}$ be the lower triangular matrices in $Y$. Let $T$ be the Cartan subgroup of $G$ consisting of diagonal matrices, and let $P=B$ be the lower triangular matrices in $G$. Then we run into the problem that the weights of $T$ acting on $Y$ are $\alpha,0,-\alpha$ where $\alpha$ is the simple root, and do not all lie in a half-space as required to apply the theorem. To rescue the example, we enlarge $G$ to $SL_2({{\mathbb C}}) \times {{\mathbb C}}^\times$, where the latter circle acts by rescaling on $Y$ and preserves $Z$. Likewise enlarge $T$ and $B$ by this rescaling circle. Now the weights are $\alpha+a,a,-\alpha+a$ where $a$ is the generator of the weight lattice of ${{\mathbb C}}^\times$. Recall that the multidegree $[Z]$ is the product of the weights [*not*]{} occurring in $Z$, in this case the one weight $\alpha+a$. Then the formula gives $d\, [G\cdot Z] = \partial_\alpha (\alpha+a) = 2$. And indeed, $G\cdot Z = Y$, so $[G\cdot Z]=1$, while the preimage in $G\times^B Z$ of a typical diagonal matrix ${{\rm diag}}(t,t^{-1})$ is $$\{ (g,z) : \mathrm{Ad}(g)\cdot z = {{\rm diag}}(t,t^{-1}) \}$$ which has $d=2!$ points, indexed by the permutations of the diagonal entries $t$ and $t^{-1}$. In the very similar example $G=SL_3({{\mathbb C}})$, with $Y,Z,P,B,T$ replaced by their $3\times 3$ counterparts, the general fiber has $3!$ points. We have $$\begin{aligned} 3! &=& \partial_{\alpha_1} \partial_{\alpha_2} \partial_{\alpha_1} (a+\alpha_1)(a+\alpha_2)(a+\alpha_1+\alpha_2) \\ &=& \sum_{w\in S_3} w \cdot \frac {(a+\alpha_1)(a+\alpha_2)(a+\alpha_1+\alpha_2)} {\alpha_1\, \alpha_2\, (\alpha_1+\alpha_2)}\end{aligned}$$ where $\alpha_1,\alpha_2$ are the simple roots of $SL_3({{\mathbb C}})$. $K$-polynomials and the proof of Theorem \[thm:Kpoly\]. ------------------------------------------------------- A $T$-equivariant coherent sheaf $\mathcal F$ on $Y$ is equivalent to a $T^*$-graded module $\Gamma$ over $Fun(Y)$. If we assume that the weights $\{\lambda_i\}$ of $T$ on $Y$ all live in a open half-space of $T^*$, then each graded piece $\Gamma_\lambda$ is finite-dimensional, and we can talk about the multigraded Hilbert series $H(\Gamma; t)$. It is a rational function, $$H(\Gamma; t) := \sum_{\lambda \in T^*} \dim(\Gamma_\lambda)\ t^\lambda = \frac{[\mathcal F]^K_Y}{\prod_{\lambda_i} (1 - t^{\lambda_i})}$$ whose numerator one calls the [[****]{}$K$-polynomial]{} of the sheaf $\mathcal F$. If $Z$ is a subscheme of $Y$, we will write $[Z]^K_Y$ for the $K$-polynomial of the structure sheaf of $Z$. It is a function on $T$, i.e. an element of the Laurent polynomial ring $K_T(Y) {\mathop{\cong}}K_T(pt)$. We need some results about $K$-polynomials, corresponding to those we used about multidegrees. The first, easily calculated from the Hilbert series definition, is that the $K$-polynomial of a linear subspace $Z\leq Y$ is the product $\prod (1-t^w)$ where $w$ varies over the weights of $Y/Z$. The analogue of Lemma \[lem:Hlocalization\] is almost word-for-word the same: \[lem:Klocalization\] Let $Z$ be a $P$-variety, and let $K_T(pt)_{frac}$ denote the field of fractions of the Laurent polynomial ring $K_T(pt)$. Then we have a formula in the localization $K_T(G\times^P Z){{\otimes}}_{K_T(pt)} K_T(pt)_{frac}$ of the equivariant $K$-ring $K_T(G\times^P Z)$: $$1 = \sum_{w\in W^P} w\cdot \frac {[Z]^K_{G\times^P Z}}{\prod_{\beta\in\Delta\setminus\Delta_P} (1-\exp(-\beta))}$$ where $[Z]^K_{G\times^P Z} \in A_T(G\times^P Z)$ is the class induced by the regularly embedded subvariety $Z$. Exactly the same proof holds, except that we need localization in torus-equivariant algebraic $K$-theory rather than Chow [@Th Théorème 2.1]. To apply this formula we need to understand the class $\kappa_!(1) \in K_T(Y)$. The pushforward $\kappa_!$ in $K$-theory is defined as the alternating sum of the higher direct images of $\kappa$, which are difficult to compute in general. An especially easy case is when $\kappa$ is a birational isomorphism, and both spaces have rational singularities; then $$\kappa_*(\mathcal O_{G\times^P Z}) = \mathcal O_{G\cdot Z}, \qquad R^i \kappa_*(\mathcal O_{G\times^P Z}) = 0\quad \forall i>0$$ so $\kappa_!(1) = [G\cdot Z]^K_Y$. Since $\kappa$ has connected fibers, by Proposition \[prop:BSshrinkdim\] the map $BS_{\vec\alpha}\cdot \iota$ is a birational isomorphism. Since $Z$ and $G\cdot Z$ have rational singularities, $(BS_{\vec\alpha}\cdot \iota)_!(1) = [G\cdot Z]^K_Y$ as just explained. Now we use Lemma \[lem:Klocalization\] to give a formula for $1 \in K_T(BS_{\vec\alpha}\cdot Z)$, and push it forward using $(BS_{\vec\alpha}\cdot \iota)_!$, where $\iota:Z\to Y$ is the inclusion. Unwinding this formula, we get the first formula claimed. (The reason we didn’t follow the same induction used in the proof of Theorem \[thm:multidegree\] is that while $Z$ and $G\cdot Z$ have rational singularities, we don’t know that the intermediate spaces constructed in Proposition \[prop:BSshrinkdim\] do (though this seems very likely).) The proof of the third formula is exactly the same as in Theorem \[thm:multidegree\], except that we need to invoke rationality of singularities. Finally, we prove the second formula from the third, using the map $G\times^B Z {\mathop{\twoheadrightarrow}}G\times^P Z$. This is a fibration with fibers $P/B$, and the map $\pi:P/B{\mathop{\twoheadrightarrow}}pt$ takes $\pi_!(1) = 1$ (the trivial line bundle case of Borel-Weil-Bott). Then we use Lemma \[lem:Klocalization\] to give a formula for $1\in K_T(G\times^B Z)$, which pushes forward to the desired formula for $[G\cdot Z]^K_Y$. (In $A_T$ rather than $K_T$, the pushforward of $1$ along $P/B{\mathop{\twoheadrightarrow}}pt$ is zero, which is why there was no analogous formula in Theorem \[thm:multidegree\].) Quiver representations {#sec:quiverreps} ====================== A [[****]{}representation]{} $V$ of a quiver $Q$ is a collection $\{V(i)\mid i\in Q_0\}$ of vector spaces and $\{V(a)\in{\mathrm{Hom}}_{\mathbb{C}}(V(ta),V(ha))\mid a\in Q_1\}$ of linear maps. We give the reference [@GR]. The path algebra ${\mathbb{C}}Q$ -------------------------------- A path of length $m>0$ is a sequence of arrows $p=a_1a_2\dotsm a_m$ such that $ha_i=ta_{i+1}$ for $1\le i\le m-1$. The tail and head of the path are given by $tp=ta_1$ and $hp=ha_m$ respectively. One should imagine that one starts at the vertex $tp=ta_1$ and walks along the arrow $a_1$ to $ha_1=ta_2$, thence along $a_2$ to $ha_2$, eventually stopping at $ha_m=hp$. For each $i\in Q_0$ there is a path of length zero also denoted $i$, with $hi=ti=i$. If $p$ and $p'$ are paths with $hp=tp'$ then their concatenation $pp'$ is a path. The [[****]{}path algebra]{} ${\mathbb{C}}Q$ of the quiver $Q$ is the associative ${\mathbb{C}}$-algebra with ${\mathbb{C}}$-basis given by the set of paths, and multiplication given by concatenation: $$p \cdot p' = \begin{cases} pp' &\text{if $hp=tp'$} \\ 0 &\text{otherwise.} \end{cases}$$ $Q_0$ forms a set of orthogonal idempotents for ${\mathbb{C}}Q$. Modules over ${\mathbb{C}}Q$ ---------------------------- Let ${\text{Mod-$\CQ$}}$ be the category of finite-dimensional [*right*]{} ${\mathbb{C}}Q$-modules. The structure of a module $V\in {\text{Mod-$\CQ$}}$ is determined as follows. From the action of $Q_0$ there is a direct sum decomposition $V \cong \bigoplus_{i\in Q_0} V(i)$ where $V(i) := V \cdot i$. The map ${\underline{\mathrm{dim}}\,}{V}: Q_0\rightarrow {{\mathbb N}}$ given by $i\mapsto \dim(V(i))$ is called the [[****]{}dimension vector]{} of $V$. For $i,j\in Q_0$ and $a\in Q_1$ we have $V \cdot i \cdot a \cdot j=0$ unless $i=ta$ and $j=ha$. Thus $a$ acts by zero on $V(i)$ for $i\not=ta$ and defines a linear map $V(a)\in {\mathrm{Hom}}_{\mathbb{C}}(V(ta),V(ha))$. So it is equivalent to work with ${\mathbb{C}}Q$-modules or with representations of $Q$. \[rem:rowvec\] We adopt the convention that matrices act on row vectors. Quiver loci and quiver polynomials ---------------------------------- We now change viewpoints, fixing a vector space and the action of the subalgebra ${\mathbb{C}}Q_0\subset {\mathbb{C}}Q$ on it, but letting the rest of the ${\mathbb{C}}Q$-module structure vary. Fix a dimension vector $d:Q_0\rightarrow{{\mathbb N}}$. Let $${\mathrm{Hom}}={\mathrm{Hom}}(Q,d)=\bigoplus_{a\in Q_1} {\mathrm{Hom}}_{\mathbb{C}}({\mathbb{C}}^{d(ta)},{\mathbb{C}}^{d(ha)})$$ be the space of all ${\mathbb{C}}Q$-module structures on the vector space $\bigoplus_{i\in Q_0} {\mathbb{C}}^{d(i)}$ where ${\mathbb{C}}^{d(i)}$ is the image of $i\in {\mathbb{C}}Q_0$. Let $GL=GL(Q,d)=\prod_{i\in Q_0} GL(d(i),{\mathbb{C}})$. The algebraic group $GL$ acts on ${\mathrm{Hom}}$ by change of basis: $(g\cdot V)(a) = g(ta) V(a) g(ha)^{-1}$ for all $g\in G$, $V\in {\mathrm{Hom}}$, and $a\in Q_1$. It is easy to check that $V,W\in{\mathrm{Hom}}$ are isomorphic as elements of ${\text{Mod-$\CQ$}}$ if and only if they are in the same $GL$-orbit. Indecomposables and multiplicities ---------------------------------- We want a nice way to index the quiver loci, which are in bijection with the isomorphism classes in ${\text{Mod-$\CQ$}}$. Let ${\mathrm{Indec}_{Q}}$ denote the set of isomorphism classes of indecomposables in ${\text{Mod-$\CQ$}}$. For simplicity of notation, we will sometimes write $U$ instead of $[U]$. For $V\in {\text{Mod-$\CQ$}}$ and $U\in {\mathrm{Indec}_{Q}}$, define the multiplicities ${\mathrm{m}}_U(V)$ of $V$ by $$\label{eq:lace} V \cong \bigoplus_{U\in{\mathrm{Indec}_{Q}}} U^{\oplus {\mathrm{m}}_U(V)}.$$ The multiplicities ${\mathrm{m}}(V)=({\mathrm{m}}_U(V)\mid U\in {\mathrm{Indec}_{Q}})$ determine $V$ up to isomorphism. Let $\Omega_{\mathrm{m}}:=\overline{GL\cdot V}$ for any $V$ with multiplicities ${\mathrm{m}}$. For the equioriented type $A$ quiver the multiplicities were in [@KMS] called the “lace array”. The Auslander-Reiten quiver {#ss:AR} --------------------------- We recall the definition of the Auslander-Reiten quiver ${\Gamma_{Q}}$ associated to the category ${\text{Mod-$\CQ$}}$ [@ARS]. A map $f$ is [[****]{}irreducible]{} if for all compositions of maps $f=gh$ with neither $g$ nor $h$ the identity, $g$ is not a split monomorphism and $h$ is not a split epimorphism. The [[****]{}Auslander-Reiten quiver ${\Gamma_{Q}}$ of $Q$]{} is the directed graph whose vertex set is ${\mathrm{Indec}_{Q}}$ with a directed edge from $[V]$ to $[W]$ if and only if there is an irreducible map $V\rightarrow W$. Extensions {#SS:ext} ---------- For $V,W\in{\text{Mod-$\CQ$}}$, call $E\in {\text{Mod-$\CQ$}}$ an [[****]{}extension of $V$ by $W$]{} if there is a short exact sequence $0 \rightarrow W \rightarrow E\rightarrow V \rightarrow 0$ of ${\mathbb{C}}Q$-modules. For each $i\in Q_0$ choose a basis of $E(i) \cong W(i)\oplus V(i)$ that consists of a basis of $W(i)$ followed by a basis of $V(i)$ and write the linear maps with respect to this basis. With our row-vector conventions of Remark \[rem:rowvec\], $E(a)$ has the form $$\label{eq:block} E(a) = \begin{pmatrix} W(a) & 0 \\ * & V(a) \end{pmatrix}.$$ Let $E(V,W)$ be the set of extensions of $V$ by $W$ with fixed underlying vector space $V\oplus W$. There is a linear isomorphism $$\label{eq:extmap} \bigoplus_{a\in Q_1} {\mathrm{Hom}}_{\mathbb{C}}(V(ta),W(ha))\rightarrow E(V,W)$$ whose $a$-th component is given by replacing the submatrix $*$ in with the element of ${\mathrm{Hom}}_{\mathbb{C}}(V(ta),W(ha))$ for $a\in Q_1$. Say that $E,E'\in E(V,W)$ are equivalent if there is a ${\mathbb{C}}Q$-module isomorphism $E\rightarrow E'$ whose restriction to $W$ is the identity and whose induced map $E/W\rightarrow E'/W$ is the identity. ${\mathrm{Ext}}^1_Q(V,W)$ is isomorphic to $E(V,W)$ modulo the above equivalence (see for example [@Ro Thm. 7.21]). The canonical resolution ------------------------ For $V,W\in{\text{Mod-$\CQ$}}$ let ${\mathrm{Hom}}_Q(V,W)$ be the space of right ${\mathbb{C}}Q$-module homomorphisms from $V$ to $W$. There is an exact sequence [@Ri] $$\label{eq:res} \begin{split} 0 \rightarrow {\mathrm{Hom}}_Q(V,W) \overset{j}\rightarrow \displaystyle{\bigoplus_{i\in Q_0}} {\mathrm{Hom}}(V(i),W(i)) &\overset{d_V^W}\rightarrow \displaystyle{\bigoplus_{a\in Q_1}} {\mathrm{Hom}}(V(ta),W(ha)) \\ &\overset{p}{\rightarrow} {\mathrm{Ext}}^1_Q(V,W) \rightarrow 0 \end{split}$$ where $j$ is inclusion, $p$ is induced by the map in and $d_V^W$ is given by $$(d_V^W(f))_a = V_a f_{ha} - f_{ta} W_a\qquad\text{for $a\in Q_1$.}$$ The exactness of gives $$\begin{aligned} \label{eq:drank} \dim {\mathrm{Hom}}_{\mathbb{C}}(V,W) &= {\mathrm{rank\,}}d_V^W+\dim {\mathrm{Hom}}_Q(V,W).\end{aligned}$$ The homological form -------------------- Let $V,W\in {\text{Mod-$\CQ$}}$. The [[****]{}homological form]{} is defined by $$\begin{aligned} {\langle V\,,\,W\rangle} = \sum_{i\ge0} (-1)^i \dim {\mathrm{Ext}}_Q^i(V,W).\end{aligned}$$ The exact sequence implies that ${\text{Mod-$\CQ$}}$ is hereditary (that is, ${\mathrm{Ext}}_Q^i(V,W)=0$ for $i\ge 2$) and its exactness gives $$\label{eq:HomExt} \begin{split} {\langle V\,,\,W\rangle} &= \dim {\mathrm{Hom}}_Q(V,W) - \dim {\mathrm{Ext}}^1_Q(V,W) \\ &= \sum_{i\in Q_0} \dim V(i) \dim W(i) - \sum_{a\in Q_1} \dim V(ta) \dim W(ha) \\ &= {\langle {\underline{\mathrm{dim}}\,}V\,,\,{\underline{\mathrm{dim}}\,}W\rangle} \end{split}$$ where, for dimension vectors $d,d':Q_0\rightarrow {{\mathbb N}}$ we write $${\langle d\,,\,d'\rangle} = \sum_{i\in Q_0} d(i)d'(i) - \sum_{a\in Q_1} d(ta)d'(ha).$$ Codimension and Ext ------------------- By for $V=W$ and the fact that ${\mathrm{Hom}}_Q(V,V)$ is the closure of the stabilizer of $V$ in $GL$, we have $$\begin{aligned} \dim GL - \dim {\mathrm{Hom}}&= {\langle V\,,\,V\rangle} = \dim {\mathrm{Hom}}_Q(V,V) - \dim {\mathrm{Ext}}^1_Q(V,V) \\ &= (\dim GL - \dim GL\cdot V) - \dim {\mathrm{Ext}}^1_Q(V,V).\end{aligned}$$ This implies that for $V\in{\mathrm{Hom}}$, we have $$\begin{aligned} {\mathrm{codim}\,}\overline{GL\cdot V} =\dim {\mathrm{Ext}}^1_Q(V,V).\end{aligned}$$ Let ${\mathrm{m}}$ be a set of multiplicities with $\Omega_{\mathrm{m}}\subset {\mathrm{Hom}}$. Then $$\label{eq:codim} {\mathrm{codim}\,}\, \Omega_{\mathrm{m}}= \sum_{U,W\in {\mathrm{Indec}_{Q}}} {\mathrm{m}}_U {\mathrm{m}}_W \,\dim {\mathrm{Ext}}^1_Q(U,W).$$ Quivers of finite type {#sec:finitequivers} ====================== Let $X_n$ be a simply-laced root system of rank $n$; it is either $A_n$ for $n\geq 1$, $D_n$ for $n\geq 4$, or $E_n$ for $n=6,7,8$, where $n$ is always the number of nodes in the Dynkin diagram. We shall also write $X_n$ for the undirected graph given by its Dynkin diagram. $$\begin{aligned} \psset{xunit=.75cm,yunit=.75cm} \begin{matrix} \pspicture(-.5,-.75)(4.5,.75) \psdots(0,0)(1,0)(2,0)(3,0)(4,0) \psline(0,0)(2,0) \psline(3,0)(4,0) \psline[linestyle=dashed](2,0)(3,0)\endpspicture \\ A_n \end{matrix} \qquad \begin{matrix} \pspicture(-.5,-.75)(4.5,.75) \psdots(0,0)(1,0)(2,0)(3,0)(4,.5)(4,-.5) \psline(0,0)(1,0) \psline[linestyle=dashed](1,0)(2,0) \psline(2,0)(3,0)(4,.5)(3,0)(4,-.5)\endpspicture \\ D_n \end{matrix}\end{aligned}$$ $$\begin{aligned} \psset{xunit=.75cm,yunit=.75cm} \begin{matrix} \pspicture(-.5,-.25)(4.5,1.25) \psdots(0,0)(1,0)(2,0)(3,0)(4,0)(2,1) \psline(0,0)(4,0) \psline(2,0)(2,1)\endpspicture \\ E_6 \end{matrix}\qquad \begin{matrix} \pspicture(-.5,-.25)(5.5,1.25) \psdots(0,0)(1,0)(2,0)(3,0)(4,0)(5,0)(2,1) \psline(0,0)(5,0) \psline(2,0)(2,1)\endpspicture \\ E_7 \end{matrix}\qquad \begin{matrix} \pspicture(-.5,-.25)(6.5,1.25) \psdots(0,0)(1,0)(2,0)(3,0)(4,0)(5,0)(6,0)(2,1) \psline(0,0)(6,0) \psline(2,0)(2,1)\endpspicture \\ E_8 \end{matrix}\end{aligned}$$ An [[****]{}orientation]{} of an undirected multigraph is a quiver obtained by choosing directions for the edges of the undirected graph. Orientations of the Dynkin diagrams of simply-laced root systems are called [[****]{}Dynkin]{} quivers. A quiver $Q$ is of [[****]{}finite type]{} if, for every dimension vector $d:Q_0\rightarrow{{\mathbb N}}$, there are finitely many isomorphism classes of representations of $Q$ with dimension vector $d$. By Gabriel’s Theorem [@G] a quiver is of finite type if and only if it is Dynkin. In this section we shall assume that $Q$ is Dynkin. Dimension vectors and roots --------------------------- We recall some well-known results of Gabriel. There is a bijection from $Q_0$ to the set of simple roots of $X_n$ given by $i\mapsto \alpha_i$. Any dimension vector $d:Q_0\rightarrow{{\mathbb N}}$ may be viewed as an element of the positive cone of roots $\bigoplus_{i\in Q_0} {{\mathbb N}}\alpha_i$, namely, $\sum_{i\in Q_0} d(i) \alpha_i$. Let $R^+$ be the set of positive roots of $X_n$.[^3] There is a bijection ${\mathrm{Indec}_{Q}}\rightarrow R^+$ given by $U\mapsto {\underline{\mathrm{dim}}\,}U$. $U$ is indecomposable if and only if ${\langle {\underline{\mathrm{dim}}\,}U\,,\,{\underline{\mathrm{dim}}\,}U\rangle}=1$. Dynkin quivers and orders on $R^+$ ---------------------------------- Let $s_i$ denote a simple reflection for the Weyl group $W(X_n)$ of $X_n$[^4] and let $w_0\in W(X_n)$ be the longest element. For $w\in W(X_n)$ let ${\mathcal{R}}(w)\subset Q_0^{\ell(w)}$ denote the set of reduced words for $w$. Given an orientation $Q$ of $X_n$ and a vertex $i\in Q_0$, let $s_i Q$ be the orientation of $X_n$ given by reversing all arrows with head $i$. Say that a reduced word ${\mathbf{a}}=a_1a_2\dotsm\in{\mathcal{R}}(w_0)$ is [[****]{}adapted]{} to the orientation $Q$ of $X_n$ if $a_j$ is a sink (the tail of no arrow) in $s_{a_{j-1}}\dotsm s_{a_2}s_{a_1} Q$ for all $j$. By [@BGP], for every orientation $Q$ of $X_n$, there is a reduced word ${\mathbf{a}}\in{\mathcal{R}}(w_0)$ that is adapted to $Q$. Each reduced word ${\mathbf{a}}=a_1a_2\dotsm\in{\mathcal{R}}(w_0)$ defines a linear ordering on $R^+$ given by $$\label{eq:rootorder} \gamma_1<\gamma_2<\dotsm$$ where $$\label{eq:rootdef} \gamma_j=s_{a_1} \dotsm s_{a_{j-1}} (\alpha_{a_j}).$$ Auslander-Reiten quiver reprise ------------------------------- There is a combinatorial recipe for the Auslander-Reiten quiver ${\Gamma_{Q}}$ of a quiver $Q$ that is an orientation of a Dynkin diagram $X_n$ of type ADE. This is well-known to the experts; see [@Be; @Z]. The vertices of ${\Gamma_{Q}}$ shall be drawn in the plane in rows indexed by the set $Q_0$ and columns indexed by ${\mathbb{Z}}_{>0}$. Let ${\mathbf{a}}\in{\mathcal{R}}(w_0)$ be adapted to $Q$. Let $\gamma_j\in R^+$ be defined as in . Let $c_1 = 1$, and $c_j = c_{j-1}$ unless for some $k<j$ with $c_k = c_{j-1}$, $\gamma_k$ is adjacent in $X_n$ to $\gamma_j$; in this case let $c_j = c_{j-1}+1$. The vertex $\gamma_j$ is drawn in row $a_j$ and column $c_j$. Draw a directed edge from $\gamma_j$ to $\gamma_k$ if $j<k$, $a_j$ and $a_k$ are adjacent in $X_n$, and $k$ is minimal with this property. Let $X_n=D_4$ with orientation $Q$ given below. $$\pspicture(0,-.75)(2,.75) \cnodeput(2,.5){A1}{3} \cnodeput(0,0){A3}{1} \cnodeput(1,0){A2}{2} \cnodeput(2,-.5){A4}{4} \ncline[arrowscale=1.5]{->}{A1}{A2} \ncline[arrowscale=1.5]{<-}{A2}{A3} \ncline[arrowscale=1.5]{<-}{A2}{A4} \endpspicture$$ One reduced word adapted to $Q$ is $213423142341$. The corresponding roots have expansions in the simple roots by the following matrix. $$\begin{aligned} \begin{pmatrix} \gamma_1 \\ \gamma_2 \\ \gamma_3 \\ \gamma_4 \\ \gamma_5 \\ \gamma_6 \\ \gamma_7 \\ \gamma_8 \\ \gamma_9 \\ \gamma_{10} \\ \gamma_{11} \\ \gamma_{12} \end{pmatrix} = \begin{pmatrix} 0&1&0&0 \\ 1&1&0&0 \\ 0&1&1&0 \\ 0&1&0&1 \\ 1&2&1&1 \\ 1&1&0&1 \\ 0&1&1&1 \\ 1&1&1&0 \\ 1&1&1&1 \\ 0&0&0&1 \\ 1&0&0&0 \\ 0&0&1&0 \end{pmatrix} \cdot \begin{pmatrix} \alpha_1 \\ \alpha_2 \\ \alpha_3 \\ \alpha_4 \end{pmatrix}\end{aligned}$$ The Auslander-Reiten quiver ${\Gamma_{Q}}$ is given by $$\pspicture(-1,.75)(5,4.25) \rput(-1,4){4} \rput(-1,3){3} \rput(-1,2){2} \rput(-1,1){1} \cnodeput(0,2){G1}{$\gamma_1$} \cnodeput(1,1){G2}{$\gamma_2$} \cnodeput(1,3){G3}{$\gamma_3$} \cnodeput(1,4){G4}{$\gamma_4$} \cnodeput(2,2){G5}{$\gamma_5$} \cnodeput(3,1){G7}{$\gamma_7$} \cnodeput(3,3){G6}{$\gamma_6$} \cnodeput(3,4){G8}{$\gamma_8$} \cnodeput(4,2){G9}{$\gamma_9$} \cnodeput(5,1){G12}{$\gamma_{12}$} \cnodeput(5,3){G10}{$\gamma_{10}$} \cnodeput(5,4){G11}{$\gamma_{11}$} \ncline[arrowscale=1.5]{->}{G1}{G2} \ncline[arrowscale=1.5]{->}{G1}{G3} \ncline[arrowscale=1.5]{->}{G1}{G4} \ncline[arrowscale=1.5]{->}{G2}{G5} \ncline[arrowscale=1.5]{->}{G3}{G5} \ncline[arrowscale=1.5]{->}{G4}{G5} \ncline[arrowscale=1.5]{->}{G5}{G6} \ncline[arrowscale=1.5]{->}{G5}{G7} \ncline[arrowscale=1.5]{->}{G5}{G8} \ncline[arrowscale=1.5]{->}{G6}{G9} \ncline[arrowscale=1.5]{->}{G7}{G9} \ncline[arrowscale=1.5]{->}{G8}{G9} \ncline[arrowscale=1.5]{->}{G9}{G10} \ncline[arrowscale=1.5]{->}{G9}{G11} \ncline[arrowscale=1.5]{->}{G9}{G12} \endpspicture$$ Since nodes $1,3,4$ have no connections in $D_4$, the orders they appear in the reduced word $2\,134\,2\,314\,2\,341$ don’t affect the shape of the Auslander-Reiten quiver. \[rem:DynkinExt\] For $Q$ an orientation of the Dynkin diagram of a simply-laced root system $X_n$ and ${\mathbf{a}}\in{\mathcal{R}}(w_0)$ a reduced word adapted to $Q$, let the positive roots (hence the indecomposables) be totally ordered as in . Then for $V,W\in{\mathrm{Indec}_{Q}}$ we have [@Ri2] $$\begin{aligned} \label{E:Extzero} {\mathrm{Ext}}^1_Q(V,W) &= 0 \qquad&\text{if $V\le W$}. \end{aligned}$$ The poset of quiver loci in ${\mathrm{Hom}}$ -------------------------------------------- \[thm:oc\] [@Bo] Let $Q$ be of finite type and $V,W\in{\text{Mod-$\CQ$}}$ with ${\underline{\mathrm{dim}}\,}{V}={\underline{\mathrm{dim}}\,}{W}$. The following are equivalent: 1. $\overline{GL\cdot V} \subset \overline{GL\cdot W}$. 2. $\dim {\mathrm{Hom}}_Q(U,V) \le \dim {\mathrm{Hom}}_Q(U,W)$ for all $U\in{\mathrm{Indec}_{Q}}$. 3. $\dim {\mathrm{Ext}}^1_Q(U,V)\ge \dim {\mathrm{Ext}}^1_Q(U,W)$ for all $U\in{\mathrm{Indec}_{Q}}$. 4. ${\mathrm{rank\,}}d_U^V \ge {\mathrm{rank\,}}d_U^W$ for all $U\in{\mathrm{Indec}_{Q}}$. Note that the latter three are equivalent for any quiver $Q$, by and . The Reineke filtration ---------------------- We recall a special case of Reineke’s filtration [@R]. Let $Q$ be an orientation of a Dynkin diagram $X_n$ of type ADE, ${\mathbf{a}}\in{\mathcal{R}}(w_0)$ adapted to $Q$, with the associated total order $\le$ on ${\mathrm{Indec}_{Q}}$. We list the elements of ${\mathrm{Indec}_{Q}}$ in descending order: ${\mathrm{Indec}_{Q}}=\{\beta_1>\beta_2>\dotsm>\beta_N\}$ where $N=|R^+|$; the decreasing indexing is for technical convenience related to our row-vector convention of Remark \[rem:rowvec\]. For short we write $I_j$ for the indecomposable instead of $I_{\beta_j}$. Let $V\in{\text{Mod-$\CQ$}}$, $d={\underline{\mathrm{dim}}\,}{V}$, $GL=GL(Q,d)$, ${\mathrm{Hom}}={\mathrm{Hom}}(Q,d)$. Let $V$ have multiplicities ${\mathrm{m}}_j(V)={\mathrm{m}}_{I_j}(V)$ as in . For $1\le j\le N$ write $W_j = I_j^{\oplus {\mathrm{m}}_j(V)}$ and $V_j = W_1 \oplus\dotsm\oplus W_j$. Let $P\subset GL$ be the parabolic subgroup such that for all $i\in Q_0$, the $i$-th component $P(i)\subset GL({\mathbb{C}}^{d(i)})$ is the stabilizer of $V_j(i)$ for all $1\le j\le N$. Note that $P$ has Levi factor $L=\prod_{i\in Q_0} \prod_{j=1}^N GL(W_j(i))\cong \prod_{j=1}^N GL(W_j)$. Let $Z,Z'\subset Y:={\mathrm{Hom}}$ be the coordinate subspaces defined by $Z'(a) = \bigoplus_{j=1}^N {\mathrm{Hom}}_{{\mathbb{C}}}(W_j(ta),W_j(ha))\subset {\mathrm{Hom}}_{{\mathbb{C}}}(V(ta),V(ha))$ and $Z(a)=\bigoplus_{1\le j\le m\le N} {\mathrm{Hom}}_{{\mathbb{C}}}(W_m(ta),W_j(ha))$. For each $a\in Q_1$, $Z'(a)$ is “block diagonal" and $Z(a)$ is “block lower triangular" inside the matrices${\mathrm{Hom}}_{{\mathbb{C}}}(V(ta),V(ha))$. We claim that $$Z = \overline{P \cdot V}$$ inside ${\mathrm{Hom}}$. By and we have ${\mathrm{codim}\,}_{{\mathrm{Hom}}(Q,W_j)} \overline{GL(W_j)\cdot W_j}={\mathrm{Ext}}^1_Q(W_j,W_j)=0,$ or equivalently, $\overline{L\cdot V} = Z'$. So it suffices to show $$Z = \overline{U \cdot Z'}$$ where $U$ is the unipotent radical of $P$. But this follows by induction from the definition of Ext in Subsection \[SS:ext\] combined with the fact that by we have $${\mathrm{Ext}}^1_Q(W_p,W_q) = 0\qquad\text{for $p<q$.}$$ The linear space $Z$ is the base of our Bott-Samelson induction. Given a quiver locus $\Omega=\overline{GL\cdot V}\subset {\mathrm{Hom}}$, we start with $Z=\overline{P \cdot V}\subset{\mathrm{Hom}}$. Then $\overline{GL\cdot Z} =\Omega$. Since $Z$ is a coordinate subspace, $[Z]\in H_T^*({\mathrm{Hom}})$ and $[{\mathcal{O}}_Z]\in K_T^*({\mathrm{Hom}})$ have simple product formulae. Applying Theorem \[thm:multidegree\] we obtain divided difference formulae for the multidegree of the quiver locus $\Omega$. By Theorem \[thm:Kpoly\], for quivers of type AD we obtain divided difference formulae for the $K$-polynomial of $\Omega$. We use an unnecessarily fine filtration. One may use a directed partition of $R^+$ as defined in [@R] to obtain a coarser filtration of $V$, which leads to a more efficient divided difference formula. Let $Q$ be the type $A_2$ quiver with dimension vector $(m,n)$. Then ${\mathrm{Hom}}(Q,d)=M_{m\times n}({{\mathbb C}})$. For each $0\le r\le \min(m,n)$ there is a quiver locus $\Omega_r\subset M_{m\times n}({{\mathbb C}})$ given by the determinantal variety of matrices of rank at most $r$. Using the reduced word $s_2 s_1s_2$ we have $W_1=I_{\alpha_1}^{\oplus (m-r)}$, $W_2 = I_{\alpha_1+\alpha_2}^{\oplus r}$, and $W_3 = I_{\alpha_2}^{\oplus (n-r)}$. The indecomposables can be realized by matrices as follows: $I_{\alpha_1}$ is a $1\times 0$ matrix, $I_{\alpha_1+\alpha_2}$ can be taken to be the $1\times 1$ identity matrix, and $I_{\alpha_2}$ is the $0\times 1$ matrix. With respect to bases adapted to the ordered direct sum $V=W_1\oplus W_2\oplus W_3$, $V\in M_{m\times n}({{\mathbb C}})$ has the $r\times r$ identity matrix in its lower left corner and zeroes elsewhere. Then $P(1)\subset GL(m)$ and $P(2)\subset GL(n)$ are block lower triangular with diagonal blocks of sizes $(m-r,r)$ and $(r,n-r)$ respectively. We have $Z=Z'$; both are equal to the linear subspace of $M_{m\times n}$ where the bottom left $r\times r$ submatrix is arbitrary and the other entries are zero. Let $T(m)\subset GL(m)$ and $T(n)\subset GL(n)$ have weights $X=(x_1,\dotsc,x_m)$ and $Y=(y_1,\dotsc,y_n)$ respectively. Since the parabolics $P(1)$ and $P(2)$ are lower triangular, the positive roots of $P(1)$ have weights $x_j-x_i$ for $1\le i<j\le m$ and those of $P(2)$ have weight $y_j-y_i$ for $1\le i<j\le n$. So for $(m,n)=(2,3)$ and $r=1$ we have $$\begin{aligned} [Z] &= (x_1-y_1)(x_1-y_2)(x_1-y_3)(x_2-y_2)(x_2-y_3) \\ [\Omega] &= {\partial}_{x_1-x_2} {\partial}_{y_1-y_2} {\partial}_{y_2-y_3} [Z] \\ &= s_2[X-Y],\end{aligned}$$ the double Schur polynomial. In general the multidegree is given by the Giambelli-Thom-Porteous formula $[\Omega_r] = s_{(m-r)\times (n-r)}[X-Y]$, where the answer is the double Schur polynomial indexed by the $(m-r)\times (n-r)$ rectangle. Beyond $ADE$ quivers ==================== Let $Q$ be a quiver, $d:Q_0 \to {{\mathbb N}}$ a dimension vector, and ${\mathrm{Hom}}$ the associated space of representations. Then as long as $Q$ has no self-loops ($ta=ha$ for some edge $a$), and no repeated edges ($ta=tb$, $ha=hb$ for two edges $a\neq b$), the weights of $T$ on ${\mathrm{Hom}}$ are all distinct. Consequently, there are only finitely many $T$-invariant subspaces in ${\mathrm{Hom}}$ (precisely $2^{\dim{\mathrm{Hom}}}$), and hence [*only finitely many*]{} $B$-invariant subspaces $Z$ to which to apply Kempf’s construction. Whereas there may be infinitely many quiver loci. This (and the fact that quiver loci can have bad singularities [@Zw section 6]) suggests that instead of quiver loci, perhaps the better-behaved objects of study are the $GL$-sweeps of the $B$-invariant subspaces. From this point of view it is merely an accident (and Reineke’s theorem) that in the $ADE$ case, the two notions coincide. [KM01]{} M. Auslander, I. Reiten, and S. O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, Cambridge, 1995. R. Bédard, On commutation classes of reduced words in Weyl groups.\ European J. Combin. **20** (1999), no. 6, 483–505. G. Bobiński, G. 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[^1]: This condition on the weights is not as restrictive as it looks. If $Z\subseteq Y$ is invariant under rescaling (i.e. is the affine cone over a projective variety), then we can extend the action of $T$ to $T\times {{\mathbb G}_m}$ where ${{\mathbb G}_m}$ acts by dilation, and now all the weights live in $T^* \times \{1\}$. If $Z$ is not already rescaling-invariant, we can replace it by the limit subscheme $Z' := \lim_{t\to 0} t\cdot Z$, and compute the more refined multidegree $[Z'] \in {\mathrm{Sym}}^\bullet((T\times {{\mathbb G}_m})^*)$. Afterwards $[Z]$ can be computed as the image of $[Z']$ in ${\mathrm{Sym}}^\bullet(T)^*$ (and this image may indeed be zero). [^2]: The term “quiver varieties” is already taken, to refer to the hyperkähler quotients $({\mathrm{Hom}}{{\otimes}}{\mathbb H}) /// GL$. [^3]: We use this notation to distinguish the root system of $X_n$ with that of the group $GL$. [^4]: Again this notation is to distinguish $s_i$ from the reflection $r_i$ in the Weyl group of $GL$.

--- abstract: 'For a cyclic group $G$ acting on a smooth variety $X$ with only one character occurring in the $G$-equivariant decomposition of the normal bundle of the fixed point locus, we study the derived categories of the orbifold $[X/G]$ and the blow-up resolution $\wY\to X/G$. Some results generalise known facts about $X=\IA^n$ with diagonal $G$-action, while other results are new also in this basic case. In particular, if the codimension of the fixed point locus equals $|G|$, we study the induced tensor products under the equivalence $\Db(\wY)\cong \Db([X/G])$ and give a ’flop-flop=twist’ type formula. We also introduce candidates for general constructions of categorical crepant resolutions inside the derived category of a given geometric resolution of singularities and test these candidates on cyclic quotient singularities.' author: - Andreas Krug - David Ploog - Pawel Sosna bibliography: - 'references.bib' title: Derived categories of resolutions of cyclic quotient singularities --- [^1] [^2] Introduction {#introduction .unnumbered} ============ For geometric, homological and other reasons, it has become commonplace to study the bounded derived category of a variety. One of the many intriguing aspects are connections, some of them conjectured, some of them proven, to birational geometry. One expected phenomenon concerns a birational correspondence $$\xymatrix{ & Z\ar_q[dl] \ar^p[dr] & \\ X \ar@{-->}[rr] & & X' }$$ of smooth varieties. Then we should have: - \[conj1\] A fully faithful embedding $\Db(X)\hookrightarrow \Db(X')$, if $q^*K_X\le p^* K_{X'}$. - \[conj2\] A fully faithful embedding $\Db(X')\hookrightarrow \Db(X)$, if $q^*K_X\ge p^* K_{X'}$. - An equivalence $\Db(X')\cong \Db(X)$, in the *flop* case $q^*K_X= p^* K_{X'}$. This is proven in many instances; see [@Bondal-Orlov2], [@Bridgelandflops], [@KawDK], [@NamikawaMukaiflops]. Another very interesting aspect of derived categories is their occurrence in the context of the McKay correspondence. Here, one of the key expectations is that the derived category of a crepant resolution $\wY\to X/G$ of a Gorenstein quotient variety is derived equivalent to the corresponding quotient orbifold: $\Db(\wY) \cong \Db([X/G]) = \Db_G(X)$. In [@BKR], this expectation is proven in many cases under the additional assumption that $\wY\cong \Hilb^{G}(X)$ is the fine moduli space of $G$-clusters on $X$. It is enlightening to view the derived McKay correspondence as an orbifold version of the conjecture on derived categories under birational correspondences described above; for more information on this point of view, see [@Kaw16 Sect. 2], where the conjecture is called the *DK-Hypothesis*. Indeed, if we denote the universal family of $G$-clusters by $\cZ\subset \wY\times X$, we have the following diagram of birational morphisms of orbifolds $$\begin{aligned}\label{orbiflop} \xymatrix@!C=2em{ & [\cZ/G]\ar_q[dl] \ar^p[dr] & \\ \wY \ar@{-->}[rr] \ar_\rho[dr] & & [X/G] \ar^\pi[dl] \\ & X/G \,.} \end{aligned}$$ Since the pullback of the canonical sheaf of $X/G$ under $\pi$ is the canonical sheaf of $[X/G]$, the condition that $\rho$ is a crepant resolution amounts to saying that is a flop of orbifolds. In many situations, a crepant resolution of $X/G$ does not exist. However, given a resolution $\rho\colon \wY\to X$, the DK-Hypothesis still predicts the behavior of the categories $\Db(\wY)$ and $\Db_G(X)$ if $\rho^*K_{X/G}\ge K_{\wwY}$ or $\rho^*K_{X/G}\le K_{\wwY}$. Another related idea is that, even though a crepant resolution does not exist in general, there should always be a *categorical crepant resolution* of $\Db(X/G)$; see [@KuzLefschetz]. The hope is to find such a categorical resolution as an admissible subcategory of the derived category $\Db(\wY)$ of a geometric resolution. Besides dimensions 2 and 3, one of the most studied testing grounds for the above, and related, ideas is the isolated quotient singularity $\IA^n/\mu_m$. Here, the cyclic group $\mu_m$ of order $m$ acts on the affine space by multiplication with a primitive $m$-th root of unity $\zeta$. In this paper, we consider the following straight-forward generalisation of this set-up. Namely, let $X$ be a quasi-projective smooth complex variety acted upon by the finite cyclic group $\mu_m$. We assume that only $1$ and $\mu_m$ occur as the isotropy groups of the action and write $S\coloneqq\Fix(\mu_m)\subset X$ for the fixed point locus. Fix a generator $g$ of $\mu_m$ and assume that $g$ acts on the normal bundle $N\coloneqq N_{S/X}$ by multiplication with some fixed primitive $m$-th root of unity $\zeta$. Then the blow-up $\wY\to X/\mu_m$ with center $S$ is a resolution of singularities; see for further details. There are four particular cases we have in mind: 1. \[basiccase\] $X=\IA^n$ with the diagonal action of any $\mu_m$. 2. $X=Z^2$, where $Z$ is a smooth projective variety of arbitrary dimension, and $\mu_2=\sym_2$ acts by permuting the factors. Then $\wY\cong Z^{[2]}$, the Hilbert scheme of two points. 3. $X$ is an abelian variety, $\mu_2$ acts by $\pm 1$. In this case, $\wY$ is known as the *Kummer resolution*. 4. $X \to Y=X/\mu_m$ is a cyclic covering of a smooth variety $Y$, branched over a divisor. Here, $n=1$ and $\wX=X$, $\wY=Y$. This case has been studied in [@KuzPerrycyclic]. First, we prove the following result in . This is probably well-known to experts, but we could not find it in the literature. Write $G \coloneqq \mu_m$. The resolution obtained by blowing up the fixed point locus in $X/G$ is isomorphic to the $G$-Hilbert scheme: $\wY \cong \Hilb^{G}(X)$. We set $n \coloneqq \codim(S\hookrightarrow X)$ and find the following dichotomy, in accordance with the DK-Hypothesis. We keep the notation from diagram . In particular, for $n=m$, we obtain new instances of BKR-style derived equivalences between orbifold and resolution. \[thm:main\] 1. The functor $\Phi \coloneqq p_*q^*\colon \Db(\wY)\to \Db_G(X)$ is fully faithful for $m\ge n$ and an equivalence for $m=n$. For $m>n$, there is a semi-orthogonal decomposition of $\Db_G(X)$ consisting of $\Phi(\Db(\wY))$ and $m-n$ pieces equivalent to $\Db(S)$. 2. The functor $\Psi \coloneqq q_*p^*\colon\Db_G(X)\to \Db(\wY) $ is fully faithful for $n\ge m$ and an equivalence for $n=m$. For $n>m$, there is a semi-orthogonal decomposition of $\Db(\wY)$ consisting of $\Psi(\Db_G(X))$ and $n-m$ pieces equivalent to $\Db(S)$. For a more exact statement with an explicit description of the embeddings of the $\Db(S)$ components into $\Db(\wY)$ and $\Db_G(X)$, see . In particular, for $m>n$, the push-forward $a_*\colon \Db(S)\to \Db_G(X)$ along the embedding $a\colon S\hookrightarrow X$ of the fixed point divisor is fully faithful. In the basic affine case \[basiccase\], the result of the theorem is also stated in [@Kaw16 Ex. 4] and there are related results in the more general toroidal case in [@Kaw16 Sect. 3]. Proofs, in the basic case, are given in [@Abuaf-catres Sect. 4] for $n\ge m$ and in [@IUspecial] for $n=2$. If $n=1$, the quotient is already smooth and we have $\wY=X/G$ — here the semi-orthogonal decomposition categorifies the natural decomposition of the orbifold cohomology; compare [@PolvdB]. The $n=1$ case is also proven in [@Lim Thm. 3.3.2]. We study the case $m=n$, where $\Phi$ and $\Psi$ are equivalences, in more detail. On both sides of the equivalence, we have distinguished line bundles. The line bundle $\reg_{\widetilde Y}(Z)$ on $\wY$, corresponding to the exceptional divisor, admits an $m$-th root $\cL$. On $[X/G]$, there are twists of the trivial line bundle by the group characters $\reg_X\otimes \chi^i$. For $i=-m+1, \dots, -1,0$, we have $\Psi(\reg_X\otimes \chi^i)\cong \cL^i$. Furthermore, we see that the functors $\Db(S)\to \Db(\wY)$ and $\Db(S)\to \Db_G(X)$, which define fully faithful embeddings in the $n>m$ and $m<n$ cases, respectively, become spherical for $m=n$ and hence induce twist autoequivalences; see for details on spherical functors and twists. We show that the tensor products by the distinguished line bundles correspond to the spherical twists under the equivalences $\Psi$ and $\Phi$. In particular, one part of is the following formula. \[thm:tensor\] There is an isomorphism $\Psi^{-1}(\Psi(\_)\otimes \cL^{-1}) \cong \TT^{\;-1}_{a_*}(\_\otimes \chi^{-1})$ of autoequivalences of $\Db_G(X)$ where the inverse spherical twist $\TT^{\;-1}_{a_*}$ is defined by the exact triangle of functors $$\TT^{\;-1}_{a_*} \to \id \to a_*(a^*(\_)^G) \to \,.$$ The tensor powers of the line bundle $\cL$ form a strong generator of $\Db(\wY)$, thus , at least theoretically, completely describes the tensor product $$\_\widehat\otimes \_ \coloneqq \Psi^{-1}(\Psi(\_)\otimes \Psi(\_))\colon \Db_G(X)\times \Db_G(X)\to \Db_G(X)$$ induced by $\Psi$ on $\Db_G(X)$. There is related unpublished work on induced tensor products under the McKay correspondence in dimensions 2 and 3 by T. Abdelgadir, A. Craw, J. Karmazyn, and A. King. In , we also get a formula which can be seen as a stacky instance of the ’flop-flop = twist’ principle as discussed in [@ADMflop]. In , we introduce a general candidate for a weakly crepant categorical resolution (see [@KuzLefschetz] or for this notion), namely the *weakly crepant neighbourhood* $\WC(\rho)\subset \Db(\wY)$, inside the derived category of a given resolution $\rho\colon\wY\to Y$ of a rational Gorenstein variety $Y$. The idea is pretty simple: by Grothendieck duality, there is a canonical section $s\colon \reg_\wwY\to \reg_\rho$ of the relative dualising sheaf, and this induces a morphism of Fourier–Mukai transforms $ t \coloneqq \rho_*(\_\otimes s)\colon \rho_* \to \rho_! \, . $ Set $\rho_+ \coloneqq \cone(t)$ and $\WC(\rho) \coloneqq \ker(\rho_+)$. Then, by the very construction, we have $\rho_{*\mid \WC(\rho)}\cong \rho_{!\mid \WC(\rho)}$ which amounts to the notion of categorical weak crepancy. There is one remaining condition needed to ensure that $\WC(\rho)$ is a categorical weakly crepant resolution: whether it is actually a smooth category; this holds as soon as it is an admissible subcategory of $\Db(\wY)$ which means that its inclusion has adjoints. We prove that, in the Gorenstein case $m\mid n$ of our set-up of cyclic quotients, $\WC(\rho)\subset \Db(\wY)$ is an admissible subcategory; see . In , we observe that there are various weakly crepant resolutions inside $\Db(\wY)$. However, a strongly crepant categorical resolution inside $\Db(\wY)$ is unique, as we show in . Our concept of weakly crepant neighbourhoods was motivated by the idea that some non-CY objects possess ‘CY neighbourhoods’’ (a construction akin to the spherical subcategories of spherelike objects in [@HKP]), i.e. full subcategories in which they become Calabi–Yau. This relationship is explained in . In the final , we construct Bridgeland stability conditions on Kummer threefolds as an application of our results; see . **Conventions.** We work over the complex numbers. All functors are assumed to be derived. We write $\kh^i(E)$ for the $i$-th cohomology object of a complex $E\in \Db(Z)$ and $\Ho^*(E)$ for the complex $\oplus_i \Ho^i(Z,E)[-i]$. If a functor $\Phi$ has a left/right adjoint, they are denoted $\Phi^L$, $\Phi^R$. There are a number of spaces, maps and functors repeatedly used in this text. For the convenience of the reader, we collect our notation at the very end of this article, on . **Acknowledgements.** It is a pleasure to thank Tarig Abdelgadir, Martin Kalck, Sönke Rollenske and Evgeny Shinder for comments and discussions. We are grateful to the anonymous referee for very careful inspection. Preliminaries {#sec:preliminaries} ============= Fourier–Mukai transforms and kernels {#sub:fmtransforms} ------------------------------------ Recall that given an object $\ke$ in $\Db(Z\times Z')$, where $Z$ and $Z'$ are smooth and projective, we get an exact functor $\Db(Z)\to \Db(Z')$, $F \mapsto p_{Z'*}(\ke\otimes p_Z^* F)$. Such a functor, denoted by $\FM_\ke$, is called a *Fourier–Mukai transform* (or FM transform) and $\ke$ is its kernel. See [@Huy] for a thorough introduction to FM transforms. For example, if $\Delta\colon Z\to Z\times Z$ is the diagonal map and $\kl$ is in $\Pic(Z)$, then $\FM_{\Delta_*\kl}(F) = F\otimes\kl$. In particular, $\FM_{\ko_\Delta}$ is the identity functor. [**Convention.**]{} We will write $\MM_\kl$ for the functor $\FM_{\Delta_*\kl}$. The calculus of FM transforms is, of course, not restricted to smooth and projective varieties. Note that $f_*$ maps $\Db(Z)$ to $\Db(Z')$ as soon as $f\colon Z\to Z'$ is proper. In order to be able to control the tensor product and pullbacks, one can restrict to perfect complexes. Recall that a complex of sheaves on a quasi-projective variety $Z$ is called *perfect* if it is locally quasi-isomorphic to a bounded complex of locally free sheaves. The triangulated category of perfect complexes on $Z$ is denoted by $\Dperf(Z)$. It is a full subcategory of $\Db(Z)$. These two categories coincide if and only if $Z$ is smooth. We will sometimes take cones of morphisms between FM transforms. Of course, one needs to make sure that these cones actually exist. Luckily, if one works with FM transforms, this is not a problem, because the maps between the functors come from the underlying kernels and everything works out, even for (reasonable) schemes which are not necessarily smooth and projective; see [@AL]. Group actions and derived categories {#sub:equivariant} ------------------------------------ Let $G$ be a finite group acting on a smooth variety $X$. Recall that a $G$-equivariant coherent sheaf is a pair $(F,\lambda_g)$, where $F \in \Coh(X)$ and $\lambda_g\colon F\isomor g^*F$ are isomorphisms satisfying a cocycle condition. The category of $G$-equivariant coherent sheaves on $X$ is denoted by $\Coh^G(X)$. It is an abelian category. The *equivariant derived category*, denoted by $\Db_G(X)$, is defined as $\Db(\Coh^G(X))$, see, for example, [@Plo] for details. Recall that for every subgroup $G'\subset G$ the restriction functor $\Res\colon \Db_G(X)\to \Db_{G'}(X)$ has the induction functor $\Ind\colon \Db_{G'}(X)\to \Db_G(X)$ as a left and right adjoint (see e.g. [@Plo Sect. 1.4]). It is given for $F\in \Db(Z)$ by $$\begin{aligned} \label{eq:induction-definition} \Ind(F) = \bigoplus_{[g]\in G'\setminus G}g^* F\end{aligned}$$ with the $G$-linearisation given by the $G'$-linearisation of $F$ together with appropriate permutations of the summands. If $G$ acts trivially on $X$, there is also the functor $\triv\colon\Db(X)\to \Db_{G}(X)$ which equips an object with the trivial $G$-linearisation. Its left and right adjoint is the functor $(\_)^G\colon \Db_G(X)\to \Db(X)$ of invariants. Given an equivariant morphism $f\colon X\to X'$ between varieties endowed with $G$-actions, there are equivariant pushforward and pullback functors, see, for example, [@Plo Sect. 1.3] for details. We will sometimes write $f_*^G$ for $(\_)^G\circ f_*$. It is also well-known that the category $\Db_G(X)$ has a tensor product and the usual formulas, e.g. the adjunction formula, hold in the equivariant setting. Finally, we need to recall that a character $\kappa$ of $G$ acts on the equivariant category by twisting the linearisation isomorphisms with $\kappa$. If $F\in \Db_G(X)$, we will write $F\otimes \kappa$ for this operation. We will tacitly use that twisting by characters commutes with the equivariant pushforward and pullback functors along $G$-equivariant maps. Semi-orthogonal decompositions {#sub:semiorth} ------------------------------ References for the following facts are, for example, [@Bon] and [@Bondal-Orlov2]. Let $\kt$ be a Hom-finite triangulated category. A *semi-orthogonal decomposition* of $\kt$ is a sequence of full triangulated subcategories $\ka_1,\ldots,\ka_m$ such that (a) if $A_i\in \cA_i$ and $A_j\in \cA_j$, then $\text{Hom}(A_i,A_j[l])=0$ for $i>j$ and all $l$, and (b) the $\cA_i$ generate $\kt$, that is, the smallest triangulated subcategory of $\kt$ containing all the $\cA_i$ is already $\kt$. We write $\kt=\sod{\ka_1,\ldots,\ka_m}$. If $m=2$, these conditions boil down to the existence of a functorial exact triangle $A_2\to T\to A_1$ for any object $T\in \kt$. A subcategory $\ka$ of $\kt$ is *right admissible* if the embedding functor $\iota$ has a right adjoint $\iota^R$, *left admissible* if $\iota$ has a left adjoint $\iota^L$, and *admissible* if it is left and right admissible. Given any triangulated subcategory $\ka$ of $\kt$, the full subcategory $\ka\orth\subseteq\kt$ consists of objects $T$ such that $\Hom(A,T[k])=0$ for all $A\in \ka$ and all $k\in \IZ$. If $\ka$ is right admissible, then $\kt=\sod{\ka\orth,\ka}$ is a semi-orthogonal decomposition. Similarly, $\kt=\sod{\ka, \lorth\ka}$ is a semi-orthogonal decomposition if $\ka$ is left admissible, where $\lorth\ka$ is defined in the obvious way. Examples typically arise from so-called exceptional objects. Recall that an object $E\in \Db(Z)$ (or any $\IC$-linear triangulated category) is called *exceptional* if $\Hom(E,E)=\IC$ and $\Hom(E,E[k])=0$ for all $k\neq 0$. The smallest triangulated subcategory containing $E$ is then equivalent to $\Db(\Spec(\IC))$ and this category, by abuse of notation again denoted by $E$, is admissible, leading to a semi-orthogonal decomposition $\Db(Z)=\sod{ E\orth, E}$. A sequence of objects $E_1,\ldots,E_n$ is called an *exceptional collection* if $\Db(Z) = \sod{ (E_1,\ldots,E_n)\orth,E_1,\ldots,E_n }$ and all $E_i$ are exceptional. The collection is called *full* if $(E_1\ldots,E_n)\orth=0$. Note that any fully faithful FM transform $\Phi\colon \Db(X)\to \Db(X')$ gives a semi-orthogonal decomposition $\Db(X')=\sod{ \Phi(\Db(X))\orth,\Phi(\Db(X))}$, because any FM transform has a right and a left adjoint, see [@Huy Prop. 5.9]. Dual semi-orthogonal decompositions ----------------------------------- Let $\cT$ be a triangulated category together with a semi-orthogonal decomposition $\cT=\sod{\cA_1,\dots,\cA_n}$ such that all $\cA_i$ are admissible. Then there is the *left-dual* semi-orthogonal decomposition $\cT = \sod{ \cB_n,\dots,\cB_1}$ given by $\cB_i \coloneqq \sod{\cA_1,\dots,\cA_{i-1}, \cA_{i+1},\dots,\cA_n}\orth$. There is also a right-dual decomposition but we will always use the left-dual and refer to it simply as the *dual* semi-orthogonal decomposition. We summarise the properties of the dual semi-orthogonal decomposition needed later on in the following \[lem:dual-sod\] Let $\cT=\sod{\cA_1,\dots,\cA_n}$ be a semi-orthogonal decomposition with dual semi-orthogonal decomposition $\cT=\sod{\cB_n,\dots,\cB_1}$. 1. $\sod{\cA_1,\dots,\cA_r}=\sod{\cB_r,\dots,\cB_1}$ and $\sod{\cA_1,\dots,\cA_r}\orth = \sod{\cB_n,\dots,\cB_{r+1}}$ for $1\le r\le n$. 2. If $\sod{\cA_1,\dots,\cA_n}$ is given by an exceptional collection, i.e. $\cA_i=\sod{E_i}$, then its dual is also given by an exceptional collection $\cB_i=\sod{F_i}$ such that $\Hom^*(E_i,F_j)=\delta_{ij}\IC[0]$. Part (i) is [@Efimov Prop. 2.7(i)]. Part (ii) is then clear. An important classical example is the following \[lem:easypdual\] There are dual semi-orthogonal decompositions $$\begin{aligned} \Db(\IP^{n-1}) &= \sod{ \reg,\reg(1),\dots,\reg(n-1) } \,, \\ \Db(\IP^{n-1}) &= \sod{ \Omega^{n-1}(n-1)[n-1],\dots,\Omega^1(1)[1],\reg } \,.\end{aligned}$$ The fact that both sequences are indeed full goes back to Beilinson, see [@Huy Sect. 8.3] for an account. The fact that they are dual is classical and follows by a direct computation, for instance using [@Bri-Ste Lem. 2.5]. The following relative version is the example of dual semi-orthogonal decompositions which we will need throughout the text. \[lem:Pdual\] Let $\nu\colon Z\to S$ be a $\IP^{n-1}$-bundle. There is the semi-orthogonal decomposition $$\begin{aligned} \Db(Z) &= \Sod{ \nu^*\Db(S), \nu^*\Db(S) \otimes \reg_\nu(1), \dots, \nu^*\Db(S)\otimes \reg_\nu(n-1) } \shortintertext{whose dual decomposition is given by} \Db(Z) &= \Sod{ \nu^*\Db(S)\otimes \Omega_\nu^{n-1}(n-1), \dots, \nu^*\Db(S)\otimes \Omega_\nu^{1}(1), \nu^*\Db(S) } \, .\end{aligned}$$ Part (i) is [@Orlov-projbund Thm. 2.6]. Part (ii) follows from . The following consequence will be used in . \[cor:sodequal\] If $m<n$, there is the equality of subcategories of $\Db(Z)$ $$\begin{aligned} & \Sod{ \nu^*\Db(S)\otimes \reg_\nu(m-n),\dots, \nu^*\Db(S)\otimes \reg_\nu(-1) } \\ =\; & \Sod{ \nu^*\Db(S)\otimes \Omega_\nu^{n-1}(n-1), \dots, \nu^*\Db(S)\otimes \Omega_\nu^{m}(m) } \, . \end{aligned}$$ Applying (i) to the dual decompositions of gives the equalities $$\begin{aligned} & \Sod{\nu^*\Db(S)\otimes \Omega_\nu^{n-1}(n-1), \dots, \nu^*\Db(S)\otimes \Omega_\nu^{m}(m)} \\ =\; & \Sod{\nu^*\Db(S),\dots,\nu^*\Db(S)\otimes\reg_\nu(m-1)}^\bot \\ =\; & \Sod{\nu^*\Db(S)\otimes\reg_\nu(m-n),\dots,\nu^*\Db(S)\otimes\reg_\nu(-1)}\,.\qedhere\end{aligned}$$ Linear functors and linear semi-orthogonal decompositions {#sec:relative} --------------------------------------------------------- Let $\cT$ be a tensor triangulated category, i.e. a triangulated category with a compatible symmetric monoidal structure. Moreover, let $\cX$ be a *triangulated module category over $\cT$*, i.e. there is an exact functor $\pi^*\colon \cT\to \cX$ and a tensor product $\otimes \colon \cT \times \cX \to \cX$, that is an assignment $\pi^*(A)\otimes E$ functorial in $A\in\cT$ and $E\in\cX$. We will take $\cT=\DperfG(Y)$ for some variety $Y$ with an action by a finite group $G$. Note that $\DperfG(Y)$ has a (derived) tensor product, and it is compatible with $G$-linearisations. For $\cX$, we have several cases in mind. If $X$ is a smooth $G$-variety $X$ with a $G$-equivariant morphism $\pi\colon X\to Y$, then we take $\cX=\Db_G(X)=\DperfG(X)$; this is a tensor triangulated category itself and $\pi^*$ preserves these structures. If $\Lambda$ is a finitely generated $\reg_Y$-algebra, then let $\cX=\Db(\Lambda)$ be the bounded derived category of finitely generated right $\Lambda$-modules with $\pi^*(A)=A\otimes_{\reg_Y} \Lambda$ and $\pi^*(A)\otimes E = A\otimes_{\reg_Y}\Lambda\otimes_\Lambda E = A\otimes_{\reg_Y}E\in\cX$. Note that if $\Lambda$ is not commutative, then $\cX$ is not a tensor category. We say that a full triangulated subcategory $\cA\subset \cX$ is *$\cT$-linear* (since in our cases we have $\cT=\Dperf(Y)$ we will also speak of *Y-linearity*) if $$\pi^*(A)\otimes E\in \cA \qquad \text{for all $A\in \cT$ and $E\in \cA$.}$$ We say that a semi-orthogonal decomposition $\cX = \sod{\cA_1,\dots,\cA_n}$ is *$\cT$-linear*, if all the $\cA_i$ are $\cT$-linear subcategories. We call a class of objects $\cS\subset \cX$ (left/right) *spanning over $\cT$* if $\pi^*\cT\otimes \cS$ is a (left/right) spanning class of $\cX$ in the non-relative sense. Recall that a subset $\cC \subset \cX$ is *generating* if $\cX=\Sod{\cC}$ is the smallest triangulated category closed under direct summands containing $\cC$. The subset $\cC \subset \cX$ is called *generating over $\cT$* if $\cC \otimes \pi^*\cT$ generates $\Db(\cX)$. Let $\cX'$ be a further tensor triangulated category together with a tensor triangulated functor $\pi'^*\colon \cT\to \cX'$. We say that an exact functor $\Phi\colon \cX\to \cX'$ is *$\cT$-linear* if there are functorial isomorphisms $$\Phi(\pi^*(A)\otimes E) \cong \pi'^*(A)\otimes \Phi(E) \qquad \text{for all $A\in \cT$ and $E\in \cX$.}$$ The verification of the following lemma is straight-forward. @anchor[currentHref]{}\[lem:relativespanninggeneral\] 1. If $\Phi\colon \cX\to \cX'$ is $\cT$-linear, then $\Phi(\cX)$ is a $\cT$-linear subcategory of $\cX'$. 2. Let $\cA\subset \Db(\cY)$ be a $\cT$-linear (left/right) admissible subcategory. Then the essential image of $\cA$ is $\Db(\cY)$ if and only if $\cA$ contains a (left/right) spanning class over $\cT$. For the following, we consider the case that $\cX=\Db(X)$ for some smooth variety $X$ together with a proper morphism $\pi\colon X\to Y$. \[lem:relativesod\] Let $\cA, \cB\subset \Db(\cX)$ be $Y$-linear full subcategories. Then $$\cA\subset \cB^\perp \iff \pi_*\sHom(B,A)=0 \quad \forall\, A\in \cA, B\in \cB \,.$$ The direction $\Longleftarrow$ follows immediately from $\Hom^*(B,A) \cong \Gamma(Y, \pi_*\sHom(B,A))$; recall that all our functors are the derived versions. Conversely, assume that there are $A\in \cA$ and $B\in \cB$ such that $\pi_*\sHom(B,A)\neq 0$. Since $\Dperf(Y)$ spans $\D(\QCoh(Y))$, this implies that there is an $E\in \Dperf(Y)$ such that $$\begin{aligned} 0 \neq \Hom^*(E, \pi_*\sHom(B,A))\cong \Gamma(Y, \pi_*\sHom(B,A)\otimes E^\vee) &\cong \Gamma(Y, \pi_*(\sHom(B,A)\otimes \pi^*E^\vee)) \\ &\cong \Gamma(Y, \pi_*\sHom(B\otimes \pi^*E, A)) \\ &\cong \Hom^*(B\otimes \pi^*E,A) \,.\end{aligned}$$ By the $Y$-linearity, we have $B\otimes \pi^*E\in \cB$ and hence $\cA\not\subset \cB^\perp$. Relative Fourier–Mukai transforms --------------------------------- Let $\pi\colon X \to Y$ and $\pi'\colon X'\to Y$ be proper morphisms of varieties with $X$ and $X'$ being smooth. We denote the closed embedding of the fibre product into the product by $i\colon X\times_YX' \into X\times X'$. We call $\Phi\colon \Db(X)\to \Db(X')$ a *relative FM transform* if $\Phi = \FM_{\iota_*\cP}$ for some object $\cP\in\Db(X\times_YX)$. It is a standard computation that a relative FM transform is linear over $Y$, with respect to the pullbacks $\pi^*$ and $\pi'^*$. Furthermore, we have $\Phi\cong p_*(q^*(\_)\otimes \cP)$ where $p$ and $q$ are the projections of the fibre diagram $$\begin{aligned}\label{diag:fibre} \xymatrix{ & X\times_Y X' \ar[dr]^{p} \ar[dl]_q & \\ X \ar[dr]_{\pi}& & X'\,. \ar[dl]^{\pi'} \\ & Y & } \end{aligned}$$ The right adjoint of $\Phi$ is given by $\Phi^R \coloneqq q_*(p^!(\_)\otimes \cP^\vee)\colon \Db(X')\to \Db(X)$. We also have the following slightly stronger statement which one could call *relative adjointness*. For $E\in \Db(X)$ and $F\in \Db(X')$, there are functorial isomorphisms $$\pi'_*\sHom(\Phi(E), F)\cong \pi_*\sHom(E,\Phi^R(F)) \,.$$ This follows by Grothendieck duality, commutativity of , and projection formula: $$\begin{aligned} \pi'_*\sHom(\Phi(E), F)\cong \pi'_*\sHom(p_*(q^*E\otimes \cP), F) &\cong \pi'_*p_*\sHom(q^*E\otimes \cP, p^!F)\\ &\cong \pi_*q_*\sHom(q^*E, p^!F\otimes \cP^\vee)\\ &\cong \pi_*\sHom(E, q_*(p^!F\otimes \cP^\vee))\\ &\cong \pi_*\sHom(E,\Phi^R( F))\,.\qedhere\end{aligned}$$ For $E,F\in \Db(X)$, using the isomorphism of the previous lemma, we can construct a natural morphism $\widetilde \Phi\colon \pi_*\sHom(E,F)\to \pi'_*\sHom(\Phi(E),\Phi(F))$ as the composition $$\begin{aligned} \label{eq:wPhi} \widetilde \Phi = \widetilde \Phi(E,F) \colon \pi_*\sHom(E,F)\to\pi'_*\sHom(E,\Phi^R\Phi(F))\cong \pi'_*\sHom(\Phi(E),\Phi(F))\end{aligned}$$ where the first morphism is induced by the unit of adjunction $F\to \Phi^R\Phi(F)$. Note that taking global sections gives back the functor $\Phi$ on morphisms, i.e. $\Phi=\Gamma(Y,\widetilde \Phi)$ as maps $$\Hom^*(E,F) \cong \Gamma(Y,\pi_*\sHom(E,F)) \to \Gamma(Y,\pi'_*\sHom(\Phi(E),\Phi(F))) \cong \Hom^*(\Phi(E),\Phi(F)) \,.$$ More generally, $\Phi$ induces functors for open subsets $U\subseteq Y$, $$\Phi_U \colon \Db(W) \to \Db(W'), \qquad\text{where } W = \pi^{-1}(U)\subseteq X \text{ and } W' = \pi'^{-1}(U)\subseteq X' ,$$ given by restricting the FM kernel $\iota_*\cP$ to $W\times W'$ and we have $\Phi_U=\Gamma(U, \widetilde\Phi)$. From this we see that $\widetilde \Phi$ is compatible with composition which means that the following diagram, for $E,F,G\in \Db(X)$, commutes $$\begin{aligned}\label{diag:relativecomposition} \xymatrix{ \pi_*\sHom(F,G) \otimes \pi_*\sHom(E,F) \ar[r]\ar[d]_{\widetilde \Phi\otimes \widetilde\Phi} & \pi_*\sHom(E,G) \ar[d]^{\widetilde \Phi} \\ \pi'_*\sHom(\Phi(F),\Phi(G)) \otimes \pi'_*\sHom(\Phi(E),\Phi(F)) \ar[r] & \pi'_*\sHom(\Phi(E),\Phi(G)) \,. } \end{aligned}$$ Relative tilting bundles {#sub:relative-tilting} ------------------------ Let $\pi\colon X \to Y$ be a proper morphism of varieties and let $X$ be smooth. Later on, $X$ and $Y$ will have $G$-actions, and $\Db(X)$ will be replaced by $\Db_G(X)$. We say that $V\in\Coh(X)$ is a *relative tilting sheaf* if $\Lambda_V \coloneqq \Lambda \coloneqq \pi_*\sHom(V,V)$ is cohomologically concentrated in degree 0 and $V$ is a spanning class over $Y$. For a more general theory of relative tilting bundles, see [@BB-tilting]. Note that $\Lambda$ is a finitely generated $\reg_Y$-algebra. We denote the bounded derived category of coherent right modules over $\Lambda$ by $\Db(\Lambda)$. It is a triangulated module category over $\Dperf(Y)$ via $\pi^*A = A\otimes_{\reg_Y}\Lambda$, and $\Lambda$ is a relative generator. In particular, for $A\in\Db(X)$ and $M\in\Db(\Lambda)$, the tensor product $A\otimes M$ is over the base $\reg_Y$. The functor $\pi_*\sHom(V,\_)\colon \Coh(X)\to \Coh(Y)$ factorises over $\Coh(\Lambda)$. Since it is left exact, we can consider its right-derived functor $\pi_*\sHom(V,\_) \colon \Db(X) \to \Db(\Lambda)$. This yields a relative tilting equivalence: \[prop:relative-tilting\] Let $V\in\Db(X)$ be a relative tilting sheaf over $Y$. Then $V$ generates $\Db(X)$ over $Y$, and the following functor defines a $Y$-linear exact equivalence: $$t_V \coloneqq \pi_*\sHom(V,\_) \colon \Db(X) \isomor \Db(\Lambda) \,.$$ The $Y$-linearity of $t_V$ is due to the projection formula $$t_V(\pi^*A\otimes E) = \pi_*(\pi^*A\otimes\sHom(V,E)) \cong A\otimes\pi_*\sHom(V,E) = A\otimes t_V(E) \,.$$ Consider the restricted functor $t'_V\colon \cV \coloneqq \sod{V\otimes\pi^*\Dperf(Y)} \to \Db(\Lambda)$. We show that $t'_V$ is fully faithful, using the adjunctions $\pi^*\dashv\pi_*$ and $\_\otimes_{\reg_Y}\Lambda \dashv \mathsf{For}$ where $\mathsf{For}\colon \Db(\Lambda) \to \Db(Y)$ is scalar restriction, the projection formula, and the $Y$-linearity of $t'_V$: $$\begin{aligned} \Hom_{\reg_X}(\pi^*A\otimes V,\pi^*B\otimes V) &\cong \Hom_{\reg_X}(\pi^*A, \pi^*B\otimes\sHom(V,V)) \\ &\cong \Hom_{\reg_Y}(A, \pi_*(\pi^*B\otimes\sHom(V,V))) \\ &\cong \Hom_{\reg_Y}( A, B\otimes\Lambda) \\ &\cong \Hom_{\Lambda}(A\otimes \Lambda, B\otimes \Lambda) \\ &\cong \Hom_{\Lambda}(t'_V(\pi^*A\otimes V), t'_V(\pi^*B\otimes V))\end{aligned}$$ Since objects of type $ \pi^*A\otimes V$ generate $\cV$, this shows that $t'_V$ is fully faithful. We have $t'_V(V)=\Lambda$. Since $\Lambda$ is a relative generator, hence a relative spanning class, of $\Db(\Lambda)$, we get an equivalence $\cV\cong\Db(\Lambda)$; see . We now claim that the inclusion $\cV\hookrightarrow \Db(X)$ has a right adjoint, namely $$t'^{-1}_Vt_V\colon \Db(X) \to \Db(\Lambda)\to \cV \,.$$ For this, take $A\in\Dperf(Y)$, $F\in \Db(X)$ and compute $$\begin{aligned} \Hom_{\reg_X}(\pi^*A\otimes V,F) \cong \Hom_{\reg_Y}(A, \pi_*\sHom(V,F)) &\cong \Hom_\Lambda(A\otimes \Lambda, t_V(F)) \\ &\cong \Hom_{\cV}(t'^{-1}_V(A\otimes \Lambda),t'^{-1}_Vt_V(F)) \\ &\cong \Hom_{\cV}(\pi^*A\otimes V,t'^{-1}_Vt_V(F)) \end{aligned}$$ where we use the projection formula, the adjunction $\Lambda\otimes_{\reg_Y}\_ \dashv \mathsf{For}$, the fact that $t'^{-1}_V$ is an equivalence, hence fully faithful, and the $Y$-linearity of $t'^{-1}_V$. Since the right-admissible $Y$-linear subcategory $\cV\subset \Db(X)$ contains the relative spanning class $V$, we get $\cV=\Db(X)$ by . This shows that $V$ is a relative generator and that $t_V=t_V'$ is an equivalence. Let $\pi'\colon X'\to Y$ be a second proper morphism and let $\Phi\colon\Db(X)\isomor\Db(X')$ be a relative FM transform. \[lem:relative-tiling–commutation\] If $$\wPhi_\Lambda \coloneqq \wPhi(V,V) \colon \Lambda_V = \pi_*\sHom(V,V) \to \pi'_*\sHom(\Phi(V),\Phi(V)) = \Lambda_{\Phi(V)}$$ is an isomorphism, then the following diagram of functors commutes: $$\begin{aligned}\label{diag:t} \xymatrix@C=4em{ \Db(X) \ar[r]^{t_V} \ar[d]_\Phi & \Db(\Lambda_V) \ar[d]^{\_\otimes_{\Lambda_V}\Lambda_{\Phi(V)}} \\ \Db(X') \ar[r]_{t_{\Phi(V)}} & \Db(\Lambda_{\Phi(V)}) } \end{aligned}$$ We first show that $\wPhi(V, E)\colon t_V(E)\to t_{\Phi(V)}(\Phi(E))$ is an isomorphism in $\Db(Y)$ for every $E\in \D(X)$. Assume first that there is an exact triangle $\pi^*A\otimes V\to E\to \pi^*B\otimes V$ for some $A,B\in \Dperf(Y)$ and consider the induced morphism of triangles $$\xymatrix{ \pi_*\sHom(V,\pi^*A\otimes V) \ar[r] \ar[d]^{\wPhi(V,\pi^*A\otimes V)} & \pi_*\sHom(V,E) \ar[r] \ar[d]^{\wPhi(V,E)} & \pi_*\sHom(V,\pi^*B\otimes V) \ar[d]^{\wPhi(V,\pi^*B\otimes V)} \\ \pi'_*\sHom(\Phi(V), \Phi(\pi^*A\otimes V)) \ar[r] & \pi'_*\sHom(\Phi(V),\Phi(E)) \ar[r] & \pi'_*\sHom(\Phi(V), \Phi(\pi^*B\otimes V)) \,.}$$ The outer vertical arrows are isomorphisms because they decompose as $$\begin{aligned} \pi_*\sHom(V,V\otimes\pi^*A) &\isomor \pi_*\sHom(V,V)\otimes A \xrightarrow[\wPhi_\Lambda]{\sim} \pi'_*\sHom(\Phi(V),\Phi(V)) \otimes A \\ &\isomor \pi'_*\sHom(\Phi(V),\Phi(V) \otimes \pi'^*A) \isomor \pi'_*\sHom(\Phi(V),\Phi(V \otimes \pi^* A)) \,.\end{aligned}$$ Therefore, the middle vertical arrow is an isomorphism as well. Since $V$ is a relative generator, we can show that $\wPhi(V,E)$ is an isomorphism for arbitrary $E\in \Db(X)$ by repeating the above argument. Using the commutativity of with $E$ plugged in for $G$ and $V$ plugged in for all other arguments, we see that $\wPhi(V,E)$ induces an $\Lambda_{\Phi(V)}$-linear isomorphism $\pi_*\sHom(V,E)\otimes_{\Lambda_V}\Lambda_{\Phi(V)}\isomor \pi'_*\sHom(\Phi(V),\Phi(E))$. \[lem:relative-tilting-fully-faithful\] The functor $\Phi$ is fully faithful if and only if $\wPhi_\Lambda\colon \Lambda_V\to \Lambda_{\Phi(V)}$ is an isomorphism. If $\Phi$ is fully faithful, the unit $\id\to \Phi^R\Phi$ is an isomorphism. Hence, $\wPhi_\Lambda$ is an isomorphism; see . Conversely, let $\wPhi_\Lambda$ be an equivalence. By , we get a commutative diagram $$\xymatrix@C=4em{ \Db(X) \ar[r]^{t_V} \ar[d]_\Phi & \Db(\Lambda_V) \ar[d]^{\_\otimes_{\Lambda_V}\Lambda_{\Phi(V)}} \\ \sod{\Phi(V)} \ar[r]_{t_{\Phi(V)}} & \Db(\Lambda_{\Phi(V)}) \,. }$$ In this diagram, the horizontal functors are tilting equivalences. The right-hand vertical functor is an equivalence, too, by assumption on $\wPhi_\Lambda$. Hence, $\Phi\colon \Db(X) \to \sod{\Phi(V)}$ is an equivalence, which implies that $\Phi\colon \Db(X) \to \Db(X')$ is fully faithful. \[lem:relative-FM\] Let $V\in\Db(X)$ be a relative tilting sheaf, $\Phi\colon\Db(X)\isomor\Db(X)$ a relative FM autoequivalence, and $\nu\colon V\isomor \Phi(V)$ an isomorphism such that $$\wPhi_\Lambda = \nu\circ\_\circ\nu^{-1} \colon \pi_*\sHom(V,V) \to \pi_*\sHom(\Phi(V),\Phi(V)) \,,$$ i.e. $\Phi_U(\phi )\circ\nu = \nu\circ\phi$ for all open subsets $U\subset Y$ and $\phi\in\Lambda_V(U)$. Then there exists an isomorphism of functors $\id\isomor \Phi$ restricting to $\nu$. We claim that, under our assumptions, the following diagram of functors commutes $$\begin{aligned}\label{diag:id} \xymatrix@C=4em{ \Db(X) \ar[r]^{t_V} \ar[d]_{\id} & \Db(\Lambda_V) \ar[d]^{\_\otimes_{\Lambda_V}\Lambda_{\Phi(V)}} \\ \Db(X) \ar[r]_{t_{\Phi(V)}} & \Db(\Lambda_{\Phi(V)}) } \end{aligned}$$ We construct a natural isomorphism $\eta\colon t_{\Phi V} \isomor t_V \otimes_{\Lambda_V}\Lambda_{\Phi V}$ as follows. For $E\in\Db(X)$, there is a natural $\reg_Y$-linear isomorphism $\pi_*\sHom(\Phi(V),E) \isomor \pi_*\sHom(V,E) \otimes_{\Lambda_V}\Lambda_{\Phi (V)}$ given by $f\mapsto f\nu \otimes 1$; the inverse map is $g\otimes1\mapsto g\nu^{-1}$. This map is linear over $\Lambda_{\Phi V}$ because, for a local section $\lambda\in \pi_*\sHom(\Phi(V), \Phi(V))$, we have by our assumption, setting $\phi=\wPhi^{-1}(\lambda)$: $$\eta(f\lambda) = f\lambda\nu \otimes 1 = f\nu \wPhi^{-1}(\lambda)\otimes1 = f\nu\otimes\lambda = (f\nu\otimes1)\lambda \,.$$ Comparing the diagrams and shows that $\Phi\cong\id$. \[cor:relative-tilting-functor-iso\] Let $V\in\Db(X)$ be a relative tilting sheaf, $\Phi_1,\Phi_2\colon\Db(X)\isomor\Db(X')$ relative FM equivalences, and $\nu\colon \Phi_1(V)\isomor \Phi_2(V)$ an isomorphism such that $\Phi_{2,U}(\phi )\circ\nu = \nu\circ \Phi_{1,U}(\phi)$ for all $\phi\in\Lambda_V(U)$ and $U\subset Y$ open. Then there exists a isomorphism of functors $\Phi_1\isomor \Phi_2$ restricting to $\nu$. Moreover, if $V=L_1\oplus\cdots\oplus L_k$ decomposes as a direct sum, then the above condition is satisfied by specifying isomorphisms $\nu_i\colon \Phi_1(L_i) \isomor \Phi_2(L_i)$ inducing functor isomorphisms $\Phi_{1,U} \isomor \Phi_{2,U}$ on the full finite subcategory $\{L_{1\mid U},\ldots,L_{k\mid U}\}$ of $\Db(\pi^{-1}(U))$. All the results of this subsection remain valid in an equivariant setting, where a finite group $G$ acts on $X$ and $\pi\colon X\to Y$ is $G$-invariant. Then the correct sheaf of $\reg_Y$-algebras is $\Lambda_V=\pi_*^G\sHom(V,V)$. Spherical functors {#sub:spherical} ------------------ An exact functor $\varphi\colon \kc\to \kd$ between triangulated categories is called *spherical* if it admits both adjoints, if the cone endofunctor $F[1] \coloneqq \cone(\id_\kc\to\varphi^R\varphi)$ is an autoequivalence of $\kc$, and if the canonical functor morphism $\varphi^R \to F\varphi^L[1]$ is an isomorphism. A spherical functor is called *split* if the triangle defining $F$ is split. The proper framework for dealing with functorial cones are dg-categories; the triangulated categories in this article are of geometric nature, and we can use Fourier–Mukai transforms. See [@ALdg] for proofs in great generality. Given a spherical functor $\varphi\colon \kc\to \kd$, the cone of the natural transformation $\TT = \TT_\varphi \coloneqq \cone(\varphi\varphi^R \to \id_\kd)$ is called the *twist around $\varphi$*; it is an autoequivalence of $\kd$. The following lemma follows immediately from the definition, since an equivalence has its inverse functor as both left and right adjoint. \[lem:spherical\] Let $\varphi\colon \kc\to \kd$ be a spherical functor and let $\delta\colon \kd\to\kd'$ be an equivalence. Then $\delta\circ\varphi\colon\cC\to\cD' $ is a spherical functor with associated twist functor $\TT_{\delta\varphi} = \delta \TT_\varphi \delta\inv$. The geometric setup {#sec:setup} =================== Let $X$ be a smooth quasi-projective variety together with an action of a finite group $G$. Let $S\coloneqq\Fix(G)$ be the locus of fixed points. Then $S\subset X$ is a closed subset, which is automatically smooth since, locally in the analytic topology, the action can be linearised by Cartan’s lemma, see [@Cartan-Quot Lem. 2]. Also note that $X/G$ has rational singularities, like any quotient singularity over $\IC$ [@Kovacs]. \[cond:main\] We make strong assumptions on the group action: 1. $G \cong \mu_m$ is a cyclic group. Fix a generator $g\in G$. 2. Only the trivial isotropic groups $1$ and $\mu_m$ occur. 3. The generator $g$ acts on the normal bundle $N\coloneqq N_{S/X}$ by multiplication with some fixed primitive $m$-th root of unity $\zeta$. Condition (ii) obviously holds if $m$ is prime. Condition (iii) can be rephrased: there is a splitting $T_{X\mid S}= T_S\oplus N_{S/X}$ because $T_S$ is the subsheaf of $G$-invariants of $T_{X\mid S}$ and we work over characteristic 0. By (iii), this is even the splitting into the eigenbundles corresponding to the eigenvalues $1$ and $\zeta$. We denote by $\chi\colon G\to\IC^*$ the character with $\chi(g)=\zeta^{-1}$. Hence, we can reformulate (iii) by saying that $G$ acts on $N$ via $\chi^{-1}$. From these assumptions we deduce the following commutative diagram $$\begin{aligned}\label{eq:maindiagram} \PandocStartInclude{geometric-diagram.tex}\PandocEndInclude{input}{615}{32} \end{aligned}$$ where $a$, $b$, $i$, and $j$ are closed embeddings and $\pi$ is the quotient morphism. The $G$-action on $X$ lifts to a $G$-action on $\wX$. Since, by assumption, $G$ acts diagonally on $N$, it acts trivially on the exceptional divisor $Z=\IP(N)$. In particular, the fixed point locus of the $G$-action on $\wX$ is a divisor. Hence, the quotient variety $\wY$ is again smooth and the quotient morphism $q$ is flat due to the Chevalley–Shephard–Todd theorem. Since the composition $\pi\circ p$ is $G$-invariant, it induces the morphism $\rho\colon \wY\to Y$ which is easily seen to be birational, hence a resolution of singularities. The preimage $\rho^{-1}(S)$ of the singular locus is a divisor in $\wY$. Hence, by the universal property of the blow-up, we get a morphism $\wY\to \Bl_SY$ which is easily seen to be an isomorphism. The resolution as a moduli space of $\boldsymbol{G}$-clusters {#sub:clusters} ------------------------------------------------------------- The result of this section might be of independent interest. Let $X$ be a smooth quasi-projective variety and $G$ a finite group acting on $X$. A *$G$-cluster* on $X$ is a closed zero-dimensional $G$-invariant subscheme $W\subset X$ such that the $G$-representation $H^0(W,\reg_W)$ is isomorphic to the regular representation of $G$. There is a fine moduli space $\Hilb^G(X)$ of $G$-clusters, called the *$G$-Hilbert scheme*. It is equipped with the *equivariant Hilbert–Chow morphism* $\tau\colon \Hilb^G(X)\to X/G, W\mapsto\supp(W)$, mapping $G$-clusters to their underlying $G$-orbits. \[prop:Hilbert-resolution\] Let $G$ be a finite cyclic group acting on $X$ such that all isotropy groups are either 1 or $G$, and such that $G$ acts on the normal bundle $N_{\Fix(G)/X}$ by scalars which means that is satisfied. Then there is an isomorphism $$\phi\colon \wY\xrightarrow \cong \Hilb^G(X) \quad \text{with} \quad \tau\circ \phi=\rho \,.$$ We use the notation from . One can identify $\wX$ with the reduced fibre product $(\wY\times_Y X)_\red$ which gives a canonical embedding $\wX\subset \wY\times X$. Under this embedding, the generic fibre of $q$ is a reduced free $G$-orbit of the action on $X$. In particular, it is a $G$-cluster. By the flatness of $q$, every fibre is a $G$-cluster and we get the classifying morphism $\phi\colon \wY \to \Hilb^G(X)$ which is easily seen to satisfy $\tau\circ \phi=\rho$. Let $s\in S$ and $z\in Z$ with $\nu(z)\in s$. Let $\ell\subset N(s)$ be the line corresponding to $z$. Then, one can check that the tangent space of the $G$-cluster $q^{-1}(i(z))\subset X$ is exactly $\ell$. Hence, the $G$-clusters in the family $\wX$ are all different so that the classifying morphism $\phi$ is injective. For the bijectivity of $\phi$, it is only left to show that the $G$-orbits supported on a given fixed point $s\in S$ are parametrised by $\IP(N(s))$. Let $\xi\subset X$ be such a $G$-cluster. In particular, $\xi$ is a length $m=|G|$ subscheme concentrated in $s$ and hence can be identified with an ideal $I\subset \reg_{X,s}/\fm_{X,s}^m$ of codimension $m$. By Cartan’s lemma, the $G$-action on $X$ can be linearised in an analytic neighbourhood of $s$. Hence, there is an $G$-equivariant isomorphism $$\reg_{X,s}/\fm_{X,s}^m \cong \IC[x_1,\dots,x_k,y_1,\dots,y_n]/(x_1,\dots,x_k,y_1,\dots,y_n)^m \eqqcolon R$$ where $G$ acts trivially on the $x_i$ and by multiplication by $\zeta^{-1}$ on the $y_i$. Furthermore, $n=\rank N_{S/X}$ and $k=\rank T_S=\dim X- n$. By assumption, $\reg(\xi)$ is the regular $\mu_m$-representation. In other words, $$\begin{aligned} \label{eq:eigendecomp} \reg(\xi) \cong R/I \cong \chi^0 \oplus \chi \oplus \dots \oplus \chi^{m-1}\end{aligned}$$ where $\chi$ is the character given by multiplication by $\zeta^{-1}$. In particular, $R/I$ has a one-dimensional subspace of invariants. It follows that every $x_i$ is congruent to a constant polynomial modulo $I$. Hence, we can make an identification $\reg(\xi)\cong R'/J$ where $J$ is a $G$-invariant ideal in $R'=\IC[y_1,\dots,y_k]/\fn^m$ where $\fn=(y_1,\dots,y_n)$. The decomposition of the $G$-representation $R'$ into eigenspaces is exactly the decomposition into the spaces of homogeneous polynomials. Hence, an ideal $J\subset R'$ is $G$-invariant if and only if it is homogeneous. Furthermore, implies that $$\dim_\IC\bigl( \fn^i/(J\cap\fn^i +\fn^{i+1}) \bigr) = 1 \quad \text{for all $i=0,\dots,m-1$}$$ which means that $\xi$ is curvilinear. In summary $\xi$ can be identified with a homogeneous curvilinear ideal $J$ in $R'$. The choice of such a $J$ corresponds to a point in $\IP((\fn/\fn^2)\dual) \cong \IP(N(s))$; see [@Goettschebook Rem. 2.1.7]. Hence, $\phi$ is a bijection and we only need to show that $\Hilb^G(X)$ is smooth. The smoothness in points representing free orbits is clear since the $G$-Hilbert–Chow morphism is an isomorphism on the locus of these points. So it is sufficient to show that $$\Hom^1_{\Db_G(X)}(\reg_\xi,\reg_\xi) = \dim X = n+k$$ for a $G$-cluster $\xi$ supported on a fixed point. Following the above arguments, we have $$\Hom^*_{\Db_G(X)}(\reg_\xi,\reg_\xi) \cong \Hom^*_{\Db_G(\IA^k\times \IA^n)}(\reg_{\xi'},\reg_{\xi'})$$ where $G$ acts trivially on $\IA^k$ and by multiplication by $\zeta$ on $\IA^n$. Furthermore, by a transformation of coordinates, we may assume that $$\xi' = V(x_1,\dots,x_k, y_1^m,y_2,\dots,y_{n}) \subset \IA^k\times \IA^n \,.$$ We have $\reg_\xi'\cong \reg_0\boxtimes \reg_\eta$ where $$\eta=V(y_1^m,y_2,\dots,y_n)\subset \IA^n \,.$$ By Künneth formula, we get $$\begin{aligned} \Hom^*_{\Db_G(\IA^k\times \IA^n)}(\reg_{\xi'},\reg_{\xi'}) &\cong \Hom^*_{\Db(\IA^k)}(\reg_0,\reg_0) \otimes \Hom^*_{\Db_G(\IA^n)}(\reg_\eta,\reg_\eta) \\ &\cong \wedge^*(\IC^k)\otimes \Hom^*_{\Db_G(\IA^n)}(\reg_\eta,\reg_\eta) \,.\end{aligned}$$ Furthermore, $\Hom^0_{\Db_G(\IA^n)}(\reg_\eta,\reg_\eta)\cong H^0(\reg_\eta)^G\cong \IC$. Hence, it is sufficient to show that $\Hom^1_{\Db_G(\IA^n)}(\reg_\eta,\reg_\eta)\cong \IC^n$. Note that $\eta$ is contained in the line $\ell=V(y_2,\dots,y_n)$. On $\ell$ we have the Koszul resolution $$0 \to \reg_\ell \xrightarrow{\cdot y_1^m} \reg_\ell \to \reg_\eta \to 0 \,.$$ Using this, we compute $$\Hom^*_{\Db(\ell)}(\reg_\eta,\reg_\eta) \cong \reg_\eta[0] \oplus \reg_\eta[-1] \,.$$ Note that the normal bundle of $\ell$, as an equivariant bundle, is given by $N_{\ell/\IA^n}\cong (\reg_\ell\otimes \chi^{-1})^{\oplus n-1}$. By [@AC Thm. 1.4], we have $$\begin{aligned} \Hom^*_{\Db(\IA^n)}(\reg_\eta, \reg_\eta) &\cong \Hom^*_{\Db(\ell)}(\reg_\eta, \reg_\eta \otimes \wedge^* N_{\ell/\IA^n}) \\ &\cong \Hom^*_{\Db(\ell)}(\reg_\eta, \reg_\eta)\otimes \wedge^*((\reg_\ell\otimes \chi^{-1})^{\oplus n-1}) \,. \end{aligned}$$ Evaluating in degree 1 gives $$\Hom^1_{\Db(\IA^n)}(\reg_\eta,\reg_\eta)\cong \reg_\eta\oplus (\reg_\eta\otimes \chi^{-1})^{\oplus n-1}\,.$$ Since, as a $G$-representation, $\reg_\eta\cong \chi^0\oplus\chi^1\oplus\dots\oplus \chi^{m-1}$, we get an $n$-dimensional space of invariants $$\Hom^1_{\Db_G(\IA^n)}(\reg_\eta,\reg_\eta) \cong \Hom^1_{\Db(\IA^n)}(\reg_\eta,\reg_\eta)^G \cong \IC^{n} \,.\qedhere$$ The following lemma is needed later in but its proof fits better into this section. \[lem:orthogonal-clusters\] Assume that $m=|G|\ge n=\codim(S\hookrightarrow X)$. Let $\xi_1,\xi_2\subset X$ be two different $G$-clusters supported on the same point $s\in S$. Then $\Hom^*_{\Db_G(X)}(\reg_{\xi_1}, \reg_{\xi_2})=0$. By the same arguments as in the proof of the previous proposition we can reduce to the claim that $$\Hom^*_{\Db_G(\IA^n)}(\reg_{\eta_1},\reg_{\eta_2}) = 0$$ where $\eta_1=V(y_1^m, y_2,\dots,y_n)$ and $\eta_2=V(y_1,y_2^m,y_3,\dots,y_n)$. Set $\ell_1=V(y_2,\dots, y_n)$, $\ell_2=V(y_1,y_3,\dots,y_n)$, $E=\gen{\ell_1,\ell_2}=V(y_3,\dots,y_n)$ and consider the diagram of closed embeddings $$\begin{aligned} \xymatrix{ & \ell_2 \ar^{\iota_2}[dr] \ar^{r}[d] & \\ \{0\} \ar^{u}[ur] \ar_{v}[dr] & E \ar^t[r] & \IA^n \, . \\ & \ell_1 \ar^{s}[u] \ar_{\iota_1}[ur] } \end{aligned}$$ where $N_t\cong (\reg_E\otimes \chi^{-1})^{\oplus n-2}$. By [@Kru4 Lem. 3.3] (alternatively, one may consult [@Gri] or [@ACHderivedint] for more general results on derived intersection theory), we get $$\label{eq:gradedHom} \begin{split} \Hom^*_{\Db(\IA^n)}(\reg_{\eta_1},\reg_{\eta_2}) &= \Hom^*_{\Db(\IA^n)}(\iota_{1*}\reg_{\eta_1},\iota_{2*}\reg_{\eta_2})\\ &\cong \Hom^*_{\Db(\ell_2)}(\iota_2^*\iota_{1*}\reg_{\eta_1},\reg_{\eta_2})\\ &\cong \Hom^*_{\Db(\ell_2)}(u_*v^*\reg_{\eta_1},\reg_{\eta_2})\otimes \wedge^*N_{t\mid \ell_2}\\ &\cong \Hom^*_{\Db(\ell_2)}(u_*v^*\reg_{\eta_1},\reg_{\eta_2})\otimes \wedge^*(\reg_{\ell_2}\otimes \chi^{-1})^{\oplus n-2} \end{split}$$ We consider the Koszul resolution $ 0\to \reg_{\ell_1}\xrightarrow{y_1^m}\reg_{\ell_1}\to \reg_{\eta_1}\to 0 $ of $\reg_{\eta_1}$. Note that this is an equivariant resolution when we consider $\reg_{\ell_1}$ equipped with the canonical linearisation since $y_1^m$ is a $G$-invariant function. Applying $u_*v^*$, we get an equivariant isomorphism $$\begin{aligned} \label{eq:uv} u_*v^*\reg_{\eta_1}\cong \reg_0\oplus \reg_0[1]\,. \end{aligned}$$ Similarly, we have the equivariant Koszul resolution $ 0\to \reg_{\ell}\otimes \chi\xrightarrow{\cdot y} \reg_{\ell} \to \reg_0 \to 0 $ of $\reg_0$, where we set $\ell\coloneqq\ell_2$ and $y\coloneqq y_2$. Applying $\Hom(\_, \reg_{\eta_2})$ to the resolution, we get $$0 \to \IC[y]/y^{m} \otimes \xrightarrow{\cdot y} \IC[y]/y^m \otimes \chi^{-1} \to 0$$ and taking cohomology yields $$\begin{aligned} \label{eq:Hom} \Hom^*_{\Db(\ell_2)}(\reg_{0},\reg_{\eta_2}) \cong \IC\gen{y^{m-1}}[0] \oplus \IC\gen{1} \otimes \chi^{-1}[-1] \cong \reg_0\otimes \chi^{-1}[0] \oplus \reg_0\otimes \chi^{-1}[-1] \,. \end{aligned}$$ Plugging and into gives $$\Hom^*_{\Db(\IA^n)}(\reg_{\eta_1},\reg_{\eta_2}) \cong \bigl(\reg_0\otimes \chi^{-1}[0] \oplus \reg_0^{\oplus 2} \otimes \chi^{-1}[-1] \oplus \reg_0 \otimes \chi^{-1}[-2]\bigr) \otimes \wedge^*(\chi^{-1})^{\oplus n-2} \,.$$ The irreducible representations occuring are $\chi^{-1},\chi^{-2},\dots,\chi^{-(n-1)}$, hence the invariants vanish (recall that $m\ge n$). Proof of the main result {#sec:main-proof} ======================== In this section, we will study the derived categories $\Db(\wY)$ and $\Db_G(X)$ in the setup described in the previous section. In particular, we will prove Theorems \[thm:main\] and \[thm:tensor\]. We set $n=\codim(S\hookrightarrow X)$ and $m=|G|$, in other words $G=\mu_m$. We consider, for $\alpha\in \IZ/m\IZ$ and $\beta\in\IZ$, the exact functors $$\begin{aligned} \Phi\coloneqq p_*\circ q^*\circ\triv &\colon \Db(\wY)\to \Db_G(X)\\ \Psi\coloneqq (-)^G\circ q_*\circ p^*&\colon \Db_G(X)\to \Db(\wY)\\ \Theta_\beta\coloneqq i_*(\nu^*(\_)\otimes \reg_\nu(\beta)) &\colon \Db(S)\to \Db(\wY)\\ \Xi_\alpha \coloneqq (a_*\circ\triv)\otimes \chi^\alpha &\colon \Db(S)\to \Db_G(X).\end{aligned}$$ With this notation, the precise version of is \[thm:mainprecise\] 1. The functor $\Phi$ is fully faithful for $m\ge n$ and an equivalence for $m=n$. For $m>n$, all the functors $\Xi_\alpha$ are fully faithful and there is a semi-orthogonal decomposition $$\Db_G(X) = \Sod{ \Xi_{n-m}(\Db(S)), \ldots, \Xi_{-1}(\Db(S)), \Phi(\Db(\wY)) } \,. $$ 2. The functor $\Psi$ is fully faithful for $n\ge m$ and an equivalence for $n=m$. For $n>m$, all the functors $\Theta_\beta$ are fully faithful and there is a semi-orthogonal decomposition $$\Db(\wY) = \Sod{ \Theta_{m-n}(\Db(S)), \ldots, \Theta_{-1}(\Db(S)), \Psi(\Db_G(X)) } \,. $$ We will see later in that $K_\wY\le \rho^* K_Y$ for $m\ge n$ and $K_\wY\ge \rho^*K_Y$ for $n\ge m$. Hence, is in accordance with the DK-Hypothesis as described in the introduction. For the proof, we first need some more preparations. Generators and linearity ------------------------ \[lem:relative-tilting\] The bundle $V \coloneqq \reg_X\otimes\IC[G] = \reg_X \otimes (\chi^0\oplus\cdots\oplus\chi^{m-1})$ is a relative tilting sheaf for $\Db_G(X)$ over $\Dperf(Y)$. If $L\in\Pic(Y)\subset\Dperf(Y)$ is an ample line bundle, then so is $\pi^*(L)$. Hence, $\Db(X)$ has a generator of the form $E \coloneqq \pi^*(\reg_Y\oplus L\oplus\cdots\oplus L^{\otimes k})$ for some $k\gg0$; see [@Orlov_gen]. In particular, $E$ is a spanning class of $\Db(X)$. Using the adjunction $\Res\dashv \Ind\dashv\Res$, it follows that $\Ind(E) \cong E\oplus E\otimes \chi \oplus \dots\oplus E\otimes \chi^{m-1}$ is a spanning class of $\Db_G(X)$. Hence, $V=\Ind\reg_X$ is a relative spanning class of $\Db_G(X)$ over $\Dperf(Y)$. Since $V$ is a vector bundle, so is $\sHom(V,V) = V\dual \otimes V$. The map $\pi$ is finite, hence $\pi_*$ is exact (does not need to be derived). Finally, taking $G$-invariants is exact because we work in characteristic 0. Altogether, $\pi_*^G\sHom(V,V)$ is a sheaf concentrated in degree 0. \[Psilinear\] The functors $\Phi$ and $\Psi$, and for all $\alpha,\beta\in\IZ$ the subcategories $$\begin{aligned} \Xi_\alpha(\Db(S)) &= a_*(\Db(S)) \otimes \chi^\alpha \subset \Db_G(X) \quad\text{and}\\ \Theta_\beta(\Db(S)) &= i_*\nu^*\Db(S) \otimes \reg_\wY(\beta) \subset \Db(\wY)\end{aligned}$$ are $Y$-linear for $\pi^*\triv\colon \Dperf(Y)\to \Db_G(X)$ and $\rho^*\colon \Dperf(Y)\to \Db(\wY)$, respectively. We first show that $\Phi$ is $Y$-linear. Recall that in our setup this means $$\Phi(\rho^*(E)\otimes F)\cong \pi^*\triv(E)\otimes \Phi(F)$$ for any $E\in \Dperf(Y)$ and $F\in \Db(\wY)$. But this holds, since $$\begin{aligned} \pi^*\triv(E) \otimes \Phi(F) &\cong \pi^*\triv(E) \otimes p_*q^*\triv(F) \\ &\cong p_*(p^*\pi^*\triv(E) \otimes q^*\triv(F)) \\ &\cong p_*(q^*\rho^*\triv(E) \otimes q^*\triv(F)) \\ &\cong p_*q^*\triv(\rho^*(E) \otimes F) \,.\end{aligned}$$ The proof that $\Psi$ is $Y$-linear is similar and is left to the reader. The $Y$-linearity of the image categories follows from (i). \[lem:relative-spanning-wY\] The set of sheaves $\cS \coloneqq \{\reg_\wwY\} \cup \{ i_{s*}\Omega^r(r) \mid s\in S, r=0,\dots, n-1\}$ forms a spanning class of $\Db(\wY)$ over $Y$, where $i_s\colon \IP^{n-1} \cong \rho^{-1}(s)\hookrightarrow \wY$ denotes the fibre embedding. We need to show that $\hat \cS\coloneqq \rho^*\Dperf(Y)\otimes \cS$ is a spanning class of $\Db(\wY)$. Let $\tilde y\in \wY\setminus Z$. Then $y=\rho(\tilde y)$ is a smooth point of $Y$. Hence, $\reg_y\in \Dperf(Y)$ and $\reg_{\tilde y}\in \rho^*\Dperf(Y)=\rho^*\Dperf(Y)\otimes \reg_\wwY\subset \hat \cS$. Thus, an object $E\in \Db(\wY)$ with $\supp E\cap(\wY\setminus Z)\neq \emptyset$ satisfies $\Hom^*(E,\hat\cS)\neq 0 \neq \Hom^*(\hat\cS, E)$; see [@Huy Lem. 3.29]. Let now $0\neq E\in \Db(\wY)$ with $\supp E\subset Z$. Then there exists $s\in S$ such that $i_s^*E\neq 0\neq i_s^!E$; see again [@Huy Lem. 3.29]. Since the $\Omega^r(r)$ form a spanning class of $\IP^{n-1}$, we get by adjunction $\Hom^*(E,\cS)\neq 0 \neq \Hom^*(\cS, E)$. On the equivariant blow-up -------------------------- Recall that the blow-up morphism $q\colon \wX\to X$ is $G$-equivariant. Let $\cL_{\wwX}\in \Pic^G(\wwX)$ (we will sometimes simply write $\cL$ instead of $\cL_\wwX$) be the equivariant line bundle $\reg_\wwX(Z)$ equipped with the unique linearisation whose restriction to $Z$ gives the trivial action on $\reg_Z(Z)\cong\reg_\nu(-1)$. We consider a point $z\in Z$ with $\nu(z)=s$ corresponding to a line $\ell\subset N_{S/X}(s)$. Then the normal space $N_{Z/\wwX}(z)$ can be equivariantly identified with $\ell$. It follows by that $N_{Z/\wwX}\cong (\cL_{\wwX}\otimes \chi^{-1})_{|Z}$ as an equivariant bundle. Hence, in $\smash{\Coh_G(\wX)}$, there is the exact sequence $$\begin{aligned} \label{eq:exact-wX} 0\to \cL_{\wX}^{-1}\otimes \chi\to\reg_\wX\to \reg_Z\to 0 \end{aligned}$$ where both $\reg_\wwX$ and $\reg_Z$ are equipped with the canonical linearisation, which is the one given by the trivial action over $Z$. \[lem:pushforward-of-p\] For $\ell=0,\dots, n-1$ we have $p_*\cL_{\wwX}^\ell=\reg_X\otimes \chi^\ell$. We have $p_*\reg_{\wwX}\cong \reg_X$, both, $\reg_\wwX$ and $\reg_X$, equipped with the canonical linearisations. Hence, the assertion is true for $\ell=0$. By induction, we may assume that $p_*\cL_{\wwX}^{\ell-1}\cong \reg_X\otimes \chi^{\ell-1}$. We tensor by $\cL_{\wwX}^\ell$ to get $$0 \to \cL_{\wX}^{\ell-1} \otimes \chi \to \cL_{\wX}^\ell \to \reg_\nu(-\ell) \to 0 \,.$$ Since $0\le \ell\le n-1$, we have $p_*\reg_\nu(-\ell)=0$. Hence, we get an isomorphism $$p_* \big( \cL_{\wX}^\ell \big) \cong p_* \big( \cL_{\wX}^{\ell-1}\otimes \chi \big) \cong p_* \big( \cL_{\wX}^{\ell-1} \big)\otimes \chi \cong \reg_X \otimes \chi^{\ell-1}\otimes \chi \cong \reg_X \otimes \chi^\ell \,.\qedhere$$ \[lem:canonical\] The smooth blow-up $p\colon\wX\to X$ has $G$-linearised relative dualising sheaf $$\omega_p \cong \cL_{\wX}^{n-1}\otimes \chi^{1-n} \in \Pic^G(\wX) \,.$$ The non-equivariant relative dualising sheaf of the blow-up is $\omega_p\cong \reg_{\wwX}((n-1)Z)$. Since $p$ is $G$-equivariant, $\omega_p$ has a unique linearisation such that $p^!=p^*(\_)\otimes \omega_p\colon \Db_G(X)\to \Db_G(\wwX)$ is the right-adjoint of $p_*\colon\Db_G(\wwX)\to \Db_G(X)$. We now compute this linearisation of $\omega_p$. As the equivariant pull-back $p^*$ is fully faithful, $p^!\colon \Db_G(X)\to \Db_G(\wwX)$ is fully faithful, too. Hence, adjunction gives an isomorphism of equivariant sheaves, $p_*\omega_p\cong p_*p^!\reg_X\cong \reg_X$. The claim now follows from . We denote by $i_s\colon \IP^{n-1} \cong \rho^{-1}(s) \hookrightarrow \wY$ the embedding of the fibre of $\rho$ and by $j_s\colon \IP^{n-1} \cong p^{-1}(s) \hookrightarrow \wX$ the embedding of the fibre of $p$ over $s\in S$. \[lem:sky\] For $s\in S$ and $r=0,\dots,n-1$, the cohomoloy sheaves of $p^*\reg_s \in \Db_G(\wX)$ are $$\CH^{-r}(p^*\reg_s) \cong j_{s*}(\Omega^r(r)\otimes \chi^r) \,.$$ It is well known that, for the underlying non-equivariant sheaves, we have $\CH^{-r}(p^*\reg_s)\cong j_{s*}\Omega^r(r)$; see [@Huy Prop. 11.12]. Since the sheaves $\Omega^r(r)$ are simple, i.e. $\End(\Omega^r(r))=\IC$, we have $\CH^{-r}(p^*\reg_s)\cong j_{s*}(\Omega^r(r)\otimes \chi^{\alpha_r})$ for some $\alpha_r\in \IZ/m\IZ$. So we only need to show $\alpha_r=r$. Let $r\in\{0,\ldots,n-1\}$. We have $p_!\cL^{-r} \cong p_*(\cL^{-r+n-1}\otimes \chi^{1-n})$ by . Since $-r+n-1 \in \{0,\ldots,n-1\}$, gives $p_!\cL^{-r}\cong \reg_X\otimes \chi^{-r}$. By adjunction, $$\IC[0] \cong \Hom^*_{\Db_G(X)}(\reg_X\otimes \chi^{-r},\reg_s\otimes \chi^{-r}) \cong \Hom^*_{\Db_G(\wwX)}(\cL^{-r}, p^*\reg_s\otimes\chi^{-r}) \,.$$ By , for $r\neq v$, we have $$\Hom^*_{\Db(\wX)}(\reg_\wX(-rZ), j_{s*}\Omega^v(v)) \cong \Hom^*_{\Db(\IP^{n-1})}(\reg(r),\Omega^v(v)) = 0 \,.$$ Using the spectral sequence in $\Db_G(\wX)$ $$E_2^{u,v} = \Hom^u(\cL^{-r},\CH^v(p^*\reg_s\otimes\chi^{-r})) \:\Rightarrow\: E^{u+v} = \Hom^{u+v}(\cL^{-r}, p^*\reg_s\otimes\chi^{-r})$$ it follows that $$\begin{aligned} \IC[0] &\cong \Hom^*_{\Db_G(\wX)}(\cL^{-r}, p^*\reg_s\otimes\chi^{-r}) \\ &\cong \Hom^*_{\Db_G(\wX)}(\cL^{-r}, \CH^{-r}(p^*\reg_s)\otimes\chi^{-r})[r] \\ &\cong \bigl( \Hom^*_{\Db(\IP^{n-1})}(\reg(r),\Omega^r(r))\otimes \chi^{\alpha_r}\otimes\chi^{-r} \bigr)^G[r] \\ &\cong \bigl( \IC[-r]\otimes\chi^{\alpha_r-r} \bigr)^G[r] \end{aligned}$$ where the last isomorphism is again due to . Comparing the first and last term of the above chain of isomorphisms, we get $\IC\cong (\chi^{\alpha_r-r})^G$ which implies $\alpha_r=r$. \[cor:Psi-sky\] Let $n\ge m$ and $\ell\in\{0,\ldots,m-1\}$. Let $\lambda\ge 0$ be the largest integer such that $\ell+\lambda m\le n-1$. Then $$\CH^*(\Psi(\reg_s\otimes \chi^{-\ell})) \cong i_{s*}\bigl( \bigoplus\limits_{t=0}^{\lambda} \Omega^{\ell+tm}(\ell+tm)[\ell+tm] \bigr) \,.$$ Since the (non-derived) functor $q_*^G\colon \Coh^G(\wX)\to \Coh(\wY)$ is exact, we have $$\CH^{-r}(\Psi(\reg_s \otimes \chi^{-\ell}))0 \cong q_*^G\bigl( \CH^{-r}(p^*\reg_s) \otimes \chi^{-\ell} \bigr)$$ and the claim follows from . On the cyclic cover ------------------- The morphism $q\colon \wX\to \wY=\wX/G$ is a cyclic cover branched over the divisor $Z$. This geometric situation and the derived categories involved are studied in great detail in [@KuzPerrycyclic]. However, we will only need the following basic facts, all of which can be found in [@KuzPerrycyclic Sect. 4.1]. \[lem:cyclic\] 1. The sheaf of invariants $q_*^G(\reg_\wwX\otimes\chi^{-1})$ is a line bundle which we denote $\cL_{\wwY}^{-1}\in \Pic(\wY)$. 2. $\cL_{\wwY}^m \cong \reg_{\wwY}(Z)$. 3. $q^G_*(\reg_\wwX\otimes \chi^{\alpha}) \cong \cL_\wwY^\alpha$ for $\alpha\in \{-m+1,\dots,0\}$. 4. $q^*\circ\triv\colon \Db(\wY) \into \Db_G(\wX)$ is fully faithful, due to $q_*^G(\reg_\wwX) \cong \reg_\wwY$. 5. $q^*(\triv(\cL_\wwY))\cong \cL_\wwX$ are isomorphic $G$-equivariant line bundles. 6. In particular, $\cL_{\wwY\mid Z}\cong\cL_{\wwX\mid Z}\cong \reg_\nu(-1)$. \[cor:Psialpha\] $\Psi(\reg_X\otimes \chi^{\alpha}) \cong \cL_{\wwY}^\alpha$ for $\alpha\in \{-m+1,\ldots,0\}$. \[lem:qcanonical\] The relative dualising sheaf of $q\colon \wX\to\wY=\wX/G$ is $\omega_q\cong \reg_\wX((m-1)Z)$. Since the $G$-action on $W \coloneqq \wX\setminus Z$ is free, we have $\omega_{q\mid W}\cong \reg_W$. Hence, $\omega_q\cong \reg_{\wX}(\alpha Z)$ for some $\alpha\in \IZ$. We have $\sHom(\reg_Z,\reg_\wX)\cong \reg_Z(Z)[-1]\cong j_*\reg_\nu(-1)[-1]$, and hence $$\begin{array}{rcll} i_*\reg_{\nu}(-1)[-1] &\cong& q_* j_*\reg_\nu(-1)[-1] \\ &\cong& q_* \sHom(\reg_Z, \reg_\wX) \\ &\cong& q_* \sHom(\reg_Z, q^*\reg_\wY) \\ &\cong& q_* \sHom(\reg_Z,q^!\cL_\wY^{-\alpha}) & \text{by \autoref{lem:cyclic}(v)} \\ &\cong& \sHom(q_*\reg_Z,\cL_\wY^{-\alpha}) & \text{by Grothendieck duality} \\ &\cong& \reg_Z(Z)\otimes \cL_\wY^{-\alpha}[-1] \\ &\cong& i_*\reg_\nu(-m+\alpha)[-1] & \text{by \autoref{lem:cyclic}(ii)+(vi)} \end{array}$$ and thus we conclude $\alpha= m-1$. As an equivariant bundle, we have $\omega_q\cong \cL_\wX^{m-1}\otimes \chi$, but we will not use this. \[lem:canonicalwY\] We have $\omega_{\wY\mid Z} \cong \reg_\nu(m-n)$. We have $\omega_{\wX\mid Z}\cong \reg_\nu(-n+1)$; compare . Furthermore, $\omega_{\wY\mid Z}\cong (q^*\omega_{\wY})_{| Z}$. Hence, $$\reg_\nu(1-m) \overset{\ref{lem:qcanonical}}\cong \omega_{q\mid Z}\cong \omega_{\wX\mid Z}\otimes \omega_{\wY\mid Z}^\vee \cong \reg_{\nu}(1-n)\otimes \omega_{\wY\mid Z}^\vee \,.\qedhere$$ The case $\boldsymbol{m\ge n}$ {#sub:m-geq-n} ------------------------------ Throughout this subsection, let $m\ge n$. \[prop:Hff\] 1. If $m>n$, then the functor $\Xi_\alpha$ is fully faithful for any $\alpha\in \IZ/m\IZ$. 2. Let $m-n\ge 2$ and $\alpha\neq \beta\in \IZ/m\IZ$. Then $$\Xi_{\beta}^R\Xi_\alpha = 0 \iff \alpha-\beta \in \{\overline{n-m+1},\overline{n-m+2},\dots,\overline{-1}\} \,.$$ Recall that $\Xi_\beta = (a_*\circ\triv(\_)) \otimes \chi^\beta \colon \Db(S)\to \Db_G(X)$. Hence, the right-adjoint of $\Xi_\beta$ is given by $\Xi_\beta^R\cong (a^!(\_)\otimes \chi^{-\beta})^G$. By [@AC Thm. 1.4 & Sect. 1.20], $$\begin{aligned} \Xi_\beta^R\Xi_\alpha &\cong \bigl(a^!a_*(\_)\otimes \chi^{\alpha-\beta}\bigr)^G \cong \bigl((\_)\otimes \wedge^*N\otimes \chi^{\alpha-\beta}\bigr)^G \\ &\cong (\_)\otimes \bigl(\wedge^*N\otimes \chi^{\alpha-\beta}\bigr)^G \end{aligned}$$ where, by , the $G$-action on $\wedge^\ell N$ is given by $\chi^{-\ell}$. We see that $(\wedge^*N)^G\cong \wedge^0N[0]\cong \reg_S[0]$; here we use that $m>n$. This shows that, in the case $\alpha=\beta$, we have $\Xi_\alpha^R\Xi_\alpha\cong \id$ which proves (i). Furthermore, since the characters occurring in $\wedge^*N$ are $\chi^0$, $\chi^{-1}$,…,$\chi^{-n}$, we obtain (ii) from $$\begin{aligned} \Xi_\beta^R\Xi_\alpha \neq 0 &\iff \bigl(\wedge^*N\otimes \chi^{\alpha-\beta}\bigr)^G\neq 0 \iff \overline{0} \in \{ \overline{\alpha-\beta},\overline{\alpha-\beta-1},\ldots,\overline{\alpha-\beta-n} \}, \text{ i.e.} \\ \Xi_\beta^R\Xi_\alpha = 0 & \iff \alpha-\beta \in \{\overline{n+1},\dots,\overline{m-1}\} = \{\overline{n-m+1},\overline{n-m+2},\dots,\overline{-1}\} \,.\qedhere\end{aligned}$$ \[cor:sod\] For $m>n$, there is a semi-orthogonal decomposition $$\Db_G(X) = \Sod{ \Xi_{n-m}(\Db(S)), \Xi_{n-m+1}(\Db(S)), \dots, \Xi_{-1}(\Db(S)), \cA },$$ where $\cA = \lorth\Sod{ \Xi_{n-m}(\Db(S)), \Xi_{n-m+1}(\Db(S)), \dots, \Xi_{-1}(\Db(S))} $. \[prop:ff\] The functor $\Phi = p_*q^*\triv \colon \Db(\wY)\to \Db_G(X)$ is fully faithful. By [@Huy Prop. 7.1], we only need to show for $x,y\in \wY$ that $$\Hom^i_{\Db_G(X)}(\Phi(\reg_x),\Phi(\reg_y)) = \begin{cases} \IC \quad & \text{if $x=y$ and $i=0$} \\ 0 \quad & \text{if $x\neq y$ or $i\notin[0,\dim X]$}. \end{cases}$$ By , $\Phi(\reg_x)=\reg_\xi$ for some $G$-cluster $\xi$. Hence, $$\Hom^0_{\Db_G(X)}(\Phi(\reg_x),\Phi(\reg_x)) \cong H^0(\reg_\xi)^G \cong \IC \,.$$ Furthermore, since $\Phi(\reg_x)$ is a sheaf, the complex $\Hom^*(\Phi(\reg_x),\Phi(\reg_x))$ is concentrated in degrees $0,\dots,\dim(X)$. It remains to prove the orthogonality for $x\neq y$. If $\rho(x)\neq\rho(y)$, the corresponding $G$-clusters are supported on different orbits. Hence, their structure sheaves are orthogonal. If $\rho(x)=\rho(y)$ but $x\neq y$, the orthogonality was shown in . \[lem:C\] The functor $\Phi$ factors through $\cA$. By , this statement is equivalent to $\Phi^R \Xi_\alpha=0$ for $\alpha\in\{n-m,\ldots,-1\}$ where $\Phi^R\colon \Db_G(X)\to \Db(\wY)$ is the right adjoint of $\Phi$. Since the composition $\Phi^R \Xi_\alpha$ is a Fourier–Mukai transform, it is sufficient to test the vanishing on skyscraper sheaves of points; see [@Kuz Sect. 2.2]. So we have to prove that $$\Phi^R \Xi_\alpha(\reg_s) \cong \Phi^R(\reg_s \otimes \chi^\alpha) = 0$$ for every $s\in S$ and every $\alpha\in \{n-m,\ldots,-1\}$. We have $\Phi^R\cong q_*^G p^!$; recall that $q_*^G$ stands for $(\_)^G\circ q_*$. By together with , we have $$\CH^{-r}(p^!\reg_s) \cong i_{s*}(\Omega^r(r+1-n) \otimes \chi^{r+1-n})$$ where the non-vanishing cohomologies occur for $r\in\{0,\dots,n-1\}$. Thus, the linearisations of the cohomologies of $p^!(\reg_s\otimes \chi^{\alpha})$ are given by the characters $\chi^\gamma$ for $\gamma \in\{\alpha+1-n,\ldots,\alpha\}$. We see that, for $\alpha\in \{n-m,\ldots,-1\}$, the trivial character does not occur in $\CH^*(p^!\reg_s\otimes \chi^\alpha)$. This implies that $q_*p^!(\reg_s\otimes \chi^\alpha)$ has vanishing $G$-invariants. We denote by $\cB \subset \Db_G(X)$ the full subcategory generated by the admissible subcategories $\Xi_\alpha(\Db(S))$ for $\alpha\in \{n-m,\dots, -1\}$ and $\Phi(\Db(\wY))$. By the above, these admissible subcategories actually form a semi-orthogonal decomposition $$\cB = \Sod{ \Xi_{n-m}(\Db(S)), \Xi_{n-m+1}(\Db(S)), \dots, \Xi_{-1}(\Db(S)), \Phi(\Db(\wY)) } \,.$$ \[prop:full\] We have the (essential) equalities $\cB = \Db_G(X)$ and $\Phi(\Db(\wY)) = \cA$. For the proof, we need the following We have $p_*\cL_{\wwX}^r\otimes \chi^{-\lambda}\in \cB$ for $r\in \IZ$ and $\lambda\in\{0,\ldots,m-n\}$. By , $\cL_{\wwX}^r\cong q^*(\triv(\cL_{\wwY}^r))$. Hence, $$p_*(\cL_{\wwX}^r) \cong p_*q^*(\triv(\cL_{\wY}^r)) = \Phi(\cL_\wY^r) \in \Phi(\Db(\wY)) \subset \cB$$ which proves the assertion for $\lambda=0$. We now proceed by induction over $\lambda$. Tensoring by $\cL_{\wX}^r\otimes \chi^{-\lambda}$ and applying $p_*$, we get the exact triangle $$\begin{aligned} \label{eq:ptriangle} p_*\cL^{r-1}_{\wX} \otimes \chi^{-(\lambda-1)} \to p_*\cL^r_{\wX}\otimes \chi^{-\lambda} \to p_*j_*\reg_Z(-r)\otimes \chi^{-\lambda} \to \end{aligned}$$ where $\reg_Z(-r)$ carries the trivial $G$-action. The first term of the triangle is an object of $\cB$ by induction. Furthermore, by diagram , we have $p_*j_*\reg_Z(-r)\cong a_*\nu_*\reg_Z(-r)$. Hence, the third term of is an object of $a_*\Db_G(S)\otimes\chi^{-\lambda}=\Xi_{-\lambda}(\Db(S))\subset \cB$. Thus, also the middle term is an object of $\cB$ which gives the assertion. The second assertion follows from the first one since, if $\cB=\Db_G(X)$ holds, both, $\Phi(\Db(\wY))$ and $\cA$, are given by the left-orthogonal complement of $$\Sod{ \Xi_{n-m}(\Db(S)), \Xi_{n-m+1}(\Db(S)), \dots, \Xi_{-1}(\Db(S))}$$ in $\Db_G(X)$. The subcategories $\Xi_\alpha(\Db(S))$ and $\Phi(\Db(\wY))$ of $\Db_G(X)$ are $Y$-linear by . Hence, for the equality $\cB=\Db_G(X)$ it suffices to show that $$\reg_X\otimes \chi^\ell\in \cB = \Sod{ \Db(S)\otimes \chi^{n-m}, \dots, \Db(S)\otimes \chi^{-1}, \Phi(\Db(\wY)) }$$ for every $\ell\in \IZ/m\IZ$; see . Combining and , we see that $$\Phi(\cL_{\wY}^\ell) \cong p_*(q^*\triv(\cL_{\wY})^\ell) \cong \reg_X\otimes \chi^\ell \quad \text{for $\ell=0,\dots, n-1$.}$$ In particular, $\reg_X\otimes\chi^\ell\in \Phi(\Db(\wY))\subset \cB$ for $\ell=0,\dots, n-1$. Setting $r=0$ in the previous lemma, we find that also $\reg_X\otimes \chi^{\ell}$ for $\ell=n-m,\dots,-1$ is an object of $\cB$. Combining the results of this subsection gives (i). The case $\boldsymbol{n\ge m}$ {#sub:n-geq-m} ------------------------------ Throughout this subsection, let $n\ge m$. Let $n>m$. Then the functors $\Theta_\beta\colon \Db(S)\to \Db(\wY)$ are fully faithful for every $\beta\in \IZ$ and there is a semi-orthogonal decomposition $$\Db(\wY) = \Sod{ \cC(m-n), \cC(m-n+1), \ldots, \cC(-1), \cD }$$ where $\cC(\ell) \coloneqq \Theta_\ell(\Db(S)) = i_*\nu^*\Db(S) \otimes \reg_\wY(\ell)$ and $$\begin{aligned} \cD &= \bigl\{ E \in \Db(\wY) \mid i^*E\in \lorth \Sod{ \nu^*\Db(S)\otimes\reg_\wY(m-n),\dots,\nu^*\Db(S)\otimes\reg_\wY(-1) } \bigr\} \\ &= \bigl\{ E \in \Db(\wY) \mid i^*E\in \Sod{ \nu^*\Db(S), \dots, \nu^*\Db(S) \otimes \reg_\wY(m-1) }\bigr\} \,.\end{aligned}$$ This follows from [@KuzLefschetz Thm. 1]. However, for convenience, we provide a proof for our special case. By construction, $\Theta_\beta^R\cong \nu_*\MM_{\reg_\nu(-\beta)} i^!$. We start with the standard exact triangle of functors $ \id\to i^!i_*\to \MM_{\reg_Z(Z)}[-1]\to $ (see e.g. [@Huy Cor. 11.4]). By , $\reg_Z(Z)\cong \reg_\nu(-m)$, and thus the above triangle induces for any $\alpha,\beta \in \IZ$ $$\nu_*\MM_{\reg_\nu(\alpha-\beta)}\nu^* \to \Theta_\beta^R\Theta_\alpha \to \nu_*\MM_{\reg_\nu(\alpha-\beta-m)}\nu^* \to \,.$$ By projection formula, we can rewrite this as $$(\_)\otimes\nu_*\reg_\nu(\alpha-\beta) \to \Theta_\beta^R\Theta_\alpha \to (\_)\otimes \nu_*\reg_\nu(\alpha-\beta-m) \to \,.$$ Now, $\nu_*\reg_\nu\cong \reg_S$ and $\nu_*\reg_\nu(\gamma)=0$ for $\gamma\in \{-n+1,\dots, -1\}$. Hence, $\Theta_\beta^R\Theta_\beta\cong \id$ and $\Theta_\beta^R\Theta_\alpha=0$ if $\alpha-\beta\in \{m-n+1,\dots,-1\}$. Therefore, we get a semi-orthogonal decomposition $$\Db(\wY) = \Sod{ \cC(m-n), \cC(m-n+1), \ldots, \cC(-1), \cD } \,.$$ The description of the left-orthogonal $\cD$ follows by the adjunction $i^*\dashv i_*$. The functor $\Psi\colon \Db_G(X)\to \Db(\wY)$ factors through $\cD$. By , the equivariant bundles $\reg_X,\reg_X\otimes\chi,\ldots\reg_X\otimes\chi^{m-1}$ generate $\Db_G(X)$ over $\Dperf(Y)$, and therefore so do the bundles $\reg_X\otimes\chi^{-m+1},\ldots,\reg_X\otimes\chi^{-1},\reg_X$ obtained by twisting with $\chi^{1-m}$. Hence, it is sufficient to prove that $\Psi(\reg_X\otimes \chi^\alpha)\in \cD$ for $\alpha \in \{-m+1,\ldots,0\}$ as $\Psi$ and $\cD$ are $Y$-linear; see . Indeed, by we have $i^*\cL_\wY^\alpha = \cL^\alpha_{\wY\mid Z}\cong \reg_\nu(-\alpha)$, hence $$\Psi(\reg_X\otimes \chi^\alpha) \smash{\overset{\ref{cor:Psialpha}}\cong} q_*^G(\reg_X\otimes \chi^\alpha) \cong \cL_\wY^{\alpha} \in \cD \quad \text{for $\alpha \in \{-m+1,\ldots,0\}$.} \qedhere$$ The functor $\Psi\colon \Db_G(X)\to \Db(\wY)$ is fully faithful. We first observe that $V \coloneqq \reg_X\otimes\IC[G] = \reg_X \otimes (\chi^0\oplus\cdots\oplus\chi^{m-1})$ is a relative tilting bundle for $\Db_G(X)$ over $\Dperf(Y)$; see . For the fully faithfulness, we follow . So we need to show that $\Psi$ induces an isomorphism $\Lambda_V=\pi_*^G\sHom(V,V) \isomor \rho_*\sHom(\Psi(V),\Psi(V))$. In turn, it suffices to consider the direct summands of $V$. Thus, let $\alpha,\beta\in\{-m+1,\ldots,0\}$ and compute $$\renewcommand{\arraystretch}{1.4} \begin{array}{rcl} \pi_*^G \sHom^*( \reg_X\otimes\chi^\alpha, \reg_X \otimes \chi^\beta) &\cong& \pi_*^G \sHom^*( \reg_X\otimes\chi^{\alpha+n-1}\otimes\chi^{1-n}, \reg_X \otimes \chi^\beta) \\ &\overset{\ref{lem:pushforward-of-p}}{\cong}& \pi_*^G \sHom^*( p_*\cL_\wX^{\alpha+n-1}\otimes\chi^{1-n}, \reg_X \otimes \chi^\beta) \\ &\overset{\ref{lem:canonical}}{\cong}& \pi_*^G \sHom^*( p_*(\cL_\wX^\alpha\otimes\omega_p), \reg_X \otimes \chi^\beta) \\ &\cong& \pi_*^G p_* \sHom^*_X(\cL_\wX^\alpha \otimes \omega_p, p^!\reg_X \otimes \chi^\beta) \\ &\cong& \pi_*^G p_* \sHom^*_\wX(\cL_\wX^\alpha, p^* \reg_X\otimes \chi^\beta) \\ &\overset{\ref{lem:cyclic}}{\cong}& \rho_* q_*^G \sHom^*_\wX(q^*q_*^G(\reg_\wX\otimes \chi^\alpha), \reg_\wX\otimes \chi^\beta) \\ &\cong& \rho_*\sHom^*_\wY(q_*^G(\reg_\wX\otimes \chi^\alpha), q_*^G(\reg_\wX\otimes \chi^\beta)) \\ &=& \rho_*\sHom^*_\wY(\Psi(\reg_\wX\otimes \chi^\alpha), \Psi(\reg_\wX\otimes \chi^\beta)) \,.\hfill\qedhere \end{array}$$ We denote by $\cE\subset \Db(\wY)$ the full subcategory generated by the admissible subcategories $\Psi(\Db_G(X))$ and $\Theta_\ell(\Db(S)) = i_*\nu^*\Db(S)\otimes\reg_\wwY(\ell)$ for $\ell\in \{m-n,\dots, -1\}$. By the above, these admissible subcategories actually form a semi-orthogonal decomposition $$\cE = \Sod{ \Theta_{m-n}(\Db(S)), \ldots, \Theta_{-1}(\Db(S)), \Psi(\Db_G(X)) } \subseteq \Db(\wY) \,.$$ \[prop:fullB\] We have the (essential) equalities $\cE = \Db(\wY)$ and $\Psi(\Db_G(X)) = \cD$. Analogously to , it is sufficient to prove the equality $\cE = \Db(\wY)$. As $\cE$ is constructed from images of fully faithful FM transforms (which have both adjoints), it is admissible in $\Db(\wY)$. Therefore, it suffices to show that $\cE$ contains a spanning class for $\Db(\wY)$. Moreover, because all functors and categories involved are $Y$-linear, it suffices to prove that the relative spanning class $\cS$ of is contained in $\cE$. We already know that $\reg_\wY\cong \Psi(\reg_X)\in\Psi(\Db_G(X))\subset \cE$. By , we get for $s\in S$ and $r\in\{m,\ldots,n-1\}$ $$i_{s*}\Omega^r(r) \in \Sod{ \Theta_{m-n}(\Db(S)), \ldots, \Theta_{-1}(\Db(S)) } \subset \cE \, .$$ By , we have, for $\ell\in\{0,\ldots,m-1\}$, an exact triangle $$E \to \Psi(\reg_s\otimes \chi^{-\ell}) \to i_{s*}\Omega^\ell(\ell)[\ell] \to$$ where $E$ is an object in the triangulated category spanned by $i_{s*}\Omega^r(r)$ for $r\in\{m,\ldots,n-1\}$. In particular, the first two terms of the exact triangle are objects in $\cE$. Hence also $i_{s*}\Omega^\ell(\ell)\in \cE$ for $\ell\in\{0,\ldots,m-1\}$. Combining the results of this subsection gives (ii). The case $\boldsymbol{m=n}$: spherical twists and induced tensor products {#m=nsection} ------------------------------------------------------------------------- Throughout this subsection, $m=n$, so that both functors $\Phi$ and $\Psi$ are equivalences. We will show that the functors $\Theta_\beta$ and $\Xi_\alpha$, which were fully faithful in the cases $n>m$ and $m>n$, respectively, are now spherical. Furthermore, the spherical twists along these functors allow to describe the transfer of the tensor structure from one side of the derived McKay correspondence to the other. We set $\Theta\coloneqq \Theta_0$ and $\Xi\coloneqq \Xi_0$. \[prop:Xispherical\] For every $\alpha\in \IZ/m\IZ$, the functor $\Xi_\alpha\colon \Db(S)\to \Db_G(X)$ is a split spherical functor with cotwist $\MM_{\omega_{S/X}}[-n]$. Since $\Xi_\alpha\cong \MM_{\chi^\alpha} \Xi$, it is sufficient to prove the assertion for $\alpha=0$; see . Following the proof of , we have $\Xi^R\Xi\cong (\_)\otimes (\wedge^*N)^G$ where $G$ acts on $\wedge^\ell N$ by $\chi^{-\ell}$. From $\rank N=n=m=\ord \chi$, we get $$(\wedge^*N)^G \cong \reg_S[0] \oplus \det N[-n] \cong \reg_S[0]\oplus\omega_{S/X}[-n] \,$$ Hence, $\Xi^R\Xi\cong \id\oplus\, C$ with $C\coloneqq \MM_{\omega_{S/X}}[-n]$. Moreover, $\Xi^R\cong C\Xi^L$ follows from $a^!\cong C a^*$. We introduce autoequivalences $\MM_{\cL}\colon \Db(\wY)\to \Db(\wY)$ and $\MM_{\chi}\colon \Db_G(X)\to \Db_G(X)$ given by the tensor products with the line bundle $\cL_{\wwY}$ and the character $\chi$, respectively. \[thm:n=m\] There are the following relations between functors: 1. \[m=n1\] $\Psi^{-1}\cong \MM_\chi \Phi \MM_{\cL^{n-1}}$; 2. $\Psi \Xi\cong \Theta$, in particular, the functors $\Theta_\beta$ are spherical too; 3. $\TT_\Theta\cong \Psi \TT_\Xi \Psi^{-1}$; 4. \[m=n4\] $\Psi^{-1} \MM_\cL \Psi \cong \MM_\chi\TT_\Xi$ and $\Psi^{-1}\MM_{\cL^{-1}}\Psi\cong \TT_\Xi^{\;-1}\MM_{\chi^{-1}}$. In the verification of (i), we use $\omega_p \cong \cL_{\wX}^{n-1}\otimes \chi$, from and $m=n$: $$\begin{aligned} \Psi^{-1}\cong \Psi^L \cong p_!q^* \overset{\ref{lem:canonical}}{\cong} p_*\MM_{\cL_{\wX}^{n-1}\otimes \chi}q^* \overset{\ref{lem:cyclic}}{\cong} \MM_\chi p_*q^* \MM_{\cL^{n-1}} \cong \MM_\chi \Phi \MM_{\cL^{n-1}} \,. \end{aligned}$$ For (ii), first note that, since the $G$-action on $Z\subset \wX$ is trivial, we have $$\Theta\cong i_*\nu^*\cong q_*j_*\nu^*\cong q_*^Gj_*\nu^*\triv \,.$$ Hence, the base change morphism $\theta \colon p^*a_*\to j_*\nu^*$ induces a morphism of functors $$\hat\theta \colon \Psi \Xi \cong q_*^Gp^*a_*\triv\to q_*^Gj_*\nu^*\triv \cong \Theta$$ which in turn is induced by a morphism between the Fourier–Mukai kernels; see [@Kuz Sect. 2.4]. Hence, it is sufficient to show that $\theta$ induces an isomorphism $\Psi \Xi(\reg_s)\cong \Theta(\reg_s)$ for every $s\in S$; see [@Kuz Sect. 2.2]. The morphism $\theta$ induces an isomorphism on degree zero cohomology $L^0p^*a_*(\reg_s)\cong \reg_{p^{-1}(a(s))}\cong j_*L^0\nu^*(\reg_s)$. But there are no cohomologies in non-zero degrees for $j_*\nu^*$ since $\nu$ is flat and $j$ a closed embedding. Furthermore, the non-zero cohomologies of $p^*a_*$ vanish after taking invariants; see . Hence, $\hat\theta(\reg_s)$ is indeed an isomorphism. The second assertion of (ii) and (iii) are direct consequences of and the formula $\Psi\Xi\cong \Theta$; see . For (iv), it is sufficient to prove the second relation, and we employ with $L_i=\reg_X\otimes\chi^i$; see also . Recall that $\TT_\Xi^{\;-1}=\cone (\id\to \Xi\Xi^L)[-1]$, and $\Xi^L\cong (\_)^Ga^*$. For $1\neq \alpha\in \IZ/n\IZ$, we get $$\Xi^L\MM_{\chi^-1}(\reg_X\otimes \chi^\alpha) \cong (\reg_S\otimes \chi^{\alpha-1})^G \cong 0 \,.$$ Hence, $\TT_\Xi^{\;-1}\MM_{\chi^{-1}}(\reg_X\otimes \chi^{\alpha})\cong \reg_X\otimes \chi^{\alpha-1}$. We have $\Xi^L(\reg_X)=\reg_S$. Therefore, $\Xi\Xi^L(\reg_X)\cong a_*\reg_S$ and $\TT_\Xi^{\;-1}(\reg_X)\cong \cI_S$. In summary, $$\TT_\Xi^{\;-1}\MM_{\chi^{-1}}(\reg_X\otimes \chi^{\alpha}) \cong \begin{cases} \reg_X\otimes \chi^{\alpha-1} \quad & \text{for $\alpha\neq 1$,} \\ \cI_S \quad & \text{for $\alpha= 1$.} \end{cases}$$ On the other hand, for $\alpha\in\{-n+1,\ldots,0\}$, we have $\Psi(\reg_X\otimes \chi^\alpha)\cong \cL^{\alpha}$; see . Hence, we have $$\Psi^{-1}\MM_{\cL^{-1}}\Psi(\reg_X\otimes \chi^\alpha) \cong \reg_X \otimes \chi^{\alpha-1} \quad \text{for $\alpha\in\{-n+2,\ldots,0\}$.}$$ For $\alpha=-n+1$, we use (i) to get $$\begin{aligned} \Psi^{-1}\MM_{\cL^{-1}}\Psi(\reg_X\otimes \chi^{1-n}) \cong \Psi^{-1}(\cL_{\wY}^{-n}) \cong \MM_\chi \Phi(\cL_{\wY}^{-1}) \overset{\ref{lem:cyclic}}\cong p_*(\cL_{\wX}^{-1}\otimes \chi) \cong \cI_S\end{aligned}$$ where we get the last isomorphism by applying $p_*$ to the exact sequence . Therefore, for every $\alpha\in \IZ/n\IZ$ we obtain isomorphisms $$\kappa_\alpha \colon F_1(L_\alpha) \coloneqq \TT_\Xi^{\;-1} \MM_{\chi^{-1}}(\reg_X\otimes \chi^\alpha) \isomor F_2(L_\alpha) \coloneqq \Psi^{-1} \MM_{\cL^{-1}} \Psi(\reg_X\otimes \chi^\alpha) \,.$$ Finally, we have to check that the isomorphisms $\kappa_\alpha$ can be chosen in such a way that they form an isomorphism of functors $\kappa\colon F_{1,V\mid \{L_0,\dots, L_{n-1}\}}\isomor F_{2,V\mid \{L_0,\dots, L_{n-1}\}}$ over every open set $V\subset Y$. Let $U \coloneqq Y\setminus S \subset Y$ the open complement of the singular locus. We claim that $F_{1,U} \cong \MM_\chi^{-1} \cong F_{2,U}$. This is clear for $F_2 = \TT_\Xi^{\;-1} \MM_\chi^{-1}$. Furthermore, the map $p\colon \wX \to X$ is an isomorphism and $q\colon \wwX \to \wwY$ is a free quotient when restricted to $W \coloneqq \pi^{-1}(U)$. Since also $\cL_\wwX = q_*^G(\reg_\wwX\otimes\chi)$, we get $\Psi_U \cong {\MM^{-1}_\chi}_{|U}$. Hence, over $W$, the $\kappa_{i\mid W}$ can be chosen functorially. By the above computations, each $\kappa_{i|W}$ is given by a section of the trivial line bundle. As $S$ has codimension at least 2 in $X$, the sections $\kappa_{i|W}$ over $W$ uniquely extend to sections $\kappa_i$ over $X$. The commutativity of the diagrams relevant for the functoriality now follows from the commutativity of the diagrams restricted to the dense subset $W$. The relations of allow to transfer structures between $\Db(\wY)$ and $\Db_G(X)$. For example, we can deduce the formula $\Psi\MM_{\chi^{-1}}\Psi^{-1} \cong \TT_\Theta \MM_{\cL^{-1}}$. Since $\reg_X\otimes \chi^{\alpha}$ for $\alpha\in \{ -(n-1),\ldots,0 \}$ form a relative generator of $\Db_G(X)$, their images $\cL^{\alpha}$ under $\Psi$ do as well. Hence, at least theoretically, our formulas give a complete description of the tensor products induced by $\Psi$ (and also $\Phi$) on both sides. Note that $\Phi$ and $\Psi$ are both equivalences, but not inverse to each other. Hence, they induce non-trivial autoequivalences $\Psi\Phi\in \Aut(\Db_G(X))$ and $\Phi\Psi\in \Aut(\Db(\wY))$. Considering the setup of the McKay correspondence as a flop of orbifolds as in diagram , it makes sense to call them *flop-flop autoequivalences*. These kinds of autoequivalences were widely studied for flops of varieties; see [@Todaspherical], [@BodBonflops], [@DonovanWemyssNCflops], [@DonovanWemysstwists], [@ADMflop]. The general picture seems to be that the flop-flop autoequivalences can be expressed via spherical and $\IP$-twists induced by functors naturally associated to the centres of the flops. This picture is called the ’flop-flop=twist’ principle; see [@ADMflop]. The following can be seen as the first instance of an orbifold ’flop-flop=twist’ principle which we expect to hold in greater generality. \[cor:flopfloptwistcor\] $\Psi\Phi \cong \TT_\Theta \MM_{\cL^{-n}} \cong \TT_\Theta \MM_{\reg_\wY(-Z)}$. Let us assume $m=n=2$ so that $\chi^{-1}=\chi$. Then, for every $k\in \IN$, we get $$\begin{aligned} \label{Scalaformula} \Phi(\cL^{-k}) \cong \cI_S^k\otimes \chi^k \end{aligned}$$ where $\cI_S^k$ denotes the $k$-th power of the ideal sheaf of the fixed point locus. Indeed, $$\begin{array}{rcl} \Phi(\cL^{-k}) \overset{\ref{thm:n=m}(i)} \cong \MM_\chi\Psi^{-1}(\cL^{-k-1}) &\cong& \MM_\chi(\Psi^{-1} \MM_{\cL^{-1}} \Psi)^k(\cL^{-1}) \\ &\overset{\ref{cor:Psialpha}}\cong& \MM_\chi(\Psi^{-1} \MM_{\cL^{-1}} \Psi)^k(\reg\otimes \chi) \\[1ex] &\overset{\ref{thm:n=m}(iv)}\cong& (\MM_\chi \TT_\Xi^{\;-1})^k(\reg_X) \\[2ex] &\cong& \cI_S^k\otimes \chi^k \,. \end{array}$$ The last isomorphism follows inductively using the short exact sequences $$0 \to \cI_S^{k+1} \to \cI_S^k \to \cI^k_S/\cI_S^{k+1} \to 0$$ and the fact that the natural action of $\mu_2$ on $\cI^k_S/\cI_S^{k+1}$ is given by $\chi^k$. Let now $S$ be a surface and $X=S^2$ with $\mu_2$ acting by permutation of the factors. Then $\wY=S^{[2]}$ is the Hilbert scheme of two points and $\cL_{\wY}$ is the square root of the boundary divisor $Z$ parametrising double points. For a vector bundle $F$ on $S$ of rank $r$, we have $$\det F^{[2]} \cong \cL_{\wY}^{-r}\otimes \cD_{\det F}$$ where $F^{[2]}$ denotes the tautological rank $2r$ bundle induced by $F$ and, for $L\in \Pic S$, we put $\cD_L \coloneqq \rho^*\pi_*(L\boxtimes L)^G\in \Pic S^{[2]}$. Hence, by the $\reg_Y$-linearity of $\Phi$, formula recovers the $n=2$ case of [@Scaladiagonal Thm. 1.8]. Categorical resolutions {#sec:categorical-resolutions} ======================= General definitions {#sub:catresdef} ------------------- Recall from [@KuzLefschetz] that a *categorical resolution* of a triangulated category $\cT$ is a smooth triangulated category $\widetilde{\cT}$ together with a pair of functors $P_*\colon \widetilde{\cT}\to \cT$ and $P^*\colon \cT^{\perf}\to \widetilde{\cT}$ such that $P^*$ is left adjoint to $P_*$ on $\cT^{\perf}$ and the natural morphism of functors $\id_{\cT^{\perf}}\to P_*P^*$ is an isomorphism. Here, $\cT^{\perf}$ is the triangulated category of perfect objects in $\cT$. Moreover, a categorical resolution $(\widetilde{\cT},P_*,P^*)$ is *weakly crepant* if the functor $P^*$ is also right adjoint to $P_*$ on $\cT^{\perf}$. For the notion of smoothness of a triangulated category see e.g. [@KuzLuntscatres]. For us it is sufficient to notice that every admissible subcategory of $\Db(Z)$ for some smooth variety $Z$ is smooth. In fact, we will always consider categorical resolutions of $\Db(Y)$, for some variety $Y$ with rational Gorenstein singularities, *inside* $\Db(\wY)$ for some fixed (geometric) resolution of singularities $\rho\colon \wY\to Y$. By this we mean an admissible subcategory $\wcT\subset \Db(\wY)$ such that $\rho^*\colon \Dperf(Y)\to \Db(\wY)$ factorises through $\wcT$. By Grothendieck duality, we get a canonical isomorphism $\reg_Y\cong \rho_*\reg_\wwY\cong \rho_*\omega_\rho$. This induces a global section $s$ of $\omega_\rho$, unique up to a global unit (i.e. scalar multiplication by an element of $\reg_Y(Y)^\times$), and hence a morphism of functors $$t \coloneqq \rho_*(\_\otimes s)\colon \rho_* \to \rho_! \,.$$ Since this morphism can be found between the corresponding Fourier–Mukai kernels, we may define the cone of functors $\rho_+ \coloneqq \cone(t) \colon \Db(\wY) \to \Db(Y)$. The *weakly crepant neighbourhood of $Y$ inside $\Db(\wY)$* is the full triangulated subcategory $$\WC(\rho) \coloneqq \ker(\rho_+) \subset \Db(\wY) \,.$$ If $\WC(\rho)$ is a smooth category (which is the case if it is an admissible subcategory of $\Db(\wY)$), it is a categorical weakly crepant resolution of singularities. By adjunction formula, $t\rho^*\colon \rho_*\rho^*\to \rho_!\rho^*$ is an isomorphism. Hence, $\rho_+\rho^*=0$ and $\rho^*\colon \Dperf(Y)\to \Db(\wY)$ factors through $\WC(\rho)$. By definition, $\rho_!$ is the left adjoint to $\rho^*$. Since $\rho_*$ and $\rho_!$ agree on $\WC(\rho)$, we also have the adjunction $\rho_*\dashv \rho^*$ on $\WC(\rho)$. We think of $\WC(\rho)$ as the biggest weakly crepant categorical resolution inside the derived category $\Db(\wY)$ of a given geometric resolution $\rho\colon \wY\to Y$. The only thing that prevents us from turning this intuition into a statement is the possibility that, for a given weakly crepant resolution $\cT\subset \Db(\wY)$, there might be an isomorphism $\rho_{*\mid \cT}\cong \rho_{!\mid\cT}$ which is not the restriction of $t$ (up to scalars). The weakly crepant neighbourhood in the cyclic setup ---------------------------------------------------- In the case of the resolution of the cyclic quotient singularities discussed in the earlier sections, $\WC(\rho)$ is indeed a categorical resolution by the following result. We use the notation of ; recall $G=\mu_m$. \[thm:cyclicWC\] Let $Y=X/G$, $\rho\colon \wY\to Y$ and $i\colon Z = \rho\inv(S)\hookrightarrow \wY$ be as in . Assume $m\mid n=\codim(S\hookrightarrow X)$ and $n>m$. Then there is a semi-orthogonal decomposition $$\WC(\rho) = \Sod{ i_*(\cE), \Psi(\Db_{\mu_m}(X)) }$$ where $$\begin{aligned} \cE = \langle &\cA(-m+1), \cA(-m+2)\dots, \cA(-1), \\ & \cA\otimes\Omega^{n-m-1}(n-m-1), \cA\otimes\Omega^{n-m-2}(n-m-2), \dots, \cA\otimes\Omega^{m}(m) \rangle\end{aligned}$$ with $\cA \coloneqq \nu^*\Db(S)$ and $\cA(i) \coloneqq \cA\otimes \reg(i)$; the $\cA\otimes \Omega^i(i)$ parts of the decomposition do not occur for $n=2m$. In particular, $\WC(\rho)$ is an admissible subcategory of $\Db(\wY)$. We first want to show that $\Psi(\Db_{\mu_m}(X))\subset \WC(\rho)$. For this, by , it is sufficient to show that $\cL_{\wwY}^a=\Psi(\reg_X\otimes \chi^a)\in \WC(\rho)$ for every $a\in\{-m+1,\ldots,0\}$. The equivariant derived category $\Db_{\mu_m}(X)$ is a strongly (hence also weakly) crepant categorical resolution of the singularities of $Y$ via the functors $\pi^*, \pi_*^{\mu_m}$; see [@Abuaf-catres Thm. 1.0.2]. Since $\Psi\circ \pi^*\cong \rho^*$ (see ), $\cC \coloneqq \Psi(\Db_{\mu_m}(X))$ is a crepant resolution via the functors $\rho^*,\rho_*$. Hence, $\rho_*\cL_{\wwY}^a\cong \rho_!\cL_\wwY^a$ for $a\in \{-m+1,\ldots,0\}$ and it is only left to show that this isomorphism is induced by $t$. Again by the $Y$-linearity of $\Psi$, we have $\rho_*\cL_\wwY^a\cong \pi_*(\reg_X\otimes \chi^a)^{\mu_m}$ which is a reflexive sheaf on the normal variety $Y$ (this follows for example by [@Hartreflexive Cor. 1.7]). By construction, $t$ induces an isomorphism over $Y\setminus S$. Since the codimension of $S$ is at least 2, $t\colon \rho_*\cL_{\wwY}^a\to \rho_!\cL_{\wwY}^a$ is an isomorphism of reflexive sheaves over all of $Y$; see [@Hartreflexive Prop. 1.6]. By (ii), we have $\Db(\wwY)\cong \sod{ \cB,\cC }$ with $$\begin{aligned} \cB \cong i_*\Sod{ \cA(m-n),\dots,\cA(-1) } \cong i_*(\Sod{ \cA,\cA(1),\dots,\cA(m-1) }\orth) \,.\end{aligned}$$ We have $\rho_*\cB=0$. It follows that $\WC(\rho) = \sod{ \cB\cap \ker(\rho_!), \cC}$. Indeed, consider an object $A\in \Db(\wwY)$. It fits into an exact triangle $ C \to A \to B \to $ with $C\in \cC$ and $B\in \cB$. From the morphism of triangles $$\xymatrix@C=2em{ \rho_*(C) \ar[r] \ar[d]^{t(C)}_\cong & \rho_*(A) \ar[r] \ar[d]^{t(A)} & \rho_*(B)=0 \ar[r] \ar[d]^{t(B)} & \\ \rho_!(C) \ar[r] & \rho_!(A) \ar[r] & \rho_!(B) \ar[r] & }$$ we see that $t(A)$ is an isomorphism if and only if $\rho_!B=0$. It is left to compute $\cB\cap \ker(\rho_!)$. Let $F\in \Db(Z)$ and $B=i_*F$. By , $$\rho_!B \cong \rho_!i_*F \cong b_*\nu_*(F\otimes\reg_\nu(m-n)) \,.$$ We see that $B\in \ker \rho_!$ if and only if $\nu_*(F\otimes\reg_\nu(m-n))=0$ if and only if $F\in \nu^*\Db(S)(n-m)\orth$. Hence, $\cB\cap \ker\rho_!= i_*(\cF^\perp)$ with $$\cF = \Sod{\cA, \cA(1), \dots, \cA(m-1), \cA(n-m)} \subset \Db(Z)$$ Carrying out the appropriate mutations within the semi-orthogonal decomposition $$\Db(Z) = \Sod{\cA(-m+1), \cA(-m+2), \dots, \cA(n-m-1), \cA(n-m)} \,,$$ we see that $\cF^\perp = \cE$; compare . Since $\cE\subset \Sod{\cA(m-n),\dots, \cA(-1)}$ is an admissible subcategory, we find that $i_*\colon \cE\to \Db(\wY)$ is fully faithful and has adjoints. Hence, $\WC(\rho)\subset \Db(\wY)$ is admissible. \[rem:SODWCN\] We have $\Db(\wY) = \Sod{ i_*(\cA\otimes \Omega^{n-1}(n-m)), \WC(\rho) }$. In other words, we can achieve categorical weak crepancy by dropping only one $\Db(S)$ part of the semi-orthogonal decomposition of $\Db(\wY)$. The discrepant category and some speculation {#sub:discrepant-category} -------------------------------------------- Let $Y$ be a variety with rational Gorenstein singularities and $\rho\colon \wY\to Y$ a resolution of singularities. Then, $\rho$ is a crepant resolution if and only if $\Db(\wY) = \WC(\rho)$; compare [@Abuaf-catres Prop. 2.0.10]. We define the *discrepant category* of the resolution as the Verdier quotient $$\disc(\rho) \coloneqq \Db(\wY)/\WC(\rho) \,.$$ By [@Neeman Remark 2.1.10], since $\WC(\rho)$ is a kernel, and hence a thick subcategory, we have $\disc(\rho)=0$ if and only if $\Db(\wY) = \WC(\rho)$. Therefore, we can regard $\disc(\rho)$ as a categorical measure of the discrepancy of the resolution $\rho\colon\wY\to Y$. In our cyclic quotient setup, where $\wY\cong \Hilb^G(X)$ is the simple blow-up resolution, we have $\disc(\rho)\cong \Db(S)$ by and [@LuntsSchnuererSOD Lem. A.8]. Hence, in this case, $\disc(\rho)$ is the smallest non-zero category that one could expect (this is most obvious in the case that $S$ is a point). This agrees with the intuition that the blow-up resolution is minimal in some way. Given a variety $Y$ with rational Gorenstein singularities, is there a resolution $\rho\colon \wY\to Y$ of minimal categorical discrepancy in the sense that, for every other resolution $\rho'\colon \wY'\to Y$, there is a fully faithful embedding $\disc(\rho)\hookrightarrow \disc(\rho')$? Often, in the case of a quotient singularity, a good candidate for a resolution of minimal categorical discrepancy should be the $G$-Hilbert scheme. At least, we can see that $\disc(\rho)$ grows if we further blow up the resolution away from the exceptional locus. \[prop:smoothblowup\] Let $\rho\colon \wY\to Y$ be a resolution of singularities and let $f\colon \wY'\to \wY$ be the blow-up in a smooth center $C\subset \wY$ which is disjoint from the exceptional locus of $\rho$. Set $\rho' \coloneqq \rho f \colon \wY'\to Y$. Then there is a semi-orthogonal decomposition $$\disc(\rho') = \Sod{\Db(C), \disc(\rho)} \,.$$ We first need the following general \[lem:Verdiersod\] Let $\cD$ be a triangulated category, $\cC\subset \cD$ a triangulated subcategory, and $\cD=\Sod{\cA,\cB}$ a semi-orthogonal decomposition so that the right-adjoint $i_\cB^!$ of the inclusion $i_\cB\colon \cB\hookrightarrow \cD$ satisfies $i_\cB^!(\cC)\subset \cB\cap \cC$. Then there is a semi-orthogonal decomposition $$\cD/\cC \cong \Sod{ \cA/(\cA\cap \cC), \cB/(\cB\cap \cC) } \,.$$ For every object $D\in \cD$, we have an exact triangle $$\begin{aligned} \label{eq:tri} i_\cB^!D \to D \to i_\cA^*D \to \end{aligned}$$ where $i_\cA^*$ is the left-adjoint to the embedding $i_\cA\colon \cA\to \cD$. Considering an object $C\in \cC$ shows that our assumption $i_\cB^!(\cC)\subset \cB\cap \cC$ implies $i_\cA^*(\cC)\subset \cA\cap \cC$. Let $C\in \cC$ and $A\in \cA$. Then, using the long exact Hom-sequence associated to the triangle , we see that every morphism $C\to A$ factors as $C\to \iota_\cA^*C\to A$. Hence, the embedding $i_\cA$ descends to a fully faithful embedding $\bar i_\cA\colon \cA/(\cA\cap \cC)\to \cD/\cC$, by [@LuntsSchnuererSOD Prop. B.2] (set $\cW=\cA$, $\cV=\cA\cap \cC$ and use (ff2) of *loc. cit.*). Similarly, we get an induced fully faithful embedding $\bar i_\cB\colon \cB/(\cB\cap \cC)\to \cD/\cC$ (use (ff2)^op^ instead of (ff2)). Now let us show that $\Hom_{\cD/\cC}\bigl(\cB/(\cB\cap \cC), \cA/(\cA\cap \cC)\bigr)=0$. For $B\in \cB$ and $A\in \cA$, a morphism $B\to A$ in $\cD/\cC$ is represented by a roof $$B \xleftarrow{\beta} D \xrightarrow{\alpha} A$$ where $\beta\colon D\to B$ is a morphism in $\cD$ with $\cone(\beta)\in \cC$ and $\alpha\colon D\to A$ is any morphism in $\cD$; see [@Neeman Def. 2.1.11]. Put $C \coloneqq \cone(\beta)[-1] \in \cC$. We apply the triangle of functors $i^*_\cA \to \id \to i^!_\cB \to$ (formally, $i^*_\cA$ has to be replaced by $i_\cA i^*_\cA$ and $i^!_\cB$ by $i_\cB i^!_\cB$) to the triangle of objects $C\to D\to B \to$ and obtain the diagram $$\xymatrix@C=2em@R=3ex{ i^!_\cB C \ar[r] \ar[d] & i^!_\cB D \ar[r] \ar[d]^\gamma & B \ar[d] \\ C \ar[r] \ar[d] & D \ar[r] \ar[d] & B \ar[d] \\ i^*_\cA C \ar[r] & i^*_\cA D \ar[r] & 0 \\ }$$ where we have used $i^!_\cB B = B$ and $i^*_\cA B = 0$. Now $i^!_\cB C \in \cC\cap\cB$ by assumption. The left column thus forces $i^*_\cA C \cong i^*_\cA D \in \cC$. We get that $\cone{\beta\gamma}\in \cC$ since $\cone{\beta}, \cone{\gamma}\in \cC$; see [@Neeman Lem. 1.5.6]. Therefore, we get another roof representing the same morphism in $\cD/\cC$, replacing $D$ by $i^!_\cB D$: $$B \xleftarrow{\beta\gamma} i^!_\cB D \xrightarrow{\alpha\gamma} A \,.$$ However, $i^!_\cB D\in\cB$ and $\Hom_\cD(\cB,\cA)=0$, so the morphism is 0 in $\cD/\cC$. Finally, we need to show that $\cA/(\cA\cap\cC)$ and $\cB/(\cB\cap\cC)$ generate $\cD/\cC$, but this is clear, because $\cA$ and $\cB$ generate $\cD$. We have a semi-orthogonal decomposition $\Db(\wY')=\Sod{\cA, \cB}$ with $\cB = f^*\Db(\wY)$ and $$\cA = \Sod{ \iota_*(g^*\Db(C)\otimes \reg_g(-c+1)),\dots, \iota_*(g^*\Db(C)\otimes \reg_g(-1))} \,.$$ Here, $c=\codim(C\into\wY)$ and $g$ and $\iota$ are the $\IP^{n-1}$-bundle projection and the inclusion of the exceptional divisor of the blow-up $f \colon \wY' \to \wY$. Let $U \coloneqq \wY\setminus C$. For $F\in \Db(\wY')$ we have $(f_*F)_{\mid U}\cong (f_!F)_{\mid U}$. Hence, if $F\in \WC(\rho')$, we must have $(f_*F)_{\mid U}\in \WC(\rho_{\mid U})$. Since $\rho$ is an isomorphism in a neighbourhood of $C$, an object $E\in \Db(\wY)$ is contained in $\WC(\rho)$ if and only if its restriction $E_{\mid U}$ is contained in $\WC(\rho_{\mid U})$. In summary, $$f_* F \in \WC(\rho) \quad\text{for every}\quad F\in \WC(\rho') \,.$$ We have $f_*\reg_{\wwY'}\cong \reg_\wwY\cong f_*\omega_f$. By the projection formula, it follows that $f_{*\mid \cB}\cong f_{!\mid \cB}$. Hence, we have $\cB\cap \WC(\rho')=f^*\WC(\rho)$ and $\cB/(\WC(\rho')\cap \cB)\cong\disc(\rho)$. Now, we can apply with $\cC=\WC(\rho')$ to get a semi-orthogonal decomposition $$\disc(\rho') = \Sod{ \cA/(\WC(\rho')\cap \cA), \disc(\rho) } \,.$$ We have $f_*(\cA)=0$, hence $\rho'_*(\cA)=0$. Accordingly, $$\WC(\rho') \cap \cA = \ker(\rho'_!) \cap \cA = \ker(f_!) \cap \cA \,.$$ The second equality is due to the fact that all objects of $f_!\cA$ are supported on $C$, where $\rho$ is an isomorphism. Now, in analogy to the computations of the proof of and , we get a semi-orthogonal decomposition $$\cA = \Sod{ \iota_*(g^*\Db(C)\otimes\Omega^{c-1}(c-1)), \ker(f_!)\cap \cA } \,.$$ Hence, $\cA/(\WC(\rho')\cap \cA) \cong \iota_*(g^*\Db(C)\otimes\Omega^{c-1}(c-1)) \cong \Db(C)$. (Non-)unicity of categorical crepant resolutions {#sub:non-unicity} ------------------------------------------------ Let $\wY\to Y$ be a resolution of rational Gorenstein singularities and let $\cD\subset \Dperf(\wY)$ be an admissible subcategory which is a weakly crepant resolution, i.e. $\rho^*\colon \Dperf(Y)\to \Db(\wY)$ factors through $\cD$ and $\rho_{*\mid \cD}\cong \rho_{!\mid \cD}$. Then every admissible subcategory $\cD'\subset\cD$ with the property that $\rho^*\colon \Dperf(Y)\to \Db(\wY)$ factors through $\cD'$ is a weakly crepant resolution, too. In particular, in our setup of cyclic quotients, there is a tower of weakly crepant resolutions of length $n-m$ given by successively dropping the $\Db(S)$ parts of the semi-orthogonal decomposition of $\WC(\rho)$. We see that weakly crepant categorical resolutions are not unique, even if we fix the ambient derived category $\Db(\wY)$ of a geometric resolution $\wY\to Y$. In contrast, *strongly crepant* categorical resolutions are expected to be unique up to equivalence; see [@KuzLefschetz Conj. 4.10]. A strongly crepant categorical resolution of $\Db(Y)$ is a module category over $\Db(Y)$ with trivial relative Serre functor; see [@KuzLefschetz Sect. 3]. For an admissible subcategory $\cD\subset \Db(\wY)$ of the derived category of a geometric resolution of singularities $\rho\colon \wY\to Y$ this condition means that $\cD$ is $Y$-linear and there are functorial isomorphisms $$\begin{aligned} \label{stronglycrepantcondition} \rho_*\sHom(A,B)^\vee \cong \rho_*\sHom(B,A)\end{aligned}$$ for $A,B\in \cD$. In our cyclic setup, $\Psi(\D_G(X))\subset \Db(\wY)$ is a strongly crepant categorical resolution; see [@KuzLefschetz Thm. 1] or [@Abuaf-catres Thm. 10.2]. We require strongly crepant categorical resolutions to be *indecomposable* which means that they do not decompose into direct sums of triangulated categories or, in other words, they do not admit both-sided orthogonal decompositions. Under this additional assumption, we can prove that strongly crepant categorical resolutions are unique if we fix the ambient derived category of a geometric resolution. \[prop:unicity\] Let $\wY\to Y$ be a resolution of Gorenstein singularities and $\cD, \cD'\subset \Db(\wY)$ admissible indecomposable strongly crepant subcategories. Then $\cD=\cD'$. The intersection $\cD\cap \cD'$ is again an admissible $Y$-linear subcategory of $\Db(\wY)$ containing $\rho^*(\Dperf(Y))$. Furthermore, condition is satisfied for every pair of objects of $\cD\cap \cD'$; so the intersection is again a strongly crepant resolution. Hence, we can assume $\cD'\subset \cD$. Let $\cA$ be the right-orthogonal complement of $\cD'$ in $\cD$, so that we have a semi-orthogonal decomposition $\cD = \Sod{\cA, \cD'}$. By , this means that $\rho_*\sHom(D,A)=0$ for $A\in \cA$ and $D\in \cD'$. But then, by , we also get $\rho_*\sHom(A,D)=0$ so that $\cD=\cA\oplus \cD'$. Connection to Calabi–Yau neighbourhoods {#sub:CY-neighbourhoods} --------------------------------------- In [@HKP], *spherelike objects* and their *spherical subcategories* were introduced and studied. The paper hinted at a role of these notions for birationality questions of Calabi–Yau varieties. One of the starting points for our project was to consider *Calabi–Yau neighbourhoods* (a generisation of spherical subcategories) as candidates for categorical crepant resolutions of Calabi–Yau quotient varieties. In this subsection, we describe the connection to the weakly crepant resolutions considered above. We recall some abstract homological notions. Let $\cT$ be a Hom-finite $\IC$-linear triangulated category and $E\in\cT$ an object. We say that $\Serre E\in\cT$ is a *Serre dual object* for $E$ if the functors $\Hom^*(E,-)$ and $\Hom^*(-,\Serre E)^\vee$ are isomorphic. By the Yoneda lemma, $\Serre E$ is then uniquely determined. Fix an integer $d$. We call the object $E$ - a *$d$-Calabi–Yau* object, if $E[d]$ is a Serre dual of $E$, - *$d$-spherelike* if $\Hom^*(E,E) = \IC\oplus \IC[-d]$, and - *$d$-spherical* if $E$ is $d$-spherelike and a $d$-Calabi–Yau object. Note a smooth compact variety $X$ of dimension $d$ is a strict Calabi–Yau variety precisely if the structure sheaf $\reg_X$ is a $d$-spherical object of $\Db(X)$ . In [@HKP] the authors show that if $E$ is a $d$-spherelike object, there exists a unique maximal triangulated subcategory of $\cT$ in which $E$ becomes $d$-spherical. In the following we will imitate this construction for a larger class of objects. Let $E\in \cT$ be an object in a triangulated category having a Serre dual $\mathsf S E$. We call $E$ a *$d$-selfdual object* if 1. $\Hom(E, E[d]) \cong \IC$, i.e. by Serre duality there is a morphism $w\colon E\to \omega(E) \coloneqq \Serre E[-d]$ unique up to scalars, and 2. the induced map $w_*\colon \Hom^*(E,E)\isomor \Hom^*(E, \omega(E))$ is an isomorphism. In particular, a $d$-selfdual object satisfies $\Hom^*(E,E)\cong \Hom(E,E)^\vee[-d]$, hence the name. If an object is $d$-spherelike, then it is $d$-selfdual; compare [@HKP Lem. 4.2]. For a $d$-selfdual object $E$, there is a triangle $E \xrightarrow{w} \omega(E)\to Q_E\to E[1]$ induced by $w$. By our assumption, we get $\Hom^*(E,Q_E)=0$. Thus, following an idea suggested by Martin Kalck after discussing [@HKP §7] with Michael Wemyss, we propose the following \[def:CYN\] The *Calabi-Yau neighbourhood* of a $d$-selfdual object $E\in\cT$ is the full triangulated subcategory $$\CY(E) \coloneqq \lorth Q_E \subseteq \cT \,.$$ \[prop:CY-neighbourhood\] If $E\in\cT$ is a $d$-selfdual object then $E\in\CY(E)$ is a $d$-Calabi-Yau object. If $T\in \CY(E)$, apply $\Hom^*(T,-)$ to the triangle $E\to \omega(E)\to Q_E$. Using the same proof as for [@HKP Thm. 4.6], we see that the Calabi-Yau neighbourhood is the maximal subcategory of $\cT$ in which a $d$-selfdual object $E$ becomes $d$-Calabi-Yau. \[prop:maximality-of-CY-neighbourhood\] If $\ku\subset \cT$ is a full triangulated subcategory and $E\in \ku$ is $d$-Calabi-Yau, then $\ku\subset \CY(E)$. \[prop:WC=CY\] Let $Y$ be a projective variety with rational Gorenstein singularities and trivial canonical bundle of dimension $d=\dim Y$ and consider a resolution of singularities $\rho\colon \wY \to Y$. Then, for every line bundle $L\in \Pic Y$, the pull-back $\rho^*L\in \Db(\wY)$ is $d$-selfdual. Furthermore, we have $$\begin{aligned} \label{WC=CY} \WC(\rho) = \bigcap_{L\in \Pic Y} \CY(\rho^*L) \,. \end{aligned}$$ Note that, by our assumption that $\omega_Y$ is trivial, we have $\omega_\wwY\cong \omega_\rho$. Hence, by Grothendieck duality, there is a morphism $w_L\colon \rho^*L\to \rho^*L\otimes \omega_\wwY$ unique up to scalar multiplication, namely $w_L=\id_{\rho^*L}\otimes s$ where $s$ is the non-zero section of $\omega_\wwY\cong \omega_\rho$; compare the previous . Furthermore, $w_{L*} \colon \Hom^*(\rho^*L,\rho^*L) \to \Hom^*(\rho^*L,\rho^*L\otimes \omega_\wwY)$ is an isomorphism, still by Grothendieck duality, which means that $\rho^*L$ is $d$-selfdual. Recall that $\WC(\rho)=\ker(\rho_+)$ where $\rho_+$ is defined as the cone $$\rho_* \xrightarrow{t} \rho_! \to \rho_+ \to \,.$$ By adjunction, we get $\WC(\rho)=\lorth(\rho^+(\Dperf(Y)))$ where $\rho^+=\rho_+^R$ is given by the triangle $$\rho^+ \to \rho^* \xrightarrow{t^R} \rho^! \to \,.$$ Note that $t^R=(\_)\otimes s$. Hence $t^R(L)=w_L\colon \rho^*L\to \rho^*L\otimes \omega_\wwY$ and $\rho^+(L)=Q_{\rho^*L}[-1]$; compare . Since the line bundles form a generator of $\Dperf(Y)$, we get for $F\in \Db(\wY)$: $$\begin{aligned} F\in \WC(\rho) &\iff F\in \lorth(\rho^+(\Dperf(Y))) \\ &\iff F\in \lorth Q_{\rho^*L} \quad \forall L\in \Pic Y \\ &\iff F\in \CY(\rho^*L) \quad \forall L\in \Pic Y \,. \qedhere\end{aligned}$$ Following the proof of , we see that, on the right-hand side of , it is sufficient to take the intersection over all powers of a given ample line bundle. In our cyclic setup, if $S$ consists of isolated points, we even have $\WC(\rho)=\CY(\reg_{\wwY})$ so that the weakly crepant neighbourhood is computed by a Calabi-Yau neighbourhood of a single object. The same should hold in general if $Y$ has isolated singularities. Stability conditions for Kummer threefolds {#sec:Kummer} ========================================== Let $A$ be an abelian variety of dimension $g$. Consider the action of $G=\mu_2$ by $\pm1$. Then the fixed point set $A[2]$ consists of the $4^g$ two-torsion points. Consider the quotient $\overline{A}$ (the *singular Kummer variety*) of $A$ by $G$, and the blow-up $K(A)$ (the *Kummer resolution*) of $\overline{A}$ in $A[2]$. This setup satisfies , with $m=2$ and $n=g$ and we get \[cor:kummersod\] The functor $\Psi\colon \Db_G(A)\to \Db(K(A))$ is fully faithful, and $$\Db(K(A)) = \Sod{ \underbrace{\Db(\pt),\dots,\Db(\pt)}_{(g-2)4^{g} \text{ times}}, \Psi(\Db_{G}(A)) } \,.$$ To explore a potentially useful consequence of this result, we need to recall that a Bridgeland stability condition on a reasonable $\IC$-linear triangulated category $\kd$ consists of the heart $\ka$ of a bounded t-structure in $\kd$ and a function from the numerical Grothendieck group of $\kd$ to the complex numbers satisfying some axioms, see [@Bri-stabcond]. \[cor:stability\] There exists a Bridgeland stability condition on $\Db(K(A))$, for an abelian threefold $A$. To begin with, by [@BMS-ab Cor. 10.3] there is a stability condition on $\Db_G(A)$. Denote by $\ka\subset\Db_G(A)$ the corresponding heart; it is a tilt of the standard heart [@BMS-ab §2]. For a two-torsion point $x\in A[2]$, we set $E_x \coloneqq \reg_{\rho^{-1}(\pi(x))}(-1)$. Then, since $g=\dim A=3$, the semi-orthogonal decomposition of is given by $$\begin{aligned} \label{eq:Kummersod} \Db(K(A)) = \Sod{ \{E_x\}_{x\in A[2]}, \Db_G(A) } \,.\end{aligned}$$ Next, we want to show that, for every $x\in A[2]$, there exists an integer $i$ such that $\Hom^{\leq i}(E_{x},\Psi(F))=0$ for all $F \in \cA \subset \Db_G(A)$. Indeed, the cohomology of any complex in the heart of the stability condition on $\Db_G(A)$, as constructed in [@BMS-ab Cor. 10.3], is concentrated in an interval of length three. The functor $\Psi$ has cohomological amplitude at most $3$, since $q_*^G \colon \Coh_G(A) \to \Coh(K(A))$ is an exact functor of abelian categories, and every sheaf on $A$ has a locally free resolution of length $\dim A=3$. This implies that the cohomology of any complex in $\Psi(\Db_{G}(A)))$ is contained in a fixed interval of length $6$. This proves the above claim. Using [@ColPol-gluestab Prop. 3.5(b)], this then implies that $\sod{ E_{x},\Psi(\Db_{G}(A)) }$ has a stability condition; compare the argument in [@BMMS Cor. 3.8]. We can proceed to show that, for $x\neq y\in A[2]$, there exists an integer $i$ such that $\Hom^{\leq i}(E_{y},\sod{ E_{x},\Psi(\Db_{G}(A)) }) = 0$ and so there is a stability condition on $\sod{E_y, E_{x},\Psi(\Db_{G}(A))}$. After $4^3$ steps we have constructed a stability condition on $\Db(K(A))$; compare . Contact: -------- ---------------------------------------------------------------------------------------------------- A. K.: Philipps-Universität Marburg, Hans-Meerwein-Stra[ß]{}e 6, Campus Lahnberge, 35032 Marburg, Germany Email: `andkrug@mathematik.uni-marburg.de` D. P.: Freie Universität Berlin, Mathematisches Institut, Arnimallee 3, 14195 Berlin, Germany Email `dploog@math.fu-berlin.de` P. S.: Universität Hamburg, Fachbereich Mathematik, Bundesstra[ß]{}e 55, 20146 Hamburg, Germany Email `pawel.sosna@math.uni-hamburg.de` -------- ---------------------------------------------------------------------------------------------------- \[notation\_recap\] [^1]: MSC 2010: 14F05, 14E16, 14E15 [^2]: Keywords: cyclic quotient singularity, McKay correspondence, derived category, categorical resolution

--- abstract: 'The error box of GRB980425 has been observed by XMM-[*Newton*]{} in March 2002, with the aim of measuring the late epoch X-ray emission of the supernova 1998bw and of clarifying its supposed association with the GRB itself. We present here the preliminary results obtained with the EPIC PN camera. Our observations confirm the association between SN 1998bw and GRB980425. The EPIC PN measurement of the SN 1998bw flux is significantly below the extrapolation of the power-law temporal trend fitted to the BeppoSAX points and implies a faster temporal decay. We propose different physical interpretations of the SN X-ray light curve, according to whether it is produced by one or more radiation components.' address: - 'INAF, Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, I-34131 Trieste, Italy' - 'ASI Science Data Center, c/o ESRIN, Via G. Galilei, I-00044 Frascati, Italy' - 'IASF-CNR, Sezione di Bologna, Via P. Gobetti 101, I-40129 Bologna, Italy' - 'IASF-CNR, Sezione di Roma, via Fosso Del Cavaliere 100, I-00133 Roma, Italy' - 'Physics Department, University of Ferrara, Via Paradiso 11, I-44100 Ferrara, Italy' - 'NASA MSFC, SD-50, Huntsville, AL 35812, USA' - 'IASF-CNR, Sezione di Palermo, via U. La Malfa 153, I-90146 Palermo, Italy' - 'Space Research Organization Netherlands, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands' author: - 'E. Pian, P. Giommi, L. Amati, E. Costa, J. Danziger$^1$, M. Feroci$^4$, M.T. Fiocchi$^2$, F. Frontera, C. Kouveliotou, N. Masetti$^3$, L. Nicastro, E. Palazzi$^3$, L. Piro$^4$, M. Tavani$^4$, and J.J.M. in ’t Zand' title: 'XMM-NEWTON OBSERVATIONS OF THE FIELD OF GAMMA-RAY BURST 980425' --- INTRODUCTION {#introduction .unnumbered} ============ The hypothesis that supernovae are the progenitors of Gamma-Ray Bursts (GRB) dates back to the epoch of first GRB discovery (Colgate 1974) and has received support in recent years from the detection of supernova features in the optical afterglows of GRBs. These are re-brightenings at rest-frame intervals of 10-15 days after the GRB (e.g., Galama et al. 2000; Bloom et al. 2002; Price et al. 2003; Masetti et al. 2003); circumburst media with wind characteristics (Berger et al. 2001; Jaunsen et al. 2001; Price et al. 2002); iron emission lines (Piro et al. 1999; Reeves et al. 2002; Butler et al. 2003); association of GRBs with star-forming regions (Fruchter et al. 1999; Frail et al. 2002). The most tempting hint of association between GRBs and SNe is obviously the similarity of the intrinsic energy of these phenomena, when collimation and beaming are taken into account in GRBs (e.g., Frail et al. 2001). While these circumstances represent only possible evidences of a GRB-SN connection, on two occasions a clear association has been found. GRB980425 and SN 1998bw have been detected within very tight temporal and angular limits: they exploded simultaneously (with an uncertainty of $\pm 1$ day) and with a maximum separation in the sky of 8$^{\prime}$ (Galama et al. 1998). More recently, prominent spectral features similar to those detected for SN 1998bw have been detected in the afterglow of GRB030329 (Stanek et al. 2003; Hjorth et al. 2003). To date, no other such compelling case of GRB-SN association has been detected, and SN 1998bw, whose brightness and kinematic conditions were exceptional, is considered a typical “hypernova”, the powerful SN explosion speculated to be at the origin of GRBs (Woosley 1993; Paczyński 1998; Zhang et al. 2003, and references therein). The field of SN 1998bw and GRB980425 was observed with the BeppoSAX Narrow Field Instruments one day, one week and 6 months after the event (Pian et al. 2000). The most sensitive of these instruments, the BeppoSAX MECS, had detected two X-ray sources within the BeppoSAX Wide Field Cameras 8$^{\prime}$-radius error box of GRB980425. The brighter one (hereafter S1) was variable and positionally consistent with the SN 1998bw and had been identified with X-ray emission from the SN. The fainter source (hereafter S2) is $\sim 4^{\prime}$ away from SN 1998bw, and therefore inconsistent with it, within the $1^{\prime}.5$ positional uncertainty of the MECS detectors (see Fig. 1 in Galama et al. 1999). No firm statement could be made about the variability of S2, given its low flux level, at the limit of the MECS sensitivity. Since the X-ray light curve of S2 may have been marginally compatible with an afterglow behavior, given the big uncertainties (see Pian et al. 2000), some reservations had remained as to whether GRB980425 and SN 1998bw were physically associated. However, the above mentioned case of GRB030329 and SN 2003dh argues strongly in favor of the GRB980425/SN 1998bw association and considerably weakens the case for association between S2 and GRB980425. The peculiarity of SN 1998bw made it imperative to further investigate its X-ray emission at late epochs. Therefore, we have re-observed its field with Chandra and XMM-[*Newton*]{} in late 2001 and March 2002, respectively, and report here the preliminary results of the latter campaign. A detailed presentation of both campaigns will be given in future papers. ![XMM-[*Newton*]{} image of the field of SN 1998bw taken with the EPIC PN chip on 28 March 2002. The size of the image is $30^{\prime} \times 30^{\prime}$; North is at the top and East to the left. Overlaid are isophotes of the BeppoSAX MECS image taken in 26-27 April 1998. The brightest source near the center of the image corresponds to SN 1998bw. About 4 arcmin SE of it, one can distinguish a number of very faint sources, the sum of which may have been detected by BeppoSAX as a single source S2 at the limit of MECS detectability.](fig1_epn_cospar.eps){width="14cm"} OBSERVATIONS AND DATA ANALYSIS {#observations-and-data-analysis .unnumbered} ============================== The field of SN 1998bw was observed by XMM-[*Newton*]{} on 28 March 2002, between 13:53:34 and 20:10:38 UT, with the European Photon Imaging Cameras (EPIC, 0.15-15 keV) PN (Strüder et al. 2001) and MOS (Turner et al. 2001), operating in full-frame mode and with the medium filter applied. The data have been cleaned and processed using the Science Analysis Software (SAS 5.3) and analyzed using standard software packages (FTOOLS 5.2). The latest calibration files released by the EPIC team have been used (update: 29 Jan 2003). Event files produced from the pipeline have been filtered from high-background time intervals and only events corresponding to pattern 0-12 for MOS and pattern 0-4 for PN have been used (see the XMM-[*Newton*]{} Users’ Handbook, Ehle et al. 2001; see also    [http://xmm.vilspa.esa.es/external]{}). The net exposure times, after data cleaning, are $\sim 13.0$ ks, $\sim 16.8$ ks, and $\sim 16.7$ ks for PN, MOS1, and MOS2, respectively. Count rates (see next Section) have been estimated by integrating the signal within circles of 30$^{\prime\prime}$ radius, which enclose $\sim$80% of the encircled energy function, and by subtracting the background signal estimated from blank sky exposures, in circles of equal area. ![X-ray light curve of SN 1998bw constructed from BeppoSAX MECS (first 4 filled circles, from Pian et al. 2000) and XMM-[*Newton*]{} EPIC PN (last filled circle) observations. The temporal origin coincides with the trigger time of GRB980425, 1998 April 25.9091 UT. For comparison, the X-ray light curve of SN 1980K is also shown (open circles, from Canizares et al. 1982; Schlegel 1994. The original data points have been converted to the 2-10 keV range, using a power-law spectrum $\propto \nu^{-\alpha}$ with index $\alpha \sim 1$).](s1_xmm.ps){width="12cm"} RESULTS AND DISCUSSION {#results-and-discussion .unnumbered} ====================== In Fig. 1 we show the EPIC PN image. At a position consistent with that of source S1 in the BeppoSAX MECS image, we detect in the EPIC PN image a source with a count rate of (9.6 $\pm$ 1.4) $\times 10^{-3}$ counts s$^{-1}$ in the 0.5-5 keV range within a circular area of 30$^{\prime\prime}$ radius. Assuming a spectrum similar to that measured by BeppoSAX for S1 (see Pian et al. 2000) and accounting for Galactic absorption by neutral hydrogen ($N_{HI} = 3.95 \times 10^{20}$ cm$^{-2}$, Dickey & Lockman 1990), we derive a flux of (3.18 $\pm$ 0.46) $\times 10^{-14}$ erg s$^{-1}$ cm$^{-2}$ between 2 and 10 keV. The signal-to-noise ratio of our XMM-[*Newton*]{} observations is not sufficient to perform a detailed spectral analysis. A number of very faint sources are visible at the location of source S2 detected by the BeppoSAX MECS in the WFC error box of GRB980425 (Fig. 1). The integral of the EPIC PN signal within the $1^{\prime}.5$ radius MECS error circle of S2 is consistent with the average flux measured by the MECS for S2, suggesting that it may be not a single source, but rather the sum of several faint sources and that its marginal variability in the BeppoSAX observations is determined by background fluctuations, or possibly by the random variations of those sources. This definitely rules out the afterglow nature of S2. From the EPIC imaging we do not detect significant contamination of the X-ray emission of SN 1998bw by its host galaxy, therefore the SN decay measured by BeppoSAX must be authentic. That was satisfactorily fitted both by a power-law $t^{-0.2}$ and by an exponential with $e$-folding time of $\sim$500 days (Pian et al. 2000). The addition of the EPIC point to the X-ray light curve of SN 1998bw (Fig. 2) shows that neither a power-law nor an exponential law are particularly satisfactory, although the exponential may be somewhat better. For this reason we favor an interpretation of the X-ray light curve as a result of the superposition of different radiation components. In fact, while the overall temporal behavior of SN 1998bw is unlike that of the very few X-ray SNe monitored at both early and late times (e.g., SN 1987A, Park et al. 2002; SN 1993J, Kohmura et al. 1994; Swartz et al. 2003; Zimmermann & Aschenbach 2003; SN 1994I, Immler et al. 1998; Immler et al. 2002), at late epochs (i.e., after day $\sim$100) it is reminiscent of that of previous X-ray SNe (e.g., SN 1980K, Canizares et al. 1982; Schlegel 1994; Schlegel 1995; SN 1994I, Immler et al. 2002). Thus, one may interpret the early X-rays of SN 1998bw as afterglow radiation, while the late epoch X-ray emission is dominated by the interaction of the SN shock with the circumstellar material, as proposed for other X-ray SNe (Kohmura et al. 1994; Schlegel 1995; Suzuki and Nomoto 1995; Fransson et al. 1996; Chevalier and Fransson 2002; Immler and Lewin 2002). In fact, if the main sequence progenitor of SN 1998bw was a star of $\sim 40 M_\odot$ as postulated by Iwamoto et al. (1998) one would expect circumstellar material in the neighbourhood of the SN as a result of an earlier mass loss wind. On the other hand, assuming that a single mechanism is responsible for X-ray emission of SN 1998bw, we may not exclude that the observed X-rays are cooling radiation from the compact remnant, provided the GRB has swept up all the surrounding material by creating an evacuated cone. Tavani (1997) has shown, in the context of X-ray afterglows of GRBs, that cooling neutron stars with “external” disturbances (e.g., a fallback) may radiate in X-rays with a temporal rate faster than a power-law. A longer diffusion time than considered by Tavani (1997) should be adopted in our case. The predicted decay rates for neutron stars with simple cooling (i.e., no fallback) are much longer than the fading time scale measured by BeppoSAX and XMM-[*Newton*]{} for SN 1998bw (Page 1998). However, exotic cooling mechanisms can significantly increase the cooling rate (Slane et al. 2002). Our observations may thus allow us to place constraints on non-standard neutron star cooling scenarios and may have important implications for determining the nature of GRB remnants. ACKNOWLEDGEMENTS {#acknowledgements .unnumbered} ================ We are grateful to Matteo Guainazzi and Matthias Ehle for their assistance with observations scheduling, and to Paolo Mazzali for his valuable comments. 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--- abstract: 'Modern robotic systems have become a substitute for humans when it’s necessary to perform risky or exhausting tasks. In such application scenarios, communications between robots and the control center are one of the major problems. The commonly used solution assumes that newer messages are more valuable. We find that it does not hold in many scenarios. In this paper, we propose a novel, resilient buffer management policy called OptSample. We make a new assumption that uniformly sampled messages are the most valuable and define an evaluation function to estimate the profit of the received message sequence. Our OptSample policy can uniformly sample messages and dynamically adjust the sample rate based on the run-time network situation. Our analysis and simulation shows that the OptSample policy can effectively prevent losing long segments of continuous messages and can improve the profit of the received messages. We implement the OptSample policy in ROS, without changing the interface or API for the applications. Our experiments show that the OptSample policy can improve the results of several application scenarios including surveillance, 3D reconstruction, and SLAM.' author: - | Yu-Ping Wang$^{1}$, Zi-Xin Zou$^{1}$, Xu-Qiang Hu$^{1}$, Dinesh Manocha$^{2}$, Lei Qiao$^{3}$, Shi-Min Hu$^{1}$\ \ The video of applications can be found at\ <https://www.youtube.com/watch?v=ji-yvbT1l-w>[^1][^2][^3][^4] bibliography: - 'main.bib' title: '**OptSample: A Resilient Buffer Management Policy for Robotic Systems Based on Optimal Message Sampling** ' --- Introduction {#sec:introduction} ============ Modern robotic systems have become a substitute for humans when it’s necessary to perform risky or exhausting tasks such as military operations, exploration, rescue operations, surveillance, or large-scale cleaning operations. In such application scenarios, a control center through which humans can monitor and operate the whole system is usually needed. However, communication issues, such as unstable network connection and limited bandwidth, cause problems. Network connections, especially wireless ones (e.g. over WiFi), are not always stable [@DBLP:conf/infocom/AggarwalSA00]. Network connections could be temporarily broken when a robot moves out of range of wireless network, switches among network source points, or is shielded by obstacles. In this case, buffering some messages at the sender is a commonly used solution. When network connections are temporarily broken, new messages are put into a buffer to wait for future transmission. However, buffers are limited in size due to limited memory. When the buffer is full, one simple solution is to discard the oldest message to make room for the new message. This solution is called Drop Oldest policy [@DBLP:conf/colcom/ChenYZW16]. This policy is reasonable when newer messages are considered more valuable than older ones. For example, obstacle avoidance algorithms should be fed with messages as new as possible. Commonly used robotic middleware, such as Robot Operating System (ROS) [@ros], employ this policy. However, we find that the assumption does not hold for many application scenarios, such as surveillance, 3D reconstruction, and Simultaneous Localization and Mapping (SLAM). In these scenarios, adjacent messages provide similar information. When network disruption occurs, repeatedly discarding the oldest message leads to losing a long segment of frames and some information would be totally lost (see Fig. \[fig:slam\] as an example). In these scenarios, we make a new assumption that uniformly sampled messages are preferred over the newest messages. Limited bandwidth is another issue. Current depth cameras are becoming increasingly higher-resolution. For example, an Intel Realsense D435 depth camera can capture $1920 \times 1080$ RGB images at 30 fps (frames per second) and $1280 \times 720$ depth images at 90 fps. Transmitting these images will need about 2.6 Gbps bandwidth while commodity WiFi routers can only provide 54 Mbps bandwith. In this case, messages come to the buffer faster than they leave. Therefore, the buffer is always full, which makes the buffer management policy more vulnerable to temporary network disruption. Explicitly sampling images to a lower frame rate provides an easy solution. However, it is difficult to decide the frame rate since it is affected by various factors including image resolutions, compression algorithm and quality, network bandwidth, and number of robots. Higher frame rates will not relieve the problem and lower frame rates will waste more information. It is an important problem to automatically decide the good frame rates for different applications.   {width="90.00000%"} [**Main Results:**]{} In this paper, we present a novel resilient buffer management policy, OptSample. When temporary network disruption occurs, rather than discarding the oldest messages, we gradually decrease the sampling rate. Messages are sampled uniformly and the sample rate is decided with run-time situations. Thus, we can avoid losing a long, continuous segment of messages and instead lose multiple, short segments of messages. Experimental results show that our OptSample policy is suitable for many application scenarios such as surveillance, 3D reconstruction, and SLAM. The main contributions of this paper include: \(1) We propose a novel buffer management policy named OptSample, which is resilient against temporary network disruption. \(2) We define an evaluation function to estimate the profit of a message sequence. We prove and show with simulation that our OptSample policy can get results bounded close to the unfeasible optimal profit (see Lemma 6). \(3) We implement our OptSample policy in ROS. The implementation is transparent to the user and can be used with all the applications with the same API. \(4) We test our OptSample policy with several applications including surveillance, 3D reconstruction and SLAM. These experiments show that using our OptSample policy can effectively improve the results of these scenarios when network disruption occurs. The rest of this paper is organized as follows. Section \[sec:motivation\] discusses a motivating scenario. Section \[sec:related\] discusses alternative solutions and related work. Section \[sec:approach\] introduces our OptSample policy and formally analyzes its performance. Section \[sec:evaluation\] evaluates our OptSample policy with simulation and practical experiments. Finally, we conclude in Section \[sec:conclusion\]. Motivating Scenario {#sec:motivation} =================== Take the rescue scenario as an example. A rescue robot traverses the accident area, captures images at 30 fps, and sends them back to the control center. Suppose that the system suffers a temporary network disruption for 5 seconds. During this time, the robot is still capturing images and has to put them into the internal buffer. If the size of the internal buffer can save 30 images, we will have to discard 120 images. The trivial solution will result in discarding the oldest 120 images. In this case, the control center cannot get any information about those 4 seconds and we will have no idea if someone needs to be rescued during that time. What makes the situation worse, if a victim is out of range of the wireless network, the robot can “see" the victim, but the control center cannot. In the above case, the assumption that newer messages are more valuable than older ones is not true. If we manage to discard 120 images uniformly from a range of 5 seconds, after the network connection is restored, the control center will still receive images as if the robot is capturing images at 6 fps. The control center can therefore be more confident that the robot did not miss any victims. Similar requirements can be found in many scenarios, such as surveillance, 3D reconstruction, and SLAM. Information provided by an image (either RGB or depth) could be largely covered by its adjacent images. Therefore, if we have to discard 50 images from 100 images, it is better to discard one message from every two images, than to discard the first 50 images. Overall, uniformly sampled messages are preferred over the newest messages in these scenarios. What we need is a new buffer management policy. Related Work {#sec:related} ============ Communication Issues in Robotic Systems --------------------------------------- Communications among robots and the control center have been a major issue in many robotic systems. Saeedi et al. reviewed multiple-robot SLAM algorithms [@DBLP:journals/jfr/GTSL16], concluding that communications is one of ten major problems in multiple-robot SLAM. There are a few important issues need to be considered when designing a multiple-robot system such as the availability and the quality of the communication channels, limited bandwidth and data rate, when and what to communicate, etc. There are two kinds of system framework, centralized system and decentralized system. In a centralized robotic system, sampling input data is the most commonly used solution to relieve limited network bandwidth. Golodetz et al. [@DBLP:journals/tvcg/GolodetzCLPMT18] aimed to reconstruct dense 3D models with collaborative robots. They explicitly pointed out the communication problem between robots and the control center. Based on some experiments and evaluation, they compressed the sensor images and sampled the image frames to about 10 fps. Dong et al. [@DBLP:journals/tog/Dong0ZTXNC19] performed collaborative scanning for dense 3D reconstruction with multiple robots. They also sampled the sensor data. New sensor data were collected when the robot moved more than a given distance or rotated more than a given angle. The given distance and angle were decided by the speed of their robots. The sensor data were sampled to roughly 1 fps, which was the processing capability of OctoMap [@DBLP:journals/arobots/HornungWBSB13]. In practice, deciding the sample rate requires considerable experimentation and parameter tweaking, and can not easily adapt to dynamic environments. Many other researchers focus on the communication issues of decentralized robotic systems [@DBLP:journals/trob/LeungBL10; @DBLP:journals/ijrr/SabattiniCS13; @DBLP:conf/iros/LazaroPPCG13; @DBLP:conf/icra/AtanasovNDP15; @DBLP:journals/arobots/PaneratiMGMKSB19]. They assume that there are no network infrastructures that can connect all robots at the same time and robots can only communicate when they are in close proximity. These systems, however, need to store information on each robot for a long period of time, and there is a high level of communication when they are in close proximity [@DBLP:conf/icra/CarloneNDBI10]. Due to limited memory and bandwidth, they can only communicate information about their estimated poses and it is not feasible to communicate information about the environment. To enable robots to communicate large amount of information with limited resources, the messages also need to be uniformly sampled, just like our assumption. Quality of Service Policies --------------------------- Quality of Service (QoS) policies are designed to meet the needs of different scenarios. These growing needs such as real-time requirements in autonomous driving inspired the upgrade from ROS to ROS2 [@ros2_qos]. ROS2 provides QoS policies by integrating with Data Distribution System (DDS) [@DBLP:conf/icdcsw/Pardo-Castellote03]. Although it introduces performance issues such as increased communication latency [@DBLP:conf/emsoft/MaruyamaKA16], it is still used by many applications. However, existing QoS policies are not suitable for our applications related to SLAM and 3D reconstruction. According to the specifications [@ros2_qos2] and evaluations [@ros2_eval], the *Reliability* configuration decides the choice of UDP or TCP connections; the *History* and *Depth* configurations decide the queue size of the buffer; the *Durability* configuration decides whether publishers keep messages locally if there are no subscribers; the *Deadline* configuration describes the real time requirement; the *Liveliness* configuration decides whether heartbeat messages are employed; the *LatencyBudget* configuration decides the minimal interval between two messages. The *LatencyBudget* configuration is the most related policy because it implies the maximal frequency. In our motivating scenarios, however, deciding the maximal frequency needs more experiments [@DBLP:journals/tvcg/GolodetzCLPMT18] and is vulnerable to temporary network disruption. Buffer Management for Other Applications ---------------------------------------- Buffer management policies have been researched for the last decade. Practice networks are modeled as Delay Tolerant Networks (DTN) [@DBLP:journals/comsur/CaoS13] or Opportunistic Networks [@DBLP:conf/icccn/SatiPG16]. A sequence of simple drop schemes such as Drop Oldest, Drop Youngest, Drop Front, and Drop Last [@DBLP:conf/colcom/ChenYZW16] has been proposed for the scenario when the buffer is full. Enhanced policies introduce the concept of *profit* and assume that each message has a different profit. The goal is to maximize the total profit of the remaining messages [@DBLP:journals/sigact/Goldwasser10]. According to the application scenarios, different types of messages may have different profits [@DBLP:conf/secon/KrifaBS08; @DBLP:conf/csa2/KimW17; @DBLP:journals/cn/Scalosub17; @DBLP:journals/pomacs/YangWH17; @DBLP:conf/apcc/KimuraM17]. For example, when transmitting video frames [@DBLP:conf/icc/WuM13], P message, B message, and I message will be will be assigned different profits. However, in our motivating scenarios, the profit of a message is not decided by itself but by the relative positions of the remaining messages in the sending message sequence. Therefore, we need a new evaluation function to estimate the profit of a message sequence. Design of The OptSample Policy {#sec:approach} ============================== In this section, we first formally define our problem. After analyzing a naive solution, we give an overview of our resilient buffer manage policy. Notation and Symbols -------------------- For better understanding, we first define some notations and symbols. - $T$ denotes the number of messages during the network disruption. $L$ denotes the size of buffer, which is also the number of messages that will be received after the network disruption. - $a_i$. Each message is assigned an integer number indicating its sending order. We use $a_i$ to denote the message number of the $i$th message in a message sequence. - $A$ denotes a message sequence, i.e. $\{a_1, a_2, ..., a_n\}$. $|A|$ denotes the number of messages of $A$, i.e. $|A| = n$. - $A_{send}$ denotes the sequence that the user intend to send and $A_{recv}$ denotes the sequence that is received. These sequences change with $T$ and we use $A_{send}^T$ and $A_{recv}^T$ to denote the sequences with a certain $T$. Note that $A_{send}^T = \{1, 2, ..., T\}$ and $|A_{recv}^T| \leq L$. Obviously, $A_{send}^T \supset A_{recv}^T$. The messages in $A_{send}^T - A_{recv}^T$ are discarded by the buffer management policy. - $\widehat{A}$ denotes the extended version of sequence $A$, i.e. $\{0, a_1, a_2, ..., a_n, T + 1\}$. - $d_i$ denotes the difference between two adjacent messages in a sequence, i.e. $d_i = a_i - a_{i-1}$. It helps evaluate the sequence. - $f(d)$ denotes the evaluation function. In our design, it takes $d_i$ as input. - $P(A)$ denotes the profit function. It takes a message sequence $A$ as input to estimate the profit of the sequence. The goal of our OptSample policy is to output an $A_{recv}$ with higher profit $P(\widehat{A_{recv}})$. Problem Definition ------------------ We call a sequence $A$ *uniformly sampled* if all $d_i$ are equal. As we have described, we assume that uniformly sampled messages are the most valuable. To formally represent this assumption, we define the evaluation function and the profit function as Eq. \[equ:uniform\]. $$\label{equ:uniform} \begin{split} P(A) & = \sum_{i=2}^{|A|} f(d_i) , \\ f(d) & = 1 + \ln{d} . \end{split}$$ [The]{} evaluation function $f$ can be replaced with any function that satisfies the two following properties: **Lemma 1**: $\forall d_1, d_2 \geq 1$, $f(d_1) + f(d_2) > f(d_1 + d_2)$. This indicates that receiving one more message between two messages is always better. **Lemma 2**: $\forall d_1, d_2, d_3, d_4 \geq 1$, if $d_1 + d_2 = d_3 + d_4$ and $|d_1 - d_2| < |d_3 - d_4|$, then $f(d_1) + f(d_2) > f(d_3) + f(d_4)$. This indicates that a message nearer to the middle of two other messages is better. Thus, we can describe the transmission problem as follows. **Problem Statement**: Knowing the buffer size $L$, with increasing network disruption time $T$, the buffer management policy computes the received sequence $A_{recv}^T$ for higher profit $P(\widehat{A_{recv}^T})$ with the following restriction. **Lemma 3**: With any $T_1 < T_2$, $A_{recv}^{T_2} \cap A_{send}^{T_1} \subseteq A_{recv}^{T_1}$. This indicates that the policy cannot re-find messages that have been discarded. The Oracle Policy ----------------- With the above definition, we can easily prove that the maximal profit $P(\widehat{A_{recv}^T}) = T + 1$ when $L = T$. In this case, all messages remain in the buffer during the network disruption. With practical $L < T$, we can also prove that the maximal profit is met when all $d_i$ are the same. This conclusion confirms our assumption. In this case, $A_{recv}^T = \{d, 2 \cdot d, 3 \cdot d, ..., L \cdot d\}$, where $d = \frac{T + 1}{L + 1}$. It is not feasible if $d$ is not an integer, but the optimal solution can be obtained with the nearest integers. This optimal solution is still not feasible because it violates Lemma 3. For example, when $L = 4$, the optimal $A_{recv}^9$ is $\{2, 4, 6, 8\}$ and the optimal $A_{recv}^{14}$ is $\{3, 6, 9, 12\}$. But, $A_{recv}^{14} \cap A_{send}^{9} = \{3, 6, 9\} \nsubseteq \{2, 4, 6, 8\}$. This indicates that we have already discarded the message $3$ and $9$ at $T = 9$ and we cannot re-find them when $T = 14$. Therefore, this optimal solution can only act as an Oracle policy. The ROS Policy -------------- With the above definition, we can also evaluate the buffer management policy of ROS. ROS employs a naive buffer management policy called *Drop Oldest* [@ros_comm_link]. When a new message arrives and the buffer is full, the oldest message is discarded to make room for the new message. As a result, the buffer management policy of ROS chooses the newest $L$ messages to transmit. Therefore, the profit of these messages is $L + \ln(T - L + 1) + 1$. Note that the last part of the value $\ln(T - L + 1) + 1$ represents the profit of the first received message, which is labeled $T - L + 1$. The $\delta$-Sample Policy -------------------------- This is a basic version of our OptSample policy. The basic idea of our OptSample policy is to uniformly sample the buffer. A direct policy can be described as follows. When the buffer becomes full, we will uniformly sample messages from the buffer so that $L / \delta$ messages remains. All $d_i$ changes from 1 to $\delta$. Newer messages are also sampled at the same rate until the buffer becomes full again. When the buffer becomes full again, we again sample messages from the buffer so that $L / \delta$ messages remain. Thus, all $d_i$ becomes $\delta^2$. We can repeat the above operations until $T$ messages are processed (either remain in the buffer or are discarded at some time). We call this policy the *$\delta$-Sample* policy. Here, $\delta$ is the key parameter that affects the result. Lower $\delta$ leads to discarding fewer input messages but also leads to more rounds of sampling messages from the buffer if $T$ is fixed. $\delta$ must be greater than 1 by its formulation, otherwise we cannot make any room for a new message. It is easy to implement when we let $\delta = 2$ which leads to the *2-Sample* policy. Whenever the buffer is full and a new message arrives, we sample a message from every two messages and the sampling rate $r$ is doubled. Fig. \[fig:2sample\] shows an example of this policy. As we can see, an obvious shortcoming of the 2-Sample policy is that half of the buffer is empty when $T=9, 17, ..., L\cdot 2^i + 1, ...$ . ![An example of the 2-Sample policy ($L = 8$). Whenever the buffer is full and a new message that is not sampled arrives, we sample a message from every two messages and the sample rate $r$ is doubled.[]{data-label="fig:2sample"}](fig/2sample.png){width="\columnwidth"} Our OptSample Policy -------------------- Our OptSample policy is one step forward based on the 2-Sample policy. To overcome the shortcoming of the 2-Sample policy, we should make only one room for new messages at a time. To avoid falling back to ROS policy of Drop Oldest, we must sample uniformly rather than simply discarding the first message in the buffer. To achieve this, we record the position we will discard at the next time when the buffer is full and move it towards the end of the buffer. When the position is at the end of the buffer, the sample rate $r$ is increased and the position is moved to the beginning of the buffer. Fig. \[fig:optsample\] shows an example. As long as no messages are taken from the buffer, the buffer is always full. ![An example of the OptSample policy ($L = 8$). When a new message arrives and the buffer is full, the highlighted position is discarded and moves to the next position. When the position is at the end of the buffer, the sampling rate $r$ is increased and the position is moved back to the beginning.[]{data-label="fig:optsample"}](fig/optsample.png){width="\columnwidth"} Analysis and Comparison ----------------------- We deduce the profit function for the above 4 policies and summarize into Table \[table:compare\]. {width="0.7\columnwidth"} We explain the formula of the profit of our OptSample policy. $r$ is the sample rate. With any $T$, the distance $d_i$ is either $r$ or $2 \cdot r$. The only exception is the distance between $a_L$ and $T + 1$. The number of $d_i = 2r$ is $l - L$ and the number of $d_i = r$ is $2L - l$. Therefore the first part of the profit $L \cdot (\ln{r} + 1) + (l - L) \cdot \ln{2}$. The profit of the last part $\ln{T + 1 - r \cdot l} + 1$ comes from the distance between $a_L = r \cdot l$ and $T + 1$. We can prove that our OptSample policy can get higher profit than the 2-Sample policy and the ROS policy. Besides, the upper bound of the profit of the OptSample policy is the same with the Oracle policy, which is much better than that of the ROS policy. The lower bound of the profit of the OptSample policy can also be bounded with Lemma 6. Overall, we consider our OptSample policy nearly optimal. **Lemma 4**: $P_{OptSample}(T, L) \geq {P_{ROS}(T, L)}$. **Lemma 5**: $P_{OptSample}(T, L) \geq {P_{2-Sample}(T, L)}$. **Lemma 6**: $min(C, \frac{L}{L + 1}) \leq \frac{P_{OptSample}(T, L)}{P_{Oracle}(T, L)} \leq 1$, with $C = 2 - \ln{2} + \ln{\ln{2}} \approx 0.94$. ![Simulation results of different buffer management policies ($L=10$). The horizontal axis is the number of messages arriving ($T$) and the vertical axis is the profit of the remaining messages. ROS (blue) performs the worst since it keeps the last $L$ messages and leaves a big gap before the first kept message. The 2-Sample policy (orange) performs better, but the profit seriously declines when a buffer sampling occurs. The Oracle (red) is an ideal but not practical policy, which give us a reference. The OptSample policy (green) performs almost the same as the Oracle.[]{data-label="fig:simulation"}](fig/simulate.png){width="\columnwidth"} Evaluation {#sec:evaluation} ========== In this section, we show the advantage of our OptSample using simulation results and its applications to surveillance, 3D reconstruction, and SLAM. Simulation ---------- In the simulation, we take $L=10$ and check the change in the profit, when a new message arrives. The result is shown in Fig. \[fig:simulation\]. From this figure, we can see that our OptSample policy performs almost the same as the ideal Oracle policy and exactly the same as some $T$. This result confirms our profit functions in Table \[table:compare\] and Lemma 4-6. Note that increasing the buffer size $L$ will increase the lower bound and the upper bound of the profit. Therefore, increasing the buffer size $L$ will generally increase the profit. Implementation and Environment Setup ------------------------------------ We implement the OptSample policy based on ROS. We modify the code of the class *TransportSubscriberLink* which is located at the *ros\_comm* project [@ros_comm] and is linked into *libroscpp.so*. The pseudo code is shown in Fig. \[fig:algorithm\]. The *Enqueue* function is implemented as part of the *TransportSubscriberLink::enqueueMessage* method and the *Dequeue* function is implemented as part of the *TransportSubscriberLink::startMessageWrite* method. ![The pseudo code of the OptSample policy. The main process (lines 8–21) is demonstrated in Fig. \[fig:optsample\]. In addition, lines 2–7 allow for half the sampling distance and lines 26–29 update the next discarded position when a message is taken.[]{data-label="fig:algorithm"}](fig/algorithm.png){width="0.8\columnwidth"} {width="80.00000%"} {width="80.00000%"} Since we implement the policy at the *roscpp* level, it is totally transparent to the users and no user code needs to be modified. Note that when the network bandwidth is not a bottleneck and no temporary disruption occurs, the buffer is never full and the OptSample policy is not triggered. Therefore, our OptSample policy does not affect normal communications. In the following experiment, the control center is equipped with Intel Core i7-8750H @2.20GHz 12x CPU, 16GB memory and GeForce GTX 1080 GPU. The control center and the robot are connected via a 54Mbps WiFi router. The ROS version is Lunar on Ubuntu 16.04. It is one of the most recent versions of ROS [@DBLP:conf/icra/DeMarinisTKKF19]. To make better comparisons, all sensor data (i.e. RGB images, depth images, and laser data) in the following experiments are recorded with *rosbag* and are replayed multiple times to get results with different policies. Application 1: Surveillance --------------------------- The first application scenario is surveillance. Fig. \[fig:arch\_video\] shows the software architecture. We deploy a laptop with a camera to monitor a “treasure." It keeps sending $640 \times 480 \times 24bits$ compressed images to the control center at 20 fps. In this case, the needed network bandwidth is about 0.8 MB/s. A malicious attacker manages to disturb the wireless network for about 4 seconds and takes the treasure during this time. The buffer size of the publisher is set to 20. The result is shown in Fig. \[fig:video\]. With native ROS, the attacker is not shown in any of the surveillance images received by the control center; with our OptSample policy, after the network disruption, the surveillance images that are captured during the network disruption arrive at the control center as if they were captured at 5 fps and the attacker is captured in a few images. Analysis tells us that with native ROS, the attacker has a time gap of about $\frac{4 \times 20 - 20}{20} \approx 3$ seconds to fulfill the theft. With OptSample, the attacker has to disturb the wireless network for about $20 \times 3 \approx 60$ seconds to give him the same time gap to fulfill the theft. This is more difficult for the attacker than to disturb the wireless network for 4 seconds and we consider that our OptSample policy makes the theft much more difficult for the attacker. ![The software architecture of the 3D reconstruction application. An ellipse represents an ROS node (with its package name and node name inside) and a rectangle represents an ROS topic (with its topic name and message type inside).[]{data-label="fig:arch_teddy"}](fig/arch_infinitam.png){width="\columnwidth"} {width="90.00000%"} ![The software architecture of the SLAM application. An ellipse represents an ROS node (with its package name and node name inside) and a rectangle represents an ROS topic (with its topic name and message type inside).[]{data-label="fig:arch_slam"}](fig/arch_rtabmap.png){width="\columnwidth"} Application 2: 3D Reconstruction -------------------------------- The second application scenario is 3D reconstruction. Fig. \[fig:arch\_teddy\] shows the software architecture. We employ InfiniTAM [@DBLP:journals/ral/KahlerPVM16] as the reconstruction algorithm. InfiniTAM is a widely-used, vision-based reconstruction algorithm and very efficient when equipped with a modern GPU. We encapsulate InfiniTAM into an ROS node. RGB images and depth images are captured with Kinect on the robot side. These images are sent to the control center via two separate ROS topics at 10 fps. In this case, the needed network bandwidth is about 1 MB/s. The control center subscribes these two topics, synchronizes them, and uses them to reconstruct a 3D model. We cut the wireless network for 5 seconds on purpose to simulate a temporary network disruption. The buffer size of the publisher is set to 20. The result is shown in Fig. \[fig:teddy\]. From Fig. \[fig:teddy\](a), we can see that, in this scenario, adjacent images provide similar information and our assumption is true. With native ROS (Fig. \[fig:teddy\](c)), a long segment of messages is lost, the reconstructed algorithm loses the correspondence of feature points, and the result does not appear good. With our OptSample policy (Fig. \[fig:teddy\](d)), however, the reconstructed model seems to be unaffected by the network disruption. In fact, the resulting model of our OptSample policy is not as dense as the ground truth, but it seems good enough. Application 3: SLAM ------------------- The third application scenario is SLAM. Fig. \[fig:arch\_slam\] shows the software architecture. We employ RTAB-Map [@DBLP:journals/jfr/LabbeM19; @rtabmap_ros] as the SLAM algorithm. RTAB-Map is one of the SLAM algorithms that has integrated with ROS as an ROS node or nodelet, and it can be visualized with rviz [@rviz]. RGB images, depth images, and laser scan data are collected on the robot side and published at about 8 fps. In this case, the needed network bandwidth is about 1.6 MB/s. The control center launches an RTAB-Map node, which subscribes and synchronizes these topics as input data. We purposefully cut the wireless network for 5 seconds for several times to simulate a temporary network disruption. The buffer size of the publisher is set to 20. The result is shown in Fig. \[fig:slam\]. From Fig. \[fig:slam\](a), we can see that, in this scenario, adjacent images provide similar information, and our assumption is true. With native ROS (Fig. \[fig:slam\](c)), the robot appears to stop and “teleport" at the control center and there are gaps in the mapped environment. However, with our OptSample policy (Fig. \[fig:slam\](d)), the path of the robot is more continuous and there are no obvious gaps in the mapped environment.   Overall, with three application scenarios, we have shown that our OptSample policy provides better resilience against network disruption. As long as the assumption that uniformly sampled messages are more valuable is true, there are more application scenarios. In addition, during the above experiments, no user code is modified. We can switch between native ROS and our OptSample policy by simply replacing the *libroscpp.so*. From the view of data flow, the above experiments also show that our OptSample policy can be easily deployed to single topic (Surveillance), two synchronized topics (3D construction), and multiple synchronized topics (SLAM). Conclusion and Future Work {#sec:conclusion} ========================== We have presented a novel buffer management policy called OptSample. By uniformly sampling the message buffer, our OptSample policy is more resilient against temporary network disruption. We explain that common solutions assume that newer messages are more valuable than older messages, but this assumption is not always true. We define an evaluation function that models the new assumption that uniformly sampled messages are more valuable. Based on this function, our analysis and simulation show that our OptSample policy is very close to the optimal Oracle policy. We have integrated our OptSample policy with ROS. The implementation is transparent to users and no user code needs to be changed. Experimental results show that our OptSample policy is more resilient than native ROS in several application scenarios. The main limitation of our OptSample policy comes from the assumption. We assume that uniformly sampled messages are the most valuable. Thus, we evaluate the profit of messages with their sequence numbers. If the content of actual messages is taken into account, the problem would be more complex but with wider adaptability. In the following appendix, we will give the proof of the lemmas.     Proof of The Lemmas =================== **Lemma 1**: $\forall d_1, d_2 \geq 1$, $f(d_1) + f(d_2) > f(d_1 + d_2)$.   $$\begin{aligned} &\because d_1, d_2 \geq 1 \\ &\therefore (d_1 - 1) \cdot (d_2 - 1) \geq 0 \\ &\therefore d_1 \cdot d_2 + 1 \geq d_1 + d_2 & (1.1) \\ &\because d_1, d_2 \geq 1 \\ &\therefore d_1 \cdot d_2 \geq 1 > \frac{1}{e - 1} \\ &\therefore e \cdot d_1 \cdot d_2 > d_1 \cdot d_2 + 1 & (1.2) \\ &\because (1.1)~and~(1.2) \\ &\therefore e \cdot d_1 \cdot d_2 > d_1 + d_2 \\ &\therefore 1 + \ln{d_1} + \ln{d_2} > \ln{(d_1 + d_2)} \\ &\therefore f(d_1) + f(d_2) > f(d_1 + d_2)\end{aligned}$$ **Lemma 2**: $\forall d_1, d_2, d_3, d_4 \geq 1$, if $d_1 + d_2 = d_3 + d_4$ and $|d_1 - d_2| < |d_3 - d_4|$, then $f(d_1) + f(d_2) > f(d_3) + f(d_4)$.   $$\begin{aligned} &\because |d_1 - d_2| < |d_3 - d_4| \\ &\therefore (d_1 - d_2)^2 < (d_3 - d_4)^2 \\ &\therefore (d_1 + d_2)^2 - 4 \cdot d_1 \cdot d_2 < (d_3 + d_4)^2 - 4 \cdot d_3 \cdot d_4 \\ &\because d_1 + d_2 = d_3 + d_4 \\ &\therefore d_1 \cdot d_2 > d_3 \cdot d_4 \\ &\therefore \ln{d_1} + \ln{d_2} > \ln{d_3} + \ln{d_4} \\ &\therefore f(d_1) + f(d_2) > f(d_3) + f(d_4)\end{aligned}$$ **Lemma 4**: $P_{OptSample}(T, L) \geq {P_{ROS}(T, L)}$.   For convenience, in the following proof, we abbreviate $P_{OptSample}(T, L)$ to $P_{4}(T, L)$ and $P_{ROS}(T, L)$ to $P_{2}(T, L)$. There exist some integers $K$ and $p$ that satisfy: $$\begin{aligned} &T = K \cdot L + p \\ &1 \leq K \\ &0 \leq p \leq L - 1 \\\end{aligned}$$ Further, there exist some integers $k$ and $q$ that satisfy: $$\begin{aligned} &K = 2^k + q \\ &0 \leq k \\ &0 \leq q \leq 2^k - 1 \\\end{aligned}$$ Let $r = 2^k$, and some integers $s$ and $t$ that satisfy: $$\begin{aligned} &q \cdot L + p = s \cdot r + t \\ &0 \leq t \leq r - 1 \\ &0 \leq s \leq L - 1\end{aligned}$$ Let $l = L + s$, then: $$\begin{aligned} T & = (r + q) \cdot L + p \\ & = r \cdot L + q \cdot L + p \\ & = r \cdot L + r \cdot s + t \\ & = r \cdot l + t\end{aligned}$$ Based on the above definition, we can rewrite $P_{OptSample}(T, L)$ as: $$\begin{aligned} &P_{4}(T, L) = L \cdot (\ln{r} + 1) + s \cdot \ln{2} + \ln{(t + 1)} + 1\end{aligned}$$ We define $D(T, L) = P_{4}(T, L) - P_{2}(T, L)$, then: $$\begin{aligned} &D(T, L) = L \cdot \ln{r} + s \cdot \ln{2} + \ln{(t + 1)} - \ln{(T + 1 - L)}\end{aligned}$$ We will prove $D(T, L) \geq 0$ in 3 cases.   **Case 1:** When $L \leq T < 2L$. By their definition, we have: $$\begin{aligned} & K = 1,~p = T - L \\ & k = 0,~q = 0,~r = 1 \\ & s = T - L,~t = 0,~l = T \\ & D(T, L) = (T - L) \cdot \ln{2} - \ln{(T + 1 - L)} \\\end{aligned}$$ Therefore, $$\begin{aligned} D'(T, L) & = \frac{\partial D(T, L)}{\partial T} = \ln{2} - \frac{1}{T + 1 - L} \\\end{aligned}$$ When $T \geq L + 1$, $D'(T, L) > 0$. Since $D(L, L) = D(L + 1, L) = 0$, we can conclude that $D(T, L) \geq 0$ for all $T$ in this case. **Case 2:** When $T \geq 2L$ and $L = 1$. Since $1 = L \leq l \leq 2L - 1 = 1$, $l = 1$ and $T = r + t$. Therefore, $$\begin{aligned} D(T, L) & = \ln{r} + \ln{(t + 1)} - \ln{(r + t)} \\ & = \ln{\frac{r \cdot t + r}{r + t}} \\ & \geq \ln{\frac{t + r}{r + t}} = 0\end{aligned}$$ **Case 3:** When $T \geq 2L$ and $L \geq 2$. In this case, $r \geq 2$, $L \leq l \leq 2 \cdot L - 1$, and $0 \leq t \leq r - 1$. Therefore: $$\begin{aligned} D(T, L) & = L \cdot \ln{r} + (l - L) \cdot \ln{2} \\ & + \ln{(t + 1)} - \ln{(r \cdot l + t + 1 - L)} \\ & \geq L \cdot \ln{r} + (l - L) \cdot \ln{2} - \ln{(r \cdot l + r - L)} \\\end{aligned}$$ We define $F(r, l) = L \cdot \ln{r} + (l - L) \cdot \ln{2} - \ln{(r \cdot l + r - L)}$. Then: $$\begin{aligned} \frac{\partial F(r, l)}{\partial r} & = \frac{L}{r} - \frac{l + 1}{r \cdot l + r - L} \\ & = \frac{(r \cdot l + r - (L + 1))(L - 1) - 1}{r \cdot (r \cdot l + r - L)} \\ & \geq \frac{(r \cdot l + t - L)(2 - 1) - 1}{r \cdot (r \cdot l + r - L)} \\ & \geq \frac{L - 1}{r \cdot (r \cdot l + r - L)} \\ & > 0 \\ \frac{\partial F(r, l)}{\partial l} & = \ln{2} - \frac{r}{r \cdot l + r - L} \\ & > \frac{1}{2} - \frac{r}{r \cdot l + r - L} \\ & = \frac{r \cdot l + r - L - 2 \cdot r}{2 \cdot (r \cdot l + r - L)} \\ & \geq \frac{r \cdot L - r - L}{2 \cdot (r \cdot l + r - L)} \\ & = \frac{(r - 1) \cdot (L - 1) - 1}{2 \cdot (r \cdot l + r - L)} \\ & \geq \frac{(2 - 1) \cdot (2 - 1) - 1}{2 \cdot (r \cdot l + r - L)} \\ & = 0\end{aligned}$$ Which means $F(r, l)$ is monotone increasing for both $r$ and $l$. In this case, $r \geq 2$ and $l \geq L$. Therefore, $$\begin{aligned} F(r, l) & \geq F(2, L) \\ & = L \cdot \ln{2} - \ln{(L + 2)} \\ & = \ln{\frac{2^L}{L + 2}}\end{aligned}$$ We define $G(L) = \frac{2^L}{L + 2}$. Then $G(2) = 1$ and $G'(L) > 0$ for $L > 0$. Therefore, $$\begin{aligned} D(T, L) & \geq F(r, l) \\ & \geq \ln{G(L)} \\ & > \ln{G(2)} = 0\end{aligned}$$ Overall, $D(T, L) \geq 0$ for all 3 cases, which means $P_{4}(T, L) \geq P_{2}(T, L)$. **Lemma 5**: $P_{OptSample}(T, L) \geq {P_{2-Sample}(T, L)}$.   We continue to use the definition of $p$, $q$, $k$, $r$, $s$, $l$, and $t$ in the proof of Lemma 4. For convenience, in the following proof, we abbreviate $P_{OptSample}(T, L)$ to $P_{4}(T, L)$ and $P_{2-Sample}(T, L)$ to $P_{3}(T, L)$. To simplify the $P_{2-Sample}(T, L)$, we further define: $$\begin{aligned} &\hat{r} = 2^{\lceil \log_2{\frac{T + 1}{L + 1}}\rceil} \\ &\hat{l} = \lfloor \frac{T}{\hat{r}} \rfloor \\ &\hat{t} = T - \hat{r} \cdot \hat{l}\end{aligned}$$ Thus, we can write $P_{2-Sample}(T, L)$ as: $$\begin{aligned} &P_{3}(T, L) = \hat{l} \cdot (\ln{\hat{r}} + 1) + \ln{(\hat{t} + 1)} + 1\end{aligned}$$ Let’s first analysis the relationship between $r$ and $\hat{r}$. 1\) We have $\frac{T + 1}{L + 1} > \frac{r}{2}$, because: $$\begin{aligned} 2T + 2 & = 2 \cdot (r \cdot l + t) + 2 \\ & > r \cdot L + r \cdot L \\ & \geq r \cdot L + r\end{aligned}$$ 2\) We have $\frac{T + 1}{L + 1} < 2 \cdot r$, because: $$\begin{aligned} T + 1 & = r \cdot l + t + 1 \\ & < r \cdot (2 \cdot L) + r \\ & < 2 \cdot r \cdot (L + 1)\end{aligned}$$ Overall, we have either $\hat{r} = r$ or $\hat{r} = 2 \cdot r$. **Case 1:** The first case $\hat{r} = r$ is satisfied when: $$\begin{aligned} & \frac{T + 1}{L + 1} \leq r \\ \Leftrightarrow~& \frac{(r + q) \cdot L + p + 1}{L + 1} \leq r \\ \Leftrightarrow~& \frac{p \cdot L + q + 1 - r}{L + 1} \leq 0 \\ \Leftrightarrow~& r \cdot s + t + 1 \leq r \\ \Leftrightarrow~& s = 0,~l = L,~t = r - 1\end{aligned}$$ In this case $\hat{r} = r$, so $\hat{l} = l$ and $\hat{t} = t$, and: $$\begin{aligned} P_{4}(T, L) & = L \cdot (\ln{r} + 1) + s \cdot \ln{2} + \ln{(t + 1)} + 1 \\ & = l \cdot (\ln{r} + 1) + \ln{(t + 1)} + 1 \\ & = \hat{l} \cdot (\ln{\hat{r}} + 1) + \ln{(\hat{t} + 1)} + 1 \\ & = P_{3}(T, L)\end{aligned}$$ **Case 2:** When $\hat{r} = 2 \cdot r$, we have: $$\begin{aligned} \hat{l} & = \lfloor \frac{T}{\hat{r}} \rfloor = \lfloor \frac{T}{2 \cdot r} \rfloor = \lfloor \frac{l}{2} \rfloor \\ & = \begin{cases} \frac{l}{2} & \quad \text{if } l \text{ is even}\\ \frac{l - 1}{2} & \quad \text{if } l \text{ is odd} \end{cases} \\ \hat{t} & = \begin{cases} t & \quad \text{if } l \text{ is even}\\ t + r & \quad \text{if } l \text{ is odd} \end{cases}\end{aligned}$$ **Case 2.1:** When $l$ is even, we have: $$\begin{aligned} P_{3}(T, L) & = \hat{l} \cdot (\ln{(\hat{r}) + 1)} + 1) + \ln{(\hat{t} + 1)} + 1 \\ & = \frac{l}{2} \cdot (\ln{(2 \cdot r)} + 1) + \ln{(t + 1)} + 1 \\ & = \frac{l}{2} \cdot (\ln{r} + 1) + \frac{l}{2} \cdot \ln{2} + \ln{(t + 1)} + 1 \\\end{aligned}$$ Since $l < 2 \cdot L$, $r \geq 1$, and $1 > \ln{2}$, we have: $$\begin{aligned} P_{4}(T, L) - P_{3}(T, L) = (L - \frac{l}{2}) \cdot (\ln{r} + 1 - \ln{2}) > 0\end{aligned}$$ **Case 2.2:** Similarly, when $l$ is odd, we have: $$\begin{aligned} P_{3}(T, L) & = \hat{l} \cdot (\ln{\hat{r}} + 1) + 1) + \ln{(\hat{t} + 1)} + 1 \\ & = \frac{l - 1}{2} \cdot (\ln{(2 \cdot r)} + 1) + \ln{(t + r + 1)} + 1 \\ & = \frac{l - 1}{2} \cdot (\ln{r} + 1) + \frac{l - 1}{2} \cdot \ln{2} \\ & + \ln{(t + r + 1)} + 1 \\\end{aligned}$$ Since $l \leq 2 \cdot L - 1$, $r \geq 1$, $1 > \ln{2}$ and $t + 1 \leq r$, we have: $$\begin{aligned} P_{4}(T, L) - P_{3}(T, L) & = (L - \frac{l + 1}{2}) \cdot (\ln{r} + 1 - \ln{2}) \\ & + \ln{r} + 1 + \ln{(t + 1)} - \ln{(t + r + 1)} \\ & > 1 - \ln{(\frac{1}{r} + \frac{1}{t + 1})} \\ & > 1 - \ln{2} > 0\end{aligned}$$ Overall, $P_{4}(T, L) - P_{3}(T, L) \geq 0$ **Lemma 6**: $min(C, \frac{L}{L + 1}) \leq \frac{P_{OptSample}(T, L)}{P_{Oracle}(T, L)} \leq 1$, with $C = 2 - \ln{2} + \ln{\ln{2}} \approx 0.94$.   We continue to use the definition of $p$, $q$, $k$, $r$, $s$, $l$, and $t$ in the proof of Lemma 4. For convenience, in the following proof, we abbreviate $P_{OptSample}(T, L)$ to $P_{4}(T, L)$ and $P_{Oracle}(T, L)$ to $P_{1}(T, L)$. We first prove the left part: **Lemma 6.1**: $min(C, \frac{L}{L + 1}) \leq \frac{P_{OptSample}(T, L)}{P_{Oracle}(T, L)}$.   We can rewrite $P_{OptSample}(T, L)$ as: $$\begin{aligned} P_{4}(T, L) = & L + 1 + l \cdot \ln{2} - L \cdot (\ln{2} + \ln{l}) \\ & + L \cdot \ln{(T - t)} + \ln{(t + 1)} & (6.1)\end{aligned}$$ Let $f(t) = L \cdot \ln{(T - t)} + \ln{(t + 1)}$. It is a monotone increasing function of $t$. Because: $$\begin{aligned} &\because t + 1 \leq r \\ &\therefore L \cdot (t + 1) \leq L \cdot r \leq l \cdot r = T - t \\ &\therefore \frac{L}{T - t} \leq \frac{1}{t + 1} \\ &\therefore \frac{\partial f(t)}{\partial t} = \frac{-L}{T - t} + \frac{1}{t + 1} \geq 0\end{aligned}$$ Therefore, $f(t) \geq f(0) = L \cdot \ln{T}$. Put it into (6.1) and we get: $$\begin{aligned} P_{4}(T, L) \geq & L + 1 + L \cdot (\ln{T} - \ln{2}) \\ & - L \cdot \ln{l} + l \cdot \ln{2} & (6.2)\end{aligned}$$ Let $g(l) = l \cdot \ln{2} - L \cdot \ln{l}$. It reaches a local minimal value when: $$\begin{aligned} \frac{\partial g(l)}{\partial l} = \ln{2} - \frac{L}{l} = 0,~i.e.~l = \frac{L}{\ln{2}}\end{aligned}$$ Put this $l$ into (6.2) and we get: $$\begin{aligned} P_{4}(T, L) & \geq L + 1 + L \cdot (\ln{T} - \ln{2}) \\ & - L \cdot \ln{L} + L \cdot \ln{\ln{2}} + L \\ & = L \cdot \ln{\frac{T}{L}} + L \cdot (2 - \ln{2} + \ln{\ln{2}}) + 1 \\ & = L \cdot ln\frac{T}{L} + L \cdot C + 1\end{aligned}$$ When $T \geq L$, $\frac{T}{L} \geq \frac{T + 1}{L + 1}$. Therefore, $$\begin{aligned} P_{4}(T, L) & \geq L \cdot \ln{\frac{T + 1}{L + 1}} + L \cdot C + 1\end{aligned}$$ We can also rewrite $P_{Oracle}(T, L)$ as: $$\begin{aligned} P_{1}(T, L) & = (L + 1) \cdot \ln{\frac{T + 1}{L + 1}} + L + 1\end{aligned}$$ When $T = L$, we have: $$\begin{aligned} \frac{P_{4}(T, L)}{P_{1}(T, L)} = \frac{L \cdot C + 1}{L + 1} > C\end{aligned}$$ And with $T$ growth, it tends to be: $$\begin{aligned} \lim\limits_{T \to \infty} \frac{P_{4}(T, L)}{P_{1}(T, L)} = \frac{L}{L + 1}\end{aligned}$$ Note that the ratio is a monotonic function, either increasing or decreasing depending on which of the two values is bigger, $C$ or $\frac{L}{L + 1}$. Therefore, we can conclude that: $$\begin{aligned} \frac{P_{4}(T, L)}{P_{1}(T, L)} \geq min(C, \frac{L}{L + 1})\end{aligned}$$ Then, we continue to prove the right part: **Lemma 6.2**: $\frac{P_{OptSample}(T, L)}{P_{Oracle}(T, L)} \leq 1$.   We can rewrite $P_{OptSample}(T, L)$ as: $$\begin{aligned} P_{4}(T, L) = & L + 1 + L \cdot (\ln{(T - t)} - \ln{2}) + \ln{(t + 1)} \\ & + l \cdot \ln{2} - L \cdot \ln{l} & (6.3) \\\end{aligned}$$ We define $g(l) = l \cdot \ln{2} - L \cdot \ln{l}$ and we have: $$\begin{aligned} g'(l) = \frac{\partial g(l)}{\partial l} = \ln{2} - \frac{L}{l}\end{aligned}$$ Note that $L \leq l \leq 2L - 1$. Let $l_0 = \frac{L}{\ln{2}}$, then $g'(l) < 0$ for $l \in [L, l_0)$ and $g'(l) > 0$ for $l \in (l_0, 2 \cdot L - 1]$. Therefore, $g(l)$ gets its maximal value when $l = L$ or $l = 2 \cdot L - 1$. We can prove that $g(L) \geq g(2 \cdot L - 1)$, because: $$\begin{aligned} &g(L) - g(2 \cdot L - 1) = \ln{2} - L \cdot \ln{(1 + \frac{1}{2 \cdot L - 1})} \\ &Let~h(L) = (1 + \frac{1}{2 \cdot L - 1})^L,~then \\ &h'(L) < 0,~h(1) = 2,~\lim\limits_{L \to \infty} h(L) = \sqrt{e} < 2 \\ &\therefore g(L) - g(2 \cdot L - 1) \geq \ln{2} - \ln{2} = 0\end{aligned}$$ Put $g(L)$ into (6.3), we have: $$\begin{aligned} P_{4}(T, L) & \leq L + 1 + L \cdot (\ln{(T - t)} - \ln{2}) + \ln{(t + 1)} + g(L) \\ & = L + 1 + L \cdot \ln{(T - t)} + \ln{(t + 1)} - L \cdot \ln{L}\end{aligned}$$ We further define $f(t) = L \cdot \ln{(T - t)} + \ln{(t + 1)}$, and we have already proved that $f(t)$ is a monotone increasing function of $t$. Since $t \leq \frac{T + 1}{L + 1} - 1$, we have: $$\begin{aligned} P_{4}(T, L) & \leq L + 1 + f(\frac{T + 1}{L + 1} - 1) - L \cdot \ln{L} \\ & = (L + 1) \cdot (\ln{\frac{T + 1}{L + 1}} + 1) \\ & = P_{1}(T, L)\end{aligned}$$ Alternative Evaluation Functions ================================ As we have stated, any functions that satisfy Lemma 1 and Lemma 2 could be employed as the evaluation function. Using different evaluation functions will lead to different results. We have tried the following alternative evaluation functions. $$\begin{aligned} f(d) & = 1 + \log_2{d} \\ f(d) & = \sqrt{d} \\ f(d) & = \frac{\arctan(d)}{\pi / 4} \\\end{aligned}$$ We can neither explain the meaning of different evaluation functions, nor decide which is appropriate. But our OptSample policy performs the best as well. [^1]: $^{1}$Yu-Ping Wang, Zi-Xin Zou, Xu-Qiang Hu, and Shi-Min Hu are with the Department of Computer Science and Technology, Tsinghua University, Beijing, China. [^2]: $^{2}$Dinesh Manocha is with the Department of Computer Science, University of Maryland, MD 20742, USA. [^3]: $^{3}$Lei Qiao is with the Beijing Institute of Control Engineering, Beijing, China. [^4]: Yu-Ping Wang is the corresponding author, e-mail: wyp@tsinghua.edu.cn.

--- abstract: | We have used extensive $V$, $I$ photometry (down to $V=20.9$) of $33615$ stars in the direction of the globular cluster M55 to study the dynamical interaction of this cluster with the tidal fields of the Galaxy. An entire quadrant of the cluster has been covered, out to $\simeq1.5$ times the tidal radius. A CMD down to about 4 magnitudes below the turn-off is presented and analysed. A large population of BS has been identified. The BS are significantly more concentrated than the other cluster stars in the inner 300 arcsec, while they become less concentrated in the cluster envelope. We have obtained luminosity functions at various radial intervals from the center and their corresponding mass functions. Both clearly show the presence of mass segregation inside the cluster. A dynamical analysis shows that the observed mass segregation is compatible with what is predicted by multi-mass King-Michie models. The global mass function is very flat with a power-law slope of $x=-1.0\pm0.4$. This suggest that M55 might have suffered selective losses of stars, caused by tidal interactions with the Galactic disk and bulge. The radial density profile of M55 out to $\sim 2\times r_t$ suggests the presence of extra-tidal stars whose nature could be connected with the cluster. author: - 'Simone R. Zaggia' - Giampaolo Piotto - Massimo Capaccioli date: 'Received 29 May 1997 / Accepted 11 July 1997' nocite: - '[@Lee77]' - '[@Shade88]' - '[@Alcaino92]' - '[@Trager95]' - '[@Bailyn95]' - '[@DJ93]' - '[@Mand96]' - '[@Ibata95]' title: 'The Stellar Distribution of the Globular Cluster M55 [^1] ' --- 10000 Introduction ============ The recent advances in our understanding of the structure and evolution of Galactic globular clusters (GCs) have been possible thanks to the advent of accurate CCD photometry. However, till few years ago, CCD photometry was limited to the internal parts of GCs due to the small fields of the detectors. All the information relative to the outer regions and to the tidal radius $r_t$, arise from visual (*by eye*) stellar counts made on Schmidt plates, especially by King and collaborators [@King68; @Trager95]. This methodology of investigation suffers from various problems and statistical biases; we list some of them: - The limiting magnitude of photographic plates, which is generally too bright to permit the investigation of the radial distribution of stars in an appropriate mass range; - the high uncertainty in the evaluation of background stellar contamination; - an insufficient crowding/completeness correction. All the more recent models of dynamical evolution need to make assumptions on the mass function, on the effects due to the radial anisotropy of the velocity distribution, and on the mass segregation which, in principle, could be determined observationally. \[M55frames\] Fostered by this lack of observational data, five years ago we started a long term project using one of the largest field CCD cameras available, EMMI at the NTT, to obtain accurate stellar photometry in two bands, $V$ and $I$, over at least a full quadrant in a number of GCs. The sample was selected taking into account the different morphological types and the different positions in the Galaxy. The principal aim was to map the stellar distribution from the central part out to the outer envelope (beyond the formal tidal radius, for a better estimate of the field stars contamination and in order to investigate on the possible presence of tidal tails), with a good statistical sampling of the stars in distinct zones of the color magnitude diagram (CMD) and of different masses. The use of CCD star counts, instead of photographic Schmidt plates for which the only advantage is still to give a larger area coverage [@Grillmair95; @LS97 for works on this subject], allows us to go deeper inside the core of the clusters, to better handle photometric errors and completeness corrections and to reach a considerably fainter magnitude level. CCDs also allows in the case of high concentration clusters to complement star counts of the central part with aperture photometry of short exposures images [@Ivo97]. An important byproduct of this study is the photometry of a significant number of stars in all the principal sectors of the CMD. This sample is of fundamental importance to test modern evolutionary stellar models [@RFP88]. In all cases the CMD extends well below the turn off of the main sequence. This permits us to estimate the effect of mass segregation for masses from the TO mass ($\sim0.8\Msun$) down to $0.6, 0.5\Msun$. So far, we have collected data for a total of 19 clusters. Same of them have already been reduced and analyzed. In this work we present the analysis of the star counts of the globular cluster =M55. Other clusters, for which we have already given a first report elsewhere [@Zaggia95; @Carla96; @Ivo95; @Alf96], will be presented in future works [@Ivo97; @Alf97]. Why the globular cluster M55? ============================= M55 is a low central concentration, $c=0.8$ [@Trager95], low metallicity, ${\rm [Fe/H]}=-1.89$ [@Zinn80], cluster located at $\simeq4.9$ kpc from the Sun [@Mand96]. Although it is a nearby object, it has received little or sporadic attention until very recently. The works of [@Mateo96] and [@Fahlman96] presented photometric datasets of M55 that have been used principally to establish the age and the tidal extension of the Sagittarius dwarf galaxy. [@Mand96] published the first deep (down to $V\simeq24.5$) photometry of the cluster (other previous studies of the stellar population of M55 are in Lee, 1977; Shade, VandenBerg, and Hartwick, 1988; and Alcaino , 1992). From the data of a field at $\simeq2$ core radii from the center, [@Mand96] estimated a new apparent distance modulus for M55, $(m-M)_V=13.90\pm0.09$, and from the luminosity function they found that the high-mass end of the mass function ($0.5<M/M_{\odot}<0.8$) is well fitted by a power law with $x=0.5\pm0.2$, whereas at the low-mass end ($M/M_{\odot}<0.4$) the mass function has a slope of $x=1.6\pm0.1$. From the dynamical point of view, M55 has been previously studied by [@Pryor91] in their papers on the mass-to-light ratio of globular clusters. Their principal conclusion is that this cluster might have a power law mass function with an exponent $x=1.35\div2.0$, with a lower limit of the mass function in the range $\simeq0.1\div0.3\Msun$ ( a total absence of low mass stars): a conclusion opposite to that found recently by [@Mand96]. \[M55tot\] An original work on M55 is in [@Irwin84]. They studied the radial star count density profile using photographic material digitalized with the *Automatic Plate Measuring System* (APM) of the Cambridge University. [@Irwin84] used a single photometric band, which did not allow them to lower the contribution of the field stars in the construction of the radial star counts. Nevertheless in this work (never repeated in other clusters), the authors reach some interesting conclusions: they claim one of the first evidences of mass segregation (even if they cannot quantify it); the central stellar luminosity function seems to be flat (with a corresponding mass function having a slope of $x\simeq0.0$) with a partial deviation from the King models. Moreover, they claim the presence of 8 short period variables, at the limit of their photometry, compatible with contact binaries of W UMa type. This last point is interesting for the presence of a large population of Blue Straggler (BS) stars in M55 to which the variables of [@Irwin84] could belong. Despite the potential interest of this nearby cluster for problems such as the dynamical evolution of globular clusters and interaction with the tidal field of the Galaxy, the existing data on M55 are so far limited and have been used to address only particular problems. Now large field CCDs offer the possibility to attack this problems in a suitable way. The following Section is dedicated to the presentation of the M55 data set and our observing strategy; in Section 4 we show the luminosity and mass function of the cluster; in Section 5 we present the analysis of the radial density profile and the conclusions. The details of the techniques adopted in the reduction and analysis of the data can be found in the appendix of the paper. The photometric data ==================== A whole quadrant of M55 was mapped (from the center out to $\sim1.5~r_t$, with $r_t=977''$ as in Trager, King, and Djorgovski, 1995), on the night of July 5 1992 with 18 EMMI-NTT fields ($7\farcm2\times7\farcm2$) in the $V$ and $I$ bands. Figure \[M55frames\] shows the field positions on the sky. For each field a $V$ and a $I$ band image were taken in succession, with exposure times of 40 and 30 seconds respectively. The night was not photometric and the observing conditions improved as we moved from the outer fields to the internal ones. Information on the various fields and on all the technicalities of the reduction and analysis are reported in the appendix of the paper. The $V$ *vs.* $(V-I)$ color magnitude diagram for a total of 33615 stars of M55 is shown in Figures \[M55tot\] and \[cmdradial\]. In total we detected 36800 objects of the cluster$+$field; $\simeq9\%$ of them were eliminated after having applied a selection in the [DAOPHOT II]{} PSF interpolation parameters as in [@Piotto90a]. Although the exposure time was relatively short, the brightest stars of the red giant branch and of the asymptotic giant branch are saturated, though they can be still used for the radial star counts. We have omitted them from the final CMD. In the following we will analyze the data using a division into three radial subsamples: inner ($r\le r_c$), intermediate ($r_c<r\le 2r_c$), and outer ($2r_c<r\le r_t$). The core radius is $r_c=143\arcsec$, as found from the radial density profile analysis ( Section 5). In Figure \[cmdradial\] we show the brightest part of the CMD of M55, divided in the three radial subsamples. A large population of blue straggler stars (BS) is clearly visible, particularly in the inner part of the cluster where the background/foreground star contamination is low. In the intermediate zone, the BS population is better defined, and the sequence seems to reach brighter magnitudes. The BS sequence of the inner part appears to be broader in color than the sequence of the intermediate radial range. Part of this broadening can be attributed to the photometric errors that are larger in the inner region than in the intermediate one. The rest of the broadening is probably natural and could be connected to the two formation mechanisms of BS stars: the outer BS stars might mainly come from merger events, while the inner BS might be the final products of collisions (see Bailyn 1995). We compared the distribution of the 95 BS with that of the 1669 sub-giant branch (SGB) stars selected in the same magnitude interval. In order to minimize the background star contamination along the SGB (very low indeed in the inner part of the cluster), we chose only the stars inside $\pm2.0\sigma$ (where $\sigma$ is the standard deviation in the mean color) from the mean position of the SGB. We subtracted the background stellar contamination estimated from the star counts in the radial zone $r>1.1~r_t$. The BS seem to be more concentrated than the corresponding SGB stars only in the inner $250\div300\arcsec$ (Figure \[figbs\]). At larger distances, the BS distribution becomes less concentrated than the comparison SGB stars. We run a 2-population Kolmogorov-Smirnov test. The test does not give a particularly high statistical significance to the result: the probability that the BS and SGB stars are *not* taken from the the same distribution is 96%. However, this possibility cannot be excluded: see [@Piotto90b] and [@DP93] for a discussion on the limits in applying this statistical test for checking population gradients. Another way to look into the same problem is to investigate the radial trend of the ratio BS/SGB, as plotted in Figure \[figbs\] (*lower panel*). Also in this case the bimodal trend is quite evident. The relative number of BS stars decreases from the center of the cluster to reach a minimum at $r\simeq 250\div300$ arcsec ($r\simeq 2r_c$), and then it rises again. Again, the statistical significance is questionable, in view of the small number of BS at $r>300$ arcsec (6 stars). Nevertheless, this possible bimodality is noteworthy. Indeed, there is a growing body of evidence that the radial distribution of BS stars in GCs might be bimodal, as shown by [@Ferraro97] for or [@Ivo97] and [@Piotto97] for . What makes our result for M55 of some interest is that this distribution has been interpreted in terms of environmental effects on the production of BS stars. However, the fact that M55 has a very low concentration (c=0.8), while M3 and NGC 1851 are high concentration clusters (c=1.85 and c=2.24 respectively, Djorgovski 1993), might make this conclusion at least questionable. Luminosity and mass function \[M55lf\] ====================================== From the CMD we have derived a luminosity function (LF) for the stars of M55. Figure \[M55fdl\] shows the LFs in the different annuli defined in the previous Section (inner, intermediate and outer). The three LFs have been normalized to the star counts of the SGB region in the magnitude interval $15.90<V<17.40$, after subtracting the contribution of the background/foreground stars scaled to the area of each annulus. In the lower part of Figure \[M55fdl\], we show also the LF of the background/foreground stars estimated from the star counts at $r>1.3~r_t$ vertically shifted for clarity. In order to reduce contamination by those stars, all the LFs have been calculated selecting the stars within $2.5\sigma$ (again, $\sigma$ is the standard deviation of the mean color) from the fiducial line of the main sequence of the cluster. The LFs do not include the HB and BS stars. The LF of the background stars has a particular shape: it suddenly drops at $M_V=4.0$. This feature has a natural explanation considering the color-magnitude distribution of the field stars around M55 and the way we selected the stars. The drop in the number of field stars is at the level of the M55 TO and as can be seen in Figure \[M55tot\], or in the lower right panel in Figure \[cmdradial\], the TO of M55 is bluer than the TO of the halo stars of the Galaxy, which are the main components of the field stars towards M55 [@Mand96]. Selecting only stars within $2.5\sigma$ of the fiducial line of M55 will naturally cause such a drop. The completeness correction, as obtained in appendix \[crowd\], has been applied to the stellar counts of each field of M55. As it is possible to see from Table \[M55tab2\], the magnitude limit varies from field to field. We have adopted the same, global, magnitude limit for all the LFs: , that of the fields with the brighter completeness limit (field 16 and 17). This limits all the LFs to $V=20.9$, corresponding to a stellar mass $m\sim0.6\Msun$, for the adopted distance modulus and a standard 15 Gyr isochrone (see next subsection). The data for the inner annuli come from the central image, which has a limiting magnitude of the corresponding LF fainter than the global value adopted here. This is due to the better seeing of the central image compared to all the other images. We adopted a brighter limiting magnitude in order to avoid problems in comparing the different LFs. \[M55fdl\] Figure \[M55fdl\] shows clearly different behaviour of the LFs below the TO: they are similar for the stars above the TO, while the LFs become steeper and steeper from the inner to the outer part of the cluster: this is a clear sign of mass segregation. For the inner LF there is also a possible reversal in slope below $M_V=5.5$. In order to verify that the difference between the three LFs is not due to systematic errors (wrong completeness correction, imperfect combination of data coming from two adjacent fields etc.), we have tested our combining procedure in several ways. In one of our tests we built LFs of two EMMI fields at the same distance from the center of the cluster: , we compared the LF of the field 2 with that of the field 6. After having corrected for the ratio between the covered areas and subtracting the field star contribution, the two LFs were consistent in all the magnitude intervals down to the completeness level of the data (that is lower than the one adopted). Having for field 2 a magnitude limit of 22.2 (see Table \[M55tab2\]) and field 6 a limit of 21.5, we also verified that for the latter our star counts are in correct proportion below the completeness level of 50%. In a second test, we generated two LFs dividing the whole cluster in two octants (dividing along the $45^\circ$ line that runs from the center of the cluster till the field 19  Figure \[M55frames\]). For each of the two slices we generated three LFs in the same radial range as in Figure \[M55fdl\]. After comparing all of them we did not find any significant difference. Therefore the differences among the three LFs in Figure \[M55fdl\] must be real. \[M55mf\] Another source of error in the LF construction is represented by the LF of the field stars. As will be shown in Section \[M55rprof\], M55 has a halo of probably unbound cluster stars. The field star LF constructed from the star counts just outside the cluster can be affected by some contamination of the cluster halo. The consequence is that we might over-subtract stars when subtracting the field LF from the cluster LF, modifying in this way the slope of the mass function (the more affected magnitudes are the faintest ones). To test this possibility, we have extracted background LFs in two different anulii outside the cluster (in terms of $r_t$, $1.0<r\le1.3$ and $r>1.3$). Comparing the two background/foreground LFs we found that the number of stars probably belonging to the cluster but outside the tidal radius must be less than $\sim25\%$ of the adopted field stars in the worst case (the faintest bins). The possible over-subtraction is not a problem for the inner and intermediate LFs, where the number of field stars (after rescaling for the covered area) is always less than $\sim3\%$ of the stars counted in each magnitude bin. For the outer LF, the total contribution of the measured field stars is larger, but it is still less than $25\%$ of the cluster stars (the worst case applies to the faintest magnitude bin): this means that the possible M55 halo star over-subtraction in the field-corrected LF is always less than $6\%$ ($25\% \times 25\%$), negligible for our purposes. Mass function of M55 \[Sm55mf\] ------------------------------- In order to build a mass function for the stars of M55, we needed to adopt a distance modulus and an extinction coefficient. [@Shade88] give $(m-M)_V=14.10$, E$(B-V)=0.14\pm0.02$, while, more recently, [@Mand96] give $(m-M)_V=13.90\pm0.07$, E$(B-V)=0.14\pm0.02$. In the absence of an independent measure made by us, we adopted the values published by [@Mand96] because they are based on the application, with updated data, of the subdwarfs fitting method. Using the LFs of the previous Section we build the corresponding mass functions using the mass-luminosity relation tabulated by [@VDB85] for an isochrone of $Z=3\times10^{-4}$ and an age of 16 Gyr [@Alcaino92]. The MFs for the three radial intervals are presented in Figure \[M55mf\]. The MFs are vertically shifted in order to make their comparison more clear. The MFs are significantly different: the slopes of the MFs increase moving outwards as expected from the effects of the mass segregation and from the LFs of Figure \[M55fdl\]. Figure \[M55mf\] clearly shows that the MF starting from the center out to the outer envelope of the cluster is flat: the index $x$ of the power law, $\xi=\xi_0m^{-(1+x)}$, best fitting the data are: $x=-2.1\pm0.4$, $x=-0.8\pm0.3$, and $x=0.7\pm0.4$ going from the inner to the outer anulii; this means that the slope of the global MF (of all the stars in M55) should be extremely flat. Indeed, the slope of the global mass function obtained from the corresponding LF of all the stars of M55 is: $x=-1.0\pm0.4$ This result agrees with the results of [@Irwin84], while the results of [@Pryor91] appear in contrast to what we have found here. Our MF in the outer radial bin can be compared with the high-mass MF of [@Mand96], obtained from a field located at $\simeq6$ arcmin from the center of M55. As already reported in Section 2, [@Mand96] obtained a deep MF for M55 (down to $M\simeq0.1\Msun$) which they describe with two power laws connected at $M\simeq 0.4\div0.5\Msun$. Their value of $x=0.5\pm0.2$ for the high-mass end of the mass function ($0.5<M/\Msun<0.8$) is in good agreement with our value of $x=0.7\pm0.4$, obtained in the same mass range for the outer radial bin. The low-mass end of the MF by [@Mand96] ($M/\Msun<0.4$) has a slope of $x=1.6\pm0.1$. The level of mass segregation of M55 is comparable to that found in by [@Richer89]. M71 shares with M55 similar structural parameters as well as positional parameters inside the Galaxy. The detailed analysis of [@Richer89] of M71 showed that this cluster should also have a large population of very low mass stars ($\sim0.1$). By fitting a multi-mass isotropic King model [@King66; @GG79] to the observed star density profile of M55, we compared the observed mass segregation effects with the one predicted by the models. Here we give a brief description of our assumptions in order to calculate the mass segregation correction from multi-mass King models. A more detailed description can be found in [@Pryor91], from which we have taken the *recipe*. The main concern in the process of building a multi-mass model is in the adoption of a realistic global MF for the cluster. For M55 we adopted a global MF divided in three parts: - a power-law for the low-mass end, $0.1<M/\Msun\le 0.5$, with a fixed slope of $x=1.6$ (as found by Mandushev   1996); - a power-law for the high-mass end, $0.5<M/\Msun\le m_{TO}$, with a variable slope $x$; - and a power-law for the mass bins of the dark-remnants where to put all the evolved stars with mass above the TO mass, $m_{TO}<M/\Msun\le 8.0$: essentially white dwarfs. Here we adopted a fixed slope of 1.35, The mass of the WDs were set according to the initial-final mass relation of [@W90]. To build the mass segregation curves we varied the MF slope $x$ (the only variable parameter of the models) of the high-mass end stars in the range $-1.0\div1.35$, finding for each slope the model best fitting the radial density profile of the cluster. Then we calculated the radial variation of $x$ for the best-fit models in the same mass range of the observed stars: $0.5<M/M_{\odot}<m_{TO}$. The radial variations of $x$ are compared with the observed MFs in Figure \[M55slx\]. The mass function slopes are shown at the right end of each curve. This plot is similar to those presented in [@Pryor86], and allows one to obtain the value of the global mass function of the cluster. The three observed points follow fairly well the theoretical curves. Also the high-mass MF slope value of [@Mand96] (open circle in Figure \[M55slx\]) is in good agreement with the models and our MFs. From these curves, we have that the slope of the high-mass end of the global MF of M55 is $x\simeq-1.0$, which is in quite good agreement with the global value of the MF found from the global LF of M55 ( previous section). In Figure \[profile\] we show the model which best fits the observed radial density profile for a global mass function with a slope $x=-1.0$. The relatively flat MF of M55 could be the result of the selective loss of main sequence stars, especially from the outer envelope of the cluster, caused by the strong tidal shocks suffered by M55 during its many passages through the Galactic disk and near the Galactic bulge [@Piotto93 for a general discussion of the problem]. A flat MF for M55 agrees well with the results of [@CPS93] who have found that the clusters with a small $R_{GC}$ and/or $Z_{GC}$ show a MF significantly flatter than the cluster in the outer Galactic halo or farther from the Galactic plane. Indeed, M55 is near to the Galactic bulge, $R_{GC}=4.7$ kpc ($R_\odot=8.0$ kpc), and to the Galactic disk $Z_{GC}=-2.0$ kpc. Figure \[m55xrz\] shows that taking into account observing errors, M55 fairly fits into the relation given by [@Manu97], which is a refined version of the one found by [@DPC93]. A different conclusion has been reached by [@Mand96] using their uncorrected (for mass segregation) value for the MF of M55. As noted by the referee, M55 lies further from the average relation defined by the other clusters: of those with a similar abscissa ($0.0\pm0.2$), M55 is the one with the lowest value of $x$. It is not possible to identify the main source of this apparent enhanced mass-loss of M55 compared to the other clusters; a possible cause can be a orbit of the cluster that deeply penetrate into the bulge of the Galaxy. This cannot be confirmed until is performed a reliable measure of the proper motion of M55. Radial density profile from star counts.\[M55rprof\] ==================================================== The CMD allows a unique way to obtain a reliable measure of the radial density profiles of GCs. In fact, the CMD allows us to sort out the stars belonging to the cluster, limiting the problems generated by the presence of the field stars. This also permits to extract radial profiles for distinct stellar masses. We have first created a profile as in [@King68], in order to compare our results with the existing data in the literature. The comparison has been done with the radial density profile of M55 published by [@Pryor91] which includes the visual star counts of King et al.. We could not compare our data with [@Irwin84] because they have not published their observations in tabular form. \[profconf\] Density profile for stars above the TO -------------------------------------- Figure \[profile\] shows the radial density profile for the TO plus SGB stars extracted from the CMD of M55 (from 1 magnitude below the TO to the brightest limit of our photometry). We have selected the stars within $2.5\sigma$ from the fiducial line of the CMD plus the contribution coming from the BS and HB stars; star counts has been limited at the magnitude $V\le18.5$. This relatively bright limit corresponds approximately to the limit of the visual star counts by [@King68] on the plate ED-2134 (in order to make the comparison easier we used the same radial bins of King). Our counts have been transformed to surface brightness and adjusted in zero point to fit the [@Pryor91] profile of M55. The agreement with the data presented by Pryor is good everywhere but in the outer parts where our CCD star counts are clearly above those of [@King68]. This difference is probably due to our better estimate of the background star contamination. In the plot we have shown also the raw star counts (crosses) prior to the background star subtraction: it can be clearly seen that our star counts go well beyond the tidal radius, $r_t=977''$, published by [@Trager95]. This allow us to estimate in a better way than in the past the stellar background contribution. The background star counts show a small radial gradient: we will discuss this point in greater detail in the next Section. Here, the minimum value has been taken as an estimate of the background level. We point out that the differences present in the central zones of the cluster could be in part due to some residual incompleteness of our star counts, to the absence in the starcounts of the brightest saturated stars, and to the difficulties in finding the center of the cluster. We searched for the center using a variant of the mirror autocorrelation technique developed by [@DJ88]. In the case of M55 we encountered some problems due to a surface density which is almost constant inside a radius of $\simeq100\arcsec$. In order to evaluate the structural parameters of M55, we have fitted the profile Figure \[profile\] with a multi-mass isotropic [@King66] model as described in the previous Section. In the following table we show the parameters of the best fitting model and we compare them with the results of [@Trager95], [@Pryor91], and [@Irwin84]: [lccc]{} Author & $c$ & $r_c$ & $r_t$\ This paper & 0.83 & $143''$ & $970''$\ Trager   & 0.76 & $170''$ & $977''$\ Pryor   & 0.80 & $140''$ & $876''$\ Irwin and Trimble & $\sim1.0$ & $\sim120''$ & $\sim1200''$\ The concentration parameter of M55 is one of the smallest known for a globular cluster. Such a small concentration implies strong dynamical evolution and indicates that the cluster is probably in a state of high disgregation [@Aguilar88; @Gnedin97]. Our value of the tidal radius is well in agreement with that of [@Trager95] who used a similar method to fit the data. [@Pryor91] give a value of $r_t$ 10% smaller than ours. We note that Pryor and Trager used the same observational data set. The difference with [@Irwin84] is probably due to the fact that the authors have not fitted their data directly but made only a comparison with a plot of King models. \[extrat\] The density profile for different stellar masses. ------------------------------------------------- Having verified the compatibility of our density profile with previously published ones, we have extracted surface density profiles for different magnitude ranges corresponding to different stellar masses. The adopted magnitude intervals have been chosen to have a significant number of stars in each bin. We used logarithmic radial binning that allows a better sampling of the stars in the outer part of the cluster. In order to lower the noise in the outer part of the profile, we have smoothed the profiles with a median static filter of fixed width of 3 points. We verified that the filtering procedure did not introduce spurious radial gradients in the density profiles. The mean masses in each magnitude bin adopted for the profiles, as obtained from the isochrone by [@VDB85] ( also Section \[Sm55mf\]), are: [cc]{} $V$ & $<m>$\ $<18$ & 0.79\ $18-19$ & 0.77\ $19-20$ & 0.71\ $20-20.9$& 0.63\ The relative profiles, without subtraction of the background stars, are shown in Figure \[profconf\] and \[extrat\]. The arrows in both figures indicate $r_c$, $2~r_c$ and $r_t$. The profiles plotted in Figure \[profconf\] are clearly different from each other: this is as expected from the mass segregation effects. To better compare the profiles, in Figure \[profconf\] they have been normalized in the radial interval $2.6<\log(r/arcsec)<2.9$ (where the profiles have a similar gradient) to the profile of the TO stars. This operation is possible because in this radial range the effects of mass segregation are small ( Figure \[M55slx\]); they are more evident within one core radius. The density profiles are consistent with the mass segregation effects that we have already seen in the mass function of the cluster. The more interesting aspect of the profiles in Figure \[profconf\] is the clear presence of a stellar radial gradient in the star counts of the background field stars. In Figure \[extrat\], we show the radial profiles of the extra cluster stars after normalization of the profiles outside $\log(r/arcsec)=3.0$. The 4 profiles are not exactly coincident outside $r_t$. Let us discuss various possible explanations for this observation: - *Errors in the completeness correction or errors in the star counts.* We repeated the extensive tests on the data made to assess the validity of the mass segregation seen in the LFs. We checked that the variation in the completeness limit of the various EMMI fields does not introduce spurious trends. In a different test, we divided the cluster in two slices along a line at $45^\circ$ from the center of the cluster up to the field 19 (  Figure \[M55frames\]), and built the radial profiles for each of the 4 magnitude bins: in all cases there were no significant differences. The radial profile of the stars in the magnitude range $18\div19$ ($M_V=4.1\div5.1$ in Figure \[M55fdl\]) has the lowest contamination of background stars, as shown by its LF in Figure \[M55fdl\]. - *A non-uniform distribution of the field stars around M55.* It is possible that the field stars around M55 are distributed in a non-uniform way. In the work by [@Grillmair95] it clearly appears that the field stars of some GCs present a non-uniform distribution around the clusters. The gradients are significant and the authors used bidimensional interpolation to the surface density of the field stars to subtract their contribution to the star counts of the clusters. In the present case, field star gradients could be a real possibility, but we cannot test it because we do not have $360^\circ$ coverage of the cluster: our coverage of M55 is only a little more than a quadrant. The Galactic position of M55 ($l\simeq-23^\circ$, $b\simeq9^\circ$) can give some possibility to this option. At this angular distance from the Galactic center the bulge and halo stars probably have a detectable radial gradient. However it remains difficult to explain the existence of the gradient also for the stars in the magnitude range $18\div19$: for them (section \[M55lf\]), as stated before, we have the lowest contamination from the field stars. We have created a surface density map of the starcounts of M55. The map was constructed using all the stars of the $2.5\sigma$-selected sample of our photometry (excluding fields 25 and 35), counting stars in square areas of approximately $9''\times9''$ and then smoothing the resulting map with a gaussian filter. The starcounts are not corrected for crowding but we stopped at $V=20.5$. The map is presented in Figure \[gray\]. The map has the same orientation as Figure \[M55frames\]. We have also overplotted contour levels to help in reading the map. Figure \[gray\] clearly shows that well outside the tidal radius of M55 (located approximately at the center of the map) there is a visible gradient in the star counts. - *A gradient generated by the presence of the dwarf spheroidal in Sagittarius* [@Ibata95]. Between the Galaxy center and M55 there is the dwarf spheroidal galaxy called Sagittarius [@Ibata95]. Sagittarius is interacting strongly with the Galaxy and probably is in the last phases of a tidal destruction by the Galactic bulge. The distance between the supposed tidal limit of this galaxy (using the contour map of Ibata 1995) and M55 is $\sim5^\circ$. In the recent work by [@Mand96] the giant sequence of the Sagittarius appears clearly overlapped with the sequence of M55. This happens only in the magnitude range $V\simeq20.0\div21.0$ where our star counts end. [@Fahlman96] showed that the SGB sequence of the Sagittarius crosses the main sequence of M55 at $V\simeq20.5\div20.7$, and at a corresponding color of $(V-I)\simeq1.1\div1.2$. Similar results were found by [@Mateo96]. This is due to the different distances of these two systems from us: $\sim4.5$ kpc for M55 and $\sim24$ kpc for Sagittarius. This implies that out star counts can be influenced by the stars of the dwarf spheroidal only in our last magnitude bin, $20\div20.9$. Our selection of stars along the CMD of M55 limits the Sagittarius stars to those effectively crossing the main sequence. In conclusion, if effectively the Sagittarius stars are present as background stars we should see them only in one of the 4 profiles, but the coincidence of the 4 profiles excludes this ipothesis. - *A halo of stars escaping from the clusters.* This possibility is more suggestive. The stellar gradient could be a possible extra-tidal extension of the cluster, similar to what [@Grillmair95] found in their sample of 12 clusters. The tidal extension could be caused by the tidal-shocks to which the cluster has been exposed during its perigalactic passages, through the Galactic disk. Another possibility is the creation of the stellar halo by stellar dynamical evaporation from the inner part of the cluster. Such mechanisms work independently of stellar mass [@Aguilar88] and so the stellar halo should have a similar gradient for all the stellar masses as in the present case. Such halos are very similar to the theoretical results obtained by [@OhLin92] and [@Grillmair95], who have obtained tidal tails for globular clusters N-bodies simulations. We believe that the probable explanation for the phenomenon shown in Figures \[profconf\] and \[extrat\] is in the presence of an extra-tidal stellar halo or tidal tail. Doubt resides in the unknown gradient of the background field stars. To resolve this we need to map the whole cluster and a large area surrounding the cluster. This would also allow us to find the exact level of field stars. Our star counts stop at 33($\simeq2\times r_t$), from the center of M55 while the tidal tails of [@Grillmair95] stop at $\simeq 2.5 \div 4\, r_t$. Consequently, we cannot correctly subtract the contribution of the field stars from our star counts. We can give only an estimate of the exponent of the power law, $f\propto r^{-\alpha}$, fitting the profiles at $ r > 1.2\,r_t$. Without subtracting any background counts $\alpha\sim0.7\pm0.3$, while subtracting different levels of background stars the slope varies in the interval $0.7<\alpha<1.7$: the highest value comes out after subtracting the outermost value of the density profiles. When it will be available a better estimate of the background/foreground level of the sky it will be possible to assign a value to the slope of the gradient of stars: actually our range, $\alpha=0.7 \div 1.7$, is in accordance with those found theoretically by [@OhLin92] and observationally by [@Grillmair95]. We are grateful to C.J. Grillmair, the referee, for his careful reading of the manuscript and his suggestions for improving the paper. The authors warmly thank Tad Pryor for making available it’s code for the generation of multi-mass King-Michie models. We thanks I. Saviane for his help in constructing the surface density map of M55. Finally, we warmly thanks Nicola Caon for doing the observations included in this work. 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Ital., 441, 667 Zinn, R., 1980, ApJS, 42, 19 Zoccali, M., Piotto, G., Zaggia, S. R., & Capaccioli, M., 1997, MNRAS, submitted [crcccccc]{} Field&Stars&Airmass& RA & DEC & & V(50%)\ & & && $V$ & $I$ &\ 01 & 13442 & 1.010 & 294.926 & $-30.948$ & 0.9 & 0.9 & 21.0/22.1\ 02 & 3176 & 1.007 & 295.062 & $-30.948$ & 0.9 & 0.9 & 22.2\ 03 & 872 & 1.004 & 295.198 & $-30.948$ & 1.1 & 1.1 & 22.1\ 04 & 495 & 1.005 & 295.333 & $-30.947$ & 1.3 & 1.1 & 21.2\ 06 & 2273 & 1.016 & 294.926 & $-30.831$ & 1.3 & 1.2 & 21.5\ 07 & 800 & 1.013 & 295.061 & $-30.831$ & 1.5 & 1.5 & 21.2\ 08 & 543 & 1.009 & 295.197 & $-30.831$ & 1.4 & 1.4 & 21.2\ 09 & 482 & 1.007 & 295.333 & $-30.831$ & 1.6 & 1.6 & 21.0\ 11 & 635 & 1.022 & 294.926 & $-30.714$ & 1.4 & 1.4 & 21.3\ 12 & 560 & 1.028 & 295.061 & $-30.714$ & 1.4 & 1.4 & 21.2\ 13 & 531 & 1.034 & 295.197 & $-30.714$ & 1.4 & 1.5 & 21.0\ 14 & 506 & 1.041 & 295.333 & $-30.714$ & 1.3 & 1.5 & 21.0\ 16 & 491 & 1.227 & 294.983 & $-30.598$ & 1.3 & 1.4 & 20.9\ 17 & 503 & 1.150 & 295.061 & $-30.598$ & 1.6 & 1.3 & 20.9\ 18 & 537 & 1.055 & 295.197 & $-30.598$ & 1.5 & 1.6 & 21.1\ 19 & 556 & 1.048 & 295.332 & $-30.598$ & 1.3 & 1.5 & 21.2\ 25 & 666 & 1.001 & 295.197 & $-31.063$ & 1.2 & 1.3 & 21.6\ 35 & 757 & 1.001 & 295.333 & $-31.063$ & 1.2 & 1.1 & 21.9\ Image reduction and analysis ============================ The images were reduced using the standard algorithms of bias subtraction, flat fielding and trimming of the overscan, without encountering particular problems. The stellar photometry was done using [DAOPHOT II]{} and [ALLSTAR]{} [@Stetson87]. The second version of [DAOPHOT]{} was particularly useful for the image analysis since we were forced to use a variable point spread function (PSF) through the images. In fact, the stellar images of the EMMI *Red Arm* together with the F/2.5 field camera presented coma aberration at the edges of the field: the resulting PSF was radially elongated. To better interpolate the PSF we also used an analytic function with 5 free parameters ( the [Penny]{} function of [DAOPHOT II]{}). In order to obtain a single CMD for all the stars found in the 18 fields we first obtained the CMD of each field matching the $V$ and I photometry. Then, we combined all the CMDs using the relative zero points determined from the overlapping regions of adjacent fields. All the CMDs were connected to the main CMD, one at a time, following a sequence aimed at maximizing the number of common stars usable for the zero point calculation. The central field CMD was used as the starting point of the combination. For the outer fields we used a minimum of 20 common stars while for the inner fields we had at least 300 stars. The mean error of the zero points was $\simeq0.05$ magnitudes, compatible with the errors calculated from the crowding experiments. Since the night was not photometric, we could not directly calibrate our data. We were only able to set an absolute zero point using the unpublished calibrated photometry of the center of M55 by Piotto (see next Section). In order to perform the photometry of the central field of the cluster, we divided it into 4 subimages of $\simeq600\times600$ pixels, to minimize the effect of the strong stellar gradients present in this image. We allowed a good overlap to be able to perform the successive combination of the photometry of the stars. In this way we also avoided two problems: we had better control of the PSF calculation and we reduced the number of stars per image to be analyzed. Thanks to the low central concentration of the cluster and the fairly good seeing of the images (even if the crowding was not completely absent), we were able to obtain complete photometry down to $V\simeq21$. Calibration of the photometry ============================= In principle, the analysis of the radial density profile does not require calibrated photometry. But this operation is necessary if we want to analyze the stellar population of the cluster, together with its stellar luminosity and mass functions. Since we could not use standards taken during the same night, we have performed a relative calibration using existing photometry of M55. For the $V$ magnitude we linked our data to Piotto’s (1994) un published photometry of the central field of M55 from images taken with the 2.2m ESO telescope. For the (V-I) we calibrated our data against [@Alcaino92] photometry. They published a CCD BVRI photometry for two different non-overlapping fields outside the center of M55, named FA and FB, with dimensions of $3\farcm1\times1\farcm9$, contained in our central field. Figure \[M55P8\] shows our $V$ zero point calculated against Piotto’s (1994) while Figure \[M55Alcv\] shows the $V$ zero point of the two fields FA and FB of [@Alcaino92]. The mean zero point for the two fields of Alcaino  gives $\Delta V_{Alcaino}=6.11\pm0.04$ which compare well with $\Delta V_{Piotto}=6.12\pm0.03$. The two are in good agreement taking into account the errors. There are no magnitude gradients. The LFs are coming from the photometry in the V-band. Before the publication of the I-band photometry by [@Mand96], the one by [@Alcaino92] was the only photometry in the literature. Unfortunately, the M55 data set of [@Mand96] does not overlap with any of our fields: it is centered just few arcmin south of our field 2. In Figure \[M55Alcv\], we show the difference between our data and those of [@Alcaino92]. In this case, the two zero points calculated for Alcaino’s fields differ by a significant amount. We do not know the origin of this discrepancy, which we believe is internal to the data of [@Alcaino92]. They could not resolve this due to the fact that fields FA and FB do not have stars in common. We believe that the problem is not in our data since both Alcaino’s fields are contained in the same subimage of the central field. Lacking other independent (V-I) calibrations, we are forced to adopt as our color zero point the mean of the two values of FA and FB: $\Delta(V-I)=0.45\pm0.05$. Crowding experiments. {#crowd} ===================== For each field, we performed a series of Monte Carlo simulations in order to establish the magnitude limit and the degree of completeness of the CMD. The magnitude limit has been defined as the level at which the completeness function reach a value of $0.5$, V(50%). This value is reported in Table \[M55tab2\] for each field. The procedure followed to generate the artificial stars for the crowding experiments is the standard one [@Piotto90b]. The completeness function used to correct our data is the combination of the results of the experiments in both the $V$ and $I$ images for each field. In the outer fields the stars were added at random positions in a magnitude range starting from $V=19$ (just 0.5 mag below the main sequence TO). In the $I$ band experiments, we used the same star positions of the $V$ experiments, with the $I$ magnitudes set according to the corresponding main sequence color. For each outer field, we performed 10 experiments with 100 stars. For the inner fields (fields number 2, 3 and 6) the experiments were 10 with 100 stars in an interval of only 1 magnitude for 5 different magnitudes (a total of 50 experiments). Moreover, in these fields the stars were added taking into account the radial density profile of the cluster. For the 4 subimages of the central field, we performed independent crowding experiments. For each subimage, we ran 10 experiments in 0.5 mag. steps in the range $19\div23$, with the stars radially distributed as the density profile of the cluster. In this way, we were able to better evaluate the level of the local completeness of the photometry. The completeness function has been calculated for each field taking into account the results of the two different experiments in $V$ and I. As an example in Figure \[M55cro\] we show the completeness functions for field number 3 (top) and 19 (bottom). The results of the experiments were fitted using the error function: $$g(x;y_0,\sigma) = 1 - \int_{-\infty}^{x} e^{\frac{(y-y_0)^2}{\sigma^2}} {\rm d}y. \label{cumgauss}$$ $y_0$ is the magnitude at which the completeness level is 50%, V(50%); $\sigma$ gives the rapidity of the decrease of the incompleteness function and is connected with the read out noise and the crowding of the image. For the star counts correction we used the interpolation with the previous equation instead of using directly the noisy results of the experiments (these were too few to lower the small number statistical noise of the results). In this way we avoid the adding of noise to the star counts. In every case we verified that the fitting function is an acceptable interpolation that gives very low residuals compared to the error distribution function. [^1]: Based on observations made at the European Southern Observatory, La Silla, Chile.

--- abstract: 'We show that any smooth stationary solution of the 3D incompressible Navier-Stokes equations in the whole space, the half space, or a periodic slab must vanish under the condition that for some $0 \le \delta \le 1<L$ and $q=6(3-\delta)/(6-\delta)$, $$\liminf_{R \to \infty} \frac 1R \|u\|^{3-\delta}_{L^{q}(R<|x|<LR)}=0.$$ We also prove sufficient conditions allowing shrinking radii ratio $L= 1+R^{-\alpha}$. Similar results hold on a slab with zero boundary condition by assuming stronger decay rates. We do not assume global bound of the velocity. The key is to estimate the pressure locally in the annuli with radii ratio $L$ arbitrarily close to 1.' author: - 'Tai-Peng Tsai[^1]' date: Dedicated to Hideo Kozono on the occasion of his 60th birthday title: 'Liouville type theorems for stationary Navier-Stokes equations' --- Introduction {#S1} ============ Consider the Liouville problem of 3D stationary incompressible Navier-Stokes equations [$$\begin{split} \label{SNS} - {\Delta}u+(u\cdot {{\nabla}})u+{{\nabla}}p=0, \quad {\mathop{\mathrm{div}}}u=0, \quad \text{in }{{\Omega}}, \end{split}$$]{} where the domain ${{\Omega}}$ is either the whole space ${{\mathbb R }}^3$, the half space ${{\mathbb R }}^3_+$ with zero boundary condition, or the slab ${{\Omega}}={{\mathbb R }}^2\times (0,1)$ with zero or periodic boundary condition (BC). In the classical setting ${{\Omega}}={{\mathbb R }}^3$, one asks if the only $H^1_{{\mathrm{loc}}}$ solution satisfying [$$\begin{split} \label{D.soln} \int_{{\Omega}}|{{\nabla}}u|^2 < \infty, \quad \lim_{|x|\to \infty} u(x)=0 \end{split}$$]{} is zero. A solution satisfying is called a *$D$-solution*. This problem has been reformulated by Seregin and Sverak to whether the only solutions satisfying [$$\begin{split} u \in H^1_{{\mathrm{loc}}}\cap L^\infty({{\Omega}}) \end{split}$$]{} are constant vectors. The same problems can be posed in other domains, and can be asked in the subclass of axisymmetric flows. We now review the literature. In the 2 dimensional case, the problem in the plane ${{\mathbb R }}^2$ is solved by Gilbarg and Weinberger [@GW]. For the 3 dimensional problem, it is not even known if a general D-solution satisfying has any explicit decay rate. The following is a list of vanishing results with extra integral or decay assumptions on the solution. Galdi [@Galdi Theorem X.9.5] proved that if $u$ is a D-solution in ${{\mathbb R }}^3$ and $u \in L^{9/2}({{\mathbb R }}^3)$, then $u = 0$. The same proof works for dimension $n\ge 4$ assuming only without additional integrability condition. This result was improved by a log factor in Chae and Wolf [@CW], assuming $$\int_{{{\mathbb R }}^3}|u|^{\frac 92} {\left\{ \ln (2+1/|u|) \right\}}^{-1}dx<\infty.$$ In [@Chae], Chae proved that a D-solution with ${\Delta}u \in L^{6/5}({{\mathbb R }}^3)$ is zero. Seregin [@Seregin] proved that a solution in ${{\mathbb R }}^3$ is 0 if $u \in L^6({{\mathbb R }}^3) \cap BMO^{-1}$. Kozono, Terasawa and Wakasugi [@KTW] showed that a D-solution $u$ in ${{\mathbb R }}^3$ is zero if either the vorticity decays like $c|x|^{-5/3}$ at infinity, or ${\| u \|}_{L^{9/2,\infty}}\le c$, with $c={\epsilon}{\| {{\nabla}}u \|}_2^{2/3}$ and ${\epsilon}$ a small constant. In [@Seregin2; @CW2], the authors prove Liouville type theorems for smooth solutions $u$ under growth conditions on the $L^s$ mean oscillation over $B_r$ of the *potential tensor* of $u$. Lin, Uhlmann and Wang [@LUW] proved the following lower bound using Carleman estimates for a bounded solution $u$ in ${{\Omega}}= {{\mathbb R }}^3 \setminus \overline B_1$: Let $M(r) = \inf _{|x|=r} \int_{B_1(x)} |u|^2$ and ${\lambda}= {\| u \|}_{W^{1,\infty}({{\Omega}})}$. Then there exist $C({\lambda})>0$ and $R_0({\lambda}, M(10))>10$ such that [$$\begin{split} \label{LUWeq2} M(r) \ge \exp (-C r^2 \log r) , \quad \forall r>R_0. \end{split}$$]{} This result does not assume any boundary condition. It implies that a bounded solution in an exterior domain in ${{\mathbb R }}^3$ must be zero if it satisfies [$$\begin{split} \label{LUWeq3} \liminf_{r \to \infty} \exp (C r^2 \log r)\,M(r)<1, \quad \forall C>0. \end{split}$$]{} It is probably the first Liouville type result with a liminf condition. Let $L^{q,l}$ denote the Lorentz spaces. Seregin and Wang [@SW] prove the vanishing of $u$ assuming either for $3<q<\infty$, $3 \le l \le \infty$ (or $q=l=3$), [$$\begin{split} \label{SW1} \liminf_{R\to \infty} R^{\frac 23 - \frac 3q} {\| u \|}_{L^{q,l}(B_R \setminus B_{R/2})}\le {\epsilon}{\| {{\nabla}}u \|}_2^{2/3}, \end{split}$$]{} with ${\epsilon}$ a small constant, or for $12/5<q<3$, $1 \le l \le \infty$, ${{\gamma}}> \frac 13 + \frac 1q$, [$$\begin{split} \label{SW2} \liminf_{R\to \infty} R^{{{\gamma}}- \frac 3q} {\| u \|}_{L^{q,l}(B_R \setminus B_{R/2})}=0. \end{split}$$]{} They don’t assume the solution is globally bounded. Note that for $q>3$ follows from $q=3$ case as [$$\begin{split} \label{SW3} R^{ - \frac 13} {\| u \|}_{L^{3}(B_R \setminus B_{R/2})} {\lesssim}R^{\frac 23 - \frac 3q} {\| u \|}_{L^{q,l}(B_R \setminus B_{R/2})}. \end{split}$$]{} For other domains, the proof of Galdi [@Galdi Theorem X.9.5] can be extended to ${{\mathbb R }}^3_+$ and slabs easily. We are not aware of any other previous results for the half space. On a slab with zero BC, Pileckas and Specovius-Neugebauer [@Pil; @PiSN] studied the asymptotic decay of solutions. They proved, under certain weighted integral assumption on the velocity $u$ and its derivatives with a force in , $u(x)$ decays like $1/|x|$. Then the vanishing of $u$ with zero force follows easily. This was extended by Carrillo, Pan, Zhang, and Zhao [@CPZZ Theorem 1.1], showing that any D-solution satisfying in a slab with zero BC is zero. There is also a rich literature on the Liouville problem for the subclass of axisymmetric solutions. As we will not discuss it here, we only refer to [@Wang; @CPZZ; @KTW2] and their references. The following is our first main result. \[th1\] Suppose $u\in H^1_{{\mathrm{loc}}}({{\mathbb R }}^3)$ is a weak solution of in ${{\Omega}}={{\mathbb R }}^3$. \(a) If for some constants $0 \le {\delta}\le 1$ and $L>1$, [$$\begin{split} \label{th1eq1} \liminf_{R \to \infty} \frac 1R {\| u \|}^{3-{\delta}}_{L^{q}(R<|x|<LR)}=0, \quad q({\delta})=\frac{3-{\delta}}{1-{\delta}/6}, \end{split}$$]{} then $u=0$. \(b) If for some constants $0 \le {\delta}\le 1$ and ${\alpha}\ge0$, [$$\begin{split} \label{th1eq2} \liminf_{R \to \infty} R^{\beta}{\| u \|}_{L^{q}(R<|x|<R+R^{1-{\alpha}})}=0, \quad q({\delta})=\frac{3-{\delta}}{1-{\delta}/6}, \end{split}$$]{} where ${\beta}={\beta}({\delta},{\alpha})= \max {\left\{ \frac{\frac {3-{\alpha}}{q}-2+3{\alpha}}{2-{\delta}},\ \frac {-1+2{\alpha}}{3-{\delta}} \right\}}$, then $u=0$. Comments on Theorem \[th1\]: 1. Part (a) is a borderline improvement of Seregin and Wang [@SW], by allowing ${{\gamma}}=\frac13+\frac1q$ in . Note that $q$ is decreasing in ${\delta}$ with lower bound $q(1)=12/5$, which is allowed in Theorem \[th1\] but excluded in . 2. Our proof is different: [@SW] is based on a Caccioppoli type inequality for the nonlinear equation, while our proof is based on pressure-independent interior estimates of the Stokes system, see Lemma \[ST\]. 3. As in , and , conditions and use $\liminf$, not limit. We do not assume $u \in L^\infty({{\Omega}})$ nor ${{\nabla}}u \in L^2({{\Omega}})$. The condition $u\in H^1_{{\mathrm{loc}}}$ implies $u \in C^\infty_{\mathrm{loc}}$. Since we do not assume a global bound of $u$, we need to estimate the pressure locally. 4. Unlike , the condition for lower $q$ does not follow from itself for higher $q$ by Hölder inequality. For example, for ${\delta}=1$ and $q=12/5$ is [$$\begin{split} \liminf_{R \to \infty} R^{-1/2} {\| u \|}_{L^{\frac{12}5}(R<|x|<LR)}=0. \end{split}$$]{} It does not follow from for ${\delta}=0$ and $q=3$ [$$\begin{split} \label{eq1.10} \liminf_{R \to \infty} R^{-1/3} {\| u \|}_{L^{3}(R<|x|<LR)}=0. \end{split}$$]{} Condition implies that, for any nonzero $H^1_{\mathrm{loc}}$ solution $u$ in ${{\mathbb R }}^3$ and any $L>1$, there are ${\epsilon}>0$ and $R_0\gg 1$ such that [$$\begin{split} \label{th1eq3} \frac 1R \int_{ R<|x|<LR} |u|^3\, dx\ge {\epsilon},\quad \forall R>R_0. \end{split}$$]{} We have similar lower bounds for other $q$ from . They are in the spirit of . 5. The main feature of part (b) is that the ratio of the outer and the inner radii is shrinking to 1 when ${\alpha}>0$. (It contains part (a) as a special case with ${\alpha}=0$.) To be able to prove it, we need explicit bounds of the *Bogovskii map* in such annuli, see Lemma \[Bog-annulus\]. For the exponent ${\beta}({\delta},{\alpha})$ with $0 \le {\delta}\le 1$ and $0 \le{\alpha}<\infty$, ${\beta}({\delta},{\alpha})= \frac {-1+2{\alpha}}{3-{\delta}}$ if and only if $5{\alpha}{\delta}-24{\alpha}-3{\delta}+6\ge 0$, in particular if $ {\alpha}\le 3/19$. 6. When ${\beta}({\delta},{\alpha})= (\frac {3-{\alpha}}{q}-2+3{\alpha})/(2-{\delta})$, one may get alternative conditions as in Theorem \[th4\] (b), by not applying Hölder inequality to bound ${\| u \|}_{q/2}$ by ${\| u \|}_q$ in . For the following three theorems, we denote a point $x \in {{\Omega}}$ as $x=(x',x_3)$ with $x'\in {{\mathbb R }}^2$. \[thmB\] Let ${{\Omega}}={{\mathbb R }}^3_+=\{(x',x_3)\in {{\mathbb R }}^3:\, x_3>0\}$. Suppose $u\in H^1_{{\mathrm{loc}}}(\overline {{{\Omega}}})$ is a weak solution of in ${{\Omega}}$ with zero boundary condition. \(a) If for some constants $0 \le {\delta}\le 1$ and $L>1$, [$$\begin{split} \label{thmBeq1} \liminf_{R \to \infty} \frac 1R {\| u \|}^{3-{\delta}}_{L^{q}(R<|x|<LR)}=0, \quad q({\delta})=\frac{3-{\delta}}{1-{\delta}/6}, \end{split}$$]{} then $u=0$. \(b) If for some constants $0 \le {\delta}\le 1$ and ${\alpha}\ge0$, [$$\begin{split} \label{thmBeq2} \liminf_{R \to \infty} R^{\beta}{\| u \|}_{L^{q}(R<|x|<R+R^{1-{\alpha}})}=0, \quad q({\delta})=\frac{3-{\delta}}{1-{\delta}/6}, \end{split}$$]{} where ${\beta}={\beta}({\delta},{\alpha})= \max {\left\{ \frac{\frac {3-{\alpha}}{q}-2+3{\alpha}}{2-{\delta}},\ \frac {-1+2{\alpha}}{3-{\delta}} \right\}}$, then $u=0$. Comments on Theorem \[thmB\]: 1. The statement and proof of Theorem \[thmB\] are similar to those of Theorem \[th1\], but we also need to estimate ${{\nabla}}u$ and $p$ on the boundary without pressure assumption. 2. Conditions and use $\liminf$, not limit. We do not assume $u \in L^\infty({{\Omega}})$ nor ${{\nabla}}u \in L^2({{\Omega}})$. The condition $u\in H^1_{{\mathrm{loc}}}(\overline {{\mathbb R }}^3_+)$ implies $u \in C^\infty_{\mathrm{loc}}(\overline {{\mathbb R }}^3_+)$, see [@Kang]. \[thmC\] Let ${{\Omega}}= {{\mathbb R }}^2 \times ({{\mathbb R }}/{{\mathbb Z}})$. We denote a point $x \in {{\Omega}}$ as $x=(x',x_3)$ with $(x',0)=(x',1)$. Suppose $u\in H^1_{{\mathrm{loc}}}({{\Omega}})$ is a weak solution of in ${{\Omega}}$. \(a) If for some constants $0 \le {\delta}\le 1$ and $L>1$, [$$\begin{split} \label{thmCeq1} \liminf_{R \to \infty} \frac 1R {\| u \|}^{3-{\delta}}_{L^{q}(R<|x'|<LR)}=0, \quad q({\delta})=\frac{3-{\delta}}{1-{\delta}/6}, \end{split}$$]{} then $u=0$. \(b) If for some constants $0 \le {\delta}\le 1$ and ${\alpha}\ge0$, [$$\begin{split} \label{thmCeq2} \liminf_{R \to \infty} R^{\tilde{\beta}} {\| u \|}_{L^{q}(R<|x'|<R+R^{1-{\alpha}})}=0, \quad q({\delta})=\frac{3-{\delta}}{1-{\delta}/6}, \end{split}$$]{} where $\tilde{\beta}= {\beta}_{ps}({\delta},{\alpha})= \max {\left\{ \frac{\frac {2-{\alpha}}{q}-2+3{\alpha}}{2-{\delta}},\ \frac {-1+2{\alpha}}{3-{\delta}} \right\}}$, then $u=0$. Comments on Theorem \[thmC\]: 1. In Theorem \[thmC\], conditions and use $\liminf$, not limit. We do not assume $u \in L^\infty({{\Omega}})$ nor ${{\nabla}}u \in L^2({{\Omega}})$. The condition $u\in H^1_{{\mathrm{loc}}}({{\Omega}})$ implies $u \in C^\infty_{\mathrm{loc}}({{\Omega}})$. 2. Note ${\beta}_{ps}({\delta},{\alpha})$ differs from ${\beta}({\delta},{\alpha})$ in Theorems \[th1\] and \[thmB\] in that the numerator $3-{\alpha}$ is replaced by $2-{\alpha}$. \[th4\] Let ${{\Omega}}= {{\mathbb R }}^2 \times (0,1)$. Suppose $u\in H^1_{{\mathrm{loc}}}(\overline {{\Omega}})$ is a weak solution of in ${{\Omega}}$ with zero boundary condition $u(x',0)=u(x',1)=0$. \(a) If for some constants $0 \le {\delta}\le 1$ and $L>1$, [$$\begin{split} \label{th4eq1} \liminf_{R \to \infty} R^{2/q} {\| u \|}^{2-{\delta}}_{L^{q}(R<|x'|<LR)}\to 0, \end{split}$$]{} where $q=q({\delta})=\frac{3-{\delta}}{1-{\delta}/6}$, then $u=0$. \(b) If for some constants $6/5 \le r \le 2$ and $L>1$, [$$\begin{split} \label{th4eq1b} \liminf_{R \to \infty} \int_0^1\int_{ R<|x'|<LR} {\left( |u|^r+ |u|^{2r} \right)}\, dx'\,dx_3=0, \end{split}$$]{} then $u=0$. Comments on Theorem \[th4\]: 1. In Theorem \[th4\] the conditions and are $\liminf$, not limit. We do not assume $u \in L^\infty({{\Omega}})$ nor ${{\nabla}}u \in L^2({{\Omega}})$. The condition $u\in H^1_{{\mathrm{loc}}}(\overline {{\Omega}})$ implies $u \in C^\infty_{\mathrm{loc}}(\overline{{\Omega}})$; see [@Kang]. 2. Its proof is different from those for Theorems \[th1\]-\[thmC\] as we cannot obtain the local pressure estimate by scaling, and we get an additional $R$ factor. As a result, we have a positive exponent for $R$ in . Moreover, we cannot vary the radii ratio $L$. 3. A D-solution satisfying in a slab with zero BC is shown to be zero by [@CPZZ Theorem 1.1]. It is extended by Theorem \[th4\] since implies : By Poincaré inequality in $x_3$ direction and zero BC, for $A_R=\{x'\in {{\mathbb R }}^2: R<|x'|<LR\}$, $$\int_0^1 \int _{A_R} |u|^2 dx \le C\int_0^1 \int _{A_R} |{{\partial}}_{x_3} u|^2 dx $$ which vanishes as $R \to \infty$ by . By regularity theory, and zero BC imply $u \in L^\infty({{\Omega}})$. We have $\int_0^1 \int _{A_R} |u|^4 dx \le {\| u \|}_{L^\infty({{\Omega}})}^2 \int_0^1 \int _{A_R} |u|^2 dx=o(1)$. 4. It is possible to prove $u=0$ assuming $\liminf_{R\to \infty}\int_0^1\int_{A_R} (|u|^r + |u|^s)=0$ with $s>2r$, by modifying the proof of part (b). We skip it to keep the presentation simple. The key to the above theorems is the estimate of the pressure $$\inf_{c\in {{\mathbb R }}} {\| p-c \|}_{L^q(E)}$$ in an annulus-like region $E$, based on integral bounds of $u$ in a slightly larger region. Here $E = B_{LR} \setminus B_R$ for ${{\Omega}}={{\mathbb R }}^3$, $E = B_{LR}^+ \setminus B_R^+$ for ${{\Omega}}={{\mathbb R }}^3_+$, $E = (B_{LR}' \setminus B_R')\times (0,R)$ for a periodic slab, and $E = (B_{LR}' \setminus B_R')\times (0,1)$ for a zero BC slab. Here $B_R$ is the ball in ${{\mathbb R }}^3$ of radius $R$ centered at the origin, $B_R^+= B_R\cap {{\mathbb R }}^3_+$, while $B_R'$ is a ball in ${{\mathbb R }}^2$. These estimates are based on Lemma \[p-est\] and the estimates of the corresponding Bogovskii maps, Lemmas \[Bog-annulus\], \[Bog-cylinder\] and \[CPZZ\]. After we prove these lemmas in §\[S2\], we will prove Theorem \[th1\] in §\[S3\], Theorem \[thmB\] in §\[S4\], Theorem \[thmC\] in §\[S5\], and Theorem \[th4\] in §\[S6\]. Bogovskii map and pressure estimate {#S2} =================================== We first recall the Bogovskii map (see [@Galdi Lemma III.3.1] and [@Tsai-book §2.8]). For a domain $E \subset {{\mathbb R }}^n$, denote $$L^q_0(E)=\{ f \in L^q(E):\ \textstyle {\int_E} f =0\}.$$ \[Bogovskii\] Let $E$ be a bounded Lipschitz domain in ${{\mathbb R }}^n$, $2 \le n \in {{\mathbb N}}$. Let $1<q<\infty$. There is a linear map $${\mathop{\mathrm{Bog}}\nolimits}: L^q_0(E) \to W^{1,q}_0(E;{{\mathbb R }}^n),$$ such that for any $f\in L^q_0(E)$, $ v = {\mathop{\mathrm{Bog}}\nolimits}f$ is a vector field that satisfies $$v\in W^{1,q}_0(E)^n,\quad {\mathop{\mathrm{div}}}v = f, \quad {\| {{\nabla}}v \|}_{L^q(E)} \le {C_{\mathrm{bg}}}(E,q) {\| f \|}_{L^q(E)},$$ where the constant ${C_{\mathrm{bg}}}$ does not depend on $f$. If $RE = \{ Rx: x\in E\}$, then ${C_{\mathrm{bg}}}(RE,q)={C_{\mathrm{bg}}}(E,q)$. This map is non-unique and we usually fix a choice that almost minimizes the constant ${C_{\mathrm{bg}}}$. Strictly speaking ${C_{\mathrm{bg}}}$ depends on this choice. The last statement ${C_{\mathrm{bg}}}(RE,q)={C_{\mathrm{bg}}}(E,q)$ is because for given ${\mathop{\mathrm{Bog}}\nolimits}$ defined on $E$, we can define ${\mathop{\mathrm{Bog}}\nolimits}_R$ on $RE$ as follows: For $\bar f \in L^q_0(RE)$, let $f(x) = \bar f(Rx)$ for $x\in E$, $v={\mathop{\mathrm{Bog}}\nolimits}f \in W^{1,q}_0(E)$, and $\bar v = {\mathop{\mathrm{Bog}}\nolimits}_R \bar f$ is given by $\bar v(y) = Rv(R^{-1} y)$. The constant ${C_{\mathrm{bg}}}$ appears in the following pressure estimate. \[p-est\] Let $E$ be a bounded Lipschitz domain in ${{\mathbb R }}^n$. Let $p \in L^q(E)$, $1<q<\infty$. Then [$$\begin{split} \label{pest-eq1} {\| p - (p)_{E} \|}_{L^q(E)} \le C_0 \sup_{ \zeta \in W^{1,q'}_0(E),\ {\| {{\nabla}}\zeta \|}_{L^{q'}(E)}=1} \int p {\mathop{\mathrm{div}}}\zeta \end{split}$$]{} where $(p)_E = \frac 1{|E|}\int _E p$ and $C_0 = 2 {C_{\mathrm{bg}}}(E,q')$. Eq.  is used in [@Sverak-Tsai] to prove Lemma \[ST\] below. Its proof follows that of [@Galdi Lemma IV.1.1] although stated differently, and is given here for completeness and to specify the constant. We may replace $p$ by $p-c$ in and hence we may assume $(p)_E=0$. Let $g=|p|^{q-2}p - (|p|^{q-2}p)_E$. Then $$\int_Eg=0, \quad {\| g \|}_{L^{q'}(E)} \le 2 {\| p \|}_{L^q(E)}^{q-1}.$$ By Lemma \[Bogovskii\], there is a solution $w \in W^{1,q'}_0(E)^n$ of $${\mathop{\mathrm{div}}}w = g, \quad {\| {{\nabla}}w \|}_{L^{q'}(E)} \le C_1{\| g \|}_{L^{q'}(E)},$$ where $C_1={C_{\mathrm{bg}}}(E,q')$. Denote $N=\sup_{ \zeta \in W^{1,q'}_0(E),\ {\| {{\nabla}}\zeta \|}_{L^{q'}(E)}=1} \int p {\mathop{\mathrm{div}}}\zeta$. Using $(p)_E=0$, $$\int_E |p|^q = \int_E pg = \int_E p {\mathop{\mathrm{div}}}w \le N {\| {{\nabla}}w \|}_{L^{q'}(E)}\le N C_1{\| g \|}_{L^{q'}(E)} \le NC_1 2 {\| p \|}_{L^q(E)}^{q-1}.$$ Thus $ {\| p \|}_{L^q(E)} \le 2NC_1$. We next construct a Bogovskii map on an annulus or a half-annulus. It is inspired by [@CPZZ Proposition 2.1] on a thin disk. See Lemma \[CPZZ\] for a variation. \[Bog-annulus\] Let $R>0$, $1<L<\infty$ and $A=B_{LR} \setminus \overline B_R$ or $A=B_{LR}^+ \setminus \overline B_R^+$ be an annulus or a half-annulus in ${{\mathbb R }}^3$. There is a linear Bogovskii map ${\mathop{\mathrm{Bog}}\nolimits}$ that maps a scalar function $f\in L^q_0(A)$, $1<q<\infty$, to a vector field $v={\mathop{\mathrm{Bog}}\nolimits}f\in W^{1,q}_0(A)$ and $${\mathop{\mathrm{div}}}v = f, \quad {\| {{\nabla}}v \|}_{L^q(A)} \le \frac{C_q}{(L-1)L^{1-1/q}}\, {\| f \|}_{L^q(A)}.$$ The constant $C_q$ is independent of $L$ and $R$. For our applications, $1<L\le 2$ and the constant becomes ${\displaystyle}\frac{C_q}{L-1}$. The construction below is the same for an annulus or a half-annulus. Since the Bogovskii map can be defined by rescaling with the same bound, we may assume $R=1$. We use spherical coordinates $\rho, \phi, {\theta}$ with $x=\rho(\sin \phi \cos {\theta}, \sin \phi \sin {\theta}, \cos \phi)$, and write a vector field $v$ as $$v=v_\rho(\rho, \phi, {\theta}) e_\rho + v_\phi(\rho, \phi, {\theta}) e_\phi + v_{\theta}(\rho, \phi, {\theta}) e_{\theta}.$$ Recall in spherical coordinates, $${\mathop{\mathrm{div}}}v = \frac1{\rho^2} {{\partial}}_\rho (\rho^2 v_\rho) + \frac1{\rho\sin \phi} {{\partial}}_\phi (\sin \phi \, v_\phi) + \frac1{\rho\sin \phi} {{\partial}}_{\theta}v_{\theta}.$$ For given $L\in (1,\infty)$, define $a\in (0,\infty)$ by $$L^2 = 3 a^2+1.$$ We define a new radial variable $\tau \in [1,2]$ by $$\tau = \frac 1a \sqrt{\rho^2 + a^2-1},\quad a^2\tau^2 = \rho^2+a^2-1, \quad \frac {d\tau}{d\rho} = \frac{\rho}{a^2\tau}.$$ It is increasing in $\rho \in [1,L]$, $\tau(\rho=1)=1$ and $\tau(\rho=L)=2$. Let $A_0=B_2 \setminus \overline B_1$ if $A=B_{L} \setminus \overline B_1$, or $A_0=B_2^+ \setminus \overline B_1^+$ if $A=B_{L}^+ \setminus \overline B_1^+$. For $f(\rho, \phi, {\theta})$ defined on $A$, we define $\bar f(\tau,\phi,{\theta})$ on $A_0$ by $$\bar f(\tau,\phi,{\theta}) = f(\rho, \phi, {\theta}).$$ We have [$$\begin{split} \int_{A_0} |\bar f|^q &= \int _1^{2} \int _0^{2\pi} \int _0^{\pi} |\bar f(\tau,\phi,{\theta})|^q \tau^2 \sin \phi\, d\phi\, d{\theta}\, d\tau \\ &= \int _1^{L} \int _0^{2\pi} \int _0^{\pi} | f(\rho,\phi,{\theta})|^q \tau^2 \sin \phi\, \frac{\rho}{a^2\tau}\, d\phi\, d{\theta}\, d\rho. \end{split}$$]{} (We replace $\int_0^\pi$ by $\int_0^{\pi/2}$ if $A_0 \subset {{\mathbb R }}^3_+$.) Thus [$$\begin{split} \label{eqB1} \int_{A_0} |\bar f|^q =\int_{A} | f|^q \frac{\tau}{a^2 \rho }. \end{split}$$]{} Fix one Bogovskii map ${\mathop{\mathrm{Bog}}\nolimits}_0$ for the domain $A_0$. Let $$\bar v= {\mathop{\mathrm{Bog}}\nolimits}_0 {\left( \frac \rho \tau \bar f \right)} =\bar v_\tau e_\tau +\bar v_\phi e_\phi + \bar v_{\theta}e_{\theta},$$ and define $v$ in $A$ from $\bar v$ by $$v_\rho(\rho, \phi, {\theta}) = \frac {a^2 \tau^2}{\rho^2} \bar v_\tau (\tau, \phi, {\theta}),\quad v_\phi(\rho, \phi, {\theta}) = \bar v_\phi (\tau, \phi, {\theta}),\quad v_{\theta}(\rho, \phi, {\theta}) = \bar v_{\theta}(\tau, \phi, {\theta}).$$ We have [$$\begin{split} {\mathop{\mathrm{div}}}v &= \frac1{\rho^2} {{\partial}}_\rho (\rho^2 v_\rho) + \frac1{\rho\sin \phi} {{\partial}}_\phi (\sin \phi \, v_\phi) + \frac1{\rho\sin \phi} {{\partial}}_{\theta}v_{\theta}\\ &= \frac1{\rho^2} \frac{d\tau}{d\rho} {{\partial}}_\tau (a^2 \tau^2 \bar v_\tau) + \frac1{\rho\sin \phi} {{\partial}}_\phi (\sin \phi \, \bar v_\phi) + \frac1{\rho\sin \phi} {{\partial}}_{\theta}\bar v_{\theta}\end{split}$$]{} Using $\frac {d\tau}{d\rho} = \frac{\rho}{a^2\tau}$, we get $${\mathop{\mathrm{div}}}v = \frac \tau \rho {\mathop{\mathrm{div}}}\bar v = \bar f = f.$$ Thus the composition $f \to \bar f \to \bar v \to v$ gives our desired Bogovskii map. Note that ${{\partial}}_\rho v_* = \frac{\rho}{a^2\tau} {{\partial}}_\tau \bar v_*$ for $*=\phi,{\theta}$. By , if $1<L< 10$, $$\int_{A} |{{\nabla}}v|^q {\lesssim}a^{2-2q} \int _{A_0} |{{\nabla}}\bar v|^q {\lesssim}a^{2-2q} \int _{A_0} | \bar f|^q {\lesssim}a^{-2q} \int _{A} | f|^q .$$ If $10 \le L < \infty$, $$\int_{A} |{{\nabla}}v|^q {\lesssim}a^{2-2q} L \int _{A_0} |{{\nabla}}\bar v|^q {\lesssim}a^{2-2q}L \int _{A_0} | \bar f|^q {\lesssim}a^{-2q} L\int _{A} | f|^q .$$ These show our desired bound, noting $a^{-2}L^{1/q} = 3(L^2-1)^{-1}L^{1/q}$. We next construct a Bogovskii map on a region enclosed by cylinders. \[Bog-cylinder\] Let $R>0$, $1<L<10$ and $E={\left( B_{LR}' \setminus \overline B_R' \right)}\times (0,R)$ where $B_R'={\left\{ x' \in {{\mathbb R }}^2: |x'|<R \right\}}$ denotes balls in ${{\mathbb R }}^2$. There is a linear Bogovskii map ${\mathop{\mathrm{Bog}}\nolimits}$ that maps a scalar function $f\in L^q_0(E)$, $1<q<\infty$, to a vector field $v={\mathop{\mathrm{Bog}}\nolimits}f\in W^{1,q}_0(E)$ and $${\mathop{\mathrm{div}}}v = f, \quad {\| {{\nabla}}v \|}_{L^q(E)} \le \frac{C_q}{L-1}\, {\| f \|}_{L^q(E)}.$$ The constant $C_q$ is independent of $L$ and $R$. The upper bound $L<10$ is only for convenience and can be changed. The construction below can be carried over to a 2D annulus $B_{LR}' \setminus \overline B_R'$. Since the Bogovskii map can be defined by rescaling with the same bound, we may assume $R=1$. We use cylindrical coordinates $r, {\theta},z$ with $x=(r \cos {\theta}, r \sin {\theta}, z)$, and write a vector field $v$ as $$v=v_r(r, {\theta}, z) e_\rho + v_{\theta}(r, {\theta}, z) e_{\theta}+ v_z(r, {\theta}, z) e_z .$$ Recall in cylindrical coordinates, $${\mathop{\mathrm{div}}}v = \frac1r v_r + {{\partial}}_r v_r + \frac 1r {{\partial}}_{\theta}v_{\theta}+ {{\partial}}_z v_z .$$ We will define a new radial variable $\tau \in [1,2]$ which is increasing in $ r \in [1,L]$, $\tau( r=1)=1$ and $\tau( r=L)=2$. For $f(r, {\theta}, z)$ defined on $E$, we define $\bar f(\tau,{\theta},z)$ on $E_0=(B_2' \setminus \overline B_1')\times (0,1)$ by $$\bar f(\tau,{\theta},z) = f(r, {\theta}, z).$$ We have [$$\begin{split} \label{eqB2} dE_0 = \tau d\tau\, d{\theta}\, dz = \frac{ \tau}{r}\frac{d \tau}{dr}\, r\, d r\, d{\theta}\, dz = \frac{ \tau}{r}\frac{d \tau}{dr} dE. \end{split}$$]{} Fix one Bogovskii map ${\mathop{\mathrm{Bog}}\nolimits}_0$ for the domain $E_0$. Let $$\bar v= {\mathop{\mathrm{Bog}}\nolimits}_0 {\left( \frac r \tau \bar f \right)} =\bar v_\tau e_\tau +\bar v_{\theta}e_{\theta}+ \bar v_z e_z ,$$ and define $v$ in $E$ from $\bar v$ by $$v_ r(r, {\theta}, z) = A(\tau) \bar v_\tau (\tau, {\theta}, z),\quad v_{\theta}(r, {\theta}, z) =B(\tau) \bar v_{\theta}(\tau, {\theta}, z),\quad v_z(r, {\theta}, z) = C(\tau) \bar v_z (\tau, {\theta}, z).$$ We want to choose $A,B,C$ suitably so that ${\mathop{\mathrm{div}}}v = D(\tau) {\mathop{\mathrm{div}}}\bar v$, i.e., [$$\begin{split} {\mathop{\mathrm{div}}}v &= ( \frac1rA +\frac {d\tau}{dr}{{\partial}}_\tau A) \bar v_\tau + \frac {d\tau}{dr} A {{\partial}}_\tau \bar v_\tau + B\frac 1r {{\partial}}_{\theta}v_{\theta}+ C{{\partial}}_z v_z \\ &= D {\left\{ \frac1\tau \bar v_\tau + {{\partial}}_\tau \bar v_\tau + \frac 1\tau {{\partial}}_{\theta}\bar v_{\theta}+ {{\partial}}_z \bar v_z \right\}}. \end{split}$$]{} Thus [$$\begin{split} \label{eqB3} D = \tau ( \frac1rA +\frac {d\tau}{dr}{{\partial}}_\tau A) = \frac {d\tau}{dr} A = \frac \tau r B = C. \end{split}$$]{} The second equality gives $$\frac 1r \frac {dr}{d\tau} + \frac 1A \frac {dA}{d\tau} = \frac {1}{\tau}.$$ Integration gives $\ln r + \ln A = \ln \tau + \ln k$ for some $k>0$, or $A= \frac{k \tau}r$. We can then solve the rest of to get $$A= \frac{k \tau}r, \quad B=k \frac {d\tau}{dr} ,\quad C=D= \frac{k \tau}r \frac {d\tau}{dr}.$$ We now choose for convenience $$\tau = 1+ \frac{r-1}{L-1}, \quad \frac {d\tau}{dr}= \frac 1{L-1}, \quad k=L-1,$$ so that $$A= \frac{k \tau}r,\quad B=1, \quad C=D=\frac{\tau}r .$$ We get $${\mathop{\mathrm{div}}}v = \frac \tau r {\mathop{\mathrm{div}}}\bar v = \bar f = f.$$ Thus the composition $f \to \bar f \to \bar v \to v$ gives our desired Bogovskii map. For $1<L< 10$, $B,C,D$ are of order $O(1)$ while $A=O(k)$. Hence $$|{{\partial}}_{\theta}v| + |{{\partial}}_z v|{\lesssim}|{{\partial}}_{\theta}\bar v| + |{{\partial}}_z \bar v|.$$ Note that ${{\partial}}_ r = \frac{ 1}{k} {{\partial}}_\tau$. Also note that $ {{\partial}}_r (\tau/r) = \frac {2-L}{(L-1)r^2}$. Thus $|{{\partial}}_r C| \le C/k$, $|{{\partial}}_r A| \le C$, and $$|{{\partial}}_r v| {\lesssim}k^{-1} ( |{{\partial}}_\tau \bar v| + |\bar v|).$$ By , $$\int_{E} |{{\nabla}}v|^q {\lesssim}k^{1-q} \int _{E_0} |{{\nabla}}\bar v|^q + | \bar v|^q {\lesssim}k^{1-q} \int _{E_0} | \bar f|^q {\lesssim}k^{-q} \int _{E} | f|^q .$$ This shows our desired bound. Whole space {#S3} =========== In this section ${{\Omega}}={{\mathbb R }}^3$ and we will prove Theorem \[th1\]. Denote $B_R=B_R(0)\subset {{\mathbb R }}^3$. We will use the following interior estimate of [@Sverak-Tsai]. See [@Kang Theorem 3.8], [@Galdi Remark IV.4.2] (not in [@Galdi93a]) and [@Tsai-book §2.6] for alternative proofs. The key is that the pressure is not needed on the right side of . \[ST\] If $(u,p)$ solves [$$\label{Stokes}\begin{split} -{\Delta}u + {{\nabla}}p = {\mathop{\mathrm{div}}}F, \quad {\mathop{\mathrm{div}}}u=0 \end{split}$$]{} in $B_{2R}\subset {{\mathbb R }}^3$, then for $1<q<\infty$ and $1\le m \le \infty$, [$$\begin{split} \label{ST-eq1} {\| {{\nabla}}u \|}_{L^q(B_{R})} + {\| p - (p)_{B_{R}} \|}_{L^q(B_{R})} \le C R^{\frac 3q-\frac 3m -1}\,{\| u \|}_{L^{m}(B_{2R})} + C {\| F \|}_{L^q(B_{2R})}, \end{split}$$]{} where $C=C(q,m)$ does not depend on $R$. We extend the above to an annulus. \[int-annulus\] Let $R>0$, $1<L\le 2$, and ${\sigma}=\frac 18(L-1)$. Denote the annuli in ${{\mathbb R }}^3$ $$A_R = B_{(L-2{\sigma})R} \setminus \overline B_{(1+2{\sigma})R}, \quad {\widehat{\mathbf A}}_R = B_{LR} \setminus \overline B_R.$$ If $(u,p)$ solves in ${\widehat{\mathbf A}}_R$, then for $1<q<\infty$ [$$\begin{split} \label{ST-eq2} {\| {{\nabla}}u \|}_{L^q(A_{R})} + {\sigma}{\| p - (p)_{A_{R}} \|}_{L^q(A_{R})} \le \frac C{{\sigma}R}\,{\| u \|}_{L^{q}({\widehat{\mathbf A}}_R)} + C {\| F \|}_{L^q({\widehat{\mathbf A}}_R)}, \end{split}$$]{} where $C=C(q)$ is uniform in $R>0$ and ${\sigma}\in (0,\frac 18]$. Note that the ${\sigma}$ factor appears in both sides of . We may assume $R=1$ by scaling. There are $N=N(L)$ points $x_j\in A_1$, $j=1,\ldots,N$, such that $$A_1 \subset \cup_{j=1}^N B_{{\sigma}}(x_j),$$ and there is a ${\sigma}$-independent upper bound for the number of overlapping of $ B_{{\sigma}}(x_j)$ for ${\sigma}\in (0,\frac18]$. Note $\cup_{j=1}^N B_{2{\sigma}}(x_j)\subset{\widehat{\mathbf A}}_1$. By Lemma \[ST\] with given $q>1$, $m=q$, and $R={\sigma}$, $$\int_{B_{{\sigma}}(x_j)} |{{\nabla}}u|^q {\lesssim}\int_{B_{2{\sigma}}(x_j)} {\left( \frac 1{{\sigma}^q} |u|^q + |F|^q \right)}. $$ Summing in $j$, $$\int_{A_1} |{{\nabla}}u|^q \le \sum_{j=1}^N \int_{B_{{\sigma}}(x_j)} |{{\nabla}}u|^q {\lesssim}\sum_{j=1}^N \int_{B_{2{\sigma}}(x_j)} {\left( \frac 1{{\sigma}^q} |u|^q + |F|^q \right)} {\lesssim}\int_{{\widehat{\mathbf A}}_1} {\left( \frac 1{{\sigma}^q} |u|^q + |F|^q \right)}.$$ This shows the estimate of ${\| {{\nabla}}u \|}_{L^q}$ in . Apply Lemma \[p-est\] to $E=A_1$, $${\| p-(p)_{A_1} \|}_{L^q(A_1)} \le C_0 \sup_{ \zeta \in W^{1,q'}_0(A_1),\ {\| {{\nabla}}\zeta \|}_{L^{q'}(A_1)}=1} \int p {\mathop{\mathrm{div}}}\zeta$$ where $C_0=2{C_{\mathrm{bg}}}(A_1,q')$. By Lemma \[Bog-annulus\], $C_0 \le C/{\sigma}$. Using the weak form of , [$$\begin{split} {\| p-(p)_{A_1} \|}_{L^q(A_1)} &\le \frac C{\sigma}\sup_{ \zeta \in W^{1,q'}_0(A_1),\ {\| {{\nabla}}\zeta \|}_{L^{q'}(A_1)}=1} \int ({{\nabla}}u+F) : {{\nabla}}\zeta \\ & \le \frac C{\sigma}{\left( {\| {{\nabla}}u \|}_{L^q(A_1)} + {\| F \|}_{L^q(A_1)} \right)}. \end{split}$$]{} This completes the proof of . By the standard regularity theory, the solution $u$ of is smooth, $u\in C^\infty_{\mathrm{loc}}$. For any scalar function $\phi \in C^\infty_c({{\mathbb R }}^3)$, we have the local energy equality [$$\begin{split} \label{LEE} \int |{{\nabla}}u|^2\phi = \frac 12 \int| u|^2 {\Delta}\phi + \frac 12 \int |u|^2 u\cdot {{\nabla}}\phi + \int (p-c) u\cdot {{\nabla}}\phi \end{split}$$]{} by testing with $u\phi$. Above $c$ is any constant. Choosing $\phi = \zeta^2$ in , $\zeta \in C^\infty_c$, we get [$$\begin{split} \label{LEE2} \int |{{\nabla}}(u\zeta)|^2 &= \int |u|^2 |{{\nabla}}\zeta |^2 + \int |u|^2 u\zeta\cdot {{\nabla}}\zeta + 2 \int (p-c) u\zeta \cdot {{\nabla}}\zeta \\ &=I_1+I_2+I_3. \end{split}$$]{} Fix $\Theta \in C^\infty({{\mathbb R }})$, $\Theta(t)=1$ for $t<0$, and $\Theta(t)=0$ for $t>1$. We now let $\zeta(x) = \Theta {\left( \frac {|x|-(L+2{\sigma})R}{4{\sigma}R} \right)}$. Then $\zeta\in C^\infty_c({{\mathbb R }}^3)$, $\zeta(x)=1$ for $|x|< R(1+2{\sigma})$, $\zeta(x)=0$ for $|x|> R(1+6{\sigma})$, and $|{{\nabla}}^k \zeta| \le C({\sigma}R)^{-k}$ for $k \in {{\mathbb N}}$. We have [$$\begin{split} |I_1|{\lesssim}({\sigma}R)^{-2} \int_{A_R} |u|^2 {\lesssim}({\sigma}R)^{-2} |A_R|^{1-2/q} {\left( \int_{A_R} |u|^q \right)}^{2/q} {\lesssim}{\left( {\sigma}^{-\frac 12-\frac1{q}} R^{\frac 12-\frac3{q}} {\| u \|}_{L^{q}({\widehat{\mathbf A}}_R)} \right)}^2. \end{split}$$]{} By Hölder and Sobolev inequalities, for $0 \le {\delta}\le 1$, [$$\begin{split} |I_2+I_3| &\le C{\| {{\nabla}}\zeta \|}_\infty \cdot {\| u\zeta \|}_{6}^{\delta}\cdot {\| |u|^{3-{\delta}}+|p-c|\cdot|u|^{1-{\delta}} \|}_{\frac1{1-{\delta}/6},A_R}\\ &\le C ({\sigma}R)^{-1} {\| {{\nabla}}(u\zeta) \|}_{2}^{\delta}\cdot {\left( {\| u \|}_{q}^2+ {\| p-c \|}_{q/2} \right)} \cdot {\| u \|}_{q,A_R}^{1-{\delta}} \end{split}$$]{} where $q=q({\delta})=\frac{3-{\delta}}{1-{\delta}/6}$. By considering as with $F=- u \otimes u$, we get from Lemma \[int-annulus\] with $q$ replaced by $q/2$ that [$$\begin{split} \label{eqHolder} {\| p - (p)_{A_{R}} \|}_{L^{q/2}(A_{R})} &\le \frac C{{\sigma}^2 R}\,{\| u \|}_{L^{q/2}({\widehat{\mathbf A}}_R)} + \frac C{{\sigma}} {\| |u|^2 \|}_{L^{q/2}({\widehat{\mathbf A}}_R)}\\ &\le \frac C{{\sigma}^2 R}\,({\sigma}R^3)^{1/q}{\| u \|}_{L^{q}({\widehat{\mathbf A}}_R)} + \frac C{{\sigma}} {\| u \|}_{L^{q}({\widehat{\mathbf A}}_R)}^2. \end{split}$$]{} Thus, choosing $c= (p)_{A_{R}}$, [$$\begin{split} |I_2+I_3| &\le C ({\sigma}R)^{-1} {\| {{\nabla}}(u\zeta) \|}_{2}^{\delta}\cdot {\left( {\sigma}^{\frac 1{q}-2}R^{\frac 3{q}-1} {\| u \|}_{L^{q}({\widehat{\mathbf A}}_R)} + {\sigma}^{-1 } {\| u \|}_{L^{q}({\widehat{\mathbf A}}_R)}^2 \right)} \cdot {\| u \|}_{q,A_R}^{1-{\delta}}. \end{split}$$]{} We now let ${\sigma}$ vary and suppose ${\sigma}= {\sigma}_0 R^{-{\alpha}}$, $0\le {\alpha}<\infty$, ${\sigma}_0>0$. Then [$$\begin{split} \label{eq3-6} |I_1|{\lesssim}{\left( R^{\frac {1+{\alpha}}2+\frac{{\alpha}-3}{q}} {\| u \|}_{L^{q}({\widehat{\mathbf A}}_R)} \right)}^2, \quad |I_2+I_3| \le C {\| {{\nabla}}(u\zeta) \|}_{2}^{\delta}\cdot J \end{split}$$]{} where $$J=R^{\frac {3-{\alpha}}{q}-2+3{\alpha}}{\| u \|}_{L^{q}({\widehat{\mathbf A}}_R)}^{2-{\delta}}+ R^{-1+2{\alpha}} {\| u \|}_{L^{q}({\widehat{\mathbf A}}_R)}^{3-{\delta}}.$$ By Young’s inequality, $$|I_2+I_3| \le \frac12 {\| {{\nabla}}(u\zeta) \|}_{2}^2 + C J ^{\frac1{1-{\delta}/2}}.$$ Thus [$$\begin{split} \label{eq3-7} \int |{{\nabla}}(u\zeta)|^2 {\lesssim}{\left( R^{\frac {1+{\alpha}}2+\frac{{\alpha}-3}{q}} {\| u \|}_{L^{q}({\widehat{\mathbf A}}_R)} \right)}^2 + J ^{\frac1{1-{\delta}/2}}. \end{split}$$]{} To make the right side go to zero, it suffices to find a sequence $R_j\to \infty$, $j\in {{\mathbb N}}$, such that [$$\begin{split} \label{eq3-8} R_j^{\beta}{\| u \|}_{L^{q}( {\widehat{\mathbf A}}_{R_j})} \to 0, \end{split}$$]{} where $ {\widehat{\mathbf A}}_{R_j} = \{ x \in {{\mathbb R }}^3:\ R_j<|x|<R_j(1+8{\sigma}_0R_j^{-{\alpha}})\}$ and $${\beta}= {\beta}({\delta},{\alpha})=\max {\left\{ \frac {1+{\alpha}}2+\frac{{\alpha}-3}{q},\ \frac{\frac {3-{\alpha}}{q}-2+3{\alpha}}{2-{\delta}},\ \frac {-1+2{\alpha}}{3-{\delta}} \right\}}.$$ Note that the first argument is never greater than the second for ${\alpha}\ge0$ and ${\delta}\in [0,1]$, and they equal only if ${\alpha}=0$. Thus $${\beta}({\delta},{\alpha})= \max {\left\{ \frac{\frac {3-{\alpha}}{q}-2+3{\alpha}}{2-{\delta}},\ \frac {-1+2{\alpha}}{3-{\delta}} \right\}}.$$ If is true, then by , $$\lim_{j \to \infty} \int_{|x|<R_j} |{{\nabla}}u|^2 =0.$$ Hence ${{\nabla}}u=0$, $u$ is a constant vector $b$, and $ R_j^{\beta}{\| u \|}_{L^{q}( {\widehat{\mathbf A}}_{R_j})} = C |b| R_j^{{\beta}+ (3-{\alpha})/q} $. Since $ {\beta}+ (3-{\alpha})/q \ge \frac {-1+2{\alpha}}{3-{\delta}} + (3-{\alpha})\frac{1-{\delta}/6}{3-{\delta}} > 0$, we get $b=0$ from . This shows both parts (a) and (b), noting that ${\beta}({\delta},0)= -\frac 1{3-{\delta}}$. Half space {#S4} ========== In this section we prove Theorem \[thmB\] for ${{\Omega}}={{\mathbb R }}^3_+$. Its boundary is ${{\Gamma}}=\{ (x',0): x'\in {{\mathbb R }}^2\}$. For $x_0 \in {{\Gamma}}$, denote $$B_R^+(x_0)= B_R(x_0) \cap {{\mathbb R }}^3_+, \quad B_R^+= B_R^+(0).$$ In addition to the interior estimate Lemma \[ST\], we will also use the following boundary estimate of Kang [@Kang]. Again, the key is that the pressure is not needed on the right side. \[Kang\] If $(u,p)$ solves [$$\label{Stokes2}\begin{split} -{\Delta}u + {{\nabla}}p = {\mathop{\mathrm{div}}}F, \quad {\mathop{\mathrm{div}}}u=0 \end{split}$$]{} in $B_{\ell R}^+ \subset {{\mathbb R }}^3_+$, with $u(x',0)=0$, $\ell>1$, then for $1<q<\infty$ and $1\le m \le \infty$, [$$\begin{split} \label{K-eq1} {\| {{\nabla}}u \|}_{L^q(B^+_{R})} + {\| p - (p)_{B^+_{R}} \|}_{L^q(B^+_{R})} \le C R^{\frac 3q-\frac 3m -1}\,{\| u \|}_{L^{m}(B^+_{\ell R})} + C {\| F \|}_{L^q(B^+_{\ell R})}, \end{split}$$]{} where $C=C(q,m,\ell)$ does not depend on $R$. We extend the above to a half annulus. \[int-halfannulus\] Let $R>0$, $1<L\le 2$, and ${\sigma}=\frac 18(L-1)$. Denote the half annuli in ${{\mathbb R }}^3$ $$A_R = B_{(L-2{\sigma})R}^+ \setminus \overline B_{(1+2{\sigma})R}^+, \quad {\widehat{\mathbf A}}_R = B_{LR}^+ \setminus \overline B_R^+.$$ If $(u,p)$ solves in ${\widehat{\mathbf A}}_R$, then for $1<q<\infty$ [$$\begin{split} \label{Kang-eq2} {\| {{\nabla}}u \|}_{L^q(A_{R})} + {\sigma}{\| p - (p)_{A_{R}} \|}_{L^q(A_{R})} \le \frac C{{\sigma}R}\,{\| u \|}_{L^{q}({\widehat{\mathbf A}}_R)} + C {\| F \|}_{L^q({\widehat{\mathbf A}}_R)}, \end{split}$$]{} where $C=C(q)$ is uniform in $R>0$ and ${\sigma}\in (0,\frac 18]$. Note that the ${\sigma}$ factor appears in both sides of . We may assume $R=1$ by scaling. We first choose $N_1=N_1(L)$ points $x^{(j)}\in \overline{ A_1} \cap {{\Gamma}}$, $j=1,\ldots,N_1$, such that $$A_1 \cap{\left\{ (x',x_3): 0 < x_3< \frac 34{\sigma}\right\}} \subset \cup_{j=1}^{N_1} B_j, \quad B_j = B_{{\sigma}}^+(x^{(j)}).$$ We then choose $N_2=N_2(L)$ points $x^{(j)}\in A_1$ with $x^{(j)}_3\ge {\sigma}$, $j=N_1+1,\ldots,N$ with $N=N_1+N_2$, such that $$A_1 \cap{\left\{ (x',x_3): \frac 34{\sigma}\le x_3 \right\}} \subset \cup_{j=N_1+1}^{N} B_j, \quad B_j = B_{{\sigma}/2}(x^{(j)}).$$ We can choose them in a way that there is a ${\sigma}$-independent upper bound for the number of overlapping of $ B_j$ for ${\sigma}\in (0,\frac18]$. We also denote $\widehat B_j = B_{2{\sigma}}^+(x^{(j)})$ for $j\le N_1$ and $\widehat B_j = B_{{\sigma}}(x^{(j)})$ for $j> N_1$. Note that $B_j$ and $\widehat B_j$ are half balls for $1\le j\le N_1$ and balls for $N_1+1\le j \le N$. Also note that $\widehat B_j \subset {\widehat{\mathbf A}}_1$. It follows that [$$\label{thmB-eq3}\begin{split} A_1 \subset \cup_{j=1}^N B_j \subset \cup_{j=1}^N \widehat B_j \subset {\widehat{\mathbf A}}_1. \end{split}$$]{} By Lemma \[Kang\] with given $q>1$, $m=q$, and $R={\sigma}$, $$\int_{B_{{\sigma}}^+(x^{(j)})} |{{\nabla}}u|^q {\lesssim}\int_{B^+_{2{\sigma}}(x^{(j)})} {\left( \frac 1{{\sigma}^q} |u|^q + |F|^q \right)}, \quad (1\le j \le N_1).$$ By Lemma \[ST\] with given $q>1$, $m=q$, and $R={\sigma}/2$, $$\int_{B_{{\sigma}/2}(x^{(j)})} |{{\nabla}}u|^q {\lesssim}\int_{B_{{\sigma}}(x^{(j)})} {\left( \frac 1{{\sigma}^q} |u|^q + |F|^q \right)}, \quad (N_1< j \le N).$$ Summing in $j$, $$\int_{A_1} |{{\nabla}}u|^q \le \sum_{j=1}^N \int_{B_j} |{{\nabla}}u|^q {\lesssim}\sum_{j=1}^N \int_{\widehat B_j } {\left( \frac 1{{\sigma}^q} |u|^q + |F|^q \right)} {\lesssim}\int_{{\widehat{\mathbf A}}_1} {\left( \frac 1{{\sigma}^q} |u|^q + |F|^q \right)}.$$ This shows the estimate of ${\| {{\nabla}}u \|}_{L^q}$ in . Apply Lemma \[p-est\] to $E=A_1$, $${\| p-(p)_{A_1} \|}_{L^q(A_1)} \le C_0 \sup_{ \zeta \in W^{1,q'}_0(A_1),\ {\| {{\nabla}}\zeta \|}_{L^{q'}(A_1)}=1} \int p {\mathop{\mathrm{div}}}\zeta$$ where $C_0=2{C_{\mathrm{bg}}}(A_1,q')$. By Lemma \[Bog-annulus\], $C_0 \le C/{\sigma}$. Using the weak form of , [$$\begin{split} {\| p-(p)_{A_1} \|}_{L^q(A_1)} &\le \frac C{\sigma}\sup_{ \zeta \in W^{1,q'}_0(A_1),\ {\| {{\nabla}}\zeta \|}_{L^{q'}(A_1)}=1} \int ({{\nabla}}u+F) : {{\nabla}}\zeta \\ & \le \frac C{\sigma}{\left( {\| {{\nabla}}u \|}_{L^q(A_1)} + {\| F \|}_{L^q(A_1)} \right)}. \end{split}$$]{} This completes the proof of . It is the same as the proof of Theorem \[th1\], with Lemma \[int-annulus\] replaced by Lemma \[int-halfannulus\]. We use the zero boundary condition when we integrate by parts and when we apply the Sobolev inequality. Periodic slab {#S5} ============= In this section we will prove Theorem \[thmC\] for the periodic slab ${{\Omega}}={{\mathbb R }}^2\times ({{\mathbb R }}/ {{\mathbb Z}})$. We can identify our velocity field $v(x)=v(x',x_3)$ as a vector field defined in ${{\mathbb R }}^3$ that is periodic in $x_3$, and the system is satisfied in the whole ${{\mathbb R }}^3$. We first extend the interior estimate, Lemma \[int-annulus\], to a region enclosed by cylinders. \[int-cyl\] Let $R>0$, $1<L\le 2$, and ${\sigma}=\frac 18(L-1)$. Denote the 3D regions $$E_R := A_R \times(0,R), \quad {\widehat{\mathbf E}}_R := {\widehat{\mathbf A}}_R \times (-2{\sigma}R, (1+2{\sigma})R),$$ where $$A_R = B_{(L-2{\sigma})R}' \setminus \overline{ B_{(1+2{\sigma})R}'}, \quad {\widehat{\mathbf A}}_R = B_{LR}' \setminus \overline{B_R' }$$ are 2D annuli, and $B_R'={\left\{ x' \in {{\mathbb R }}^2: |x'|<R \right\}}$. If $(u,p)$ solves the Stokes system in ${\widehat{\mathbf E}}_R$, then for $1<q<\infty$ [$$\begin{split} \label{ST-cyl} {\| {{\nabla}}u \|}_{L^q(E_{R})} + {\sigma}{\| p - (p)_{E_{R}} \|}_{L^q(E_{R})} \le \frac C{{\sigma}R}\,{\| u \|}_{L^{q}({\widehat{\mathbf E}}_R)} + C {\| F \|}_{L^q({\widehat{\mathbf E}}_R)}, \end{split}$$]{} where $C=C(q)$ is uniform in $R>0$ and ${\sigma}\in (0,\frac 18]$. Its proof is identical to the proof of Lemma \[int-annulus\], with the reference to Lemma \[Bog-annulus\] replaced by Lemma \[Bog-cylinder\]. By the standard regularity theory, $u\in C^\infty_{\mathrm{loc}}$. Using the periodic BC, we still have the local energy equality for any scalar function $\zeta \in C^\infty_c( {{\Omega}})$ and any constant $c$, [$$\begin{split} \label{LEE3} \int_{{\Omega}}|{{\nabla}}(u\zeta)|^2 &= \int_{{\Omega}}|u|^2 |{{\nabla}}\zeta |^2 + \int_{{\Omega}}|u|^2 u\zeta\cdot {{\nabla}}\zeta + 2 \int_{{\Omega}}(p-c) u\zeta \cdot {{\nabla}}\zeta \\ &=I_1+I_2+I_3. \end{split}$$]{} By considering as with $F=- u \otimes u$, we get from Lemma \[int-cyl\] with $q$ replaced by $q/2$ that $${\| p - (p)_{E_{R}} \|}_{L^{q/2}(E_{R})} \le \frac C{{\sigma}^2 R}\,{\| u \|}_{L^{q/2}({\widehat{\mathbf E}}_R)} + \frac C{{\sigma}} {\| |u|^2 \|}_{L^{q/2}({\widehat{\mathbf E}}_R)}.$$ Raising to the $q/2$-th power and using the periodicity with $L\le 2 \ll R$, we get [$$\begin{split} R\int_0^1 \int_{A_R} |p-(p)_{E_R}|^{q/2} &\le \frac {CR}{{\sigma}^q R^{q/2}} \int_0^1 \int_{{\widehat{\mathbf A}}_R}|u|^{q/2} + \frac {CR}{{\sigma}^{q/2} }\int_0^1 \int_{{\widehat{\mathbf A}}_R}|u|^q. \end{split}$$]{} Thus, with ${{\Omega}}_R:= {\widehat{\mathbf A}}_R\times(0,1)$, [$$\label{p-ARper-est}\begin{split} {\| p - (p)_{E_{R}} \|}_{L^{q/2}(A_{R}\times(0,1))} &\le \frac C{{\sigma}^2 R}\,{\| u \|}_{L^{q/2}({{\Omega}}_R)} + \frac C{{\sigma}} {\| u \|}^2_{L^{q}({{\Omega}}_R)} \\ &\le \frac C{{\sigma}^2 R}\,({\sigma}R^2)^{1/q}{\| u \|}_{L^{q}({{\Omega}}_R)} + \frac C{{\sigma}} {\| u \|}^2_{L^{q}({{\Omega}}_R)}. \end{split}$$]{} Here we use $|{{\Omega}}_R|=C {\sigma}R^2$, which is different from $|A_R|=C{\sigma}R^3$ in the proofs of Theorems \[th1\] and \[thmB\]. Fix $Z \in C^\infty_c({{\mathbb R }}^2)$, $Z(x')=1$ for $|x'|< 1+ 2{\sigma}$, $Z(x')=0$ for $|x'|>L-2{\sigma}$. Thus ${{\nabla}}Z$ is supported in $\overline{A_1}$. Let $\zeta(x) =\zeta_R(x)= Z(\frac {x'}R)$ in the local energy equality , with ${{\nabla}}\zeta_R$ supported in $\overline{A_R}\times ({{\mathbb R }}/{{\mathbb Z}})$. Note $\zeta$ does not depend on $x_3$. Choose $c= (p)_{E_{R}}$. By Hölder and Sobolev inequalities, for $0 \le {\delta}\le 1$ and $q=q({\delta})=\frac{3-{\delta}}{1-{\delta}/6}$, [$$\begin{split} |I_2+I_3| &\le C{\| {{\nabla}}\zeta \|}_\infty \cdot {\| u\zeta \|}_{6}^{\delta}\cdot {\| |u|^{3-{\delta}}+|p-c|\cdot|u|^{1-{\delta}} \|}_{\frac1{1-{\delta}/6},A_R \times(0,1)}\\ &\le C ({\sigma}R)^{-1} {\| {{\nabla}}(u\zeta) \|}_{2}^{\delta}\cdot {\left( {\| u \|}_{q}^2+ {\| p-c \|}_{q/2,\, A_R\times(0,1)} \right)} \cdot {\| u \|}_{q,\, A_R\times(0,1)}^{1-{\delta}}. \end{split}$$]{} By , $$|I_2+I_3| \le C ({\sigma}R)^{-1} {\| {{\nabla}}(u\zeta) \|}_{2}^{\delta}\cdot {\left( \frac 1{{\sigma}^2 R}\,({\sigma}R^2)^{1/q}{\| u \|}_{L^{q}({{\Omega}}_R)} + \frac 1{{\sigma}} {\| u \|}^2_{L^{q}({{\Omega}}_R)} \right)} \cdot {\| u \|}_{q,{{\Omega}}_R}^{1-{\delta}}.$$ We now let ${\sigma}$ vary and suppose ${\sigma}= {\sigma}_0 R^{-{\alpha}}$, $0\le {\alpha}<\infty$, ${\sigma}_0>0$. Then we have [$$\begin{split} |I_1| &{\lesssim}({\sigma}R)^{-2} \int_{{{\Omega}}_R} |u|^2 {\lesssim}({\sigma}R)^{-2} ({\sigma}R^2)^{1-2/q} {\| u \|}_{q,{{\Omega}}_R}^2\\ &= {\left( {\sigma}^{-\frac 12-\frac1{q}} R^{-\frac2{q}} {\| u \|}_{L^{q}({{\Omega}}_R)} \right)}^2 =C{\left( R^{\frac {\alpha}2+\frac{{\alpha}-2}{q}} {\| u \|}_{L^{q}({{\Omega}}_R)} \right)}^2. \end{split}$$]{} We also have by Young’s inequality, [$$\begin{split} |I_2+I_3| &\le C {\| {{\nabla}}(u\zeta) \|}_{2}^{\delta}\cdot J \le \frac12 {\| {{\nabla}}(u\zeta) \|}_{2}^2 + C J ^{\frac1{1-{\delta}/2}}, \\ J&=R^{\frac {2-{\alpha}}{q}-2+3{\alpha}}{\| u \|}_{L^{q}({{\Omega}}_R)}^{2-{\delta}}+ R^{-1+2{\alpha}} {\| u \|}_{L^{q}({{\Omega}}_R)}^{3-{\delta}}. \end{split}$$]{} Thus [$$\begin{split} \label{eq5-4} \int |{{\nabla}}(u\zeta)|^2 {\lesssim}{\left( R^{\frac {\alpha}2+\frac{{\alpha}-2}{q}} {\| u \|}_{L^{q}({{\Omega}}_R)} \right)}^2 + J ^{\frac1{1-{\delta}/2}}. \end{split}$$]{} To make the right side go to zero, it suffices to find a sequence $R_j\to \infty$, $j\in {{\mathbb N}}$, such that [$$\begin{split} \label{eq5-8} R_j^{\beta}{\| u \|}_{L^{q}( {{\Omega}}_{R_j})} \to 0, \end{split}$$]{} where $ {{\Omega}}_{R_j} = \{ x=(x',x_3) \in {{\Omega}}:\ R_j<|x'|<R_j(1+8{\sigma}_0R_j^{-{\alpha}})\}$ and $${\beta}= {\beta}_{ps}({\delta},{\alpha})=\max {\left\{ \frac {{\alpha}}2+\frac{{\alpha}-2}{q},\ \frac{\frac {2-{\alpha}}{q}-2+3{\alpha}}{2-{\delta}},\ \frac {-1+2{\alpha}}{3-{\delta}} \right\}}=\max \{{\beta}_1,{\beta}_2,{\beta}_3\}.$$ Numerically ${\beta}_1 \le{\beta}_3$ for ${\alpha}\le 2$, and ${\beta}_1 \le{\beta}_2$ for ${\alpha}\ge 0.2$. Thus we can drop ${\beta}_1$, $${\beta}_{ps}({\delta},{\alpha})=\max {\left\{ \frac{\frac {2-{\alpha}}{q}-2+3{\alpha}}{2-{\delta}},\ \frac {-1+2{\alpha}}{3-{\delta}} \right\}}.$$ If is true, then by , $$\lim_{j \to \infty} \int_{|x'|<R_j} |{{\nabla}}u|^2 =0.$$ Hence ${{\nabla}}u=0$, $u$ is a constant vector $b$, and $ R_j^{\beta}{\| u \|}_{L^{q}( {{\Omega}}_{R_j})} = C |b| R_j^{{\beta}+ (2-{\alpha})/q} $. Since $ {\beta}+ (2-{\alpha})/q \ge \frac {-1+2{\alpha}}{3-{\delta}} + (2-{\alpha})\frac{1-{\delta}/6}{3-{\delta}} > 0$, we get $b=0$ from . This shows both parts (a) and (b), noting that ${\beta}_{ps}({\delta},0)= -\frac {1}{3-{\delta}}$. Zero BC slab {#S6} ============ In this section we prove Theorem \[th4\]. The domain is ${{\Omega}}={{\mathbb R }}^2\times (0,1)$ and the vector field $u$ satisfies the zero boundary condition $u(x',0)=u(x',1)=0$. Its proof is different from those for Theorems \[th1\]-\[thmC\] as we cannot obtain the local pressure estimate by scaling. We also cannot vary the radii ratio $L$. Denote $B_R'=\{ x' \in {{\mathbb R }}^2: \ |x'|<R\}$ and $B_R'(x_0')=\{ x' \in {{\mathbb R }}^2: \ |x'-x_0'|<R\}$. For given $L>1$, let ${\sigma}=\frac 18(L-1)$. Denote the 2-D annuli $$A_R = B_{(L-2{\sigma})R}' \setminus B_{(1+2{\sigma})R}', \quad {\widehat{\mathbf A}}_R = B_{LR}' \setminus B_R' .$$ We will use a variation of [@CPZZ Proposition 2.1]: \[CPZZ\] There is a (linear) *Bogovskii map* ${\mathop{\mathrm{Bog}}\nolimits}_R$ on $E=A_R \times (0,1)$, $R \ge 1$, that maps $f\in L^q_0(E)=\{ f \in L^q(E):\ \int_E f =0\}$ to $v = {\mathop{\mathrm{Bog}}\nolimits}_R f \in W^{1,q}_0(E;{{\mathbb R }}^3)$ satisfying $${\mathop{\mathrm{div}}}v = f, \quad {\| {{\nabla}}v \|}_{L^q(E)} \le C_{q,L} R {\| f \|}_{L^q(E)}.$$ The key is that the constant grows linearly in $R$. The proof is the same as [@CPZZ Proposition 2.1] which is formulated for $E=B_R'\times(0,1)$ and $q=2$. The idea is to fix ${\mathop{\mathrm{Bog}}\nolimits}_1$ for $E_1=A_1\times (0,1)$, and for given $f(x) \in L^q_0(E)$, define $\bar f(\bar x) = f(x)$ for $\bar x \in E_1$ with $x=(R\bar x_1, R\bar x_2, \bar x_3)$, let $\bar v = {\mathop{\mathrm{Bog}}\nolimits}_1 \bar f$, and then $v={\mathop{\mathrm{Bog}}\nolimits}_R f$ is defined by $v(x) = (R\bar v_1(\bar x), R\bar v_2(\bar x), \bar v_3(\bar x))$. We may assume $L\le 2$ and let $A_R$ and ${\widehat{\mathbf A}}_R$ be defined as above. By the standard regularity theory, $u\in C^\infty_{\mathrm{loc}}(\overline {{\Omega}})$. We still have the local energy equality for any scalar function $\phi \in C^\infty_c({{\Omega}})$. There are $N_0=N_0(L)\in {{\mathbb N}}$ and $R_0=R_0(L)\gg 1$ such that, if $R \ge R_0$, then there are $N\le N_0 R^2$ points $x_j\in A_R$, $j=1,\ldots,N$, such that [$$\label{th4-eq3}\begin{split} A_R \subset \cup_{j=1}^N B_{1/2}'(x_j) \subset \cup_{j=1}^N B_{1}'(x_j) \subset {\widehat{\mathbf A}}_R, \end{split}$$]{} and there is an $R$-independent upper bound for the number of overlapping of $ B_j'(x_j)$. Unlike in the proofs of the previous theorems, the choice of $x_j$ is not by scaling, and the radii do not depend on $L$. We can rewrite as with $F=-u\otimes u$. By Lemma \[Kang\] and zero BC at $x_3=0$, for $q=m=r>1$ and $\ell=\sqrt2$, $${\| {{\nabla}}u \|}_{L^r(B_{1/2}'(x_j) \times (0,\frac 12))}^r {\lesssim}{\| u \|}_{L^{r}(B_{1}'(x_j) \times (0,1))}^r + {\| u \|}_{L^{2r}(B_{1}'(x_j) \times (0,1))}^{2r} .$$ In applying Lemma \[Kang\] we have used the inclusions $$B_{\frac 12}'(x_j) \times (0,\tfrac 12) \subset B^+_{\sqrt 2/2} ((x_j,0)) \subset B^+_{1} ((x_j,0)) \subset B_{1}'(x_j) \times (0,1).$$ Similarly, by Lemma \[Kang\] and zero BC at $x_3=1$, $${\| {{\nabla}}u \|}_{L^r(B_{1/2}'(x_j) \times (\frac12,1))}^r {\lesssim}{\| u \|}_{L^{r}(B_{1}'(x_j) \times (0,1))}^r + {\| u \|}_{L^{2r}(B_{1}'(x_j) \times (0,1))}^{2r} .$$ Denote $E_R=A_R\times (0,1)$ and ${{\Omega}}_R={\widehat{\mathbf A}}_R\times (0,1)$. Summing the two estimates, summing in $j$ and using , we get [$$\label{th4-er4}\begin{split} {\| {{\nabla}}u \|}_{L^r(E_R)}^r {\lesssim}{\| u \|}_{L^{r}( {{\Omega}}_R)}^r + {\| u \|}_{L^{2r}({{\Omega}}_R)}^{2r} . \end{split}$$]{} By Lemma \[p-est\] and Lemma \[CPZZ\] with $p_R =\frac 1{|A_R|}\int_0^1\int_{A_R} p$, $${\| p-p_R \|}_{L^r(E_R)} \le CR \sup_{ \zeta \in W^{1,r'}_0(E_R),\ {\| {{\nabla}}\zeta \|}_{L^{r'}(E_R)}=1} \int p {\mathop{\mathrm{div}}}\zeta.$$ Using weak form of Stokes system , and then , [$$\label{th4-eq5}\begin{split} {\| p-p_R \|}_{L^r(E_R)} &{\lesssim}R {\left( {\| {{\nabla}}u \|}_{L^r(E_R)} + {\| u\otimes u \|}_{L^r(E_R)} \right)} {\lesssim}R {\left( {\| u \|}_{L^{r}({{\Omega}}_R )} + {\| u \|}_{L^{2r}({{\Omega}}_R)}^{2} \right)} . \end{split}$$]{} By the same argument between and and same $\zeta_R$, with ${\alpha}=0$ and ${\sigma}$ ignored being a constant, we get for $r>1$, [$$\begin{split} \label{eq6.4} \int_{{\Omega}}|{{\nabla}}(u\zeta)|^2 {\lesssim}{\left( R^{-1/r} {\| u \|}_{L^{2r}({{\Omega}}_R)} \right)}^2 + J ^{\frac1{1-{\delta}/2}} \end{split}$$]{} where $$J= {\left( {\| u \|}_{L^{r}({{\Omega}}_R )} + {\| u \|}_{L^{2r}({{\Omega}}_R)}^{2} \right)} {\| u \|}_{L^{s}({{\Omega}}_R )}^{1-{\delta}}, \quad \frac{1-{\delta}}s = 1-\frac {\delta}6 - \frac 1r.$$ Note that the factor $R^{-1}$ from $|{{\nabla}}\zeta|$ cancels the factor $R$ in . If we set $s=2r$, then $s=2r=q({\delta})=\frac{3-{\delta}}{1-{\delta}/6}$. Using ${\| u \|}_{L^{q/2}({{\Omega}}_R )} {\lesssim}R^{2/q} {\| u \|}_{L^{q}({{\Omega}}_R )}$, becomes $$\int_{{\Omega}}|{{\nabla}}(u\zeta)|^2 {\lesssim}{\left( R^{-2/q} {\| u \|}_{L^q({{\Omega}}_R)} \right)}^2 + {\left( R^{2/q} {\| u \|}^{2-{\delta}}_{L^{q}({{\Omega}}_R )} + {\| u \|}_{L^{q}({{\Omega}}_R)}^{3-{\delta}} \right)} ^{\frac1{1-{\delta}/2}}.$$ Thus, if condition holds, there is a sequence $R_j \to \infty$ as $j \to \infty$ such that $R_j^{2/q} {\| u \|}^{2-{\delta}}_{L^{q}({{\Omega}}_{R_j} )}\to 0$, then the above shows $\lim_{j \to \infty}\int_{|x'|<R_j} |{{\nabla}}u|^2=0$, and hence $u \equiv0$. Alternatively, if condition holds that there is a sequence $R_j \to \infty$ as $j \to \infty$ such that $ {\| u \|}_{L^{r}({{\Omega}}_{R_j} )} + {\| u \|}_{L^{2r}({{\Omega}}_{R_j})} \to 0$, and suppose $r \le s \le 2r$, then $J \to 0$, $\lim_{j \to \infty}\int_{|x'|<R_j} |{{\nabla}}u|^2=0$, and hence $u \equiv0$. The condition $r \le s \le 2r$ is equivalent to $$r_1({\delta}) = \frac{3(3-{\delta})}{6-{\delta}} \le r \le r_2({\delta}) = \frac{6(2-{\delta})}{6-{\delta}}.$$ Both $r_1,r_2$ are decreasing in ${\delta}$ with $r_1(0)=3/2$, $r_2(0)=2$, and $r_1(1)=r_2(1)=6/5$. Hence for any $r \in [6/5,2]$, we can find ${\delta}\in [0,1]$ such that $r_1({\delta}) \le r \le r_2({\delta})$. This shows part (b). [XX]{} B. Carrillo, X. Pan, Qi S. Zhang, and N. Zhao, Decay and vanishing of some D-solutions of the Navier-Stokes equations, arXiv:1808.10386 D. Chae, Liouville-type theorems for the forced Euler equations and the Navier-Stokes equations, Comm. Math. Phys. 326 (2014), no. 1, 37-48. D. Chae and J. Wolf, On Liouville type theorems for the steady Navier-Stokes equations in ${{\mathbb R }}^3$, J. Differential Equations 261 (2016), no. 10, 5541-5560. D. Chae and J. Wolf, On Liouville type theorem for the stationary Navier-Stokes equations. Calc. Var. Partial Differential Equations 58 (2019), no. 3, Paper No. 111, 11 pp. G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I. Linearized steady problems. Springer Tracts in Natural Philosophy, 38. Springer-Verlag, New York, 1994. G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems. Second edition. Springer Monographs in Mathematics. Springer, New York, 2011. D. Gilbarg and H. F. Weinberger, Asymptotic properties of steady plane solutions of the Navier-Stokes equations with bounded Dirichlet integral, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 2, 381-404. K. Kang, On regularity of stationary [S]{}tokes and [N]{}avier-[S]{}tokes equations near boundary, J. Math. Fluid Mech. 6 (2004), no. 1, 78-101. H. Kozono, Y. Terasawa and Y. Wakasugi, A remark on Liouville-type theorems for the stationary Navier-Stokes equations in three space dimensions, J. Funct. Anal. 272 (2017), no. 2, 804-818. H. Kozono, Y. Terasawa and Y. Wakasugi, Asymptotic properties of steady solutions to the 3D axisymmetric Navier-Stokes equations with no swirl, https://arxiv.org/abs/2004.13471 C.-L. Lin, G. Uhlmann, and J.-N. Wang, Asymptotic behavior of solutions of the stationary Navier-Stokes equations in an exterior domain. Indiana Univ. Math. J. 60 (2011), no. 6, 2093-2106. K. Pilekcas, On the asymptotic behavior of solutions of a stationary system of Navier-Stokes equations in a domain of layer type. (Russian) Mat. Sb. 193 (2002), no. 12, 69–104; translation in Sb. Math. 193 (2002), no. 11-12, 1801-1836 K. Pileckas and M. Specovius-Neugebauer, Asymptotics of solutions to the Navier-Stokes system with nonzero flux in a layer-like domain. Asymptot. Anal. 69 (2010), no. 3-4, 219-231. G. Seregin, Liouville type theorem for stationary Navier-Stokes equations, Nonlinearity 29 (2016), no. 8, 2191-2195. G. Seregin, Remarks on Liouville type theorems for steady-state Navier-Stokes equations. Algebra i Analiz 30 (2018), no. 2, 238-248; reprinted in St. Petersburg Math. J. 30 (2019), no. 2, 321-328. G. Seregin and W. Wang, Sufficient conditions on Liouville type theorems for the 3D steady Navier-Stokes equations. Algebra i Analiz 31 (2019), no. 2, 269-278; reprinted in St. Petersburg Math. J. 31 (2020), no. 2, 387-393. V. [Š]{}ver[á]{}k and T.-P. Tsai, On the spatial decay of 3-[D]{} steady-state [N]{}avier-[S]{}tokes flows, Comm. Partial Differential Equations **25** (2000), no. 11-12, 2107–2117. T.-P. Tsai, *Lectures on Navier-Stokes Equations*. Graduate Studies in Mathematics, 192. American Mathematical Society, Providence, RI, 2018. W. Wang, Remarks on Liouville type theorems for the 3D steady axially symmetric Navier-Stokes equations. J. Differential Equations 266 (2019), no. 10, 6507-6524. [^1]: Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada. ttsai@math.ubc.ca. The work of Tsai was partially supported by NSERC grant RGPIN-2018-04137.

--- abstract: | For a graph $G$, Chartrand et al. defined the rainbow connection number $rc(G)$ and the strong rainbow connection number $src(G)$ in “G. Charand, G.L. John, K.A. Mckeon, P. Zhang, Rainbow connection in graphs, Mathematica Bohemica, 133(1)(2008) 85-98”. They raised the following conjecture: for two given positive integers $a$ and $b$, there exists a connected graph $G$ such that $rc(G)=a$ and $src(G)=b$ if and only if $a=b\in\{1,2\}$ or $ 3\leq a\leq b$”. In this short note, we will show that the conjecture is true.\ [**Keywords:**]{} edge-colored graph, (strong) rainbow coloring, (strong) rainbow connection number.\ [**AMS Subject Classification 2000:**]{} 05C15, 05C40 author: - | [Xiaolin Chern, Xueliang Li]{}\ [Center for Combinatorics and LPMC-TJKLC]{}\ [Nankai University, Tianjin 300071, P.R. China]{}\ [E-mail: xiaolin\_chern@yahoo.cn; lxl@nankai.edu.cn]{} title: | **A solution to a conjecture on the\ rainbow connection number[^1]** --- Introduction ============ All graphs in this paper are finite, undirected, simple and connected. We follow the notation and terminology of \[1\]. Let $c$ be a coloring of the edges of a graph $G$, i.e., $c: \ E(G)\longrightarrow \{1,2,\cdots,k\},\,k\in \mathbb{N}$. A path is called a rainbow path if no two edges of the path have the same color. The graph $G$ is called rainbow connected (with respect to $c$) if for every two vertices of $G$, there exists a rainbow path connecting them in $G$. If by coloring $c$ the graph $G$ is rainbow connected, then the coloring $c$ is called a rainbow coloring of $G$. If $k$ colors are used in $c$, then $c$ is a rainbow $k$-coloring of $G$. The minimum number $k$ for which there exists a rainbow $k$-coloring of $G$, is called the rainbow connection number of $G$, denoted by $rc(G)$. Let $c$ is a rainbow coloring of a graph $G$. If for every pair $u$ and $v$ of distinct vertices of the graph $G$, the graph $G$ contains a rainbow $u$-$v$ geodesic (a shortest path in $G$ between $v$ and $u$), then $G$ is called strongly rainbow connected. In this case, the coloring $c$ is called a strong rainbow coloring of $G$. If $k$ colors are used, then $c$ is a strong rainbow $k$-coloring of $G$. The minimum number $k$ satisfying that $G$ is strongly rainbow connected, i.e., the minimum number $k$ for which there exists a strong rainbow $k$-coloring of $G$, is called the strong rainbow connection number of $G$, denoted by $src(G)$. Thus for every connected graph $G$, $rc(G)\leq src(G)$. Recall that the diameter of $G$ is defined as the largest distance between two vertices of $G$, denoted $diam(G)$. Then $diam(G)\leq rc(G)\leq src(G)$. The following results were obtained in \[2\] by Chartrand et al. \[prop1\] Let $G$ be a nontrivial connected graph of size $m$. Then\ 1. $rc(G)=1$ if and only if $src(G)=1$ .\ 2. $rc(G)=2$ if and only if $src(G)=2$.\ 3. $diam(G)\leq rc(G)\leq src(G)$ for every connected graph $G$. ------------------------------------------------------------------------ Chartrand et al. also considered the problem that, given any two integers $a$ and $b$, whether there exists a connected graph $G$ such that $rc(G)=a$ and $src(G)=b$ ? and they got the following result. \[th1\] Let $a$ and $b$ be positive integers with $a\geq 4$ and $b \geq (5a-6)/3$. Then there exists a connected graph $G$ such that $rc(G)=a$ and $src(G)=b$. ------------------------------------------------------------------------ Then, combining Proposition \[prop1\] and Theorem \[th1\], they got the following result. \[cor1\] Let $a$ and $b$ be positive integers. If $a=b$ or $3\leq a<b$ and $b\leq \frac{5a-6}{3}$, then there exists a connected graph $G$ such that $rc(G)=a$ and $src(G)=b$. ------------------------------------------------------------------------ Finally, they thought the question that whether the condition $b\leq \frac{5a-6}{3}$ can be deleted ? and raised the following conjecture: \[con1\] Let $a$ and $b$ be positive integers. Then there exists a connected graph $G$ such that $rc(G)=a$ and $src(G)=b$ if and only if $a=b\in\{1,2\}$ or $ 3\leq a\leq b$. ------------------------------------------------------------------------ This short note is to give a confirmative solution to this conjecture. Proof of the conjecture ======================= [**Proof of Conjecture \[con1\]:**]{} From Proposition \[prop1\] one can see that the condition is necessary. For the sufficiency, when $a=b\in\{1,2\}$, from Corollory \[cor1\] the conjecture is true. So, we just need to consider the situation $ 3\leq a\leq b$. Let $n=3b(b-a+2)$, and let $H_{n}$ be the graph consisting of an $n$-cycle $C_n: \ v_1, v_2, \cdots, v_n$ and another two vertices $w$ and $v$, each of which joins to every vertex of $C_n$. Let $G$ be the graph constructed from $H_n$ of order $n+2$ and the path $P_{a-1}: u_1, u_2,\cdots, u_{a-1}$ on $a-1$ vertices by identifying $v$ and $u_{a-1}$. First, we will show $rc(G)=a$. Because diam($G$)=$a$, by Proposition \[prop1\] we have $rc(G)\geq a$. It remains to show $rc(G)\leq a$. Note that $n=3b(b-a+2)\geq 18$. Define a coloring $c$ for the graph $G$ by the following rules: $$c(e)=\left\{ \begin{array}{ll} i & if \,\,e\,=\,\, u_{i}u_{i+1}\,\, for\,\, 1\leq i \leq a-2,\\ a-1 & if \,\,e\,=\,\,v_{i}v \,and\, i\,\, is\,\, odd,\\ a & if\,\, e\,=\,v_{i}v\,\, and\,\, i\, \,is\, even,\\ a & if\,\, e\,=\,v_{i}w\,\, and\,\,1\leq i \leq n\\ 1 & otherwise. \end{array} \right.$$ Since $c$ is a rainbow $a$-coloring of the edges of $G$, it follows that $rc(G)\leq a$. This implies $rc(G)=a$. Next, we will show $src(G)=b$. We first show $src(G)\leq b$, by giving a strong rainbow $b$-coloring $c$ for the graph $G$ as follows: $$c(e)=\left\{ \begin{array}{ll} i & if \,\,e\,=\,\, u_{i}u_{i+1}\,\, for\,\, 1\leq i \leq a-2,\\ a-2+i & if\, \,e\,=\,\,v_{3b(i-1)+j}v \,\,for\,\,1\leq i \leq b-a+2\,\, and\,\, 1\leq j \leq 3b,\\ i & if\,\, e\,=\,\,v_{3(j-1)b+3(i-1)+k}w\,\, for\,\, 1\leq j \leq b-a+2\,\,and\,\,1\leq i \leq b\,\,\\ &and\,\,1\leq k\leq 3,\\ 1 & if\,\, e\,=\,\,v_{3(i-1)+1}v_{3(i-1)+2}\,\,for\,\, 1\leq i\leq b(b-a+2),\\ 2 & if\,\, e\,=\,\,v_{3(i-1)+2}v_{3(i-1)+3}\,\,for\,\, 1\leq i\leq b(b-a+2),\\ 3 & otherwise\\ \end{array} \right.$$ It remains to show $src(G)\geq b$. By contradiction, suppose $rc(G)<b$. Then there exists a strong rainbow $(b-1)$-coloring $c: \ E(G)\rightarrow \{1,2,\cdots,b-1\}$. For every $v_i \ (1\leq i\leq n)$, $d(v_i,u_1)=a-1$, and the path $v_ivu_{a-2}\cdots u_{1}$ is the only path of length $a-1$ connecting $v_i$ and $u_1$, and so $v_ivu_{a-2}\cdots u_{1}$ is a rainbow path. Without loss of generality, suppose $c(u_2u_1)=1,\,c(u_3u_2)=2,\cdots,\,c(u_{a-1}u_{a-2})=a-2$. Then $c(v_iv)\in\{a-1,a,\cdots,b\}$, for $1\leq i\leq n$. We first consider the set of edges $A=\{v_iv,1\leq i\leq n\}$, and so $|A|=n$. Thus there exist at least $\lceil\frac{n}{b-a+1} \rceil \geq 3b+1$ edges in $A$ colored the same. Suppose there exist $m$ edges $v_{j_1}v,\cdots,v_{j_m}v,(1\leq j_1<j_2<\cdots<j_m\leq n)$ colored the same and $m\geq \lceil\frac{n}{b-a+1} \rceil \geq 3b+1$. Second, we consider the set of edges $B=\{v_{j_1}w,\cdots,v_{j_m}w\}$. Since $c(v_{j_i}w)\in \{1,2,\cdots,b-1\}$, for $1\leq i\leq m$, then there exist at least $\lceil\frac{m}{b-1} \rceil \geq \lceil\frac{3b+1}{b-1}\rceil \geq 4$ edges colored the same. Thus from $B$ we can choose $4$ edges of the same color. Since $n\geq 18$, from the corresponding vertices on the cycle $C_n$ of the four edges chosen above, we can get two vertices such that their distance on the cycle $C_n$ is more than $3$. Without loss of generality, we assume that the two vertices are $v_{1}^{'},v_{2}^{'}$ and their distance in graph $G$ is $2$. Then the geodesic between $v_{1}^{'}$ and $v_{2}^{'}$ in graph $G$ is either $v_{1}^{'}wv_{2}^{'}$ or $v_{1}^{'}vv_{2}^{'}$. However, neither $v_{1}^{'}wv_{2}^{'}$ nor $v_{1}^{'}vv_{2}^{'}$ is a rainbow path. Thus the coloring $c$ is not a strong rainbow coloring of $G$, a contradiction. Therefore $src(G)\leq b$ and so $src(G)=b$. The proof is thus complete. ------------------------------------------------------------------------ [20]{} J.A. Bondy, U.S.R. Murty. *Graph Theory*, Springer, Heidelberg, 2008. G. Chartrand, G.L. Johns, K.A. MeKeon, P. Zhang. Rainbow connection in graphs. *Math.Bohem.*, 133(1)(2008) 85-98. [^1]: Supported by NSFC.

--- abstract: 'Deformation-induced lateral migration of a bubble slowly rising near a vertical plane wall in a stagnant liquid is numerically and theoretically investigated. In particular, our focus is set on a situation with a short clearance $c$ between the bubble interface and the wall. Motivated by the fact that numerically and experimentally measured migration velocities are considerably higher than the velocity estimated by the available analytical solution using the Faxén mirror image technique for $a/(a+c)\ll 1$ (here $a$ is the bubble radius), when the clearance parameter $\varepsilon(= c/a)$ is comparable to or smaller than unity, the numerical analysis based on the boundary-fitted finite-difference approach solving the Stokes equation is performed to complement the experiment. The migration velocity is found to be more affected by the high-order deformation modes with decreasing $\varepsilon$. The numerical simulations are compared with a theoretical migration velocity obtained from a lubrication study of a nearly spherical drop, which describes the role of the squeezing flow within the bubble-wall gap. The numerical and lubrication analyses consistently demonstrate that when $\varepsilon\leq 1$, the lubrication effect makes the migration velocity asymptotically $\mu V_{B1}^2/(25\varepsilon \gamma)$ (here, $V_{B1}$, $\mu$, and $\gamma$ denote the rising velocity, the dynamic viscosity of liquid, and the surface tension, respectively).' author: - 'KAZUYASU SUGIYAMA$^1$ and FUMIO TAKEMURA$^{2, 1}$' title: On the lateral migration of a slightly deformed bubble rising near a vertical plane wall --- Introduction ============ Recent technical progress in generating microbubbles (e.g. [@gar2006; @mak2006]), including potentials as actuator and sensor, has enhanced the range of applications, e.g. additives to reduce a turbulence friction ([@ser2005]), drug delivery capsules ([@sho2004]), and contrast agents ([@cor2001]). In many situations, a bubble encounters a boundary wall during its transport process, and a hydrodynamic interaction occurs as characterized by the inter-scale between the bubble and the wall. In practice, it is of primary importance that the bubble undergoes a repulsive or attractive force in the wall-normal direction, which causes a lateral migration ([@lea1980; @mag2003; @hib2007]) and determines the bubble distribution, when translating parallel to the wall. As the simplest model system, one may raise a phenomenon of a spherical bubble rising near a vertical infinite plane wall in a creeping (Stokes) flow. However, there is no mechanism to generate the lateral migration force, as kinematic reversibility is ensured by symmetry of the boundary and by linearity in the Stokes equation ([@lea1992], chapter 4). In fact, the migration force stems from nonlinearities in the advective momentum transport ([@cox1968; @ho1974; @vas1976; @vas1977; @cox1977; @mcl1993; @che1994; @bec1996; @mag2003]) and/or the interfacial deformability ([@cha1965; @cha1979; @sha1988; @uij1993; @uij1995; @mag2003; @wan2006]), to break the symmetry. For a tank-treading vesicle translating parallel to the wall, the migration force was theoretically obtained by [@oll1997], in which the vesicle shape is prescribed as a strongly non-spherical ellipsoid, and the theory was experimentally validated by [@cal2008]. For a bubble or drop, the shape cannot be prescribed since it obeys the Laplace law and depends on the surrounding fluid flow. The theoretical success in solving the nontrivial problem of the deformation-induced migration of the bubble or drop was made by [@mag2003] using the Faxén mirror image technique and the Lorentz reciprocal theorem. However, [@wan2006] performed a numerical study on the motion of a drop with the same viscosity as the surrounding fluid in a linear shear flow by means of a boundary element method, and pointed out that the theory considerably underestimates the migration velocity or erroneously predicts the lateral motion, despite consistent predictions of the rising velocity and the interfacial deformation. Recently, [@tak2009] experimentally measured the lateral migration velocity of slightly deformed bubbles in a wall-bounded shear flow, and found a clear discrepancy between the experimental and theoretical values of the deformation-induced transverse force. Then, they computed the quasi-steady evolution of deformable bubbles moving in a wall-bounded linear shear flow at zero Reynolds number using a spectral boundary element method developed by [@dim2007], and found that the measured deformation-induced lift force agrees quantitatively well with the computational prediction. Motivated by their conclusions, we revisited experimental data of the bubble migration in a quiescent liquid obtained by [@tak2002], and analysed the data on the conditions that the clearance between the bubble interface and the wall is comparable to or shorter than the bubble radius, which was not considered there. The results revealed that the discrepancy between the migration velocities of the experiment and the theory increased as the bubble was closed to the wall, as detailed below. In this paper, we focus on the bubble motion in a quiescent liquid to simplify the subject. Let us consider the migration velocity $V_{B3}(={\bm V}_{B}\cdot {\bm e}_{3})$ of a bubble rising near a vertical plane wall at a distance $d$ between the bubble centroid and the wall as schematically illustrated in figure \[fig:schem\](a). The bubble has an equivalent radius $a$ to that of a sphere with the same volume. Introducing an interfacial deflection $f(\theta,\phi)$ from a sphere, we write the distance from the bubble centroid to the interface as $a+f$. The experimental results used here were measured under the condition that the Reynolds number ${\rm Re}=2\rho aV_{B1}/\mu$ (here $V_{B1}(={\bm V}_B\cdot{\bm e}_{1}$), $\rho$, and $\mu$ respectively denote the rising velocity, the density, and the dynamic viscosity of liquid) is unity or less ([@tak2002]). The pure lateral migration velocities induced by the deformation $V_{B3}$ were calculated from the measured values substituting the velocities induced by the inertia effects. Following a Stokes flow theory for the deformation-induced migration ([@mag2003]), we can characterize the system using two parameters, i.e., a clearance parameter $\varepsilon(= c/a)$ and a capillary number ${\rm Ca}=\mu V_{B1}/\gamma$ (or a Bond number ${\rm Bo}=\rho a^2 g/\gamma$ as used in [@mag2003]). Here $\gamma$ and $g$ respectively denote the surface tension and the acceleration of gravity. Further, as long as ${\rm Ca}\ll 1$, we may use ${\rm Ca}$ as a perturbation parameter, and reduce the ${\rm Ca}$-dependent system to another, in which $V_{B1}$, $V_{B3}/{\rm Ca}$ and $f/{\rm Ca}$ are dependent only upon $\varepsilon$, under the infinitesimal deformation assumption. Figure \[fig:migvel\_comp\_exp\] shows the migration velocity $V_{B3}$ away from the wall normalized by ${\rm Ca}V_{B1}$ as a function of $\kappa (\equiv (1+\varepsilon)^{-1})$. (It should be noticed that although $\kappa$ as well as $\varepsilon$ are measures of the distance between the wall and the bubble, hereafter $\kappa$ is also used to make some equations for the wide gap case ($\kappa\ll 1$) simple.) The measured velocity is found to be much higher than the analytical solution especially for the large $\kappa$. A possible inference drawn from this result is that there exists an additional ingredient to generate repulsive force for narrow bubble-wall gap, which is not covered by the theory of [@mag2003]. As the most crucial restriction involved in the mirror image technique, we can raise an assumption that the bubble-wall distance is much longer than the bubble radius. However, with regard to an inertia effect on the lateral velocity of a rigid sphere, [@tak2004] experimentally demonstrated that the mirror image approach has robust applicability in prediction beyond the wide-gap precondition. At this moment, we cannot conclude whether the discrepancy in the deformation-induced migration velocity comes from the erroneous prediction due to the miscalculation or the contradictory conditions in the boundary element computation and the experiment with the theoretical assumptions. To complement the experiment and to gain further insight into the deformation-induced migration, we investigate the migration behavior with attention to shortness of a bubble-wall clearance $c(=d-a)$. As in [@mag2003], using two Stokes flow solutions for a spherical bubble translating parallel and perpendicular to the wall, we apply the Lorentz reciprocal theorem to evaluating the migration velocity. We carry out numerical simulations using a boundary-fitted grid, which can accurately implement the boundary conditions and release the constraint of the sufficiently wide bubble-wall gap in the mirror image technique. In addition, comparisons with a theoretical migration velocity for $\varepsilon\ll 1$ obtained from a lubrication study of a nearly spherical drop moving near a tilted plane ([@hod2004]), in which the secondary flow due to the change in the boundary geometry caused by the bubble deformation is responsible for the wall-normal force, are made to shed more light on the short clearance effect. Numerical simulation ==================== General formulation {#sec:gf} ------------------- To clarify the physical mechanism of the repulsive force, we numerically address the bubble migration. In a similar manner to [@mag2003], instead of directly solving the flow field with the deformed bubble, we employ the Lorentz reciprocal theorem to determine the lateral migration force and velocity through coupling two flow fields around a spherical bubble translating parallel and perpendicular to the wall. In the subsequent developments, the basic equations and the involved variables are nondimensionalized using $a$, $V_{B1}$ and $\mu$. We suppose that the bubble quasi-steadily rises near an infinite flat plate in a stagnant incompressible liquid, and both the Reynolds and capillary numbers are sufficiently smaller than unity. Hence, the system is described by the steady Stokes equation for solenoidal velocity vectors, i.e. $$\nabla\cdot{\bm U}=\nabla\cdot{\bm u}=0, \ \ \ -\infty<x_1<\infty,\ -\infty<x_2<\infty,\ -1-\varepsilon\le x_3<\infty, \label{eq:cont_st}$$ $$\nabla\cdot{\bm \Sigma}=\nabla\cdot{\bm \sigma}=0, \ \ \ -\infty<x_1<\infty,\ -\infty<x_2<\infty,\ -1-\varepsilon\le x_3<\infty, \label{eq:mom_st}$$ where $({\bm U},{\bm \Sigma})$ and $({\bm u},{\bm \sigma})$ are the velocity and stress fields for the bubble translating respectively parallel and perpendicular at a speed of unity to the wall. The ${\rm Ca}$ dependence of the interfacial deflection is given by $f(\theta,\phi;{\rm Ca})\ ={\rm Ca}f^{({\rm Ca})}(\theta,\phi)$. The bubble deformation obeys the Laplace law for the infinitesimal deflection $|f|\ll 1$ with ${\rm Ca}\ll 1$, $$\left(\nabla_s^2+2\right)f^{({\rm Ca})}= -{\bm n}\cdot{\bm\Sigma}\cdot{\bm n} +3x_1 \langle x_1{\bm n}\cdot{\bm \Sigma}\cdot{\bm n}\rangle_{S_B} \ \ {\rm at}\ \ \sqrt{x_1^2+x_2^2+x_3^2}=1, \label{eq:laplace_law}$$ where ${\bm n}$ represents the normal unit vector pointing outwards the liquid, $\nabla_s(=\nabla-{\bm n}({\bm n}\cdot\nabla))$ is the nabla operator along the tangential directions on the bubble surface, $\langle ...\rangle_{S_B}$ is the area average taken over the bubble surface, and $x_1$ is the coordinates in the upward direction from the origin at the bubble centroid. Kinematic and free-slip conditions are imposed on the bubble surface, i.e. $$\left. \begin{array}{rl} {\bm n}\cdot{\bm U}&=0,\\ ({\bm n}\cdot{\bm\Sigma})\times{\bm n}&=0,\\ {\bm n}\cdot{\bm u}&=0,\\ ({\bm n}\cdot{\bm\sigma})\times{\bm n}&=0,\\ \end{array} \right\} \ \ {\rm at}\ \ \sqrt{x_1^2+x_2^2+x_3^2}=1, \label{eq:bc_bubsurf_st}$$ where we take the reference frames viewed from the bubble. On the plane wall, we impose the no-slip condition $$\left. \begin{array}{rl} {\bm U}&=-{\bm e}_1,\\ {\bm u}&={\bm e}_3, \end{array} \right\} \ \ {\rm at}\ \ x_3=-1-\varepsilon. \label{eq:wall_st}$$ Sufficiently far from the bubble, the velocity vectors approach the uniform velocities $$\left. \begin{array}{rl} {\bm U}&\rightarrow -{\bm e}_1,\\ {\bm u}&\rightarrow {\bm e}_3,\\ \end{array} \right\} \ \ {\rm as}\ \ \sqrt{x_1^2+x_2^2+x_3^2}\rightarrow \infty. \label{eq:far_st}$$ Thanks to the reciprocal theorem ([@lea1980]; (35) of [@mag2003]), the deformation-induced lateral force $F_M={\rm Ca}F_M^{({\rm Ca})}$ to cancel the migration velocity and to maintain the wall-parallel motion is expressed as $$F_{M}^{({\rm Ca})}= \oint_{S_B}\!\!\!\!\!{\rm d}^2{\bm x}\ {\cal L}(f^{({\rm Ca})}), \label{eq:fmca_st}$$ where $S_B$ denotes the bubble surface, and the operator ${\cal L}$ is given by $$\begin{split} &{\cal L}= {\bm n}\!\cdot{\bm \sigma}\!\cdot{\bm n} \left( \frac{\partial{\bm U}}{\partial n}\!\cdot\!{\bm n}-{\bm U}\!\cdot\!\nabla_s \right) -{\bm u}\!\cdot\!\biggl( \frac{\partial {\bm \Sigma}}{\partial n}\!\cdot\!{\bm n} -{\bm \Sigma}\!\cdot\!\nabla_s \biggr) \\& -\left\{ {\bm n}\cdot{\bm\Sigma}\cdot{\bm n} -3x_1\langle x_1 {\bm n}\cdot{\bm\Sigma}\cdot{\bm n}\rangle_{S_B} \right\}{\bm u}\cdot\nabla_s. \end{split}$$ The migration velocity $V_{B3}={\rm Ca}V_{B3}^{({\rm Ca})}$ is expressed as $$\frac{V_{B3}^{({\rm Ca})}}{V_{B1}}=\frac{F_{M}^{({\rm Ca})}}{F_{DC}}, \label{eq:vb3ca}$$ where $$F_{DC}=\oint_{S_B}\!\!\!{\rm d}^2{\bm x}\ {\bm e}_3\cdot{\bm\sigma}\cdot{\bm n} \label{eq:fdc}$$ denotes the drag force acting on the bubble translating perpendicular to the plane wall. Simulation method ----------------- The basic equations are numerically solved by the second-order finite-difference method discretized on the bipolar coordinates $(\xi,\eta)$ grid, which is boundary-fitted on both the bubble surface and the plane wall (see figure \[fig:schem\_bipolar\](a) in Appendix \[appendix\_a\]). We take care of the mass and momentum conservations in a discretized form. For technical detail on the discretization, see Appendix \[appendix\_a\]. The number of grid points is $N_\xi\times N_\eta = 200 \times 200$, and the grid is non-uniform and refined near the wall and the bubble surface. The computational procedure is based on a Simplified-Marker-And-Cell method ([@ams1970]) with a first-order Eulerian implicit time marching scheme. Such an unsteady scheme enables us to check whether the computation converges to the fully developed state through temporal changes in the budgets of the momentum and kinetic-energy transports. To avoid a problem associated with singularities in the discretization near the axis, we follow a method proposed by [@fuk2002]. For each run, we confirm that drag forces, numerically evaluated on the bubble surface, for both the perpendicular and parallel motions are in good agreement with the respective kinetic-energy dissipation rates, numerically integrated over the entire computational domain, normalized by the translational velocities with an error of less than $0.040\%$. Such an agreement between the surface and bulk quantities indicates that the computation is well converged and reaches to the steady state in view of the momentum and kinetic-energy budgets. Further, the drag force for the perpendicular motion shows quantitative agreement with the infinite series solution of [@bar1968] with an error of less than $0.043\%$. The drag force for the parallel motion approaches the wide-gap solution of [@mag2003] with increasing $\varepsilon$. To make sure of numerical stability and accuracy, we set the clearance parameter in a range of $10^{-3}\leq \varepsilon\leq 9$. We performed the convergence tests with varying the size $R_{\rm max}$ of the computational domain, the number $N_\xi$ of nodes in the gap between the bubble and the wall, and the number $N_\eta$ of nodes describing the bubble surface. We confirmed that the relative errors in the migration velocity $V_{B3}$ and the drag forces for the perpendicular and parallel bubble motions to those obtained on the base meshes $N_\xi\times N_\eta = 200 \times 200$ for various $\varepsilon$ decrease with increasing the size $R_{\rm max}$ and the resolutions $N_\xi$ and $N_\eta$. From the convergence behaviors, we deduce to obtain the migration velocities $V_{B3}$ with errors of much less than $1\%$ using the present base meshes, which are accurate enough for the subsequent discussion. Migration velocity ------------------ As shown in the inset of figure \[fig:migvel\_comp\_exp\], the ratio of the simulation to the analytical migration velocity becomes close to unity as $\kappa (=(1+\varepsilon)^{-1})$ approaches zero, indicating the simulation result is consistent with the analytical solution ([@mag2003]) for small $\kappa (< \sim 0.3)$. Such a consistency between the different approaches may deny the erroneous prediction of the migration velocity in [@mag2003], of which the possibility was pointed out by [@wan2006]. By contrast, the simulated migration velocity for the bubble closer to the wall with the clearance shorter than the bubble radius (i.e., for $\kappa > \sim 0.5$ presumably beyond the theoretical precondition $\kappa\ll 1$) is considerably higher than the analytical solution. Although the simulation result reveals lower velocity than the experimental one, the tendency of the higher velocity than the theoretical one for the narrow gap is qualitatively similar to that in the experiment. Thus, the present simulation also indicates the presence of the additional narrow-gap repulsive force. For an undeformed spherical bubble at small but non-zero Reynolds numbers $0<{\rm Re}\ll 1$, using the solutions to the equation set (\[eq:cont\_st\])–(\[eq:far\_st\]), we can also evaluate the inertia effect on the migration velocity $V_{B3}={\rm Re}\ V_{B3}^{({\rm Re})}$ from $$F_M^{({\rm Re})}= \frac{1}{2} \int_{{\cal V}}\!\!\!{\rm d}^3{\bm x}\ ({\bm e}_3-{\bm u})\cdot\{({\bm U}\cdot\nabla){\bm U}\}, \label{eq:fmre_st}$$ $$\frac{V_{B3}^{({\rm Re})}}{V_{B1}}=\frac{F_M^{({\rm Re})}}{F_{DC}}, \label{eq:vb3re_st}$$ where ${\cal V}$ stands for the entire volume of liquid around the bubble. The force expression (\[eq:fmre\_st\]) is theoretically justified for the case that the wall is placed within a Stokes expansion region, i.e., $O({\rm Re})< \kappa$ ([@vas1976]). Figure \[fig:migvel\_inert\] shows the inertia-driven migration velocity as a function of $\kappa$. The simulated profile is globally consistent with the analytical solution ([@mag2003]) even for the narrow gap $\kappa\sim 1$, as opposed to the profile of the deformation-induced migration velocity in figure \[fig:migvel\_comp\_exp\]. It should be noticed that the expression (\[eq:fmca\_st\]) of the deformation-induced lateral force is written in a surface integral form, while the inertia-driven force (\[eq:fmre\_st\]) in a volume integral form. The overall agreement in the inertia-driven migration velocity indicates that capturing the bulk velocity distributions is important for predicting the migration velocity and can be robustly attained by the mirror image technique over the wide range of $\kappa$. The wide range agreement with the theories ([@vas1976; @mag2003]) was also experimentally demonstrated for the sedimenting rigid particle in a range of $0.1<{\rm Re}< 1$ by [@tak2004] as long as the wall is placed in the Stokes expansion region. Contrastingly, as shown in figure \[fig:migvel\_comp\_exp\], the larger discrepancy between the deformation-induced migration velocities of the simulation and the theory with increasing $\kappa$ indicates that the migration velocity is sensitive to the local effect leading to the additional narrow-gap repulsive force, which may not be covered by the mirror image technique. Interfacial deformation ----------------------- To demonstrate the narrow-gap effect, we investigate the bubble deformation. We here examine the scaled interfacial deflection $\hat{f}^{({\rm Ca})}(\theta)=f(\theta,\phi)/({\rm Ca}\ \cos\phi)$. In the experiment, we estimated $f(\theta,\phi)$ taking a circumference of the bubble on the plane $x_2=0$. Figure \[fig:scale\_deform\] shows the angular profile of the deflection $-\hat{f}^{({\rm Ca})}$ for various $\kappa (=(1+\varepsilon)^{-1})$. As shown in figure \[fig:scale\_deform\](a) for the relatively wide gap $\varepsilon=0.67$, the analytical solution for $\kappa\ll 1$ ([@mag2003]) is consistent with the measured and simulated deflections. Note that the agreement between the theoretical and simulated deflections is confirmed to be better in the wider separation. For the narrower gap ($\varepsilon=0.10$, $0.25$), by contrast, the analytical solution of the deflection magnitude is smaller than the measured one especially in the wall neighborhood ($\theta\sim 0$), which may be related to the considerable underestimation of the migration velocity as shown in figure \[fig:migvel\_comp\_exp\]. Contrastingly, the present simulation quantitatively captures the local near-wall profile of the measured deflection as well as the global magnitude. To quantify the local effect of such a large discrepancy in $\hat{f}^{({\rm Ca})}$ on the migration velocity, we here describe the deflection in an expansion form $$\hat{f}^{({\rm Ca})}(\theta)= \sum_{n=2}^\infty \hat{f}_n^{({\rm Ca})} P_n^1(\cos\theta),$$ where $P_n^1$ represents the associated Legendre polynomial. Figure \[fig:deformationr\_kap\] shows the modal deflections $\hat{f}_n^{({\rm Ca})}$ for $2\leq n\leq 5$ as a function of $\kappa$. The simulation results are consistent with the measured deflections for all the shown modes. In predicting the bubble migration, the wide-gap theory ([@mag2003]) assumes that the mirror image primarily induces the deformation of the mode $n=2$. The leading-order of the $n=2$ deflection is $\hat{f}_2^{({\rm Ca})}=\kappa^2/4$ in the limit of $\kappa\rightarrow 0$. Considering the higher-order effect with respect to $\kappa$, [@mag2003] derived $\hat{f}_n^{({\rm Ca})}=\frac{1}{4}\kappa^2$ $\{1+\frac{3}{8}\kappa$ $(1+\frac{3}{8}\kappa+\frac{73}{64}$ $\kappa^2)\}$. For $\kappa < \sim 0.7$, such a higher-order $\kappa$ correction is responsible for the enhancement of the $n=2$ deflection from the leading-order one as seen in the better agreement with the measured and simulated deflections. However, the correction is not sufficient for the narrower gap $\kappa> \sim 0.7$, and thus the theoretical underestimation of the $n=2$ deflection becomes more serious with $\kappa$. Further, the theory does not cover the considerable increase in the higher-order $n\geq 3$ deflections with $\kappa$ as demonstrated by both the measurement and the simulation. The modal deflection $\hat{f}_n^{({\rm Ca})}$ is linked to the migration velocity and force as decomposed into $$V_{B3}^{({\rm Ca})}=\sum_{n=2}^\infty V_{B3,n}^{({\rm Ca})},\ \ F_M^{({\rm Ca})} = \sum_{n=2}^\infty F_{M ,n}^{({\rm Ca})},$$ which are $$\frac{V_{B3,n}^{({\rm Ca})}}{V_{B1}}=\frac{F_{M ,n}^{({\rm Ca})}}{F_{DC}}, \label{eq:vb3nca_fm}$$ $$F_{M ,n}^{({\rm Ca})}= \oint_{S_B}\!\!\!\!\!{\rm d}^2{\bm x}\ {\cal L}\left( \hat{f}_n^{({\rm Ca})}P_n^1(\cos\theta)\cos\phi \right).$$ Figure \[fig:modal\](a) shows the contribution of the modal deflection to the migration velocity for the modes $n=2$ and $n=3$. For small $\kappa$, the simulation result is consistent with the analytical solution ([@mag2003]), which considers only the $n=2$ deformation to cause the bubble migration. The inset shows that for small $\kappa$, the contribution of the mode $n=2$ is proportional to $\kappa^2$, while that of the mode $n=3$ to $\kappa^5$, whose exponent is not trivially proven but may be predictable extending the regular perturbation to the higher-order. The difference in the exponent ensures that the relative contribution of the mode $n=3$ to $n=2$ becomes more significant with $\kappa$. It should be noticed that although we confirmed that the migration force contribution $F_{M, 2}^{({\rm Ca})}$ of the mode $n=2$ increases as $\kappa\rightarrow 1$ (i.e., $\varepsilon\rightarrow 0$), the velocity contribution $V_{B3, 2}^{({\rm Ca})}$ reduces as shown in figure \[fig:modal\](a). It is because the slope of the $n=2$ migration force, $-{\rm d}\log F_{M, 2}^{({\rm Ca})}/{\rm d}\log \varepsilon$, in a logarithmic plot is more gentle than that of the drag force, $-{\rm d}\log F_{DC}/{\rm d}\log \varepsilon\rightarrow 1$ in the denominator of (\[eq:vb3nca\_fm\]) as $\varepsilon\rightarrow 0$. Figure \[fig:modal\](b) shows the modal contribution $V_{B3,n}^{({\rm Ca})}$ compared with the migration velocity $V_{B3}^{({\rm Ca})}$. The contribution of the mode $n=2$ monotonically decreases with $\kappa$. The contribution of the mode $n=3$ increases with $\kappa$ in the range of $\kappa< \sim 0.8$, while decreases in the greater $\kappa$. It is because the higher modal contributions for $n\geq 4$ become no longer disregarded. Moreover, the fact that all the shown modal contributions decay as $\kappa\rightarrow 1$ indicates that the further higher-order contributions become considerable, and the regular perturbation approach with respect to $\kappa$ is no longer effective. Comparison with small deformation theory in the lubrication limit {#sec:abe_lub} ----------------------------------------------------------------- Figures \[fig:scale\_deform\], \[fig:deformationr\_kap\] and \[fig:modal\] imply that for small $\varepsilon$, the bubble deformation is preferentially enhanced within the narrow bubble-wall gap, and then its squeezing effect promotes the bubble migration. To shed more light on the role of the hydrodynamics in the gap, comparisons with a small deformation theory in the lubrication limit will be made. It should be noticed that [@hod2004] have performed a lubrication study for a nearly spherical drop near a tilted plane in so-called ‘slipping’ regime, and derived the deformation-induced normal force. One can also access a relevant physical picture in theoretical studies on the lift force on an elastic body induced by its deformation ([@sek1993; @sko2004; @sko2005; @urz2007]). Following the spirit of the lubrication theory, we evaluate the migration velocity in the limit of $\varepsilon\rightarrow 0$. The basic equations for the lubrication analysis and the solutions are shown in Appendix \[appendix\_b\]. To be recalled and to be used for comparison with the simulation results, the preconditions and the perturbed quantities are detailed here. We prescribe the wall-parallel velocity $V_{B1}$, and employ standard lubrication assumption, i.e., $\varepsilon \ll 1$. We also suppose small capillary number ${\rm Ca}\ll 1$. As implied in (\[eq:def\_f\_lub\]), the deflection is $O({\rm Ca}\varepsilon^{-1/2}a)$, which has to be sufficiently smaller than the gap $\varepsilon a$ if the tilt angle of the near-wall interface from the plane wall is supposed to be small. Here we adopt an additional constraint $\delta\equiv\varepsilon^{-3/2}{\rm Ca}\ll 1$. For inner coordinates $(R,Z,\phi)$ (here $\varepsilon^{1/2}R=r$ and $\varepsilon^{-1}Z=z$ as illustrated in figure \[fig:schem\](b), (see e.g. [@gol1967; @one1967])), a parabolic profile $Z=H(R)\equiv 1+R^2/2$ represents the interface within the inner region $r\sim \varepsilon^{1/2}$, if the deformation is absent. Using $\varepsilon$, we write the velocity vector, the pressure, and the deflection $f^{\rm (Ca)}$ of the interface in an expansion form with respect to $\delta$ $$\begin{split} u_r(r,\phi,z)=& \hat{U}_r^{(0)}(R,Z)\cos\phi+ \delta\hat{U}_r^{(f)}(R,Z) \\& +\varepsilon\hat{U}_r^{(1)}(R,Z)\cos\phi +\delta\hat{U}_r^{(f,2)}(R,Z)\cos2\phi +O(\delta^2)+O(\varepsilon\delta)+..., \label{eq:ur_lub} \end{split}$$ $$\begin{split} u_\phi(r,\phi,z)=& \hat{U}_\phi^{(0)}(R,Z)\sin\phi\\ &+\varepsilon\hat{U}_\phi^{(1)}(R,Z)\sin\phi +\delta\hat{U}_\phi^{(f,2)}(R,Z)\sin2\phi +O(\delta^2)+O(\varepsilon\delta)+..., \end{split}$$ $$\begin{split} u_z(r,\phi,z)=& \varepsilon^{1/2} \left(\hat{U}_z^{(0)}(R,Z)\cos\phi+ \delta\hat{U}_z^{(f)}(R,Z)\right.\\ &\left.+\varepsilon\hat{U}_z^{(1)}(R,Z)\cos\phi +\delta\hat{U}_z^{(f,2)}(R,Z)\cos2\phi +O(\delta^2)+O(\varepsilon\delta)+...\right), \end{split}$$ $$\begin{split} p(r,\phi,z)=& \varepsilon^{-3/2} \left(\hat{P}^{(0)}(R)\cos\phi +\delta \hat{P}^{(f)}(R)\right.\\ &\left.+\varepsilon\hat{P}^{(1)}(R,Z)\cos\phi +\delta\hat{P}^{(f,2)}(R,Z)\cos2\phi +O(\delta^2)+O(\varepsilon\delta)+...\right), \label{eq:p0_lub} \end{split}$$ $$f^{\rm (Ca)}(r,\phi)= \varepsilon^{-1/2} \left( \hat{F}(R)\cos\phi +O(\delta)+O(\varepsilon) \right), \label{eq:def_f_lub}$$ whose scaling relations are suitable to all equations in Appendix \[appendix\_b\]. It should be noted that the terms with the superscript $(0)$, $(1)$ or $(f,2)$ in (\[eq:ur\_lub\])–(\[eq:p0\_lub\]) are proportional to $\cos\phi$, $\sin\phi$, $\cos 2\phi$ or $\sin 2\phi$, and thus provide no wall-normal force. Here we make clear the physical domain of validity of the condition $\delta\ll 1$ when the rising velocity $V_{B1}$, which is needed to evaluate ${\rm Ca}(=\mu V_{B1}/\gamma)$, is unknown. Following the analysis by [@one1967], we write the drag force $F_D$ acting on the spherical bubble translating parallel to the wall in a form $F_{D}=6\pi\mu a V_{B1}(A\log \varepsilon+B)$, where $A$ and $B$ are independent of $\varepsilon$. Considering the free-slip boundary condition on the bubble surface, we can analytically find $A=-1/5$. From our numerical data, we approximately estimate $B=0.6$. Hence, from the force balance $F_D=4\pi \rho a^3g/3$, for $\varepsilon\ll 1$, we evaluate the rising velocity $$V_{B1}= \frac{1}{(-\log\varepsilon+3)}\frac{10\rho a^2 g}{9\mu},$$ and obtain the following relation for the bubble radius $a$ to satisfy the condition $\delta\ll 1$. $$a \ll \sqrt{\frac{9\gamma}{10\rho g}}\varepsilon^{3/4}(-\log \varepsilon+3)^{1/2}.$$ To assure us of the appearance of the lubrication effect, figure \[fig:scale\_pres\] shows the profiles of the pressure $\hat{P}^{(0)}(=3R/(5H^2))$ and deflection $-\hat{F}(=3\log H/(5R))$ in the lubrication limit (see (\[eq:p\_lub\]) and (\[eq:f\_lub\]), respectively), which are compared with the simulation results of the scaled interfacial pressure $\varepsilon^{3/2} P/\cos\phi$ and the scaled deflection $\varepsilon^{1/2}\hat{f}^{({\rm Ca})}$ near the wall as a function of $\varepsilon^{-1/2}\theta$. For sufficiently small $\varepsilon$, the simulation data of the scaled pressure collapse onto the curve (\[eq:p\_lub\]) as shown in figure \[fig:scale\_pres\](a). Therefore, the scaled deflection profile reasonably approaches the lubrication solution (\[eq:f\_lub\]) with decreasing the bubble-wall gap as shown in figure \[fig:scale\_pres\](b). As plotted as the dashed-dotted curve in figure \[fig:scale\_deform\], the deflection based on (\[eq:def\_f\_lub\]) and (\[eq:f\_lub\]) is consistent with the measured and simulated deflections in the wall neighbour ($\theta\sim 0$). Hence, the discrepancy between the deflections of the narrow-gap experiment and the wide-gap theory ([@mag2003]) is attributable to the lubrication effect. From [@bar1968], the drag force acting on the bubble translating perpendicular to the plane wall is $F_{DC}\rightarrow 3\pi \mu /(2\varepsilon)$ as $\varepsilon\rightarrow 0$. Substituting this relation and (\[eq:fm\_lub\]) (i.e., $F_M$ in the lubrication limit) into (\[eq:vb3ca\]), we obtain the asymptotic solution of the migration velocity $$\frac{V_{B3}}{V_{B1}}=\frac{F_M}{F_{DC}} \rightarrow \frac{{\rm Ca}}{25}\varepsilon^{-1} \ \ \ {\rm as}\ \ \ \varepsilon\rightarrow 0. \label{eq:vb3_lub}$$ To make comparisons with the asymptotic solutions, figures \[fig:migvel\_comp\_exp\] and \[fig:migvel\_comp\_lub\] show the scaled migration velocity as functions of the inverse distance $\kappa (=(1+\varepsilon)^{-1})$ and the clearance parameter $\varepsilon$, respectively. The inset of figure \[fig:migvel\_comp\_lub\] shows the scaled force (\[eq:fm\_lub\]). The simulation results are consistent with two asymptotic behaviors based on the lubrication theory for $\varepsilon\ll 1$ as well as the mirror image technique for $\varepsilon \gg 1$ (i.e., $\kappa\ll 1$). As seen from figure \[fig:migvel\_comp\_lub\], the theories provide the different exponents of the migration velocity scaling with respect to $\varepsilon$, namely, $V_{B3}/V_{B1}\propto {\rm Ca}\ \varepsilon ^{-2}$ for $\varepsilon\gg 1$ and $V_{B3}/V_{B1}\propto {\rm Ca}\ \varepsilon ^{-1}$ for $\varepsilon\ll 1$. The puzzling finding in [@tak2009] that the theory of [@mag2003] accurately predicts the deformation but fails to predict quantitatively the deformation-induced migration velocity is explained from the fact that the ratio of the simulation result of $V_{B3}$ to the theoretical prediction is more sensitively dependent upon $\varepsilon$ than that of $\hat{f}_2^{({\rm Ca})}$ when the lubrication effect becomes relevant. For instance, for $\varepsilon=0.4$, $\varepsilon=0.2$ and $\varepsilon=0.1$, the simulation-to-theory ratios of $\hat{f}_2^{({\rm Ca})}$ are respectively $1.004$, $1.1$ and $1.3$ (figure \[fig:deformationr\_kap\]), while those of $V_{B3}$ are respectively $2.5$, $4.3$ and $7.3$ (figure \[fig:migvel\_comp\_exp\]). The lubrication effect is likely to compensate the large discrepancy between the migration velocities of the experiment and the wide-gap theory revealed in [@tak2009]. However, although the quantitative agreement between the interfacial deflections of the experiment and the simulation is shown in figures \[fig:scale\_deform\] and \[fig:deformationr\_kap\], the migration velocity in the experiment is still considerably higher than the simulated one. Its cause is not clear at this moment, and further joint researches among theory, numerics and experiment are needed to resolve this discrepancy problem. Conclusion ========== We numerically and theoretically investigated deformation-induced lateral migration of a bubble slowly rising near a vertical plane wall in a stagnant liquid. We focused on a situation with a short clearance $c$ between the bubble interface and the wall. We demonstrated that the wide-gap theory ([@mag2003]), which considers the $n=2$ deformation mode, describes the deformation-induced lift force as long as the bubble-wall gap is sufficiently wide ($a/(a+c)\ll 1$, here $a$ is the bubble radius). For the narrow-gap case with the clearance parameter $\varepsilon(= c/a)$ smaller than unity, we found that the higher-order $n\geq 3$ deformation modes crucially enhance the migration velocity, and the lubrication effect ([@hod2004]) appears to induce the migration velocity, which scales asymptotically like $V_{B3}\rightarrow {\rm Ca}\ \varepsilon^{-1}V_{B1}/25$ as $\varepsilon\rightarrow 0$. This contrasts with the case of the inertia-driven migration, to which the wide-gap theory demonstrated a robust applicability in prediction over a wide range of $\varepsilon$. The present simulation consistently served as bridge between the wide- and narrow-gap theories (see figure \[fig:migvel\_comp\_lub\]) as long as the bubble deformation is assumed to be infinitesimal. However, in spite of the qualitative success of the simulation in revealing the narrow-gap repulsive force, the deformation-induced migration velocity in the experiment is considerably higher by a factor of about 3 than the simulated one, as shown in figure \[fig:migvel\_comp\_exp\]. The experiment may inevitably involve unknown factors such as unsteadiness, imperfection from the infinite plate-fluid system, and measurement uncertainty, which cannot be captured by the simulation. Nevertheless, we have not expected such a large discrepancy because firstly the quantitative agreement between the interfacial deflections of the experiment and the simulation has been confirmed in figures \[fig:scale\_deform\] and \[fig:deformationr\_kap\], secondly the consistent inertia-driven migration velocity of a rigid sphere with the available theories ([@vas1976; @mag2003]) has been obtained by [@tak2004] using the same experimental setup, and thirdly considerable uncertainty seems not to be introduced into such a simple system as illustrated in figure \[fig:schem\]. Further joint researches among theory, numerics and experiment are needed to resolve this problem. As a possible factor causing the inconsistency, we raise a difference between the bubble deformation levels of the experiment and the present analysis. As stated in §\[sec:abe\_lub\], the infinitesimal deformation assumption in the lubrication limit is justified only for the case that $\delta(=\varepsilon^{-3/2}{\rm Ca})\ll 1$. Beyond this limitation, unexplored hydrodynamic ingredients possibly become important on the bubble migration. For the experimental data shown in figure \[fig:migvel\_comp\_exp\] and figure \[fig:migvel\_comp\_lub\], the maximum value of $\delta$ is $0.74$, which is less than but comparable to unity. Therefore, the bubble deformation is finite rather than infinitesimal, and is likely to induce the higher-order force, which is possibly comparable to or stronger than the leading migration force evaluated with the infinitesimal deformation theories. From the theoretical viewpoint, a tiny bubble experiment, which results in a tiny capillary number and thus a tiny deformation, is favorable for comparative study. However, such an experiment has often resulted in an undetectably low migration velocity and made an accurate measurement difficult. To overcome such a dilemma, the highly accurate boundary element computations (e.g. [@wan2006; @dim2007]) for various deformation levels would be helpful to complement the infinitesimal deformation theories. We thank Shu Takagi for fruitful discussion. Finite-difference descriptions of the basic equation set in bipolar coordinates {#appendix_a} ===============================================================================     We describe the basic equation set in the bipolar coordinates (see e.g. [@one1964; @hap1973] Appendix A-19) as illustrated in figure \[fig:schem\_bipolar\](a). The coordinates $(r,z)$ in figure \[fig:schem\] (b) are $$r=\frac{k\sin\eta}{{\cal D}},\ \ z=\frac{k\sinh\xi}{{\cal D}},$$ where ${\cal D}=\cosh\xi-\cos\eta$ and $k=\sqrt{\varepsilon(\varepsilon+2)}$. The bubble surface is located at $\xi=\alpha\equiv\log(1+\varepsilon+k)$ as shown in figure \[fig:schem\_bipolar\](a). The gradient of a scalar function $q$ is written as $$\nabla q= \frac{{\bm e}_\xi}{h_\xi}\frac{\partial q}{\partial\xi}+ \frac{{\bm e}_\eta}{h_\eta}\frac{\partial q}{\partial\eta}+ \frac{{\bm e}_\phi}{h_\phi}\frac{\partial q}{\partial\phi},$$ where ${\bm e}$ represents the unit vector, and its subscript the corresponding component. $h$ denotes the scale factor, defined by e.g. $h_\xi=\{(\partial x_1/\partial_\xi)^2$ $+(\partial x_2/\partial_\xi)^2$ $+(\partial x_3/\partial_\xi)^2\}^{1/2}$ (see e.g. [@bat1967], Appendix 2). Each component is explicitly given by $$h_\xi=h_\eta=\frac{k}{{\cal D}}, \ \ \ h_\phi=r.$$ The vector and pressure field $({\bm U},P)$ for the bubble translating parallel to the wall is written using the quantities with hat in a form (e.g. [@sug2008]) $${\bm U}= \left\{ \left( {\bm e}_\xi\hat{U}_\xi+{\bm e}_\eta\hat{U}_\eta \right)\cos\phi+ {\bm e}_\phi\hat{U}_\phi \sin\phi -{\bm e}_1 \right\},\ \ \ P=\hat{P}\cos\phi,$$ of which the Fourier expansion reduces the three-dimensional problem to the two-dimensional one. The vector and pressure field $({\bm u},p)$ for the bubble translating perpendicular to the wall is $${\bm u}= {\bm e}_\xi\hat{u}_\xi+{\bm e}_\eta\hat{u}_\eta+{\bm e}_3,\ \ \ p=\hat{p}.$$ It should be noticed that as opposed to the reference frames in §\[sec:gf\], we take those for $(\hat{U}_\xi, \hat{U}_\eta, \hat{U}_\phi)$ and $(\hat{u}_\xi, \hat{u}_\eta)$ viewed from the plane wall for convenience of the simulations. Consequently, $\hat{U}_\xi=\hat{U}_\eta=\hat{U}_\phi=\hat{u}_\xi=\hat{u}_\eta=0$ on the plane wall and on the boundary sufficiently far from the bubble. We follow a conventional staggered grid arrangement ([@har1965]), where the velocity component is located on the corresponding cell interface, and the pressure at the cell centre, as shown in figure \[fig:schem\_bipolar\](b). The basic equations are discretized by the second-order finite-difference scheme. To numerically guarantee the mass and momentum conservations, and to accurately conduct the numerical integration in computing the drag and migration forces, we use the exact values of the grid width, the cell interfacial area, and the control volume in the bipolar coordinates. To this end, for the integral of the scale factors $$g_\xi(\xi,\eta)= \int_{\alpha}^\xi\!\!\!{\rm d}\bar{\xi}\ h_\xi(\bar{\xi},\eta),\ \ \ g_{\xi\phi}(\xi,\eta)= \int_{\alpha}^\xi\!\!\!{\rm d}\bar{\xi}\ h_\xi(\bar{\xi},\eta)h_\phi(\bar{\xi},\eta),$$ $$g_\eta(\xi,\eta)= \int_{\pi}^\eta\!\!\!{\rm d}\bar{\eta}\ h_\eta(\xi,\bar{\eta}),\ \ \ g_{\eta\phi}(\xi,\eta)= \int_{\pi}^\eta\!\!\!{\rm d}\bar{\eta}\ h_\eta(\xi,\bar{\eta})h_\phi(\xi,\bar{\eta}),$$ $$g_{\xi\eta}(\xi,\eta)= \int_{\alpha}^\xi\!\!\!{\rm d}\bar{\xi} \int_{\pi}^\eta\!\!\!{\rm d}\bar{\eta}\ h_\xi(\bar{\xi},\bar{\eta})h_\eta(\bar{\xi},\bar{\eta}),\ \ g(\xi,\eta)= \int_{\alpha}^\xi\!\!\!{\rm d}\bar{\xi} \int_{\pi}^\eta\!\!\!{\rm d}\bar{\eta}\ h(\bar{\xi},\bar{\eta}),$$ we use the exact expressions $$g_\xi=\frac{2k}{\sin\eta}\biggl( \tan^{-1}\left(\frac{{\cal D}+{\cal C}}{\cal S}\right) -\tan^{-1}\left(\frac{{\cal D}_\alpha+{\cal C}_\alpha}{\cal S_\alpha}\right) \biggr),$$ $$g_\eta=-\frac{2k}{\sinh\xi} \tan^{-1}\left(\frac{{\cal D}+{\cal C}}{\cal S}\right),$$ $$\begin{split} g_{\xi\eta}=&k^2\biggl\{ \left(\frac{1}{\sinh^2\xi}-\frac{1}{\sin^2\eta}\right)\tan^{-1}\left(\frac{{\cal D}+{\cal C}}{\cal S}\right) \\& - \left(\frac{1}{\sinh^2\alpha}-\frac{1}{\sin^2\eta}\right)\tan^{-1}\left(\frac{{\cal D}_\alpha+{\cal C}_\alpha}{\cal S_\alpha}\right) -\left( \frac{{\cal C}+2}{2{\cal S}}-\frac{{\cal C}_\alpha+2}{2{\cal S}_\alpha} \right)\biggr\}, \end{split}$$ $$g_{\eta\phi}=k^2\left( -\frac{1}{{\cal D}}+\frac{1}{\cosh\xi+1} \right),$$ $$g_{\xi\phi}=k^2\left\{ \frac{2\cos\eta}{\sin^2\eta} \left( \tan^{-1}\left(\frac{{\cal D}+{\cal C}}{\cal S}\right)-\tan^{-1}\left(\frac{{\cal D}_\alpha+{\cal C}_\alpha}{\cal S_\alpha}\right) \right) +\frac{\sinh\xi}{{\cal D}\sin\eta}-\frac{\sinh\alpha}{{\cal D}_\alpha\sin\eta} \right\},$$ $$\begin{split} g=&-\frac{k^3}{\sin^3\eta}\left\{ \left( \tan^{-1}\left(\frac{{\cal D}+{\cal C}}{\cal S}\right)-\tan^{-1}\left(\frac{{\cal D}_\alpha+{\cal C}_\alpha}{\cal S_\alpha}\right) \right)\cos\eta+\frac{{\cal S}}{2{\cal D}}-\frac{{\cal S}_\alpha}{2{\cal D}_\alpha} \right\} \\& -\frac{k^3}{12}\left( \tanh^3\frac{\xi}{2}-\tanh^3\frac{\alpha}{2} \right) +\frac{k^3}{4}\left( \tanh\frac{\xi}{2}-\tanh\frac{\alpha}{2} \right), \end{split}$$ where $$\begin{split} &h=h_\xi h_\eta h_\phi,\ \ \ {\cal C}=\cosh\xi\cos\eta-1,\ \ \ {\cal S}=\sinh\xi\sin\eta, \\& {\cal D}_\alpha=\cosh\alpha-\cos\eta,\ \ \ {\cal C}_\alpha=\cosh\alpha\cos\eta-1,\ \ \ {\cal S}_\alpha=\sinh\alpha\sin\eta. \end{split}$$ We introduce the finite-difference operators $\delta_i$ and $\delta_j$, of which the indices $i$ and $j$ correspond to discretized coordinates along the respective directions $\xi$ and $\eta$, such as $$\left. \begin{array}{rcl} \delta_i(q)|_{i,j}&=&q_{i+\frac{1}{2},j}-q_{i-\frac{1}{2},j}, \\ \delta_j(q)|_{i,j}&=&q_{i,j+\frac{1}{2}}-q_{i,j-\frac{1}{2}}, \\ \delta_i\delta_j(q)|_{i,j}&=& q_{i+\frac{1}{2},j+\frac{1}{2}}- q_{i+\frac{1}{2},j-\frac{1}{2}}- q_{i-\frac{1}{2},j+\frac{1}{2}}+ q_{i-\frac{1}{2},j-\frac{1}{2}}. \end{array} \right\}$$ Using these operators, we write the divergence of the velocity vector ${\bm U}$ in (\[eq:cont\_st\]) as $$\left.\widehat{\nabla\cdot{\bm U}}\right|_{i,j} \left(\equiv \frac{\nabla\cdot{\bm U}}{\cos\phi}\right) = \frac{\left( \delta_i(\delta_j(g_{\eta\phi})\hat{U}_\xi)|_{i,j} + \delta_j(\delta_i(g_{\xi\phi})\hat{U}_\eta)|_{i,j} + \hat{U}_\phi\delta_i\delta_j(g_{\xi\eta})|_{i,j} \right)}{\delta_i\delta_j(g)|_{i,j}},$$ and the components of the Stokes equation (\[eq:mom\_st\]) for the bubble translating parallel to the plane wall as $$\left. \begin{array}{rcl} 0&=&\displaystyle \frac{\delta_i(-\hat{P}+\widehat{\nabla\cdot{\bm U}})|_{i+\frac{1}{2},j}}{\delta_i(g_\xi)|_{i+\frac{1}{2},j}} +\frac{-\delta_j(r\hat{\Omega}_\phi)|_{i+\frac{1}{2},j}+\hat{\Omega}_\eta\delta_j(g_\eta)|_{i+\frac{1}{2},j}}{\delta_j(g_{\eta\phi})|_{i+\frac{1}{2},j}}, \\ 0&=&\displaystyle \frac{\delta_j(-\hat{P}+\widehat{\nabla\cdot{\bm U}})|_{i,j+\frac{1}{2}}}{\delta_j(g_\eta)|_{i,j+\frac{1}{2}}} +\frac{\delta_i(r\hat{\Omega}_\phi)|_{i,j+\frac{1}{2}}-\hat{\Omega}_\xi\delta_i(g_\xi)|_{i,j+\frac{1}{2}}}{\delta_i(g_{\xi\phi})|_{i,j+\frac{1}{2}}}, \\ 0&=&\displaystyle \frac{(\hat{P}-\widehat{\nabla\cdot{\bm U}})|_{i,j}}{r|_{i,j}} +\frac{-\delta_i(\delta_j(g_\eta)\hat{\Omega}_\eta)|_{i,j}+\delta_j(\delta_i(g_\xi)\hat{\Omega}_\xi)|_{i,j}} {\delta_i\delta_j(g_{\xi\eta})|_{i,j}}, \end{array} \right\}$$ where $\hat{\Omega}$ denotes the vorticity, of which each component is $$\left. \begin{array}{rcl} \hat{\Omega}_\xi|_{i,j+\frac{1}{2}}&=&\displaystyle \frac{\delta_j(r\hat{U}_\phi)|_{i,j+\frac{1}{2}}+\hat{U}_\eta\delta_j(g_\eta)|_{i,j+\frac{1}{2}}} {\delta_j(g_{\eta\phi})|_{i,j+\frac{1}{2}}}, \\ \hat{\Omega}_\eta|_{i+\frac{1}{2},j}&=&\displaystyle \frac{-\delta_i(r\hat{U}_\phi)|_{i+\frac{1}{2},j}-\hat{U}_\xi\delta_i(g_\xi)|_{i+\frac{1}{2},j}} {\delta_i(g_{\xi\phi})|_{i+\frac{1}{2},j}}, \\ \hat{\Omega}_\phi|_{i+\frac{1}{2},j+\frac{1}{2}}&=&\displaystyle \frac{\delta_i(\delta_j(g_\eta)\hat{U}_\eta)|_{i+\frac{1}{2},j+\frac{1}{2}}- \delta_j(\delta_i(g_\xi)\hat{U}_\xi)|_{i+\frac{1}{2},j+\frac{1}{2}}} {\delta_i\delta_j(g_{\xi\eta})|_{i+\frac{1}{2},j+\frac{1}{2}}}. \end{array} \right\}$$ Replacing $\hat{U}_\xi$, $\hat{U}_\eta$ and $\hat{P}$ respectively by $\hat{u}_\xi$, $\hat{u}_\eta$ and $p$ with $\hat{U}_\phi=\hat{\Omega}_\xi=\hat{\Omega}_\eta=0$, we readily obtain the governing equation for the field $(\hat{u}_\xi,\hat{u}_\eta,\hat{p})$. We write the kinematic and free-slip conditions (\[eq:bc\_bubsurf\_st\]) as $$\hat{U}_\xi|_{N_\alpha+\frac{1}{2},j}= \widehat{{\bm e}_r\cdot{\bm e}_\xi}|_{N_\alpha+\frac{1}{2},j}, \ \ \hat{u}_\xi|_{N_\alpha+\frac{1}{2},j}= -{\bm e}_z\cdot{\bm e}_\xi|_{N_\alpha+\frac{1}{2},j},$$ $$\hat{\Sigma}_{\xi\eta}|_{N_\alpha+\frac{1}{2},j+\frac{1}{2}} =\hat{\Sigma}_{\xi\phi}|_{N_\alpha+\frac{1}{2},j} =\hat{\sigma}_{\xi\eta}|_{N_\alpha+\frac{1}{2},j+\frac{1}{2}}=0,$$ where the index $N_\alpha+\frac{1}{2}$ denotes the node at the bubble surface $\xi=\alpha$, $$\widehat{{\bm e}_r\cdot{\bm e}_\xi}= \frac{\delta_j(g_{\xi\eta}^{(\xi)})}{\delta_j(g_{\eta\phi})}, \ \ \ {\bm e}_z\cdot{\bm e}_\xi= -\frac{\delta_j(r^2)}{2\delta_j(g_{\eta\phi})},$$ $$g_{\xi\eta}^{(\xi)} =\int_{\pi}^\eta\!\!\!{\rm d}\bar{\eta} \left(-\frac{k^2{\cal S}\sin\bar{\eta}}{{\cal D}^3}\right) = k^2\Biggl\{ \frac{1}{\sinh^2\xi} \tan^{-1}\left(\frac{{\cal D}+{\cal C}}{{\cal S}}\right) +\frac{{\cal C}\sin\eta}{2{\cal D}^2\sinh\xi} \Biggr\},$$ $$\hat{\Sigma}_{\xi\eta}= \frac{1}{\delta_i\delta_j(g_{\xi\eta})} \left\{ \overline{\delta_i(g_\xi )}^j\delta_j(\hat{U}_\xi )-\delta_i\delta_j(g_\xi )\overline{\hat{U}_\xi }^j+ \overline{\delta_j(g_\eta)}^i\delta_i(\hat{U}_\eta)-\delta_i\delta_j(g_\eta)\overline{\hat{U}_\eta}^i \right\},$$ $$\hat{\Sigma}_{\xi\phi}= \frac{1}{\delta_i(g_{\xi\phi})} \left\{ \overline{r}^i\delta_i(\hat{U}_\phi) -\delta_i(r)\overline{\hat{U}_\phi}^i -\delta_i(g_\xi)\hat{U}_\xi \right\},$$ $$\hat{\sigma}_{\xi\eta}= \frac{1}{\delta_i\delta_j(g_{\xi\eta})} \left\{ \overline{\delta_i(g_\xi )}^j\delta_j(\hat{u}_\xi )-\delta_i\delta_j(g_\xi )\overline{\hat{u}_\xi }^j+ \overline{\delta_j(g_\eta)}^i\delta_i(\hat{u}_\eta)-\delta_i\delta_j(g_\eta)\overline{\hat{u}_\eta}^i \right\},$$ and the overline stands for the interpolation such as $$\overline{q}^i|_{i,j}=\frac{q_{i+\frac{1}{2},j}+q_{i-\frac{1}{2},j}}{2},\ \ \ \overline{q}^j|_{i,j}=\frac{q_{i,j+\frac{1}{2}}+q_{i,j-\frac{1}{2}}}{2}.$$ We write the area integral on the bubble surface in a summation form $$\oint_{S_B}\!\!\!\!\!{\rm d}{\bm x}^2\ q \equiv \int_0^{2\pi}\!\!\!{\rm d}\phi \int_0^{\pi}\!\!\!{\rm d}\eta\ h_\eta h_\phi\ q|_{\xi=\alpha} = \sum_{j=1}^{N_j} \left(\delta_j(g_{\eta\phi}) \int_0^{2\pi}\!\!\!{\rm d}\phi\ q\right)_{N_\alpha+\frac{1}{2},j},$$ where $N_j$ is the number of grid points in the $\eta$-direction. The drag force $F_{DC}$ in (\[eq:fdc\]) is given by $$F_{DC}= 2\pi \sum_{j=1}^{N_j} \left(\delta_j(g_{\eta\phi}) {\bm e}_z\cdot{\bm e}_\xi \overline{\hat{\sigma}_{\xi\xi}}^i \right)_{N_\alpha+\frac{1}{2},j},$$ where $$\hat{\sigma}_{\xi\xi}= -\hat{p}+ \frac{2\overline{\delta_j(g_{\eta\phi})}^i\delta_i(\hat{u}_\xi)}{\delta_i\delta_j(g)} +\frac{2\overline{r}^j\delta_i\delta_j(g_\xi)\overline{\hat{u}_\eta}^j}{\delta_i\delta_j(g)}.$$ For the deflection $\hat{f}=f^{({\rm Ca})}/(a\cos\phi)$, the Laplace law (\[eq:laplace\_law\]) is expressed as $$a_{n,j}^{(f)}\hat{f}|_{j+1}+a_{s,j}^{(f)}\hat{f}|_{j-1}-a_{p,j}^{(f)}\hat{f}|_{j} =S^{(f)}|_j,$$ where $$\left. \begin{array}{l} \displaystyle a_{n,j}^{(f)}=\frac{r_{N_\alpha+\frac{1}{2},j+\frac{1}{2}}} {\overline{\delta_j(g_{\eta})}^i|_{N_\alpha+\frac{1}{2},j+\frac{1}{2}} \delta_j(g_{\eta\phi})|_{N_\alpha+\frac{1}{2},j}},\\ \displaystyle a_{s,j}^{(f)}=\frac{r_{N_\alpha+\frac{1}{2},j-\frac{1}{2}}} {\overline{\delta_j(g_{\eta})}^i|_{N_\alpha+\frac{1}{2},j-\frac{1}{2}} \delta_j(g_{\eta\phi})|_{N_\alpha+\frac{1}{2},j}},\\ \displaystyle a_{p,j}^{(f)}=\frac{ a_{n,j}^{(f)}(\widehat{{\bm e}_r\!\cdot\!{\bm e}_\xi})_{N_\alpha+\frac{1}{2},j+1}+ a_{s,j}^{(f)}(\widehat{{\bm e}_r\!\cdot\!{\bm e}_\xi})_{N_\alpha+\frac{1}{2},j-1} }{ (\widehat{{\bm e}_r\!\cdot\!{\bm e}_\xi})_{N_\alpha+\frac{1}{2},j }}, \end{array} \right\},$$ $$S^{(f)}=-\overline{\hat{\Sigma}_{\xi\xi}}^i+\frac{ \widehat{{\bm e}_r\!\cdot\!{\bm e}_\xi} \sum_{j=1}^{N_j}(\delta_j(g_{\eta\phi})\widehat{{\bm e}_r\!\cdot\!{\bm e}_\xi}\overline{\hat{\Sigma}_{\xi\xi}}^i)_j}{ \sum_{j=1}^{N_j}(\delta_j(g_{\eta\phi})\widehat{{\bm e}_r\!\cdot\!{\bm e}_\xi})^2_j},$$ $$\hat{\Sigma}_{\xi\xi}= -\hat{P}+ \frac{2\overline{\delta_j(g_{\eta\phi})}^i\delta_i(\hat{U}_\xi)}{\delta_i\delta_j(g)} +\frac{2\overline{r}^j\delta_i\delta_j(g_\xi)\overline{\hat{U}_\eta}^j}{\delta_i\delta_j(g)}.$$ The deformation-induced lateral force $F_M^{({\rm Ca})}$ in (\[eq:fmca\_st\]) is given by $$\begin{split} &F_M^{({\rm Ca})} = \pi\sum_{j}\biggl[ \frac{ \delta_{j}(g_{\eta\phi}) \overline{\hat{\sigma}_{\xi\xi}}^i \delta_i(\hat{U}_\xi-\widehat{{\bm e}_r\!\cdot\!{\bm e}_\xi}) \hat{f} }{\delta_i(g_\xi)} + \frac{ r\delta_i\delta_j(g_\xi) \overline{\overline{\hat{\sigma}_{\xi\xi}}^i}^j \overline{(\hat{U}_\eta-\widehat{{\bm e}_r\!\cdot\!{\bm e}_\eta})}^i \overline{\hat{f}}^j }{\overline{\delta_i(g_\xi)}^j} \\& + \delta_j(g_{\eta\phi})\overline{\hat{\sigma}_{\xi\xi}}^i \overline{\left(\frac{(\hat{U}_\phi+1)}{r}\right)}^i \hat{f} - r\overline{\overline{\hat{\sigma}_{\xi\xi}}^i}^j \overline{(\hat{U}_\eta-\widehat{{\bm e}_r\!\cdot\!{\bm e}_\eta})}^i \delta_j(\hat{f}) \\& - \frac{ \overline{\delta_j(g_{\eta\phi})}^j \overline{(\hat{u}_\eta+{\bm e}_z\!\cdot\!{\bm e}_\eta)}^i \delta_i(\hat{\Sigma}_{\xi\eta}) \overline{\hat{f}}^j}{\delta_i(g_\xi)} + \frac{r \delta_i\delta_j(g_\xi) \overline{(\hat{u}_\eta+{\bm e}_z\!\cdot\!{\bm e}_\eta)}^i }{\overline{\delta_i(g_\xi)}^j} \left( \overline{\overline{\hat{\Sigma}_{\xi \xi }}^i}^j- \overline{\overline{\hat{\Sigma}_{\eta\eta}}^i}^j \right)\overline{\hat{f}}^j \\& + r \overline{(\hat{u}_\eta+{\bm e}_z\!\cdot\!{\bm e}_\eta)}^i \overline{\overline{\hat{\Sigma}_{\eta\eta}}^i}^j \delta_j(\hat{f}) - \frac{ \overline{(\hat{u}_\eta+{\bm e}_z\!\cdot\!{\bm e}_\eta)}^i }{r} \overline{ \left( \delta_j(g_{\eta\phi}) \hat{\Sigma}_{\eta\phi} \right)}^i \overline{\hat{f}}^j \\& + r\overline{(\hat{u}_\eta+{\bm e}_z\!\cdot\!{\bm e}_\eta)}^i \overline{S^{(f)}}^j \delta_j(\hat{f}) \biggr]_{N_\alpha+\frac{1}{2}}, \end{split}$$ where $$\widehat{{\bm e}_r\cdot{\bm e}_\eta}=\frac{\delta_i(g_{\xi\eta}^{(\eta)})}{\delta_i(g_{\xi\phi})}, \ \ \ {\bm e}_z\cdot{\bm e}_\eta=\frac{\delta_i(r^2)}{2\delta_i(g_{\xi\phi})},$$ $$\begin{split} g_{\xi\eta}^{(\eta)} &=\int_{\alpha}^\xi\!\!\!{\rm d}\bar{\xi}\ \frac{k^2{\cal C}\sin\eta}{{\cal D}^3} = k^2\Biggl\{ -\frac{1}{\sin^2\eta} \left( \tan^{-1}\left(\frac{{\cal D}+{\cal C}}{{\cal S}}\right)- \tan^{-1}\left(\frac{{\cal D}_\alpha+{\cal C}_\alpha}{{\cal S}_\alpha}\right) \right) \\& -\left( \frac{{\cal C}+2}{2{\cal S}}-\frac{{\cal C}_\alpha+2}{2{\cal S}_\alpha} \right) -\left( \frac{{\cal C}\sin\eta}{2{\cal D}^2\sinh\xi} -\frac{{\cal C}_\alpha\sin\eta}{2{\cal D}_\alpha^2\sinh\alpha} \right) \Biggr\}, \end{split}$$ $$\hat{\Sigma}_{\eta\eta}= -\hat{P}+ \frac{2\overline{\delta_i(g_{\xi\phi})}^j\delta_j(\hat{U}_\eta)}{\delta_i\delta_j(g)} +\frac{2\overline{r}^i\delta_i\delta_j(g_\eta)\overline{\hat{U}_\xi}^i}{\delta_i\delta_j(g)},$$ $$\hat{\Sigma}_{\eta\phi}= \frac{1}{\delta_j(g_{\eta\phi})} \left\{ \overline{r}^j\delta_j(\hat{U}_\phi) -\delta_j(r)\overline{\hat{U}_\phi}^j -\delta_j(g_\eta)\hat{U}_\eta \right\}.$$ Small deformation theory in the lubrication limit {#appendix_b} ================================================= The governing equations for $\hat{U}_r^{(0)}$, $\hat{U}_\phi^{(0)}$, $\hat{U}_z^{(0)}$, $\hat{P}^{(0)}$, $\hat{U}_r^{(f)}$, $\hat{U}_z^{(f)}$ and $\hat{P}^{(f)}$ in (\[eq:ur\_lub\])–(\[eq:def\_f\_lub\]) are written as $$\frac{1}{R}\frac{\partial(R \hat{U}_r^{(0)})}{\partial R}+ \frac{\hat{U}_\phi^{(0)}}{R}+ \frac{\partial \hat{U}_z^{(0)}}{\partial Z}= \frac{1}{R}\frac{\partial(R \hat{U}_r^{(f)})}{\partial R}+ \frac{\partial \hat{U}_z^{(f)}}{\partial Z}=0, \label{eq:cont}$$ $$\begin{split} &-\frac{\partial \hat{P}^{(0)}}{\partial R}+\frac{\partial^2 \hat{U}_r^{(0)}}{\partial Z^2} =\frac{\hat{P}^{(0)}}{R}+\frac{\partial^2 \hat{U}_\phi^{(0)}}{\partial Z^2} =\frac{\partial \hat{P}^{(0)}}{\partial Z} \\=& -\frac{\partial \hat{P}^{(f)}}{\partial R}+\frac{\partial^2 \hat{U}_r^{(f)}}{\partial Z^2} =\frac{\partial \hat{P}^{(f)}}{\partial Z} =0, \end{split} \label{eq:mom}$$ with the no-slip boundary condition on the plane wall $${\bm u}=0\ \ \ {\rm at}\ \ \ Z=0, \label{eq:noslip_wall}$$ and the free-slip and kinematic boundary conditions on the bubble surface. The deformed interface is located on the curve where $H-\delta\hat{F}\cos\phi-Z=0$ holds. From this relation, the normal unit vector ${\bm n}$ pointing outwards the liquid on the bubble surface is approximated by $${\bm n}(R,\phi)= -{\bm e}_z +{\bm e}_r\varepsilon^{1/2}\left( R-\delta\frac{{\rm d}\hat{F}}{{\rm d}R}\cos\phi \right) +{\bm e}_\phi \varepsilon^{1/2}\delta\frac{\hat{F}}{R}\sin\phi+... \label{eq:approx_n}$$ Applying the Taylor expansion to a function $q=q^{(0)}+\delta\ q^{(f)}+...$ in terms of the deflection around the undeformed interface, one obtains a relation on the deformed interface $Z=H-\delta\hat{F}\cos\phi$ $$\begin{split} q|_{\rm interface} =& q|_{Z=H}^{(0)} + \delta \left( -\hat{F}\cos\phi\left. \frac{\partial q^{(0)}}{\partial Z}\right|_{Z=H} +q|_{Z=H}^{(f)} \right)+... \label{eq:approx_q} \end{split}$$ Taking (\[eq:approx\_n\]) and (\[eq:approx\_q\]) into account, one writes the kinematic condition on the bubble surface as $$\begin{aligned} &R\hat{U}_r^{(0)}-\hat{U}_z^{(0)}-R \\=& R\hat{U}_r^{(f)}-\hat{U}_z^{(f)} -\frac{1}{2}\frac{{\rm d}\hat{F}}{{\rm d}R}\hat{U}_r^{(0)} +\frac{\hat{F}}{2R}\hat{U}_\phi^{(0)} -\frac{\hat{F}R}{2}\frac{\partial \hat{U}_r^{(0)}}{\partial Z} \nonumber\\& +\frac{\hat{F}}{2}\frac{\partial \hat{U}_z^{(0)}}{\partial Z} + \frac{1}{2} \left( \frac{{\rm d}\hat{F}}{{\rm d}R} +\frac{\hat{F}}{R} \right)=0 \ \ {\rm at}\ \ Z=H, \label{eq:bc_kin1}\end{aligned}$$ and the free-slip boundary condition as $$\begin{aligned} \frac{\partial \hat{U}_r^{(0)}}{\partial Z}= \frac{\partial \hat{U}_\phi^{(0)}}{\partial Z}= \frac{\partial \hat{U}_r^{(f)}}{\partial Z} -\frac{\hat{F}}{2}\frac{\partial^2 \hat{U}_r^{(0)}}{\partial Z^2}= 0 \ \ \ {\rm at}\ \ \ Z=H. \label{eq:bc_fsr0}\end{aligned}$$ The vertical drag force acting on the bubble, which is involved in the second term in the right-hand-side of (\[eq:laplace\_law\]), is of order $\log\varepsilon$ as determined for a motion of a rigid sphere ([@gol1967; @one1967]) by means of the matched asymptotic expansion technique and the normal stress on the bubble surface is dominated by the pressure $p = O(\varepsilon^{-3/2})$ as compared with $\partial_r u_r = O(\varepsilon^{-1/2})$, $u_r/r = O(\varepsilon^{-1/2})$, $u_\phi/r = O(\varepsilon^{-1/2})$ and $\partial_z u_z = O(\varepsilon^{-1/2})$, which are related to the viscous stresses. Hence, the Laplace law (\[eq:laplace\_law\]) is simplified into $$\frac{{\rm d}}{{\rm d}R}\left( \frac{1}{R} \frac{{\rm d}(R\hat{F})}{{\rm d}R} \right)=\hat{P}^{(0)}, \label{eq:laplace1}$$ with no singularity conditions $\hat{F}=0$ at $R=0$ and $\hat{F}\rightarrow 0$ as $R\rightarrow \infty$. [As obtained by [@hod2004], the leading-order and perturbed solutions are]{} $$\begin{split} &\hat{U}_r^{(0)}=\frac{(6-9R^2)}{20H}\left( \frac{Z^2}{H^2}-\frac{2Z}{H}\right), \ \ \ \hat{U}_\phi^{(0)}=-\frac{3}{10}\left( \frac{Z^2}{H^2}-\frac{2Z}{H}\right), \\& \hat{U}_z^{(0)}= \frac{(4R-R^3)Z^3}{5H^4}+\frac{(-42R+3R^3)Z^2}{20H^3}, \label{eq:u_lub} \end{split}$$ $$\hat{P}^{(0)}=\frac{3R}{5H^2}, \label{eq:p_lub}$$ $$\hat{F}=-\frac{3\log H}{5R}. \label{eq:f_lub}$$ $$\begin{split} \hat{U}_r^{(f)}(R,Z)=& A_2(R)Z^2+A_1(R)Z,\\ \hat{U}_z^{(f)}(R,Z)=& -\frac{1}{3R}\frac{{\rm d}(RA_2)}{{\rm d}R}Z^3 -\frac{1}{2R}\frac{{\rm d}(RA_1)}{{\rm d}R}Z^2, \label{eq:uz1} \end{split}$$ $$\hat{P}^{(f)}(R)=\int_\infty^R\!\!\!{\rm d}\bar{R}\ 2A_2(\bar{R}), \label{eq:p1}$$ where $$A_1=\frac{9(14-R^2)\log H}{100H^3R}, \label{eq:a1}$$ $$A_2=-\frac{9(4-R^2)\log H}{50H^4R}. \label{eq:a2}$$ Substituting (\[eq:a2\]) into (\[eq:p1\]) estimates the asymptotic order of the perturbed pressure $$\varepsilon^{-3/2}\delta \hat{P}^{(f)}= \frac{12\varepsilon^{-3}{\rm Ca} \log(R^2/2)}{25R^6}+O(R^{-6}) \ \ \ {\rm for}\ \ \ R\gg 1,$$ which is $O({\rm Ca}\log\varepsilon\ (\varepsilon^{1/2}R)^{-6})$ in the overlapping region $R\sim \varepsilon^{-1/2}$, and thus to be matched with the pressure of $O({\rm Ca}\log\varepsilon\ r^{-6})$ in the outer region ([@one1967]). This outer pressure may contribute to the lateral force of $O({\rm Ca}\log\varepsilon)$, which is larger than that for the wide-gap case, corresponding to $O({\rm Ca})$. Nevertheless, as discussed in [@urz2007], the rapid decay of the perturbed pressure for $R\gg 1$ indicates that the contribution of the outer pressure to the lateral force is negligibly smaller than that of the inner pressure. Therefore, to evaluate the leading-order migration force, one does not have to solve the outer problem. (In fact, the order of the lateral force (\[eq:fm\_lub\]) evaluated only in the inner region is confirmed to be $O({\rm Ca}\varepsilon^{-2})$ and larger than the outer contribution $O({\rm Ca}\log\varepsilon)$.) The leading pressure $\varepsilon^{-3/2}\hat{P}^{(0)}\cos\phi$ with no deformation is locally dominant but does not contribute to the lateral force due to its azimuthal cosine dependence. The viscous stress contribution to the lateral force is $O(\varepsilon)$ smaller than the pressure in the inner region. The deformation-induced lateral force $F_M$ to cancel the migration velocity and to maintain the wall-parallel motion is expressed as $$F_M\approx \oint_{\rm contact}\!\!\!\!\!\!\!\!\!\!{\rm d}^2{\bm x}\ \varepsilon^{-3/2}\delta\hat{P}^{(f)}. \label{eq:intp01}$$ The surface integral for a function $q$ on the contact side is taken from the axis $R=0$ to the overlapping region, i.e., $$\oint_{\rm contact}\!\!\!\!\!\!\!\!\!\!{\rm d}^2{\bm x}\ q =\varepsilon \int_0^{2\pi}\!\!\!{\rm d}\phi\int_0^{\frak R}\!\!\!{\rm d}R\ R\ q(R,\phi), \label{eq:surf_int_contact}$$ where ${\frak R}=O(\varepsilon^{-1/2})$. For ${\frak R} \gg 1$, one obtains an asymptotic relation $$\begin{split} \int_0^{\frak R}\!\!\!\!\!\!{\rm d}R\ R\ \hat{P}^{(f)}(R) =& -\int_0^{\frak R}\!\!\!\!\!\!{\rm d}R\ R^2 A_2(R) -{\frak R}^2\int_{\frak R}^{\infty}\!\!\!\!\!\!{\rm d}R\ A_2(R) \\=& \frac{3}{100}+\frac{6\log({\frak R}^2/2)}{25{\frak R}^4} +\frac{1}{5{\frak R}^4}+O({\frak R}^{-6}\log {\frak R}), \label{eq:int_pf_lar} \end{split}$$ of which only the first term does not vanish as ${\frak R}\rightarrow \infty$. As obtained by [@hod2004], consequently, the asymptotic solution of the deformation-induced lateral force in the lubrication limit is $$\begin{split} F_M\rightarrow& \lim_{{\frak R}\rightarrow\infty} \varepsilon \int_0^{2\pi}\!\!\!\!\!\!{\rm d}\phi \int_0^{\frak R}\!\!\!\!\!\!{\rm d}R\ R\ \varepsilon^{-3/2}\delta\hat{P}^{(f)}(R) \\=& \frac{3\pi{\rm Ca}}{50}\varepsilon^{-2} \ \ {\rm as}\ \ \varepsilon\rightarrow 0. \label{eq:fm_lub} \end{split}$$ [99]{} natexlab\#1[\#1]{} 1970 A simplified MAC technique for incompressible fluid flow calculations. [*J. Comput. Phys.*]{} [**6**]{}, 322–325. 1968 The slow unsteady settling of a fluid sphere toward a flat fluid interface. [*Chem. Engrg. Sci.*]{} [**23**]{}, 193–210. 1967 [*An Introduction to Fluid Dynamics*]{}, Cambridge University Press. 1996 Sedimentation of a sphere near a plane wall: weak non-Newtonian and inertial effects. [*J. 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--- address: - | School of Engineering and Computing Sciences, Durham University\ Stockton Road, Durham DH1 3LE, Great Britain - | Department of Computer Science, KU Leuven\ Celestijnenlaan 200A, B-3001 Leuven, Belgium - | Vrije Universiteit Brussel\ Pleinlaan 2, 1050 Elsene, Belgium - | Laboratoire de Physique et Chimie de l’Environnement et de l’Espace (LPC2E)\ CNRS, Université d’Orléans\ 45071 Orléans Cedex 2, France author: - 'T. Weinzierl' - 'B. Verleye' - 'P. Henri' - 'D. Roose' bibliography: - './paper.bib' title: 'Two Particle-in-Grid Realisations on Spacetrees' --- Particle-in-cell ,spacetree ,particle sorting ,AMR , Lagrangian-Eulerian methods ,communication-avoiding

--- abstract: 'This is a continuation of  [@HRS2] in which we proved irreducibility of spaces of rational curves on a general hypersurface $X_d\subset {\mathbb{P}}^n$ of degree $d<\frac{n+1}{2}$. In this paper, we prove that if $d^2 + d + 2 \leq n$ and if $d\geq 3$, then the spaces of rational curves are themselves rationally connected.' address: - | Department of Mathematics\ Harvard University\ Cambridge MA 02138 - | Department of Mathematics\ Massachusetts Institute of Technology\ Cambridge MA 02139 author: - Joe Harris - Jason Starr bibliography: - 'my.bib' title: 'Rational curves on hypersurfaces of low degree, II' --- Statement of results ====================  \[sec-results\] In  [@HRS2], it is proved that if $X_d \subset {\mathbb{P}}^n$ is a general hypersurface of degree $d < \frac{n+1}{2}$, then each space $\text{RatCurves}^e(X)$ parametrizing smooth rational curves of degree $e$ on $X$, is itself an integral, local complete intersection scheme of the expected dimension $(n+1-d)e+(n-4)$. More precisely, it is proved that for every stable $A$-graph $\tau$ and every flag $f\in \text{Flag}(\tau)$, the Behrend-Manin stack ${\overline{{\mathcal}M}}(X,\tau)$ is an integral, local complete intersection stack of the expected dimension $\text{dim}(X,\tau)$, and the evaluation morphism $\text{ev}_f:{\overline{{\mathcal}M}}(X,\tau) \rightarrow X$ is flat of the expected fiber dimension $\text{dim}(X,\tau) - \text{dim}(X)$. Since $\text{RatCurves}^e(X)$ is a Zariski open set in the stack ${\overline{{\mathcal}M}_{0,0}({X,e})}$, the result on $\text{RatCurves}^e(X)$ follows. After establishing irreducibility and the dimension of the spaces $\text{RatCurves}^e(X)$, the next question is to determine the Kodaira dimension of $\text{RatCurves}^e(X)$. For a general Fano hypersurface $X_d \subset {\mathbb{P}}^n$ with $d\leq n$, determining the Kodaira dimensions of $\text{RatCurves}^e(X)$ is subtle. For instance for smooth cubic threefolds $X_3 \subset {\mathbb{P}}^4$, $X$ has a nontrivial intermediate Jacobian $J(X)$, and the Abel-Jacobi maps $\text{RatCurves}^e(X) \rightarrow J(X)$ is dominant for $e \geq 4$. So the Kodaira dimension is at least $0$; conjecturally the fibers of the Abel-Jacobi map are rationally connected so that the Kodaira dimension is exactly $0$. On the other hand for $d =1,2$, it is a theorem of Kim and Pandharipande  [@KP theorem 3], that each of the spaces $\text{RatCurves}^e(X)$ is itself a rational variety, and thus has Kodaira dimension $-\infty$. In this paper we present the following generalization of  [@KP theorem 3]: \[thm-thm1\] Given positive integers $(d,n)$ with $d^2 + d + 2 \leq n$ and $d\geq 3$, for $X_d \subset {\mathbb{P}}^n$ a general hypersurface of degree $d$, each of the spaces ${\overline{{\mathcal}M}_{0,0}({X,e})}$ is rationally connected. Thus $\text{RatCurves}^e(X)$ has a rationally connected compactification. To remind the reader, a variety $V$ is rationally connected if given two general points $p,q\in V$, there is a map $f:{\mathbb{P}}^1 \rightarrow V$ whose image contains $p$ and $q$. This property is strictly weaker than rationality, and it is unknown whether this property is the same as unirationality. It is a priori a much simpler property to check. And any rationally connected variety has Kodaira dimension $-\infty$, therefore each of the schemes $\text{RatCurves}^e(X)$ has Kodaira dimension $-\infty$. The proofs rely on  [@K theorem IV.3.7]: given a smooth variety $V$ and a morphism $f:{\mathbb{P}}^1 \rightarrow V$ such that $f^* T_V$ is an ample vector bundle, then $V$ is rationally connected. This criterion also works for smooth Deligne-Mumford stacks $V$. For readers not versed in stacks, this is a moot point – every morphism of ${\mathbb{P}}^1$ into a stack constructed in this paper can be deformed to a map contained in the fine moduli locus of the stack. For a Behrend-Manin stack, ${\overline{{\mathcal}M}}(X,\tau)$, a morphism $f:{\mathbb{P}}^1 \rightarrow {\overline{{\mathcal}M}}(X,\tau)$ is equivalent to a fibered surface $\pi:\Sigma \rightarrow {\mathbb{P}}^1$, with some collection of sections $\sigma_1,\dots,\sigma_r:{\mathbb{P}}^1 \rightarrow \Sigma$, and a map $f:\Sigma \rightarrow X$. Assuming that $f:{\mathbb{P}}^1 \rightarrow {\overline{{\mathcal}M}}(X,\tau)$ maps into the *unobstructed locus*, the bundle $f^*T_{{\overline{{\mathcal}M}}(X,\tau)}$ can be computed from a universal construction applied to the datum $(\pi:\Sigma \rightarrow {\mathbb{P}}^1,\sigma_1,\dots,\sigma_r,f)$. Thus, to prove ${\overline{{\mathcal}M}}(X,\tau)$ is rationally connected, we are reduced to finding a datum $(\pi:\Sigma \rightarrow {\mathbb{P}}^1,\sigma_1,\dots,\sigma_r,{\mathbb{P}}^1)$ satisfying certain properties. In the proof of the induction step, we will use the following technical hypotheses: \[hyp-1\] For each contraction of genus $0$ stable $A$-graphs, $\phi:\sigma\rightarrow \tau$, the codimension of the image of the corresponding morphism of Behrend-Manin stacks ${\overline{{\mathcal}M}}(X,\sigma) \rightarrow {\overline{{\mathcal}M}}(X,\tau)$ equals $\text{dim}(X,\tau) - \text{dim}(X,\sigma)$. In particular, by [@HRS2 proposition 7.4], for a general $X_d\subset {\mathbb{P}}^n$ with $d<\frac{n+1}{2}$, each stack ${\overline{{\mathcal}M}}(X,\sigma)$ has the expected dimension, and hypothesis \[hyp-1\] holds for $X$. \[hyp-1.5\] The general fiber of the evaluation map $\text{ev}:{\overline{{\mathcal}M}_{0,1}({X,1})} \rightarrow X$ is irreducible. In particular, for a general pair $([X],[p])$ consisting of a hypersurface $X\subset {\mathbb{P}}^n$ of degree $d$ and a point $p\in X$, the associated fiber of $\text{ev}$ is a subvariety $Z\subset {\mathbb{P}}^{n-1}$ which is a complete intersection of a general sequence of hypersurfaces $Y_1,\dots, Y_d$ in ${\mathbb{P}}^n$ with $\text{deg}(Y_i)=i$. By the Bertini-Kleiman theorem, a general such complete intersection is smooth and connected if $d < n-1$. \[hyp-1.75\] For each integer $e\geq 0$, the locus in ${\overline{{\mathcal}M}_{0,1}({X,e})}$ parametrizing stable maps with nontrivial automorphism group has codimension at least $2$. Of course any stable map with nontrivial automorphism group has an irreducible component which is a multiple cover of its image. In light of [@HRS2 proposition 7.4], a simple parameter count shows that for a general hypersurface $X_d \subset {\mathbb{P}}^n$ with $d\leq \frac{n+1}{2}$, hypothesis \[hyp-1.75\] is satisfied. Acknowledgments --------------- We are very grateful to Johan de Jong and Steven Kleiman for many useful conversations. Deformation ample =================  \[sec-DA\] Unless stated otherwise, all schemes will be finite type, separated schemes over ${\text{Spec }}{\mathbb{C}}$. All absolute fiber products will be fiber products over ${\text{Spec }}{\mathbb{C}}$. The following lemma is completely trivial.  \[lem-gend\] Let $B$ be a connected, proper, prestable curve of arithmetic genus $0$ and let $E$ be a locally free sheaf on $E$ which is generated by global sections. Then $H^1(B,E)$ is zero. Moreover if $p\in B$ is any point and ${{\mathcal}I}_p\subset {\mathcal O}_B$ is the corresponding ideal sheaf, then $H^1(B,{{\mathcal}I}_p\cdot E)$ is zero. Since $E$ is generated by global sections, there is a short exact sequence of the form: $$\begin{CD} 0 @>>> K @>>> {\mathcal O}_B^{\oplus N} @>>> E @>>> 0 \end{CD}$$ Since $B$ is a curve, $H^2(B,K)=0$. Therefore we have a surjection $H^1(B,{\mathcal O}_B)^{\oplus N} \rightarrow H^1(B,E)$. Since $B$ is connected of arithmetic genus $0$, $H^1(B,{\mathcal O}_B)$ is zero. Hence $H^1(B,E)$ is zero. Now for any point $p\in B$ we have a short exact sequence: $$\begin{CD} 0 @>>> {{\mathcal}I}_p\cdot E @>>> E @>>> E|_p @>>> 0 \end{CD}$$ This gives rise to a long exact sequence in cohomology: $$\begin{CD} H^0(B,E) @>>> E|_p @>>> H^1(B,{{\mathcal}I}_p\cdot E) @>>> H^1(B,E) \end{CD}$$ By the last paragraph, $H^1(B,E)$ is zero. And since $E$ is generated by global sections, $H^0(B,E)\rightarrow E|_p$ is surjective. Therefore $H^1(B,{{\mathcal}I}_p \cdot E)$ is zero. Suppose given a connected, proper, prestable curve $B$ of arithmetic genus $0$ and a locally free sheaf $E$ of positive rank on $B$. Any pair $({{\mathcal}B},{{\mathcal}E})$ consisting of a flat family of connected, proper prestable curves over a DVR, say $\pi:{{\mathcal}B} \rightarrow {\text{Spec }}R$ along with a locally free sheaf ${{\mathcal}E}$ on ${{\mathcal}B}$ such that the closed fiber of $\pi$ is isomorphic to $B$, such that the generic fiber $B_\eta$ of $\pi$ is smooth, and such that the restriction of ${{\mathcal}E}$ to $B\subset {{\mathcal}B}$ is isomorphic to $E$ will be called a *smoothing* of the pair $(B,E)$. We want to know when $(B,E)$ satisfies the condition that for every smoothing $({{\mathcal}B},{{\mathcal}E})$, the restriction ${{\mathcal}E}_\eta$ of ${{\mathcal}E}$ to the generic fiber is an ample locally free sheaf. Certainly if $E$ is ample, this is true, but $E$ need not be ample for this condition to hold: e.g. if $E$ is an invertible sheaf such that the total degree of $E$ is positive, then every smoothing of $(B,E)$ will have ample generic fiber. Although it is not the most general criterion, we find the following notion to be useful and it is the one we use in the remainder of the paper.  \[def-DA\] Let $B$ be a connected, proper, prestable curve of arithmetic genus $0$. A locally free sheaf $E$ on $B$ with positive rank is *deformation ample* if 1. $E$ is generated by global sections, and 2. $H^1(B,E(K_B))$ is zero, where ${\mathcal O}_B(K_B)$ is the dualizing sheaf of $B$.  \[rmk-DA\] 1. Conditions (1) and (2) above are independent. 2. If $E$ is invertible, then $E$ is deformation ample iff the restriction of $E$ to every irreducible component has nonnegative degree and the restriction to at least one irreducible component has positive degree. 3. For a general $E$, one can determine whether $E$ is deformation ample in terms of the splitting type of the restriction of $E$ to each irreducible component together with the patching isomorphisms at the nodes of $B$.  \[defn-fDA\] Let $T$ be a scheme and let $\pi:B\rightarrow T$ be a family of prestable curves of arithmetic genus $0$. A locally free sheaf $E$ on $B$ with positive rank is *$\pi$-relatively deformation ample* (or simply deformation ample if $\pi$ is understood) if 1. the canonical map $\pi^*\pi_*E\rightarrow E$ is surjective, and 2. $R^1\pi_*(E(K_\pi))$ is zero, where ${\mathcal O}_B(K_\pi)$ is the relative dualizing sheaf of $\pi$.  \[lem-bcDA\] With notation as in definition \[defn-fDA\], suppose $f:T'\rightarrow T$ is a morphism of schemes and let $\pi':B'\rightarrow T'$ be the base-change of $\pi$ by $f$. Let $E'$ be the pullback of $E$ by the projection $g:B'\rightarrow B$. If $E$ is $\pi$-relatively deformation ample, then $E'$ is $\pi'$-relatively deformation ample. If $f'$ is surjective, the converse also holds. For the main direction, by  [@EGA4 section 8.5.2, proposition 8.9.1], it suffices to consider the case when $T$ and $T'$ are Noetherian affine schemes. There is a canonical map of ${\mathcal O}_{T'}$-modules $\alpha:f^*\pi_* E\rightarrow (\pi')_*g^* E$. There is a commutative diagram: $$\begin{CD} (\pi')^*f^*\pi_* E @> = >> g^* \pi^* \pi_* E \\ @V (\pi')^*\alpha VV @VVV \\ (\pi')^*(\pi')_* E' @>>> E' \end{CD}$$ Since $\pi^*\pi_*E \rightarrow E$ is surjective, we conclude that $g^*\pi^*\pi_* E\rightarrow E'$ is surjective. Therefore also $(\pi')^*(\pi')_*E' \rightarrow E'$ is surjective. Now $R^2\pi_* E(K)$ is identically zero. So by [@H theorem III.12.11(b)], we conclude that for every closed point $t\in T$, we have $H^1(B_t,E(K)|_{B_t}) = 0$. By  [@H prop. III.9.3], we conclude that for every closed point $t'\in T'$, we have $H^1(B'_{t'},E'(K')|_{B'_{t'}}) = 0$. So by [@H theorem III.12.11(a)] and Nakayama’s lemma, we conclude that $R^1\pi'_*(E'(K'))$ is identically zero. This shows that $E'$ is $\pi'$-relatively deformation ample. For the converse in case $T'\rightarrow T$ is surjective, we may reduce to the case that $T$ is a Noetherian affine scheme. By the argument above, we conclude that for each closed point $t'\in T'$, we have $H^1(B'_{t'},E'(K')|_{B'_{t'}}) = 0$. Since $T'\rightarrow T$ is surjective, we conclude by  [@H prop. III.9.3] that for each closed point $t\in T$ we have $H^1(B_t,E(K)|_{B_t}) = 0$. So by  [@H theorem III.12.11(a)] and Nakayama’s lemma, we conclude that $R^1\pi_*(E(K))$ is identically zero. It remains to show that $E$ is $\pi$-relatively generated by global sections. For each point $t\in T$, there is some point $t'\in T'$ mapping to $t$. Since $E'|_{B'_{t'}}$ is generated by global sections, it follows that $E|_{B_t}$ is generated by global sections. So, by lemma \[lem-gend\], we conclude that $H^1{\left}( B_t, E|_{B_t} {\right})$ is zero. So by  [@H theorem III.12.11(a)] and Nakayama’s lemma, we conclude that $R^1\pi_*(E)$ is identically zero. Since $E$ is flat over $T$, it follows by a standard argument that for any coherent ${\mathcal O}_T$-module ${{\mathcal}F}$, we have that $R^1\pi_*( \pi^*{{\mathcal}F}\otimes E)$ is also zero. In particular, applying the long exact sequence of higher direct images to the short exact sequence: $$\begin{CD} 0 @>>> \pi^*{{\mathcal}I}_t\otimes E @>>> E @>>> E|_{B_t} @>>> 0 \end{CD}$$ we conclude that $\pi_*(E) \rightarrow H^0{\left}( B_t, E|_{B_t} {\right})$ is surjective. Since $E|_{B_t}$ is generated by global sections for each $t\in T$, we conclude that $E$ is $\pi$-relatively generated by global sections. \[lem-smDA\] With notation as in definition \[defn-fDA\], if $\pi$ is smooth, then $E$ is $\pi$-relatively deformation ample iff $E$ is $\pi$-relatively ample. Both properties are local on $T$ and can be checked after étale, surjective base-change of $T$. Thus we may assume that $\pi:B\rightarrow T$ is isomorphic to $\pi_1:T\times {\mathbb{P}}^1 \rightarrow T$. Define $F=(\pi_1)_*(E\otimes \pi_2^*{\mathcal O}_{{\mathbb{P}}^1}(-1))$. There is a natural map $\alpha: (\pi_1)^*F \otimes\pi_2^*{\mathcal O}_{{\mathbb{P}}^1}(1)\rightarrow E$. Suppose that $E$ is deformation ample. Then the claim is that $\alpha$ is surjective. To prove this, it suffices to prove the following: 1. For each geometric point $t$ of $T$ with residue field $\kappa$, we have $H^1({\mathbb{P}}^1_\kappa, E|_{B_t}\otimes {\mathcal O}_{{\mathbb{P}}^1}(-1))$ is trivial, 2. $\pi^* F|_{B_t} = H^0({\mathbb{P}}^1_\kappa,E|_{B_t}\otimes {\mathcal O}_{{\mathbb{P}}^1}(-1))$, and 3. the map $H^0({\mathbb{P}}^1_\kappa,E|_{B_t}\otimes {\mathcal O}_{{\mathbb{P}}^1}(-1))\otimes {\mathcal O}_{{\mathbb{P}}^1}(1) \rightarrow E|_{B_t}$ is surjective. Now by Grothendieck’s lemma  [@H exercise V.2.6], $E|_{B_t}$ splits as a direct sum ${\mathcal O}_{{\mathbb{P}}^1}(a_1)\oplus \dots\oplus {\mathcal O}_{{\mathbb{P}}^1}(a_r)$ for some integers $a_1\leq \dots \leq a_r$. By lemma \[lem-bcDA\], we know that $E|_{B_t}$ is deformation ample. It follows that $a_1\geq 1$. Thus $H^1(B_t, E|_{B_t}\otimes {\mathcal O}_{{\mathbb{P}}^1}(-1))=0$ and $(1)$ is established. Combined with [@H theorem III.12.11(b)], also $(2)$ follows. Finally, for $a_i \geq 1$, we clearly have $H^0({\mathbb{P}}^1_\kappa, {\mathcal O}_{{\mathbb{P}}^1}(a_i-1))\otimes {\mathcal O}_{{\mathbb{P}}^1}(1)\rightarrow {\mathcal O}_{{\mathbb{P}}^1}(a_i)$ is surjective. Thus $(3)$ holds and the claim is proved. Now $(\pi_1)^*F\otimes (\pi_2)^*{\mathcal O}_{{\mathbb{P}}^1}(1)$ is $\pi_1$-relatively ample. Since $E$ is a quotient of $(\pi_1)^*F\otimes (\pi_2)^*{\mathcal O}_{{\mathbb{P}}^1}(1)$, we conclude that $E$ is also $\pi_1$-relatively ample. The converse result follows in the same way.  \[lem-opDA\] With notation as in lemma \[defn-fDA\], there exists an open subscheme $i:U\rightarrow T$ with the following property: for every morphism $f:T'\rightarrow T$, $f(T')$ is contained in $U$ iff $E'$ is $\pi'$-relatively deformation ample. By [@EGA4 section 8.5.2,proposition 8.9.1], we may reduce to the case that $T$ and $T'$ are Noetherian affine schemes. Let $Z_1\subset T$ be the closed subset with is the image under $f$ of the support of $\text{coker}(\pi^*\pi_* E\rightarrow E)$. Let $Z_2\subset T$ be the closed subset which is the support of $R^1\pi_*(E(K_\pi))$. Let $i:U\rightarrow T$ be the open complement of $Z_1\cup Z_2$. Suppose $f:T'\rightarrow T$ is a morphism of schemes. By  [@H theorem III.12.11, prop. III.9.3] and Nakayama’s lemma, $R^1\pi'_*(E'(K'))$ is identically zero iff for each $t'\in T'$ with $t=f(t')$, we have $H^1(B_t,E(K)|_{B_t})$ is zero, i.e. if $t$ is contained in the complement of $Z_2$. So $R^1\pi'_*(E'(K'))$ is identically zero iff $f(T)$ is contained in the complement of $Z_2$. If $R^1\pi'_*(E'(K'))$ is zero, then also $R^1\pi'_*(E')$ is zero (by the same argument as in the proof of lemma \[lem-bcDA\]). So using  [@H theorem III.12.11, prop. III.9.3] again, $E'$ is $\pi'$-relatively generated by global sections iff for each $t'\in T'$ with $t=f(t')$, we have $E_t$ is generated by global sections, i.e. $t$ is in the complement of $Z_1$. So we conclude that $E'$ is $\pi'$-relatively deformation ample iff $f(T)$ is contained in the complement of $Z_1\cup Z_2$.  \[lem-secDA\] With notation as in definition \[defn-fDA\]: 1. If $E_1\rightarrow E_2$ is a morphism of locally free sheaves on $B$ whose cokernel is torsion in every fiber (in particular, if the morphism is surjective), if $E_2$ is nonzero, and if $E_1$ is $\pi$-relatively deformation ample, then also $E_2$ is $\pi$-relatively deformation ample. 2. Given a short exact sequence of nonzero locally free sheaves on $B$, $$\begin{CD} 0 @>>> E_1 @>>> E_2 @>>> E_3 @>>> 0 \end{CD}$$ if $E_1$ and $E_3$ are $\pi$-relatively deformation ample, then $E_2$ is also $\pi$-relatively deformation ample. 3. If $E$ is $\pi$-relatively deformation ample, then for each integer $n\geq 1$, also $E^{\otimes n}$ is $\pi$-relatively deformation ample. First we prove $(1)$. Let $Q$ denote the cokernel, which is assumed to be torsion. Now $R^1\pi_*$ is right exact on coherent sheaves, $R^1\pi_*(Q(K))$ is zero (because $Q(K)$ is torsion in fibers), and $R^1\pi_*(E_1(K))=0$ by assumption. So we conclude that also $R^1\pi_*(E_2(K))=0$. Let $I\subset E_2$ denote the image sheaf of $E_1\rightarrow E_2$. The surjective composition map $$\begin{CD} \pi^*\pi_*E_1 @>>> E_1 @>>> I \end{CD}$$ factors through $\pi^*\pi_* I\rightarrow I$. Therefore $I$ is $\pi$-relatively generated by global sections. In particular, $R^1\pi_* I$ is zero (because $R^1\pi_* {\mathcal O}_B$ is zero). Since $Q$ is fiberwise torsion, it is $\pi$-relatively generated by global sections. Now we have a short exact sequence of coherent sheaves: $$\begin{CD} 0 @>>> I @>>> E_2 @>>> Q @>>> 0. \end{CD}$$ Applying the long exact sequence of higher direct images, and using that $R^1\pi_*I$ is zero, locally on $B$ all global sections of $Q$ lift to global sections of $E_2$. Since also $I$ is $\pi$-relatively generated by global sections, we conclude that $E_2$ is $\pi$-relatively generated by global sections. Thus $E_2$ is $\pi$-relatively deformation ample. Next we prove $(2)$. The long exact sequence of higher direct images and the vanishings $R^1\pi_*(E_1(K))=R^1\pi_*(E_3(K))=0$ imply that $R^1\pi_*(E_2(K))$ is also trivial. Now $E_1$ and $E_3$ are $\pi$-relatively generated by global sections an so (locally on $T$) they are quotients of a trivial sheaf ${\mathcal O}_B^{\oplus N}$. Since $R^1\pi_*{\mathcal O}_B$ is zero, and since $R^1\pi_*$ is right exact for short exact sequences of coherent sheaves, we conclude that $R^1\pi_*E_1 = R^1\pi_* E_3 = 0$. Applying the long exact sequence of higher direct images to the short exact sequence above, we conclude that $\pi_*(E_2)\rightarrow \pi_*(E_3)$ is surjective. Since $E_3$ is $\pi$-relatively generated by global sections, we conclude that $\pi^*\pi_*E_2\rightarrow E_2 \rightarrow E_3$ is surjective. Therefore the map from $E_1$ to the cokernel of $\pi^*\pi_*E_2\rightarrow E_2$ is surjective. But since $E_1$ is $\pi$-relatively generated by global sections, this map is zero and we conclude $\pi^*\pi_*E_2\rightarrow E_2$ is surjective. So $E_2$ is $\pi$-relatively deformation ample. Finally we prove $(3)$ by induction on $n$. It suffices to consider the case when $T$ is affine. For $n=1$ it is trivial. Suppose $n>1$ and suppose the result is known for all smaller values of $n$. In particular, $E^{\otimes(n-1)}$ is $\pi$-relatively generated by global sections, and we have a natural surjection $$\pi_*\pi^*(E^{\otimes(n-1)})\otimes_{{\mathcal O}_B} E \rightarrow E^{\otimes n}.$$ Now we can find a surjective map ${\mathcal O}_T^{\oplus r}\rightarrow \pi_*(E^{\otimes(n-1)})$. Thus we have a surjection $E^{\oplus r}\rightarrow E^{\otimes n}$. By $(2)$ above and induction, we have that $E^{\oplus r}$ is $\pi$-relatively deformation ample. By $(1)$ above, we conclude that $E^{\otimes n}$ is $\pi$-relatively deformation ample, and we have proved $(3)$ by induction on $n$.  \[lem-DAcrit\] Suppose now that $T={\text{Spec }}k$ for some algebraically closed field $k$. If $E$ satisfies the hypotheses 1. For every irreducible component $B_i\subset B$ we have $E|_{B_i}$ is generated by global sections, and 2. for some nonempty closed subcurve $B'\subset B$, $E|_{B'}$ is deformation ample, then $E$ is deformation ample. Let $\delta$ be the number of irreducible components of $B$ which aren’t contained in $B'$. We prove the result by induction on $\delta$. If $\delta=0$, then $B=B'$ and there is nothing to prove. Thus suppose that $\delta >0$ and suppose the result has been proved for all smaller values of $\delta$. Let $B_1\subset B$ be an irreducible component of $B$ which is not in $B'$, let $B_2\subset B$ denote the union of all irreducible components other than $B_1$, and suppose that $B_1$ intersects $B_2$ in precisely one node $p$ of $B$. By the induction assumption $E|_{B_2}$ is deformation ample. First we prove that $E$ is generated by global sections. Let $F\subset E$ be the image of $H^0(B,E)\otimes_k {\mathcal O}_B \rightarrow E$. We have a short exact sequence of coherent sheaves: $$\begin{CD} 0 @>>> E|_{B_1}(-p) @>>> E @>>> E|_{B_2} @>>> 0. \end{CD}$$ Since $E|_{B_1}$ is a locally free sheaf on ${\mathbb{P}}^1$ generated by global sections, Grothendieck’s lemma and cohomology of line bundles on ${\mathbb{P}}^1$ imply that $H^1(B,E|_{B_1}(-p))=0$. Thus all the global sections of $E|_{B_2}$ lift to global sections of $E$, i.e. $F\rightarrow E|_{B_2}$ is surjective. So $E/F$ is supported on $B_1$ and thus is a quotient of $E|_{B_1}$. Since $E|_{B_1}$ is generated by global sections, also $E/F$ is generated by global sections. We have a short exact sequence $$\begin{CD} 0 @>>> F @>>> E @>>> E/F @>>> 0. \end{CD}$$ Since $H^1(B,{\mathcal O}_B)=0$, also $H^1(B,H^0(B,E)\otimes_k {\mathcal O}_B) = 0$. Since $H^1(B,*)$ is right exact, $H^1(B,F)=0$. Thus all the global sections of $E/F$ lift to global sections of $E$. This can only hold if $E/F=0$, i.e. if $E$ is generated by global sections. Next we prove that $H^1(B,E(K_B))=0$. We have a short exact sequence of sheaves: $$\begin{CD} 0 @>>> E(K_B)|_{B_2}(-p) @>>> E(K_B) @>>> E(K_B)|_{B_1} @>>> 0. \end{CD}$$ This gives an exact sequence of vector spaces: $$\begin{CD} H^1(B,E(K_B)|_{B_2}(-p)) @>>> H^1(B,E(K_B)) @>>> H^1(B,E(K_B)|_{B_1}) @>>> 0. \end{CD}$$ By standard results on dualizing sheaves and finite morphisms, we have $$K_{B_2}=\textit{Hom}_{{\mathcal O}_B}({\mathcal O}_{B_2},K_B) = K_B|_{B_2}(-p).$$ Therefore $H^1(B,E(K_B)|_{B_2}(-p)) = H^1(B_2,E|_{B_2}(K_{B_2}))$, which is zero by the induction assumption. Similarly, $E(K_B)|_{B_1}$ equals $E|_{B_1}(-1)$ (identifying $B_1$ with ${\mathbb{P}}^1$). Since $E|_{B_1}$ is generated by global sections, it follows by Grothendieck’s lemma and cohomology of line bundles on ${\mathbb{P}}^1$ that $H^1(B_1,E|_{B_1}(-1))$ is trivial. Therefore we conclude that $H^1(B,E(K_B))$ is trivial. So $E$ is deformation ample, and the result is proved by induction on $\delta$. \[rmk-DAcrit\] A particular case of lemma \[lem-DAcrit\] is when $B'$ is one irreducible component of $B$. Then the lemma says that a locally free sheaf on $B$ which is *generically ample* in the sense of Lazarsfeld  [@F] is deformation ample. Some deformation theory {#sec-def} ======================= In the next section we will need some deformation theory of stable maps. \[not-cpx\] Suppose $T$ is a scheme and suppose $$\zeta = ((\pi:B\rightarrow T,\sigma_1,\dots, \sigma_r), g:B \rightarrow X)$$ is a family of marked prestable maps to a smooth scheme $X$. Denote by $L_\zeta$ the complex of coherent sheaves on $B$ $$\begin{CD} -1 & & 0 \\ f^*\Omega_X @> (df)^\dagger >> \Omega_\pi(\sigma_1 + \dots + \sigma_r) \end{CD}$$ Denote by $L_\zeta^\vee$ the object $$L_\zeta^\vee := \mathbb{R}\textit{Hom}_{{\mathcal O}_B}(L_\zeta,{\mathcal O}_B)$$ in the derived category of $B$. The relevance of the complex $L_\zeta^\vee$ is the following. \[lem-cpx\] Suppose $X$ is a smooth projective scheme. Let $\pi:{{\mathcal}B}\rightarrow {\overline{{\mathcal}M}_{g,r}({X,\beta})}$ denote the universal curve, let $\sigma_i: {\overline{{\mathcal}M}_{g,r}({X,\beta})} \rightarrow {{\mathcal}B}$ denote the universal sections, and let $f:{{\mathcal}B} \rightarrow X$ denote the universal map, i.e. $$\zeta = {\left}( {\left}( \pi:{{\mathcal}B} \rightarrow {\overline{{\mathcal}M}_{g,r}({X,\beta})}, \sigma_1, \dots, \sigma_r {\right}), f:{{\mathcal}B} \rightarrow X {\right})$$ is the universal family of stable maps. There is an *obstruction theory* for ${\overline{{\mathcal}M}_{g,r}({X,\beta})}$ in the sense of [@BF definition 4.4] of the form $$\phi:{\left}( \mathbb{R}\pi_*(L_\zeta^\vee) [1] {\right})^\vee \rightarrow L_{{\overline{{\mathcal}M}_{g,r}({X,\beta})}}.$$ Essentially this follows from  [@BF] and  [@B]. \[rmk-cpx\] Explicitly, if $\zeta = ((B,p_1,\dots,p_r),f:B\rightarrow X)$ is a stable map (i.e $T = {\text{Spec }}C$), then the space of first order deformations of $\zeta$ is given by ${\mathbb{E}\text{xt}_{{\mathcal O}_{B}}}^1(L_\zeta,{\mathcal O}_B)$ and the obstruction group is a subgroup of ${\mathbb{E}\text{xt}_{{\mathcal O}_{B}}}^2(L_\zeta,{\mathcal O}_B)$. In particular, if ${\mathbb{E}\text{xt}_{{\mathcal O}_{B}}}^2(L_\zeta,{\mathcal O}_B)$ vanishes, then ${\overline{{\mathcal}M}_{g,r}({X,\beta})}$ is smooth at the point $[\zeta]$. Contracting unstable components {#subsec-unstable} ------------------------------- In this subsection we wish to investigate the relationship between ${\mathbb{E}\text{xt}_{{\mathcal O}_{B}}}(L_\zeta, {\mathcal O}_B)$ and ${\mathbb{E}\text{xt}_{{\mathcal O}_{B'}}}(L_\zeta',{\mathcal O}_B')$ where $$\zeta = ((B,p_1,\dots,p_r),f:B \rightarrow X)$$ is a prestable map, where $$(B',p'_1,\dots, p'_r,q'_1,\dots,q'_s)$$ is a prestable curve, where $$h:(B',p'_1,\dots, p'_r) \rightarrow (B,p_1,\dots, p_r)$$ is a map which contracts some of the unstable components of $(B',p_1,\dots,p_r)$, where $f' = f\circ h$ and where $\zeta'$ is the prestable map $$\zeta' = ((B',p'_1,\dots, p'_r,q'_1, \dots, q'_s), f':B' \rightarrow X).$$ Any morphism $h:B'\rightarrow B$ as above can be factored as a sequence of *elementary* maps. We begin by analyzing the case $\zeta = (B,f:B\rightarrow X)$ is a prestable map without marked points, $h:B'\rightarrow B$ contracts a single unstable component, and $\zeta' = (B', f' = f\circ h:B' \rightarrow X)$ (again without marked points). We call such maps either *type I* or *type II*, depending on whether the image under $h$ of the contracted component is a smooth point of $B$ or a node of $B$. After this, we analyze the case where $\zeta = (B,(p_1,\dots,p_n),f:B\rightarrow X)$ is a marked prestable map, where $B'$ is the same prestable map, but with one extra marked point, and where $h:B'\rightarrow B$ is the identity map. We call such a map *type III*. To simplify calculations, we replace $L_\zeta$ by a quasi-isomorphic complex as follows. Choose a regular embedding $i:B\rightarrow S$ of $B$ into a smooth surface $S$. Then the morphism $(f,i):B \rightarrow X \times S$ is a regular embedding. Let $N_{(f,i)}$ denote the normal bundle of the embedding. There are induced morphisms $\alpha_f: N_{(f,i)}^\vee \rightarrow f^* \Omega_X$ and $\beta_i: N_{(f,i)}^\vee \rightarrow i^* \Omega_S$ which are the components of the canonical morphism $N_{(f,i)}^\vee \rightarrow (f,i)^* \Omega_{X \times S}$. Define the complex $L_{(f,i)}$ to be $$\begin{CD} -1 & & 0 \\ N_{(f,i)}^\vee @> \beta_i >> i^* \Omega_S \end{CD}$$ There is a map of complexes $\gamma_{(f,i)}:L_{(f,i)} \rightarrow L_\zeta$ defined by the commutative diagram $$\begin{CD} N_{(f,i)}^\vee @> \beta_i >> i^* \Omega_S \\ @V \alpha_f VV @VV (di)^\dagger V \\ f^* \Omega_X @> (df)^\dagger >> \Omega_B \end{CD}$$ \[lem-qism\] The morphism $\gamma_{(f,i)}: L_{(f,i)} \rightarrow L_\zeta$ is a quasi-isomorphism of complexes. This is an easy local argument which is left as an exercise for the reader. As a corollary of the lemma, we see that $L_\zeta^\vee$ is represented in $D({\mathcal O}_{B})$ by the complex $L_{(f,i)}^\vee$ defined to be $$\begin{CD} 0 & & 1 \\ i^* T_S @> (\beta_i)^\vee >> N_{(f,i)} \end{CD}$$ Now suppose that $p\in B \subset S$ is a point, define $g:S'\rightarrow S$ to be the blow up of $S$ at $p$ with exceptional divisor $E$, and define $i':B' \rightarrow S'$ to be the reduced total transform of $B$. We break up our analysis according to the type of behavior of $p\in B$. If $p\in B$ is a smooth point, we call this *type (I)*. If $p\in B$ is a node, we call this *type (II)*. We further decompose each type as follows. If $p\in B$ is a smooth point which lies on a stable component, we say this is *type (Ia)*. If $p\in B$ is a smooth point which lies on an unstable component, we say this is *type (Ib)*. If $p\in B$ is a node, and the first order deformations of $\zeta$ smooth the node, we say this is *type (IIa)*. If the first order deformations of $\zeta$ don’t smooth the node, we say this is *type (IIb)*. For type (I), we have $g^* B$ equals $B'$ as Cartier divisors. For type (II), we have $g^* B$ equals $B' + E$ as Cartier divisors. For both types, we define $h:B' \rightarrow B$ to be the restriction of $g$ and we define $f':B' \rightarrow B$ to be $f' = f\circ h$. Of course we have $g_*{\mathcal O}_{S'} = {\mathcal O}_S$ and $R^{k>0}g_* {\mathcal O}_{S'}$ is zero. For type (I), we have ${\mathcal O}_{S'}(-B') = g^* {\mathcal O}_S(-B)$. So by the projection formula we have $g_*{\mathcal O}_{S'}(-B') = {\mathcal O}_S(-B)$ and $R^{k>0}g_*{\mathcal O}_{S'}(-B')$ is zero. Also we have that $g_*{\mathcal O}_{S'}(E) = {\mathcal O}_S$ and $R^{k>0}g_* {\mathcal O}_{S'}(E)$ is zero. So also for type (II) we have $g_*{\mathcal O}_{S'}(-B') = {\mathcal O}_S(-B)$ and $R^{k>0}g_* {\mathcal O}_{S'}(-B')$ is zero. Using the resolution of ${\mathcal O}_{B'}$ $$\begin{CD} 0 @>>> {\mathcal O}_{S'}(-B') @>>> {\mathcal O}_{S'} @>>> {\mathcal O}_{B'} @>>> 0 \end{CD}$$ we conclude that $h_* {\mathcal O}_{B'} = {\mathcal O}_B$ and $R^{k>0}h_* {\mathcal O}_{B'}$ is zero. In other words, the canonical morphism ${\mathcal O}_B \rightarrow \mathbb{R}h_* {\mathcal O}_{B'}$ is a quasi-isomorphism. From this it follows by the projection formula that the canonical morphism $$L_{(f,i)}^\vee \rightarrow \mathbb{R}h_* \mathbb{L}h^* (L_{(f,i)}^\vee)$$ is a quasi-isomorphism. Therefore the pullback morphisms $$\mathbb{H}^k(B,L_\zeta^\vee) \rightarrow \mathbb{H}^k(B',g^*L_\zeta^\vee)$$ are isomorphisms. Now there is a canonical morphism ${\mathcal O}_{S'}(B') \rightarrow g^* {\mathcal O}_S(B)$. For type (I), this morphism is an isomorphism. For type (II), this morphism is injective and the cokernel is $g^*{\mathcal O}_S(B)|_E$, i.e. ${\mathcal O}_E \otimes_{\mathbb{C}}M$ where $M= {\mathcal O}_S(B)|_p$ is a one-dimensional vector space. For type (I), we conclude that the canonical morphism $N_{(f',i')} \rightarrow h^* N_{(f,i)}$ is an isomorphism. For type (II), we conclude that there is an exact sequence: $$0 \rightarrow \textit{Tor}_1^{{\mathcal O}_{S'}}({\mathcal O}_{B'},{\mathcal O}_E)\otimes_{\mathbb{C}}M \rightarrow N_{(f',i')} \rightarrow h^* N_{(f,i)} \rightarrow {\mathcal O}_E \otimes_{\mathbb{C}}M \rightarrow 0$$ For both types, denote by $\Gamma \subset B'$ the subcurve which is the union of all components other than $E$, and let $D = E\cap \Gamma$. From the resolution of ${\mathcal O}_E$ $$\begin{CD} 0 @>>> {\mathcal O}_{S'}(-E) @>>> {\mathcal O}_S @>>> {\mathcal O}_E @>>> 0 \end{CD}$$ we have the relation $$\textit{Tor}_1^{{\mathcal O}_{S'}}({\mathcal O}_{B'},{\mathcal O}_E) = {{\mathcal}I}_{\Gamma}\otimes_{{\mathcal O}_{S'}}{\mathcal O}_{S'}(-E) = {\mathcal O}_E(-E)(-D).$$ where ${{\mathcal}I}_{\Gamma}$ is the ideal sheaf of ${\mathcal O}_{B'}$ defining $\Gamma \subset B'$, i.e. ${\mathcal O}_E(-D)$. So in case $(2)$, we have an exact sequence: $$0 \rightarrow {\mathcal O}_E(-E)(-D)\otimes_{\mathbb{C}}M \rightarrow N_{(f',i')} \rightarrow h^*N_{(f,i)} \rightarrow {\mathcal O}_E \otimes_{\mathbb{C}}M \rightarrow 0.$$ For both types, we have a short exact sequence of ${\mathcal O}_{S'}$-modules $$\begin{CD} 0 @>>> g^* \Omega_S @> (dg)^\dagger >> \Omega_{S'} @>>> \Omega_E @>>> 0. \end{CD}$$ Applying $\mathbb{R}\textit{Hom}_{{\mathcal O}_{S'}}(*,{\mathcal O}_{S'})$, we have an exact sequence $$\begin{CD} 0 @>>> T_{S'} @>>> g^* T_S @>>> \textit{Ext}^1_{{\mathcal O}_{S'}}({\mathcal O}_E,{\mathcal O}_{S'}) @>>> 0 \end{CD}$$ Using the resolution of ${\mathcal O}_E$ in the last paragraph, we have that $$\textit{Ext}^1_{{\mathcal O}_{S'}}({\mathcal O}_E,{\mathcal O}_{S'}) = {\mathcal O}_E(E).$$ So we have an exact sequence $$\begin{CD} 0 @>>> T_{S'} @>>> g^* T_S @>>> T_E(E) @>>> 0 \end{CD}$$ Using our *Tor* result from the last paragraph, we have a short exact sequence $$0 \rightarrow T_E(-D) \rightarrow (i')^*T_{S'} \xrightarrow{dg} h^* i^* T_S \rightarrow T_E(E) \rightarrow 0.$$ Notice this holds in both cases. The maps $N_{(f',i')} \rightarrow h^* N_{(f,i)}$ and $(i')^*T_{S'} \rightarrow h^* i^* T_S$ considered in the last two paragraphs are compatible with $\alpha_{i'}$ and $h^*\alpha_i$. So we have an induced map of complexes $dg: L_{\zeta'}^\vee \rightarrow h^* L_{\zeta}^\vee$. Define $I\hookrightarrow h^* L_\zeta^\vee$ to be the image complex of $dg:L_{\zeta'}^\vee \rightarrow h^* L_{\zeta}^\vee$. For type (I), we define two complexes of coherent sheaves on $B'$, $K_I$ and $Q_I$, by $K_I = T_E(-D)[0]$ and $Q_I = T_E(E)[0]$. For type (II), we define complexes of coherent sheaves on $B'$, $K_{II}$ and $Q_{II}$ by $$\begin{aligned} K_{II} = T_E(-D)[0] \oplus{\left}( {\mathcal O}_E(-E)(-D)\otimes_{\mathbb{C}}M{\right})[-1] \\ Q_{II} = T_E(E)[0] \oplus {\left}({\mathcal O}_E \otimes_{\mathbb{C}}M{\right})[-1]\end{aligned}$$ Then we have exact sequences of complexes $$\begin{CD} 0 @>>> K @>>> L_{\zeta'}^\vee @> dg >> I @>>> 0 \\ 0 @>>> I @>>> h^* L_{\zeta}^\vee @>>> Q @>>> 0 \end{CD}$$ For type (I), we have that $\mathbb{H}^0(B',K_I) = H^0(E,T_E(-D))$ is $2$-dimensional, because $T_E(-D) \cong {\mathcal O}_E(1)$. And $\mathbb{H}^{k > 0}(B',K_1)$ is zero. Similarly $\mathbb{H}^0(B',Q_I) = H^0(E,T_E(E))$ is $2$-dimensional and $\mathbb{H}^{k>0}(B',Q_I)$ is zero. Therefore we have a long exact sequence of hypercohomology groups: $$\begin{aligned} 0 \rightarrow H^0(E,T_E(-D)) \rightarrow \mathbb{H}^0(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^0(B, L_{(f,i)}^\vee) \rightarrow \dots \\ \dots \rightarrow H^0(E, T_E(E)) \rightarrow \mathbb{H}^1(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^1(B, L_{(f,i)}^\vee ) \rightarrow 0 \\ 0 \rightarrow \mathbb{H}^2(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^2(B, L_{(f,i)}^\vee ) \rightarrow 0\end{aligned}$$ For type (Ia), the map $$\mathbb{H}^0(B,L_{(f,i)}^\vee) \rightarrow H^0(E,T_E(E))$$ is the zero map. So we have proved the following lemma. \[lem-def1a\] Suppose that $h:B'\rightarrow B$ is type (Ia). Then we have exact sequences: $$\begin{aligned} 0 \rightarrow H^0(E,T_E(-D)) \rightarrow \mathbb{H}^0(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^0(B, L_{(f,i)}^\vee) \rightarrow 0 \\ 0 \rightarrow H^0(E, T_E(E)) \rightarrow \mathbb{H}^1(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^1(B, L_{(f,i)}^\vee ) \rightarrow 0 \\ 0 \rightarrow \mathbb{H}^2(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^2(B, L_{(f,i)}^\vee ) \rightarrow 0\end{aligned}$$ So the canonical map from the Lie algebra of infinitesimal automorphisms of $\zeta'$ to the Lie algebra of infinitesimal automorphisms of $\zeta$ is surjective with $2$-dimensional kernel, the canonical map from the space of first order deformations of $\zeta'$ to the space of first order deformations of $\zeta$ is surjective with $2$-dimensional kernel, and the obstruction space of $\zeta'$ equals the obstruction space of $\zeta$. For type (Ib), the map $$\mathbb{H}^0(B,L_{(f,i)}^\vee) \rightarrow H^0(E,T_E(E))$$ has a $1$-dimensional image; we will call it $N$. We have proved the following lemma. \[lem-def1b\] Suppose that $h:B'\rightarrow B$ is type (Ib). Then there is a $1$-dimensional subspace $N\subset H^0(E,T_E(E))$ such that we have exact sequences: $$\begin{aligned} 0 \rightarrow H^0(E,T_E(-D)) \rightarrow \mathbb{H}^0(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^0(B, L_{(f,i)}^\vee) \rightarrow N \rightarrow 0 \\ 0 \rightarrow H^0(E, T_E(E))/N \rightarrow \mathbb{H}^1(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^1(B, L_{(f,i)}^\vee ) \rightarrow 0 \\ 0 \rightarrow \mathbb{H}^2(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^2(B, L_{(f,i)}^\vee ) \rightarrow 0\end{aligned}$$ So the canonical map from the Lie algebra of infinitesimal automorphisms of $\zeta'$ to the Lie algebra of infinitesimal automorphisms of $\zeta$ has both $1$-dimensional kernel and cokernel, the canonical map from the space of first order deformations of $\zeta'$ to the space of first order deformations of $\zeta$ is surjective with $1$-dimensional kernel, and the obstruction space of $\zeta'$ equals the obstruction space of $\zeta$. Next we consider type (II). Then ${\mathcal O}_E(-E)(-D)$ is isomorphic to ${\mathcal O}_E(-1)$. Since $H^0(E,{\mathcal O}_E(-1)) = H^1(E,{\mathcal O}_E(-1)) = 0$, the terms ${\mathcal O}_E(-E)(-D)\otimes_{\mathbb{C}}M[-1]$ do not contribute to the hypercohomology of $K_{II}$. And $T_E(-D)$ is isomorphic to ${\mathcal O}_E$. So we have $\mathbb{H}^0(B',K_{II}) = H^0(E,T_E(-D))$ is $1$-dimensional and $\mathbb{H}^{k>0}(B',K_{II})$ is zero. For $Q_{II}$ both terms contribute to the cohomology. We have $T_E(E)$ is isomorphic to ${\mathcal O}_E(1)$ so that $\mathbb{H}^0(B',Q_{II}) = H^0(E,T_E(E))$ is $2$-dimensional, and $\mathbb{H}^1(B',Q_{II}) = M = {\mathcal O}_S(B)|_p$ is $1$-dimensional. Therefore we have a long exact sequence in hypercohomology: $$\begin{aligned} 0 \rightarrow H^0(E,T_E(-D)) \rightarrow \mathbb{H}^0(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^0(B,L_{(f,i)}^\vee) \rightarrow \dots \\ \dots \rightarrow H^0(E,T_E(E)) \rightarrow \mathbb{H}^1(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^1(B,L_{(f,i)}^\vee) \rightarrow \dots \\ \dots \rightarrow M \rightarrow \mathbb{H}^2(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^2(B,L_{(f,i)}^\vee) \rightarrow 0\end{aligned}$$ Geometrically, every infinitesimal automorphism of $\zeta$ lifts to an infinitesimal automorphism of $\zeta'$, so that $\mathbb{H}^0(B,L_{(f,i)}^\vee)\rightarrow H^0(E,T_E(E))$ is the zero map. Similarly, the map $\mathbb{H}^1(B,L_{(f,i)}^\vee) \rightarrow {\mathcal O}_S(B)|_p$ is nonzero iff there are deformations of $\zeta$ which smooth the node $p$ to first order. So we have the following lemma: \[lem-def2a\] Suppose that $h:B'\rightarrow B$ is type (IIa). Then we have exact sequences: $$\begin{aligned} 0 \rightarrow H^0(E,T_E(-D)) \rightarrow \mathbb{H}^0(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^0(B,L_{(f,i)}^\vee) \rightarrow 0 \\ 0 \rightarrow H^0(E,T_E(E)) \rightarrow \mathbb{H}^1(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^1(B,L_{(f,i)}^\vee) \rightarrow {\mathcal O}_S(B)|_p \rightarrow 0 \\ 0 \rightarrow \mathbb{H}^2(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^2(B, L_{(f,i)}^\vee) \rightarrow 0 \end{aligned}$$ So the canonical map from the space of first order deformations of $\zeta'$ to the space of first order deformations of $\zeta$ has both $1$-dimensional kernel and cokernel, and the obstruction space to $\zeta'$ equals the obstruction space to $\zeta$. \[lem-def2b\] Suppose that $h:B'\rightarrow B$ is type (IIb). Then we have exact sequences: $$\begin{aligned} 0 \rightarrow H^0(E,T_E(-D)) \rightarrow \mathbb{H}^0(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^0(B,L_{(f,i)}^\vee) \rightarrow 0 \\ 0 \rightarrow H^0(E,T_E(E)) \rightarrow \mathbb{H}^1(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^1(B,L_{(f,i)}^\vee) \rightarrow 0 \\ 0 \rightarrow {\mathcal O}_S(B)|_p \rightarrow \mathbb{H}^2(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^2(B, L_{(f,i)}^\vee) \rightarrow 0 \end{aligned}$$ So the canonical map from the space of first order deformations of $\zeta'$ to the space of first order deformations of $\zeta$ is surjective with $1$-dimensional kernel, and the map from the obstruction space of $\zeta'$ to the obstruction space of $\zeta$ is surjective and has a $1$-dimensional kernel. Finally, we consider the case when $h:B'\rightarrow B$ is the identity map, but there is one extra marked point $q\in B'$ which is not in $B$; we call this *type (III)*. We further break this up as follows. If $q\in B$ lies on an unstable component, we call this *type (IIIa)*. If $q\in B$ lies on a stable component, we call this *type (IIIb)*. For type (III), there is a canonical short exact sequence of complexes: $$\begin{CD} 0 @>>> L_\zeta @>>> L_\zeta' @>>> \Omega_B(q)|_q[0] @>>> 0 \end{CD}$$ Of course we have: $$\begin{aligned} {\mathbb{E}\text{xt}_{{\mathcal O}_{B}}}^0(\Omega_B(q)|_q[0],{\mathcal O}_B) = 0, \\ {\mathbb{E}\text{xt}_{{\mathcal O}_{B}}}^1(\Omega_B(q)|_q[0],{\mathcal O}_B) = T_B|_q, \\ {\mathbb{E}\text{xt}_{{\mathcal O}_{B}}}^2(\Omega_B(q)|_q[0],{\mathcal O}_B) = 0.\end{aligned}$$ The induced map $\mathbb{E}\text{xt}^0_{{\mathcal O}_B}(L_\zeta,{\mathcal O}_B) \rightarrow T_B|_q$ is nonzero iff $q$ lies on an unstable component of $\zeta$. Thus we have the following lemma: \[lem-def3a\] Suppose that $h:B'\rightarrow B$ is type (IIIa). Then we have exact sequences: $$\begin{aligned} 0 \rightarrow {\mathbb{E}\text{xt}_{{\mathcal O}_{B'}}}^0(L_{\zeta'},{\mathcal O}_{B'}) \rightarrow {\mathbb{E}\text{xt}_{{\mathcal O}_{B}}}^0(L_\zeta,{\mathcal O}_B) \rightarrow T_B|_q \rightarrow 0 \\ 0 \rightarrow {\mathbb{E}\text{xt}_{{\mathcal O}_{B'}}}^1(L_{\zeta'},{\mathcal O}_{B'}) \rightarrow {\mathbb{E}\text{xt}_{{\mathcal O}_{B}}}^1(L_\zeta,{\mathcal O}_B) \rightarrow 0 \\ 0 \rightarrow {\mathbb{E}\text{xt}_{{\mathcal O}_{B'}}}^2(L_{\zeta'},{\mathcal O}_{B'}) \rightarrow {\mathbb{E}\text{xt}_{{\mathcal O}_{B}}}^2(L_\zeta,{\mathcal O}_B) \rightarrow 0 \end{aligned}$$ So the Lie algebra of infinitesimal automorphisms of $\zeta'$ has codimension $1$ in the Lie algebra of infinitesimal automorphisms of $\zeta$, the space of first order deformations of $\zeta'$ equals the space of first order deformations of $\zeta$, and the obstruction space of $\zeta'$ equals the obstruction space of $\zeta$. \[lem-def3b\] Suppose that $h:B'\rightarrow B$ is type (IIIb). Then we have exact sequences: $$\begin{aligned} 0 \rightarrow {\mathbb{E}\text{xt}_{{\mathcal O}_{B'}}}^0(L_{\zeta'},{\mathcal O}_{B'}) \rightarrow {\mathbb{E}\text{xt}_{{\mathcal O}_{B}}}^0(L_\zeta,{\mathcal O}_B) \rightarrow 0 \\ 0 \rightarrow T_B|_q \rightarrow {\mathbb{E}\text{xt}_{{\mathcal O}_{B'}}}^1(L_{\zeta'},{\mathcal O}_{B'}) \rightarrow {\mathbb{E}\text{xt}_{{\mathcal O}_{B}}}^1(L_\zeta,{\mathcal O}_B) \rightarrow 0 \\ 0 \rightarrow {\mathbb{E}\text{xt}_{{\mathcal O}_{B'}}}^2(L_{\zeta'},{\mathcal O}_{B'}) \rightarrow {\mathbb{E}\text{xt}_{{\mathcal O}_{B}}}^2(L_\zeta,{\mathcal O}_B) \rightarrow 0 \end{aligned}$$ So the Lie algebra of infinitesimal automorphisms of $\zeta'$ equals the Lie algebra of infinitesimal automorphisms of $\zeta$, the canonical map from the space of first order deformations of $\zeta'$ to the space of first order deformations of $\zeta$ is surjective with $1$-dimensional kernel, and the obstruction space of $\zeta'$ equals the obstruction space of $\zeta'$. Combining lemma \[lem-def1a\] through lemma \[lem-def3b\], one can analyze the associated maps of vector spaces ${\mathbb{E}\text{xt}_{{\mathcal O}_{B'}}}^k(L_{\zeta'},{\mathcal O}_{B'}) \rightarrow {\mathbb{E}\text{xt}_{{\mathcal O}_{B}}}^k(L_\zeta,{\mathcal O}_B)$ for any morphism $h:B'\rightarrow B$ which removes some subset of marked points from $B'$ and then contracts some subset of the unstable components. Gluing stable curves {#subsec-glue} -------------------- Just as one has an obstruction theory for ${\overline{{\mathcal}M}_{g,r}({X,\beta})}$ of the form ${\left}( {\mathbb R}\pi_*{\left}(L_\zeta^\vee{\right})[1]{\right})^\vee$, also for any stable $A$-graph $\tau$, one has an analogous obstruction theory for each of the Behrend-Manin stacks ${\overline{{\mathcal}M}}(X,\tau)$ (c.f. [@BM] for the definition of ${\overline{{\mathcal}M}}(X,\tau)$). We will not describe this obstruction theory here. In [@B], a *relative obstruction theory* for the morphism ${\overline{{\mathcal}M}}(X,\tau) \rightarrow {\mathfrak M}(\tau)$ is given from which an absolute obstruction theory for ${\overline{{\mathcal}M}}(X,\tau)$ can be deduced. Suppose that $\tau$ is a stable $A$-graph, and suppose that $\{f_1,f_2\}$ is a disconnecting edge of $\tau$. Let $\tau_1\subset \tau$ be the maximal connected subgraph which contains $f_1$ and not $f_2$, and let $\tau_2\subset \tau$ be the maximal connected subgraph which contains $f_2$ and not $f_1$. So we have forgetful $1$-morphisms ${\overline{{\mathcal}M}}(X,\tau) \rightarrow {\overline{{\mathcal}M}}(X,\tau_i)$ for $i=1,2$. \[lem-glue1\] Suppose that $\zeta:T \rightarrow {\overline{{\mathcal}M}}(X,\tau)$ is a $1$-morphism and let $\zeta_i:T \rightarrow {\overline{{\mathcal}M}}(X,\tau_i)$, $i=1,2$ be the composition of $\zeta$ with the forgetful $1$-morphism above. Suppose that for each point $t\in T$, the stable maps $\zeta_1(t)\in {\overline{{\mathcal}M}}(X,\tau_1)$ and $\zeta_2(t) \in {\overline{{\mathcal}M}}(X,\tau_2)$ are unobstructed (in the sense that the obstruction groups described above are zero) and the evaluation morphism $\text{ev}_{f_1}:{\overline{{\mathcal}M}}(X,\tau_1) \rightarrow X$ is smooth at $\zeta_1(t)$. Let $T_{\text{ev}_{f_1}}$ denote the dual of the sheaf of relative differentials of $\text{ev}_{f_1}$. Then also $\zeta(t)\in {\overline{{\mathcal}M}}(X,\tau)$ is unobstructed, and there is a short exact sequence: $$\begin{CD} 0 @>>> \zeta_1^*T_{\text{ev}_{f_1}} @>>> \zeta^* T_{{\overline{{\mathcal}M}}(X,\tau)} @>>> \zeta_2^* T_{{\overline{{\mathcal}M}}(X,\tau_2)} @>>> 0 \end{CD}$$ The proof essentially follows from the fact that ${\overline{{\mathcal}M}}(X,\tau)$ is an open substack of the $2$-fiber product: $${\overline{{\mathcal}M}}(X,\tau_1) \times_{\text{ev}_{f_1},X,\text{ev}_{f_2}} {\overline{{\mathcal}M}}(X,\tau_2).$$ The details are left to the reader. Now suppose that $\phi:\tau \rightarrow \sigma$ is the contraction of stable $A$-graphs which contracts the edge $\{f_1,f_2\}$. The induced $1$-morphism ${\overline{{\mathcal}M}}(X,\tau) \rightarrow {\overline{{\mathcal}M}}(X,\sigma)$ is unramified of codimension at most $1$. In some circumstances, it is the normalization of a Cartier divisor. \[lem-glue2\] With the same notation as in lemma \[lem-glue1\], suppose that $\tau$ is a genus $0$ tree. For $i=1,2$, let the domain of $\zeta_i$ be given by $\pi_i:C_i \rightarrow T$, let $g_i:C_i \rightarrow X$ be the map of $\zeta_i$, and let $s_i:T \rightarrow C_i$ be the section corresponding to the flag $f_i$ of $\tau_i$. Denote by $T_{\pi_i}$ the dual of the sheaf of relative differentials of $\pi_i$. Suppose that for every point $t\in T$, ${\overline{{\mathcal}M}}(X,\tau_2)$ is unobstructed at $\zeta_2(t)$, and suppose that $g_1^*T_X$ is $\pi_1$-relatively generated by global sections. Then for each point $t\in T$, ${\overline{{\mathcal}M}}(X,\sigma)$ is unobstructed at $\zeta(t)$, the morphism ${\overline{{\mathcal}M}}(X,\tau) \rightarrow {\overline{{\mathcal}M}}(X,\sigma)$ is a regular embedding of codimension $1$ at $\zeta(t)$, and we have a short exact sequence: $$\begin{CD} 0 @>>> \zeta^*T_{{\overline{{\mathcal}M}}(X,\tau)} @>>> \zeta^* T_{{\overline{{\mathcal}M}}(X,\sigma)} @>>> s_1^*T_{\pi_1} \otimes s_2^* T_{\pi_2} @>>> 0 \end{CD}$$ Let $\pi:C\rightarrow T$ be the family of curves obtained by identifying the section $s_1$ of $\pi_1$ and the section $s_2$ of $\pi_2$. Let $s:T \rightarrow C$ be the section corresponding to $s_1$ and $s_2$. Let $g:C \rightarrow X$ be the map obtained by gluing $g_1$ and $g_2$. First we prove that for any point $t\in T$ we have that ${\overline{{\mathcal}M}}(X,\sigma)$ is unobstructed at $\zeta(t)$ and the morphism ${\overline{{\mathcal}M}}(X,\tau) \rightarrow {\overline{{\mathcal}M}}(X,\sigma)$ is a regular embedding of codimension $1$. The two statements together are equivalent to the statement that there are first order deformations of the map $\zeta(t)\in {\overline{{\mathcal}M}}(X,\sigma)$ which smooth the node $s(t)\in C_t$. Now there is an exact sequence for $\zeta(t)\in {\overline{{\mathcal}M}}(X,\sigma)$: $$T_{{\overline{{\mathcal}M}}(X,\sigma)}|_{\zeta(t)} \rightarrow \text{Ext}^1{\left}( \Omega_{C_t},{\mathcal O}_{C_t} {\right}) \rightarrow H^1{\left}( C_t,g^*T_X {\right}) \rightarrow \text{Obs}(\zeta(t)) \rightarrow 0$$ Here $\text{Obs}(\zeta)$ is the obstruction group to ${\overline{{\mathcal}M}}(X,\sigma)$ at $\zeta$. Similarly, we have an exact sequence for $\zeta_2(t)$: $$\begin{CD} T_{{\overline{{\mathcal}M}}(X,\tau_2)}|_{\zeta_2(t)} @>>> \text{Ext}^1{\left}( \Omega_{(C_2)_t} {\right}) @>>> H^1{\left}( (C_2)_t, g_2^*T_X {\right}) @>>> 0 \end{CD}$$ We have a short exact sequence of sheaves on $C_t$: $$\begin{CD} 0 @>>> g_1^*T_X{\left}(-s(t){\right}) @>>> g^*T_X @>>> g_2^*T_X @>>> 0 \end{CD}$$ By assumption, $g_1^*T_X$ is generated by global sections. Thus by lemma \[lem-gend\], we conclude that $H^1{\left}(C_t, g_1^*T_X{\left}(-s(t) {\right}) {\right})$ is zero. Thus we have an identification of $H^1{\left}( C_t,g^*T_X {\right})$ and $H^1{\left}( (C_2)_t, g_2^*T_X {\right})$. Also, $\text{Ext}^1(\Omega_{C_t},{\mathcal O}_{C_t})$ is canonically isomorphic to the product over all nodes $q\in C_t$ of $T'_q\otimes T''_q$ where $T'_q$ and $T''_q$ are the tangent spaces to the two branches of $C_t$ at $q$. We have the analogous result for $(C_2)_t$. Via these identifications, we have a commutative diagram: $$\begin{CD} \text{Ext}^1{\left}( \Omega_{C_t},{\mathcal O}_{C_t} {\right}) @>>> H^1{\left}( C_t, g^* T_X {\right}) \\ @V p VV @VV = V \\ \text{Ext}^1{\left}( \Omega_{(C_2)_t}, {\mathcal O}_{(C_2)_t} {\right}) @>>> H^1{\left}( (C_2)_t, g_2^* T_X {\right}) \end{CD}$$ where $p$ is the canonical projection. Choose any section $$\phi:T'_{s(t)}\otimes T''_{s(t)} \rightarrow \text{Ext}^1{\left}( \Omega_{C_t}, {\mathcal O}_{C_t} {\right})$$ and any section $$\psi:\text{Ext}^1{\left}( \Omega_{(C_2)_t}, {\mathcal O}_{(C_2)_t} {\right}) \rightarrow \text{Ext}^1{\left}( \Omega_{C_t},{\mathcal O}_{C_t} {\right})$$ of the canonical projections. Choose any element $u\in T'_{s(t)}\otimes T''_{s(t)}$ and consider the image $\overline{\phi(u)}$ of $\phi(u)$ in $H^1{\left}( C_t, g^* T_X {\right})$. Since the map $$\begin{CD} \text{Ext}^1{\left}( \Omega_{(C_2)_t}, {\mathcal O}_{(C_2)_t} {\right}) @>>> H^1{\left}( (C_2)_t, g_2^* T_X {\right}) \end{CD}$$ is surjective, we can find some element $v\in \text{Ext}^1{\left}( \Omega_{(C_2)_t}, {\mathcal O}_{(C_2)_t} {\right})$ such that $\overline{\psi(v)}$ equals $\overline{\phi(u)}$. Consider $\phi(u)-\psi(v)$. This has image $0$ in $H^1{\left}( C_t, g^*T_X {\right})$. Therefore we conclude that it is in the image of $T_{{\overline{{\mathcal}M}}(X,\sigma)}|_{\zeta(t)}$. So we conclude that $u$ is in the image of the projection map $T_{{\overline{{\mathcal}M}}(X,\sigma)} \rightarrow T'_{s(t)}\otimes T''_{s(t)}$, i.e. this projection map is surjective. So the deformations of $\zeta(t)$ smooth the node $s(t)$. Therefore ${\overline{{\mathcal}M}}(X,\sigma)$ is smooth at $\zeta(t)$ and the morphism ${\overline{{\mathcal}M}}(X,\tau) \rightarrow {\overline{{\mathcal}M}}(X,\sigma)$ is a regular embedding of codimension $1$ at $\zeta(t)$. Finally, the short exact sequence above is just the globalized version of the projection map $T_{{\overline{{\mathcal}M}}(X,\sigma} \rightarrow T'_{s(t)}\otimes T''_{s(t)}$ appearing in the last paragraph. Properties of families of stable maps {#sec-props} ===================================== In this section we introduce some definitions and lemmas regarding properties of families of stable maps. Recall, to prove that ${\overline{{\mathcal}M}_{0,0}({X,e})}$ is rationally connected, we have to find a *very free* $1$-morphism $\zeta:{\mathbb{P}}^1 \rightarrow {\overline{{\mathcal}M}_{0,0}({X,e})}$, i.e. a $1$-morphism whose image is contained in the smooth locus and such that $\zeta^*T_{{\overline{{\mathcal}M}_{0,0}({X,e})}}$ is an ample vector bundle. Our proof that such a $1$-morphism exists is by induction, where the induction step consists of attaching a $1$-parameter family of lines to our $1$-parameter family of degree $e$ stable maps. To make the induction argument work, we need a bit more than a very free $1$-morphism $\zeta:{\mathbb{P}}^1 \rightarrow {\overline{{\mathcal}M}_{0,0}({X,e})}$. The property we need is what we call a *very positive* $1$-morphism. Additionally, we need that our $1$-parameter family of lines has a property which we call *very twisting*. Finally, because of an operation we perform on $1$-morphisms which we call *modification*, and which we introduce in the next section, we need to consider the case of $1$-morphisms with reducible domain, i.e. $\zeta:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e})}$ where $B$ is a connected, prestable curve of arithmetic genus $0$. This is the level of generality in which we make all our definitions. Given a genus $0$ stable map $\zeta = ((B,p_1,\dots,p_n),f:B\rightarrow X)$, we say $\zeta$ is *very stable* if the unmarked prestable map $(B,f:B\rightarrow X)$ is stable. \[not-1mor\] Given a closed subscheme $X\subset {\mathbb{P}}^n$ and a scheme $B$, a $1$-morphism $\zeta:B\rightarrow {\overline{{\mathcal}M}_{g,r}({X,e})}$ is equivalent to a datum: $$\zeta = {\left}( {\left}(p_\zeta:\Sigma_\zeta \rightarrow B, \sigma_{\zeta,1},\dots, \sigma_{\zeta,r} {\right}), g_\zeta {\right}).$$ Here $p_\zeta:\Sigma_\zeta\rightarrow B$ is a family of prestable curves, $\sigma_{\zeta,i}:B\rightarrow \Sigma_\zeta$ is a collection of sections, $g_\zeta:\Sigma_\zeta\rightarrow X$ is a morphism of schemes and we denote $h_{\zeta,i}=g_\zeta\circ \sigma_{\zeta,i}$. When there is no risk of confusion, we will suppress the $\zeta$ subscripts.  \[defn-twisting\] Suppose $\pi:B\rightarrow T$ is a family of prestable, geometrically connected curves of arithmetic genus $0$. Suppose given a $1$-morphism $\zeta:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$, i.e. a datum $$\zeta = {\left}( p:\Sigma \rightarrow B, \sigma: B \rightarrow \Sigma, g:\Sigma \rightarrow X {\right})$$ such that $X$ is smooth along $g(\Sigma)$. The $1$-morphism $\zeta:B\rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ is *twisting* if 1. The data $(\pi:B\rightarrow T, h:B \rightarrow X)$ is a family of stable maps to $X$, i.e. a $1$-morphism $\xi:T\rightarrow {\overline{{\mathcal}M}_{0,0}({X,e})}$ for some $e\geq 0$. 2. The image of $\xi:T\rightarrow {\overline{{\mathcal}M}_{0,0}({X,e})}$ is contained in the *very unobstructed* locus of ${\overline{{\mathcal}M}_{0,0}({X})}$. 3. The image of $\zeta:T \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ is contained in the *very unobstructed* locus of the evaluation morphism $\text{ev}:{\overline{{\mathcal}M}_{0,1}({X,1})} \rightarrow X$. 4. Denoting by $T_{\text{ev}}$ the dual of the sheaf of relative differentials $\Omega_{\text{ev}}$, the pullback bundle $\zeta^*T_{\text{ev}}$ is $\pi$-relatively generated by global sections. 5. Denoting by $\text{pr}:{\overline{{\mathcal}M}_{0,1}({X,1})}\rightarrow {\overline{{\mathcal}M}_{0,0}({X,1})}$ the projection map, and by $T_{\text{pr}}$ the dual of the sheaf of relative differentials $\Omega_{\text{pr}}$, the pullback bundle $\zeta^*T_{\text{pr}}$, i.e. the line bundle $\sigma^*{\mathcal O}_\Sigma(\sigma)$, is $\pi$-relatively generated by global sections.  \[defn-verytwisting\] With notation as in definition \[defn-twisting\], a morphism $\zeta:T\rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ is *very twisting* if it is twisting and if $\zeta^*T_{\text{ev}}$ is $\pi$-relatively deformation ample.  \[rmk-twisting\] Regarding the definitions above: 1. In $(2)$ and $(3)$ of defintion \[defn-twisting\], *very unobstructed* means that the naive obstruction group vanishes. For $(2)$ this means that for each $t\in T$ and the corresponding stable map $(h_t:B_t\rightarrow X)$, the following group vanishes: $$\begin{CD} & -1 & & 0 \\ \mathbb{E}\text{xt}^1_{B_t}( & h_t^*\Omega_X @>>> \Omega_{B_t},\ & {\mathcal O}_{B_t}) \end{CD}$$ 2. It is easy to see that $\zeta^* T_{\text{pr}}$ is just $\sigma^*{\mathcal O}_{\Sigma}(\sigma)$. 3. Observe the product morphism $(p,g):\Sigma\rightarrow B \times X$ is a regular embedding. Denote by ${{\mathcal}N}$ the normal bundle of this regular embedding. Then $(3)$ is equivalent to the condition that $R^1p_*{\left}({{\mathcal}N}(-\sigma){\right})$ is trivial. In this case $\zeta^*T_{\text{ev}}$ is the locally free sheaf $p_*{\left}({{\mathcal}N}(-\sigma){\right})$. 4. Since the prestable family of maps $(\pi:B \rightarrow T, \xi:B \rightarrow X)$ is stable, clearly also $(\pi:B \rightarrow T, \zeta: B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})})$ is stable. 5. There are some degree conditions implicit in these definitions. The total degree of $\sigma^*{\mathcal O}_{\Sigma}(\sigma)$ is simply $s=2e-e'$ where $e$ is the degree of $h:B \rightarrow {\mathbb{P}}^N$ and $e'$ is the degree of $g:\Sigma \rightarrow {\mathbb{P}}^N$ (both degrees with respect to ${\mathcal O}_{{\mathbb{P}}^N}(1)$). So if $\zeta$ is twisting, we have that $2e \geq e'$ and if $\zeta$ is very twisting, we have that $2e > e'$. 6. Additionally, given a twisting family $\zeta$ a point $b\in B$, and a deformation of the line $g_b:\Sigma_b \rightarrow X$ which continues to contain $h(b)$, there must be a deformation of the whole family $\zeta$ giving rise to the deformation of $\Sigma_b$ and which does not deform $h:B \rightarrow X$. In particular, if $h:B \rightarrow X$ is also an embedded line, then the map of the surface $g:\Sigma \rightarrow X$ must have degree $1$ or $2$ and must deform along with a line which intersects $h(B)$ in a fixed point. If $X \subset {\mathbb{P}}^n$ is a hypersurface of low degree $d > 1$ such that to a general line $h:B \rightarrow X$ there is a corresponding twisting family $\zeta$ (what we refer to as a *twistable line* below), then we must have that $h:B \rightarrow X$ is an embedding of a smooth quadric surface. The condition on such $X$ that a general line is *twistable* is essentially that, given two general intersecting lines $B$ and $L$ in $X$, there is a smooth quadric surface $\Sigma$ in $X$ which contains both $B$ and $L$. In a later section we will see that this condition does hold for a general hypersurface $X\subset {\mathbb{P}}^n$ of degree $d$ when $d^2 \leq n+1$. \[lem-twisttogether\] Suppose $B = B_1 \cup B_2$ is a prestable, geometrically connected curve of arithmetic genus $0$ where $B_1$ and $B_2$ are connected subcurves such that $B_1 \cap B_2$ is a single node of $B$. Suppose given $\zeta:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ such that $\zeta|_{B_i} : B_i \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ is twisting for $i=1,2$. Then $\zeta$ is twisting. If, in addition, at least one of $\zeta_i$ is very twisting, then $\zeta$ is very twisting. This is an easy consequence of lemma \[lem-DAcrit\]. \[lem-twistopen\] Let $\pi:B\rightarrow T$ be a family of prestable, geometrically connected curves of arithmetic genus $0$ and let $\zeta:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ be a morphism. There is an open subscheme $U_{\text{twist}}\subset T$ (resp. $U_{vtwist}\subset T$) with the following property: for any morphism of schemes $f:T'\rightarrow T$, the pullback family $f^*\pi:f^*B \rightarrow T'$ and $f^*\zeta: f^*B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ is twisting (resp. very twisting) iff $f(T')\subset U_{\text{twist}}$ (resp. $f(T')\subset U_{\text{vtwist}}$). By [@B lemma 1] there is an universal open subscheme $U_1\subset T$ over which $(\pi:B\rightarrow T, h:B\rightarrow X)$ is a family of stable maps. It is clear that $U$, if it exists, must also be contained in the complement of the support of $$\mathbb{R}^1{\left}(\pi_*\textit{Hom}_{{\mathcal O}_B}{\right}){\left}(h^*\Omega_X \rightarrow \Omega_\pi, {\mathcal O}_B{\right}),$$ and in the complement of the image under $\pi$ of the supports of the sheaves: $$\begin{aligned} R^1p_*{\left}({{\mathcal}N}(-\sigma){\right}), \\ \text{coker}{\left}( \pi^*\pi_* \sigma^*{\mathcal O}_{\Sigma}(\sigma) \rightarrow \sigma^*{\mathcal O}_{\Sigma}(\sigma) {\right}).\end{aligned}$$ Let $U_2$ denote the complement of these sets in $U_1$. On $U_2$ all of the conditions to be twisting (resp. very twisting) are satisfied except the condition that $\zeta^*T_{\text{ev}}$ is $\pi$-relatively generated by global sections (resp. $\pi$-relatively deformation ample). So we define $U_{\text{twist}}$ to be the complement in $U_2$ of the image under $\pi$ of the cokernel of the morphism $$\pi^*\pi_* \zeta^*T_{\text{ev}}\rightarrow \zeta^*T_{\text{ev}}.$$ And, using lemma \[lem-opDA\], we define $U_{\text{vtwist}}$ to be the universal open subscheme of $U_2$ over which $\zeta^*T_{\text{ev}}$ is $\pi$-relatively deformation ample. It follows immediately from the construction that $U_{\text{twist}}$ and $U_{\text{vtwist}}$ have the desired universal properties. \[defn-twistable\] Suppose $(\pi:B\rightarrow T, h:B\rightarrow X)$ is a family of genus $0$ stable maps, i.e. a $1$-morphism $\xi:T\rightarrow {\overline{{\mathcal}M}_{0,0}({X,e})}$ for some $e \geq 0$. We say $\xi:T \rightarrow {\overline{{\mathcal}M}_{0,0}({X,e})}$ is *twistable* (resp. *very twistable*) if there exists a surjective étale morphism $u:T'\rightarrow T$ and a morphism $\zeta:u^*B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ with $h_\zeta = u^* h$ such that $\zeta$ is twisting (resp. very twisting). \[prop-twistopen\] Let $\xi = (\pi:B\rightarrow T, h:B\rightarrow X)$ be a $1$-morphism $\xi:T \rightarrow {\overline{{\mathcal}M}_{0,0}({X,e})}$. There is an open subscheme $U_{t-able}\subset T$ (resp $U_{vt-able}\subset T$) such that for each morphism of schemes $f:T'\rightarrow T$, the pullback $(f^*\pi: f^*B \rightarrow T', f^*h: f^*B \rightarrow X)$ is twistable (resp. very twistable) iff $f(T') \subset U_{t-able}$ (resp. $f(T')\subset U_{vt-able}$). It suffices to check that if $t_0\in T$ is a geometric point such that $h_{t_0}:B_{t_0} \rightarrow X$ is twistable (resp. very twistable), then there is an étale neighborhood of $t_0\in T$ over which $\xi$ is twistable (resp. very twistable). Denote by $\zeta_0:B_{t_0} \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ the twisting morphism. We consider ${\overline{{\mathcal}M}_{0,1}({X,1})}$ as a projective scheme via the Plücker and Segré embeddings of ${\mathbb{G}}(1,n) \times {\mathbb{P}}^n \hookrightarrow {\mathbb{P}}^{\frac{n(n+1)^2}{2}-1}$. Let $\beta$ denote the degree of the stable map $\zeta_0$. Define ${{\mathcal}M} = T \times {\overline{{\mathcal}M}_{0,0}({{\overline{{\mathcal}M}_{0,1}({X,1})}, \beta})}$, i.e. ${{\mathcal}M}$ parametrizes pairs $(t,\zeta)$ where $t\in T$ is a point and where $\zeta:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ is a genus $0$ stable map of degree $\beta$. Denote the universal stable map by $$\begin{aligned} \rho:{{\mathcal}B} \rightarrow {\overline{{\mathcal}M}_{0,0}({{\overline{{\mathcal}M}_{0,1}({X,1})},\beta})}, \\ \zeta: {{\mathcal}B} \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}.\end{aligned}$$ As in notation \[not-1mor\], let $p:\Sigma \rightarrow {{\mathcal}B}$ be the pullback by $\zeta$ of the universal curve over ${\overline{{\mathcal}M}_{0,1}({X,1})}$, let $\sigma:{{\mathcal}B} \rightarrow \Sigma$ be the pullback of the universal section, let $g:\Sigma \rightarrow X$ be the pullback of the universal map, and let $h = g \circ \sigma$. So we have a family of prestable maps $$\widetilde{\xi} = {\left}( \rho:{{\mathcal}B} \rightarrow {\overline{{\mathcal}M}_{0,0}({{\overline{{\mathcal}M}_{0,1}({X,1})}, \beta})}, h: {{\mathcal}B} \rightarrow X {\right}).$$ By [@B lemma 1] there is a maximal open substack ${{\mathcal}U}_e \subset {\overline{{\mathcal}M}_{0,0}({{\overline{{\mathcal}M}_{0,1}({X,1})}, \beta})}$ over which $\widetilde{\xi}$ is stable of degree $e$. By assumption, $(t_0,\zeta_0)$ is in $T \times {{\mathcal}U}_e$. In the last paragraph we constructed a $1$-morphism $$(1_T,\widetilde{\xi}): T \times {{\mathcal}U}_e \rightarrow T \times {\overline{{\mathcal}M}_{0,0}({X,e})}.$$ We also saw that $(t_0,\zeta_0)$ is in the domain of this $1$-morphism. The claim is that $(1_T,\widetilde{\xi})$ is smooth on a neighborhood of $(t_0,\zeta_0)$, i.e. that $\widetilde{\xi}: {{\mathcal}U}_e \rightarrow {\overline{{\mathcal}M}_{0,0}({X,e})}$ is smooth at $\zeta_0$. First we will show that ${{\mathcal}U}_e$ is smooth at $\zeta_0$. The space of first order deformations and the obstruction space of ${{\mathcal}U}_e$ at $\zeta_0$ are given by $$\mathbb{E}\text{xt}^i_{B_{t_0}}(L^\cdot_{\zeta_0},{\mathcal O}_{B_{t_0}}),$$ for $i=1,2$ respectively, where $L^\cdot_{\zeta_0}$ is the complex $$\begin{CD} -1 & & 0 \\ \zeta_0^* \Omega_{{\overline{{\mathcal}M}_{0,1}({X,1})}} @> d(\zeta_0)^\dagger >> \Omega_{B_{t_0}}. \end{CD}$$ Now the induced morphism $\xi_0:B_{t_0} \rightarrow X$ by $\xi_0 = \text{ev} \circ \zeta_0$ also has an associated complex $L^\cdot_{\xi_0}$: $$\begin{CD} -1 & & 0 \\ \xi_0^* \Omega_X @> d(\xi_0)^\dagger >> \Omega_{B_{t_0}}. \end{CD}$$ There is a morphism of complexes: $$\begin{aligned} \gamma: L^\cdot_{\xi_0} \rightarrow L^\cdot_{\zeta_0} \\ \gamma^0 = \text{id} : \Omega_B \rightarrow \Omega_B \\ \gamma^{-1} = \zeta_0^*{\left}( d(\text{ev})^\dagger {\right}): \zeta_0^* \text{ev}^* \Omega_X \rightarrow \zeta_0^* \Omega_{{\overline{{\mathcal}M}_{0,1}({X,1})}}.\end{aligned}$$ There is also a morphism of complexes: $$\delta: L^\cdot_{\zeta_0} \rightarrow \zeta_0^* \Omega_{\text{ev}}[1],$$ where $\delta^{-1}:\zeta_0^* \Omega_{{\overline{{\mathcal}M}_{0,1}({X,1})}} \rightarrow \zeta_0^*\Omega_{\text{ev}}$ is the pullback of the canonical surjection. And the triple: $$\begin{CD} L^\cdot_{\xi_0} @> \gamma >> L^\cdot_{\zeta_0} @> \delta >> \zeta_0^* \Omega_{\text{ev}}[1] \end{CD}$$ is an exact triangle. Thus there is a corresponding long exact sequence of $\mathbb{E}\text{xt}$’s. Condition $(2)$ of definition \[defn-twisting\] says that $\mathbb{E}\text{xt}^1_{B_{t_0}}(L^\cdot_{\xi_0},{\mathcal O}_{B_{t_0}})$ is zero. By condition $(4)$ of the definition, $\zeta_0^* T_{\text{ev}}$ is generated by global sections. Since $B_{t_0}$ is connected of arithmetic genus $0$, we have that $H^1(B_{t_0},{\mathcal O}_{B_{t_0}})$ is zero. So for any trivial bundle, $H^1$ is zero. Since $H^2$ vanishes on all coherent sheaves, we conclude that for any sheaf generated by global sections, $H^1$ is zero. Thus we have that the group $$\mathbb{E}\text{xt}^2(\zeta_0^*\Omega_{\text{ev}}, {\mathcal O}_{B_{t_0}}) = H^1(B_{t_0}, T_{\text{ev}}),$$ is also zero. By the long exact sequence, we conclude that $\mathbb{E}\text{xt}^1_{B_{t_0}}( L^\cdot_{\zeta_0}, {\mathcal O}_{B_{t_0}} )$ is also zero. So the obstruction group vanishes and ${{\mathcal}U}_e$ is smooth at $\zeta_0$. By condition $(2)$ the image point $\xi_0\in {\overline{{\mathcal}M}_{0,0}({X,e})}$ is a smooth point of ${\overline{{\mathcal}M}_{0,0}({X,e})}$. Thus to prove that $\widetilde{\xi}: {{\mathcal}U}_e \rightarrow {\overline{{\mathcal}M}_{0,0}({X,e})}$ is smooth, it suffices to prove that derivative map $d(\widetilde{\xi})$ is surjective on the space of first order deformations. The map $$d(\widetilde{\xi}): \mathbb{E}\text{xt}^1_{B_{t_0}}(L^{\cdot}_{\zeta_0}, {\mathcal O}_{B_{t_0}}) \rightarrow \mathbb{E}\text{xt}^1_{B_{t_0}}(L^\cdot_{\xi_0}, {\mathcal O}_{B_{t_0}}),$$ is precisely the map occurring in the long exact sequence of $\mathbb{E}\text{xt}$’s from the paragraph above. By the long exact sequence, the cokernel of this map is a subgroup of $H^1(B_{t_0},T_{\text{ev}})$, and this is zero as we have seen. Therefore $\widetilde{\xi}$ is smooth at $\zeta_0\in {{\mathcal}U}_e$. Consider the morphism $(1_T,\xi): T\rightarrow T \times {\overline{{\mathcal}M}_{0,0}({X,e})}$. We can form the fiber product ${{\mathcal}M}$ of $(1_T,\xi)$ with the morphism $(1_T,\widetilde{\xi}): T\times {{\mathcal}U}_e \rightarrow T \times {\overline{{\mathcal}M}_{0,0}({X,e})}$. The fiber product ${{\mathcal}M}$ exactly parametrizes triples $(t,\zeta,\theta)$ where $t\in T$ is a point, $\zeta:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ is a point in ${{\mathcal}U}_e$, and $\theta:\xi_t \rightarrow \widetilde{\zeta}$ is an equivalence of objects in the groupoid ${\overline{{\mathcal}M}_{0,0}({X,e})}({\text{Spec }}\kappa(t))$. The projection map $\text{pr}_1:{{\mathcal}M} \rightarrow T$ is smooth at $(t_0,\zeta_0)$ by the last paragraph. So we can find an étale morphism $f:M \rightarrow {{\mathcal}M}$ of a scheme to ${{\mathcal}M}$ whose image contains $(t_0,\zeta_0)$, and such that $M\rightarrow T$ is smooth. Thus there is an étale morphism $u:T' \rightarrow T$ and a section $z:T' \rightarrow M$. Define $\zeta:T'\rightarrow {{\mathcal}U}_e$ to be the composition $\text{pr}_2\circ g \circ z$. We also denote by $\zeta:B' \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ the pullback by $\zeta:T'\rightarrow {{\mathcal}U}$ of the universal stable map. As $\widetilde{\xi}(\zeta): B' \rightarrow {\overline{{\mathcal}M}_{0,0}({X,e})}$ is equivalent to $u^* \xi: u^*B \rightarrow {\overline{{\mathcal}M}_{0,0}({X,e})}$, after replacing $T'$ by an étale, cover, we may suppose that $B'=u^*B$ as $T'$-schemes, and $\widetilde{\xi}(\zeta) = u^*\xi$. Now the fiber of $\zeta:u^*B\rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ over any preimage of $(t_0,\zeta_0)$ is twisting. So by lemma \[lem-twistopen\], up to replacing $T'$ by a Zariski open subscheme, we may suppose that $\zeta:u^*B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ is twisting. Similarly, if $(t_0,\zeta_0)$ is very twisting, we may suppose that $\zeta$ is very twisting. So we conclude that on the Zariski open subscheme of $T$ which is the image of $u:T'\rightarrow T$, the family $\xi:B \rightarrow {\overline{{\mathcal}M}_{0,0}({X,e})}$ is twistable (resp. very twistable). Since this holds for every point $t_0\in T$ where $\xi_0$ is twistable, the lemma is proved. \[lem-twisttogether2\] Suppose given two families $$\xi_i = {\left}( {\left}( \pi_i:B_i \rightarrow T, \sigma_i:T \rightarrow B_i {\right}), h_i:B_i \rightarrow X {\right}), i=1,2.$$ such that for each $(\pi_i:B_i \rightarrow T, h_i: B_i \rightarrow X)$ is twistable, and such that $h_1\circ \sigma_1 = h_2 \circ \sigma_2$. For each $t\in T$, assume that the locus of free lines in $X$ passing through $h_1\circ\sigma_1(t) = h_2\circ \sigma_2(t)$ is irreducible. Let us denote by $$\xi = {\left}( \pi:B \rightarrow T, h:B \rightarrow X {\right})$$ the family obtained by taking $B$ to be the connected sum of $B_1$ and $B_2$ where the section $\sigma_1$ is identified with the section $\sigma_2$. Then $\xi$ is a twistable family. Moreover, if at least one of $\xi_1, \xi_2$ is very twistable, then $\xi$ is very twistable. This follows essentially by lemma \[lem-twisttogether\]. First of all, using proposition \[prop-twistopen\], it suffices to prove the result when $T={\text{Spec }}k$ for some algebraically closed field $k$. We suppose that we are in this case. For each of $i=1,2$, let ${{\mathcal}M}_1$ denote the fiber product constructed in the proof of proposition \[prop-twistopen\], i.e. ${{\mathcal}M}_1$ parametrizes pairs $(\zeta_i,\theta_i)$ where $\zeta_i:B_i \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ is a twisting family (resp. very twisting family) such that the induced map $$\widetilde{\zeta}_i = {\left}( (B_i,\sigma_i),g_i\circ \rho_i:B_i \rightarrow X {\right})$$ is stable, and where $\theta_i:\xi_i \rightarrow \widetilde{\zeta}_i$ is an equivalence of objects. Since each of $\xi_i$ is twistable, we see that each of ${{\mathcal}M}_i$ is nonempty. By the proof of proposition \[prop-twistopen\], each of ${{\mathcal}M}_i$ is smooth. By the definition of twisting families, for each $i=1,2$ the morphism $$e_i: {{\mathcal}M}_i \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})},\ \zeta_i\mapsto \zeta_i(\sigma_i)$$ has image contained in the unobstructed locus of $\text{ev}: {\overline{{\mathcal}M}_{0,1}({X,1})}\rightarrow X$. Let $P\subset {\overline{{\mathcal}M}_{0,1}({X,1})}$ be the preimage under $\text{ev}$ of the point $p=h_1(\sigma_1)=h_2(\sigma_2)$. The image of $e_i$ is contained in the smooth locus of $P$. The claim is that $e_i:{{\mathcal}M}_i \rightarrow P$ is smooth. By [@K proposition I.2.14.2], the obstruction space at a point $\zeta_i$ is contained in the cohomology group $H^1{\left}( B_i, \zeta_i^* T_{\text{ev}}(-\sigma_i) {\right})$. By the definition of a twisting family, $\zeta_i^* T_{\text{ev}}$ is generated by global sections. Thus, by lemma \[lem-gend\], the cohomology group above is zero. Since the obstruction space vanishes, we conclude that $e_i$ is smooth. Since both $e_1:{{\mathcal}M}_1 \rightarrow P$ and $e_2:{{\mathcal}M}_2 \rightarrow P$ are smooth, both have nonempty, open image. And $P$ is irreducible by assumption. Therefore the image of $e_1$ and the image of $e_2$ intersect. If we choose a family $\zeta_1\in {{\mathcal}M}_1$ and $\zeta_2\in {{\mathcal}M}_2$ such that $e_1(\zeta_1)=e_2(\zeta_2)$, then we can glue $\zeta_1$ and $\zeta_2$ to obtain a morphism $\zeta:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ such that $\zeta|_{B_1}=\zeta_1$ and $\zeta|_{B_2} = \zeta_2$. By lemma \[lem-twisttogether\], we conclude that $\zeta$ is twisting. Moreover, if at least one of $\zeta_i, i=1,2$ is very twisting, then $\zeta$ is very twisting. And $\widetilde{\zeta} = \xi$. This shows that $\xi$ is twistable, and it is very twistable if at least one of $\xi_i, i=1,2$ is very twistable. \[hyp-2\] Let $U \subset {\overline{{\mathcal}M}_{0,1}({X,1})}$ denote the preimage of $U_{\text{t-able}} \subset {\overline{{\mathcal}M}_{0,0}({X,1})}$ under $\text{pr}$. The evaluation morphism $\text{ev}:U \rightarrow X$ has Zariski dense image. \[defn-pos\] Suppose $\pi:B \rightarrow T$ is a family of prestable, geometrically connected curves of arithmetic genus $0$. A $1$-morphism $\zeta:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e})}$ is *positive* (resp. *very positive*) if: 1. The data $(\pi:B \rightarrow T, h:B \rightarrow X)$ is a family of stable maps to $X$, i.e. a $1$-morphism $\xi:T \rightarrow {\overline{{\mathcal}M}_{0,0}({X,\epsilon})}$ for some $\epsilon\geq 0$. 2. The image of $\xi:T \rightarrow {\overline{{\mathcal}M}_{0,0}({X,\epsilon})}$ is contained in the *very unobstructed* locus of ${\overline{{\mathcal}M}_{0,0}({X,\epsilon})}$. 3. The image of $\text{pr}\circ \zeta: T\rightarrow {\overline{{\mathcal}M}_{0,0}({X,e})}$ is contained in the *very unobstructed* locus of ${\overline{{\mathcal}M}_{0,0}({X,e})}$. 4. The pullback bundle $(\text{pr}\circ \zeta)^* T_{{\overline{{\mathcal}M}_{0,0}({X,e})}}$ is $\pi$-relatively deformation ample. 5. The pullback line bundle $\sigma^*{\mathcal O}_{\Sigma}(\sigma)$ is $\pi$-relatively generated by global sections (resp. $\pi$-relatively ample). \[rmk-pos\] Regarding the definition above: 1. This definition is very similar to definition \[defn-twisting\]. It differs in that $e$ need not equal $1$ and that we only require $\text{pr}\circ \zeta$ to have image in the very unobstructed locus, instead of requiring $\zeta$ to have image in the very unobstructed locus of $\text{ev}$. 2. Consider the case when $T = {\text{Spec }}\kappa$ for some field $\kappa$, and suppose that $B$ is smooth, i.e. $B \cong {\mathbb{P}}^1_\kappa$. If $\zeta:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e})}$ is positive, then the morphism $\text{pr} \circ \zeta: B \rightarrow {\overline{{\mathcal}M}_{0,0}({X,e})}$ is *very free* in the sense of Debarre [@De p. 86]. \[lem-posopen\] Let $\pi:B \rightarrow T$ be a family of prestable, geometrically connected curves of arithmetic genus $0$ and let $\zeta:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ be a $1$-morphism. There is an open subscheme $U_{\text{pos}} \subset T$ (resp. $U_{\text{v-pos}}\subset T$) with the following property: for any morphism of schemes $f:T'\rightarrow T$, the pullback family $f^*\pi: f^*B \rightarrow T'$ and $f^*\zeta: f^*B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ is positive (resp. very positive) iff $f(T')\subset U_{\text{free}}$ (resp. $f(T')\subset U_{\text{v-free}}$). The proof is almost identical to the proof of lemma \[lem-twistopen\]. \[lem-posdef\] Suppose that $T={\text{Spec }}k$ is a point and $\zeta:B\rightarrow {\overline{{\mathcal}M}_{0,1}({X,e})}$ is a positive $1$-morphism whose image is contained in the locus of very stable maps. 1. If $B$ is smooth, then $\zeta:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e})}$ is *free* in the sense of Kollár  [@K definition II.3.11]. If $\zeta$ is very positive, then $\zeta$ is *very free*. 2. In any case, the $1$-morphism is *unobstructed* in the sense of Kollár [@K definition I.2.6]; in particular it is the specialization of a positive $1$-morphism $\zeta_\eta:B_\eta \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e})}$ with $B_\eta$ geometrically connected and smooth, and whose image is contained in the locus of very stable maps. Suppose that the image of $\zeta$ lies in the locus of very stable maps. By the relative version of lemma \[lem-def3a\], we have that the image of $\zeta$ is in the smooth locus of ${\overline{{\mathcal}M}_{0,1}({X,e})}$. And we have a short exact sequence: $$\begin{CD} 0 @>>> \sigma^* {\mathcal O}_{\Sigma}(\sigma) @>>> \zeta^* T_{{\overline{{\mathcal}M}_{0,1}({X,e})}} @>>> (\text{pr}\circ \zeta)^* T_{{\overline{{\mathcal}M}_{0,0}({X,e})}} @>>> 0 \end{CD}$$ By condition $(4)$ of definition  \[defn-pos\], $(\text{pr}\circ \zeta)^* T_{{\overline{{\mathcal}M}_{0,0}({X,e})}}$ is deformation ample. And by condition $(5)$ of definition  \[defn-pos\], $\sigma^*{\mathcal O}_{\Sigma}(\sigma)$ is generated by global sections (resp. deformation ample). Therefore $\zeta^* T_{{\overline{{\mathcal}M}_{0,1}({X,e})}}$ is generated by global sections. And if $\zeta$ is very positive, then $\zeta^* T_{{\overline{{\mathcal}M}_{0,1}({X,e})}}$ is deformation ample by $(2)$ of lemma  \[lem-secDA\]. So if $B$ is smooth, then $\zeta$ is free, and it is very free if $\zeta$ is very positive. This proves $(1)$. Since $\zeta^* T_{{\overline{{\mathcal}M}_{0,1}({X,e})}}$ is generated by global sections, by lemma  \[lem-gend\], $H^1(B,\zeta^* T_{{\overline{{\mathcal}M}_{0,1}({X,e})}})$ is zero. Therefore $\zeta$ is unobstructed in the sense of Kollár. Now let $\pi: {{\mathcal}B} \rightarrow {\text{Spec }}R$ be a smoothing of $B$, i.e. a flat family of proper, geometrically connected, prestable curves of arithmetic genus $0$ over a DVR such that the special fiber is isomorphic to $B$ and such that the general fiber $B_\eta$ is smooth. By  [@K theorem I.2.10], the projection of the relative Hom-scheme, $$\textit{Hom}_{{\text{Spec }}R}({{\mathcal}B}, {\text{Spec }}R \times {\overline{{\mathcal}M}_{0,1}({X,e})}) \rightarrow {\text{Spec }}R,$$ is smooth at $[\zeta]$. Therefore, after making some finite, flat base change ${\text{Spec }}R' \rightarrow {\text{Spec }}R$ we may suppose that $\zeta$ is the specialization of a $1$-morphism $\zeta_R:{{\mathcal}B} \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e})}$. By lemma  \[lem-posopen\], we have that $\zeta_R$ is positive, in particular $\zeta_\eta:B_\eta \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e})}$ is positive. Since the locus of very stable maps in ${\overline{{\mathcal}M}_{0,1}({X,e})}$ is open, we conclude that the image of $\zeta_R$ is contained in this locus. Now we come to the main notion of this section. \[defn-inducts\] Suppose $\pi:B \rightarrow T$ is a family of prestable, geometrically connected curves of arithmetic genus $0$. An *inducting pair* of degree $e$ is a pair $${\left}( \zeta_1:B\rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}, \overline{\zeta}_e:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e})} {\right}),$$ such that: 1. $\zeta_1$ is very twisting, 2. $\overline{\zeta}_e$ is very positive and the image of $\overline{\zeta}_e$ is contained in the locus of very stable maps, and 3. the two morphisms $h_{\zeta_1}:B \rightarrow X$ and $h_{\overline{\zeta}_e}: B \rightarrow X$ are equal. \[lem-inducting\] Let $\pi:B \rightarrow T$ be a family of prestable, geometrically connected curves of arithmetic genus $0$, and let $${\left}( \zeta_1:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}, \overline{\zeta}_e:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e})} {\right}),$$ be a pair of $1$-morphisms such that $h_{\zeta_1} = h_{\overline{\zeta}_e}$. Then there is an open subscheme $U_{\text{induct}}\subset T$ with the following property: for any morphism of schemes $f:T'\rightarrow T$, the pullback of $(\zeta_1,\overline{\zeta}_e)$ is inducting iff $f(T')\subset U$. We just define $U_{\text{induct}}$ to be the intersection of the open subset $U_{\text{vtwist}}\subset T$ as in lemma \[lem-twistopen\] for $\zeta_1$ and the open subset $U_{\text{v-pos}}\subset T$ as in lemma \[lem-posopen\] for $\overline{\zeta}_e$. We finally come to our last definition. \[defn-inductable\] Suppose $(\pi:B\rightarrow T, \overline{\zeta}_e:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e})})$ is a very positive family whose image is contained in the locus of very stable maps. We say $\overline{\zeta}_e$ is *inductable* if there is a surjective étale morphism $u:T' \rightarrow T$ and a morphism $\zeta_1:u^*B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ with $h_{\zeta_1} = u^* h_{\overline{\zeta}_e}$ such that $(\zeta_1,\overline{\zeta}_e)$ is an inducting pair. \[lem-inductable\] Let $(\pi:B \rightarrow T, \overline{\zeta}_e:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e})})$ be a very free family. There is an open subscheme $U_{i-able} \subset T$ such that for each morphism of schemes $f:T'\rightarrow T$, the pullback $(f^*\pi: f^*B \rightarrow T', f^*\overline{\zeta}_e: f^* B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e})})$ is inductable iff $f(T')\subset U_{i-able}$. We simply apply proposition \[prop-twistopen\] to $$\xi := (\pi:B \rightarrow T, h_{\overline{\zeta}_e}: B \rightarrow X).$$ The induction argument {#sec-induct} ====================== In this section we will show that given an inductable $1$-morphism $\overline{\zeta}_e: B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e})}$, this gives rise to an inductable $1$-morphism $\overline{\zeta}_{e+1}: B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e+1})}$. The basic idea is, given an inducting pair $(\zeta_1,\overline{\zeta}_{e+1})$ to form the family of connected sums. This isn’t quite an inductable $1$-morphism, but after deforming and then performing a simple operation which we call a *modification*, we do obtain an inductable $1$-morphism $\overline{\zeta}_{e+1}$. \[not-divs\] We will follow [@QDiv] in our notation of the tautological divisors on ${\overline{{\mathcal}M}_{0,1}({{\mathbb{P}}^N,e})}$. Specifically, in $A^1({\overline{{\mathcal}M}_{0,1}({{\mathbb{P}}^N,e})})_{\mathbb{Q}}$ we denote by $\Delta_{(e_1,e_2)}$ the ${\mathbb{Q}}$-divisor whose general point parametrizes a reducible embedded curve with one irreducible component of degree $e_1$, one irreducible component of degree $e_2$ and where the marked point is on the first irreducible component. We denote by ${{\mathcal}L}$ the divisor class $\text{ev}^*{\mathcal O}_{{\mathbb{P}}^N}(1)$. And we denote by ${{\mathcal}H}$ the divisor which parametrizes stable maps whose image in ${\mathbb{P}}^r$ intersects a given codimension $2$ linear space. Given a closed subscheme $X\subset {\mathbb{P}}^N$, we also denote by $\Delta_{(e_1,e_2)}$, ${{\mathcal}L}$ and ${{\mathcal}H}$ the pullbacks of the divisors given above by the induced $1$-morphism ${\overline{{\mathcal}M}_{0,1}({X,e})} \rightarrow {\overline{{\mathcal}M}_{0,1}({{\mathbb{P}}^N,e})}$. Before proceeding to the main result of this section, we describe an operation which we will perform repeatedly in the proof. Suppose that $\zeta:B \rightarrow {\overline{{\mathcal}M}_{g,r}({X,e})}$ is a family of stable maps as in notation  \[not-1mor\]. Suppose $b\in B$ is a point whose image $\zeta(b)$ is a stable map $$((\Sigma_b, p_1, \dots, p_r), g_b:\Sigma_b \rightarrow X).$$ Suppose that $L\subset \Sigma_b$ is an irreducible component which is not contracted by $g_b$ and $p_i\in L$ is one of the marked points. For simplicity assume that $L$ contains no nodes, in particular this is the case when $L$ has genus $0$. Let $M\subset \Sigma_b$ denote all the irreducible components of $\Sigma_b$ other than $L$. Let $R=(r_1,\dots, r_l)$ denote the set of intersection points of $L$ and $M$ with some ordering. Let $D = (p_{j_1},\dots, p_{j_m})$ denote the set of marked points which lie on $L$ other than $p_i$ and let $E=(p_{k_1},\dots, p_{k_n})$ denote the set of marked points which lie on $M$. Form the product surface $L\times L$ with diagonal $\Delta:L \rightarrow L\times L$ and let $u:\Lambda \rightarrow L$ denote the blowing up of $L\times L$ along the set of points $\Delta(R\cup D)$. For each point $p \in R\cup D$, let $F_{p} \subset \Lambda$ be the proper transform of $L \times \{ p \}$. Let $F_{p_i}$ denote the proper transform of the diagonal $\Delta(L) \subset L \times L$. Consider $\text{pr}_1\circ u: \Lambda \rightarrow L$ as a family of prestable curves parametrized by $L$. Then the data $$\widetilde{\zeta}_{\Lambda} = (\text{pr}_1 \circ u: \Lambda \rightarrow L, (\sigma_{p_i}, \sigma_p | p\in R\cup D) ),$$ where $\sigma_p:L \rightarrow F_p$ is the unique isomorphism such that $\text{pr}_1\circ \sigma_p$ is the identity, gives a family of prestable marked curves parametrizing by $L$, it is essentially the constant family $L\times (L,\{p_i\}\cup R \cup D) \rightarrow L$ except that we are allowing $p_i$ to vary among all points in $L$ and then blowing up to obtain a stable family. Next we form the constant family of prestable marked curves parametrized by $L$: $$\widetilde{\zeta}_{L\times M} = (\text{pr}_1: L \times M \rightarrow L, (s_p | p \in R \cup E)$$ where $s_p: L \rightarrow L\times M$ is simply $s_p(t)=(t,p)$. We can glue $\widetilde{\zeta}_{\Lambda}$ and $\widetilde{\zeta}_{L\times M}$ as follows. For each $p\in R$, we identify the section $\sigma_p$ of $\widetilde{\zeta}_{\Lambda}$ with the section $s_p$ of $\widetilde{\zeta}_{L\times M}$. Here the identification is the unique one compatible with projection to $L$. Let us denote the new family of prestable marked curves by $$(\rho:\Pi \rightarrow L, (\phi_j: L \rightarrow \Pi | j=1,\dots, r)$$ where $\Pi$ is the surface obtained by gluing $\Lambda$ and $L\times M$ as above, and where $$\phi_j = {\left}\{ \begin{array}{ll} \sigma_{p_j} & ,p_j\in L \\ s_{p_j} & ,p_j\in M \end{array} {\right}.$$ Notice that there is a unique morphism $\text{pr}_2:\Pi \rightarrow \Sigma_b$ whose restriction to $\Lambda$ is $\text{pr}_2\circ u: \Lambda \rightarrow L \subset \Sigma_b$ and whose restriction to $L \times M$ is $\text{pr}_2: L \times M \rightarrow M\subset \Sigma_b$. We form a family of stable maps parametrized by $L$, $$\widetilde{\zeta}_\Pi = {\left}( {\left}( \rho: \Pi \rightarrow L, \phi_1,\dots, \phi_r {\right}), g_b \circ \text{pr}_2: \Pi \rightarrow X {\right}).$$ Notice that the family $\widetilde{\zeta}_\Pi$ is stable, and if we remove the section $\phi_i$ and stabilize, we just get the constant family parametrized by $L$ whose image is the stabilization of $\zeta(b)$ upon removing $p_i$. Also, the image $\widetilde{\zeta}_\Pi(p_i)$ is precisely $\zeta(b)$. Let $\tilde{B}$ be the connected sum of $B$ and $L$ where $b\in B$ is identified with $p_i\in L$. Since $\widetilde{\zeta}_\Pi(p_i) = \zeta(b)$, we may form a $1$-morphism $\widetilde{\zeta}: \widetilde{B} \rightarrow {\overline{{\mathcal}M}_{g,r}({X,e})}$ such that $\widetilde{\zeta}$ restricted to $B$ is $\zeta$, and $\widetilde{\zeta}$ restricted to $L$ is $\widetilde{\zeta}_\Pi$. \[not-modif\] Given a $1$-morphism $\zeta:B \rightarrow {\overline{{\mathcal}M}_{g,r}({X,e})}$, a point $b\in B$, an irreducible component $L\subset \Sigma_b$, and a marked point $p_i\in L$ as above, we call the $1$-morphism $\widetilde{\zeta}:\widetilde{B} \rightarrow {\overline{{\mathcal}M}_{g,r}({X,e})}$ constructed in the last paragraph the *modification* of $\zeta$ determined by $b\in B$, by $L$ and by $p_i$. \[lem-modif\] Suppose given a $1$-morphism $\zeta:B \rightarrow {\overline{{\mathcal}M}_{0,r}({X,e})}$, a point $b\in B$, an irreducible component $L\subset \Sigma_b$ which is not contracted, and a marked point $p_i\in L$ such that when we remove $p_i$, the resulting stable map is a smooth point of ${\overline{{\mathcal}M}_{0,r-1}({X,e})}$ (this condition is equivalent to the condition that the image of $\widetilde{\zeta}_\Pi$ is contained in the smooth locus of ${\overline{{\mathcal}M}_{0,r}({X,e})}$). Then we have the vanishing $$H^1 {\left}( L,\widetilde{\zeta}_\Pi^*T_{{\overline{{\mathcal}M}_{0,r}({X,e})}} {\right}) = 0.$$ In particular, if $\zeta:B \rightarrow {\overline{{\mathcal}M}_{0,r}({X,e})}$ is a free morphism of a rational curve into the smooth locus, then there are deformations of $\widetilde{\zeta}:\widetilde{B} \rightarrow {\overline{{\mathcal}M}_{0,r}({X,e})}$ which smooth the node of $\widetilde{B}$. This is an application of the deformation theory of section  \[sec-def\]. Let $\tau$ denote the dual graph of $\zeta(b)$ and let $\psi:\tau \rightarrow \tau'$ be the combinatorial morphism of graphs which removes the tail associated to $p_i$. The morphism ${\overline{{\mathcal}M}}(X,\psi):{\overline{{\mathcal}M}}(X,\tau) \rightarrow {\overline{{\mathcal}M}}(X,\tau')$ is smooth along the image of $\widetilde{\zeta}_\Pi$ and the pullback of the vertical tangent bundle is simply $T_L$. So $\widetilde{\zeta}_\Pi^* T_{{\overline{{\mathcal}M}}(X,\tau)}$ is generated by global sections. The pullback of the normal sheaf ${{\mathcal}N}$ of ${\overline{{\mathcal}M}}(X,\tau) \rightarrow {\overline{{\mathcal}M}_{0,r}({X,e})}$ is the direct sum over all $r_j\in R$ of $N_{F_j/\Lambda} \otimes_{{\mathbb{C}}} T_{r_j} M$. As the normal bundle of $N_{F_j/\Lambda}$ is just ${\mathcal O}_L(-1)$, we conclude that $H^1(L,{{\mathcal}N}) = 0$. We have a short exact sequence: $$\begin{CD} 0 @>>> \widetilde{\zeta}_\Pi^*T_{{\overline{{\mathcal}M}}(X,\tau)} @>>> \widetilde{\zeta}_\Pi^*T_{{\overline{{\mathcal}M}_{0,r}({X,e})}} @>>> {{\mathcal}N} @>>> 0 \end{CD}$$ In the corresponding long exact sequence of cohomology, $H^1$ of the first and third terms vanishes. Therefore we have the vanishing result. Suppose that $\zeta:B \rightarrow {\overline{{\mathcal}M}_{0,r}({X,e})}$ is a free $1$-morphism of a rational curve into the smooth locus of ${\overline{{\mathcal}M}_{0,r}({X,e})}$. Then $\zeta^*T_{{\overline{{\mathcal}M}_{0,r}({X,e})}}$ is generated by global sections, so $H^1(B,\zeta^* T_{{\overline{{\mathcal}M}_{0,r}({X,e})}}(-b))$ is zero by lemma \[lem-gend\] We have a short exact sequence: $$\begin{CD} 0 @>>> \zeta^* T_{{\overline{{\mathcal}M}_{0,r}({X,e})}}(-b) @>>> \widetilde{\zeta}^* T_{{\overline{{\mathcal}M}_{0,r}({X,e})}} @>>> \widetilde{\zeta}_\Pi^* T_{{\overline{{\mathcal}M}_{0,r}({X,e})}} @>>> 0 \end{CD}$$ In the corresponding long exact sequence of cohomology, $H^1$ of the first and third terms vanishes. Therefore $H^1(\widetilde{B},\widetilde{\zeta}^* T_{{\overline{{\mathcal}M}_{0,r}({X,e})}} )$ vanishes. This cohomology group is the obstruction to smoothing the node, therefore there are deformations of $\widetilde{\zeta}:\widetilde{B}\rightarrow {\overline{{\mathcal}M}_{0,r}({X,e})}$ which smooth the node of $\widetilde{B}$. \[rmk-modif\] In case the line bundle $\sigma^*{\mathcal O}_{\Sigma}(\sigma)$ is generated by global sections and $L$ contains only one node of $\Sigma_b$, we have a simpler proof of the deformation result. We have a short exact sequence of sheaves on $\Sigma$: $$\begin{CD} 0 @>>> {\mathcal O}_{\Sigma}(\sigma) @>>> {\mathcal O}_{\Sigma}(\sigma + L) @>>> N_{L/\Sigma}(p_i) @>>> 0 \end{CD}$$ Since $L$ contains only one node, $N_{L/\Sigma}\cong {\mathcal O}_L(-1)$, so the last term is isomorphic to ${\mathcal O}_L$. So ${\mathcal O}_{\Sigma}(\sigma+L)$ is generated by global sections. A small deformation $\sigma'$ of $\sigma + L$ in the linear series $|\sigma + L|$ will be a section of $\pi:\Sigma \rightarrow B$, and the stabilization of the $1$-morphism $B \rightarrow {\overline{{\mathcal}M}_{0,r}({X,e})}$ which removes the section $\sigma$ from $\zeta$ and replaces it by $\sigma'$ will be a small deformation of $\widetilde{\zeta}$ which smooths the node of $\widetilde{B}$. Now we come to the main theorem of this section, which we use for the induction step in the proof of theorem \[thm-thm1\]. \[thm-induction\] Suppose that $X$ satisfies hypothesis \[hyp-1\], hypothesis \[hyp-1.5\], hypothesis \[hyp-1.75\], and hypothesis \[hyp-2\]. For each integer $e\geq 1$, if there exists an inductable map $\overline{\zeta}_e: B_e \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e})}$, then there exists an inductable map $\overline{\xi}_{e+1}: B_{e+1} \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e+1})}$. More precisely, suppose that $(\zeta_1,\overline{\zeta}_e)$ is an inducting pair. Let us denote: $$\begin{aligned} s=\text{deg}(\zeta_1)^*(2{{\mathcal}L} - {{\mathcal}H}), \\ \overline{s}=\text{deg}(\overline{\zeta}_e)^*(2{{\mathcal}L} - {{\mathcal}H})\end{aligned}$$ Then for each $k=1,\dots,\overline{s}$, there is an inducting pair $(\xi^k_1,\overline{\xi}^k_{e+1})$ satisfying the following. 1. We have $$\text{deg}{\left}((\xi^k_1)^*{{\mathcal}H}{\right}) = \text{deg}{\left}(\zeta_1^*{{\mathcal}H} {\right}) + 2k.$$ 2. We have $$\text{deg}{\left}( (\xi^k_1)^*{{\mathcal}L} {\right}) = \text{deg}{\left}( (\overline{\xi}^k_{e+1})^*{{\mathcal}L} {\right}) = \text{deg}{\left}( \zeta_1^*{{\mathcal}L} {\right}) + k.$$ 3. We have $$\text{deg}{\left}( (\overline{\xi}^k_{e+1})^*{{\mathcal}H} {\right}) = \text{deg} {\left}( \zeta_1^*{{\mathcal}H} {\right}) + \text{deg} {\left}( \overline{\zeta}_e^*{{\mathcal}H} {\right}).$$ 4. For each $e_1 + e_2 = e$ with both $e_1, e_2 \geq 2$, we have $$\text{deg}{\left}( (\overline{\xi}^k_{e+1})^* \Delta_{(e_1+1,e_2)} {\right}) = \text{deg}{\left}( \overline{\zeta}_{e}^* \Delta_{(e_1,e_2)} {\right}).$$ 5. If $e > 1$, we have $$\text{deg}{\left}( (\overline{\xi}^k_{e+1})^* \Delta_{(e,1)} {\right}) =\text{deg}{\left}( \overline{\zeta}_{e}^* \Delta_{(e-1,1)} {\right})+ s+ k,$$ and we have $$\text{deg}{\left}( (\overline{\xi}^k_{e+1})^* \Delta_{(1,e)} {\right}) = \overline{s}-k.$$ 6. If $e=1$, we have $$\text{deg}{\left}( (\overline{\xi}^k_{2})^* \Delta_{(1,1)} {\right}) = s+\overline{s}.$$ By lemma  \[lem-posdef\] and lemma  \[lem-inductable\], we may suppose that $B_e$ is smooth. Moreover, in this case $\overline{\zeta}_e:B_e \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e})}$ is free (in fact very free). Therefore, we may suppose that $\overline{\zeta}_e(B_e)$ is in general position: for any finite collection of codimension $2$ subvarieties $(Z_\alpha | \alpha=1,\dots,M)$ and any finite collection of divisors $(D_\beta | \beta=1,\dots, N)$, we may suppose that $\overline{\zeta}_e(B_e)$ is disjoint from each $Z_\alpha$ and has $0$-dimensional intersection with each $D_\beta$. Let us denote the family of stable maps $\overline{\zeta}_e$ by: $${\left}( \overline{p}:\overline{\Sigma} \rightarrow B, \overline{\sigma}:B \rightarrow \overline{\Sigma}, \overline{g}: \overline{\Sigma} \rightarrow X {\right}).$$ And let us denote the family of stable maps $\zeta_1$ by $${\left}( p:\Sigma \rightarrow B, \sigma:B \rightarrow \Sigma, g: \Sigma \rightarrow X {\right}).$$ The basic idea is to form the connected sum of the surfaces $\Sigma$ and $\overline{\Sigma}$ glued along the sections $\sigma$ and $\overline{\sigma}$. The actual family $\overline{\zeta}_{e+1}$ is a bit more complicated. Define $\pi':\Sigma'\rightarrow B$ to be the family of curves obtained by taking the connected sum of $\Sigma$ and $\overline{\Sigma}$ glued along the sections $\sigma$ and $\overline{\sigma}$. Here $\pi'$ is the unique morphism such that $\pi'|_{\Sigma} = \pi$ and $\pi'|_{\overline{\Sigma}} = \overline{\pi}$. Define $g':\Sigma' \rightarrow X$ to be the unique morphism such that $g'|_{\Sigma} = g$ and $g'|_{\overline{\Sigma}} = \overline{g}$. Then $\zeta' = (\pi':\Sigma'\rightarrow B, g':\Sigma' \rightarrow X)$ is a family of stable maps in the boundary divisor $\Delta_{e,1}$ of ${\overline{{\mathcal}M}_{0,0}({X,e+1})}$. Moreover, $\zeta'$ clearly factors through the Behrend-Manin stack ${\overline{{\mathcal}M}}(X,\tau) \rightarrow X$ where $\tau$ is the genus $0$ stable $A$-graph with two vertices $v_1, v_2$ with $\beta(v_1)=1$ and $\beta(v_2)=e$. By lemma \[lem-glue1\], we have a short exact sequence: $$\begin{CD} 0 @>>> \zeta_1^* T_{\text{ev}} @>>> (\zeta')^* T_{{\overline{{\mathcal}M}}(X,\tau)} @>>> \overline{\zeta}_e^* T_{{\overline{{\mathcal}M}_{0,1}({X,e})}} @>>> 0 \end{CD}$$ Since $\zeta_1$ is very twisting, by definition  \[defn-verytwisting\] we have that $\zeta_1^* T_{\text{ev}}$ is ample. Since $\overline{\zeta}_e$ is very positive, by lemma  \[lem-posdef\], we have that $\overline{\zeta}_e^* T_{{\overline{{\mathcal}M}_{0,1}({X,e})}}$ is ample. Therefore $(\zeta')^* T_{{\overline{{\mathcal}M}}(X,\tau)}$ is ample. By lemma \[lem-glue2\], we have a short exact sequence: $$0 \rightarrow (\zeta')^* T_{{\overline{{\mathcal}M}}(X,\tau)} \rightarrow (\zeta')^* T_{{\overline{{\mathcal}M}_{0,0}({X,e+1})}} \rightarrow \sigma^* {\mathcal O}_{\Sigma}(\sigma) \otimes \overline{\sigma}^* {\mathcal O}_{\overline{\Sigma}}(\overline{\sigma}) \rightarrow 0$$ By definition  \[defn-verytwisting\] and definition  \[defn-pos\], we have that both $\sigma^*{\mathcal O}_{\Sigma}(\sigma)$ and $\overline{\sigma}^* {\mathcal O}_{\overline{\Sigma}}(\overline{\sigma})$ are ample. Therefore their tensor product is ample, and we conclude that $(\zeta')^* T_{{\overline{{\mathcal}M}_{0,0}({X,e+1})}}$ is ample. Denote by $\overline{s}$ the self-intersection of $\overline{\sigma}\subset \overline{\Sigma}$, i.e. the degree of the invertible sheaf $\overline{\sigma}^*{\mathcal O}_{\overline{\Sigma}}(\overline{\sigma})$. Notice that we have $$\overline{s} = \text{deg}{\left}( 2\overline{\zeta}_e^*{{\mathcal}L} - \overline{\zeta}_e^*{{\mathcal}H} {\right}).$$ Let $\sigma':B \rightarrow \overline{\Sigma}$ be a general member of the linear series of $|\overline{\sigma}|$. Since $\overline{\sigma}^*{\mathcal O}_{\overline{\Sigma}}(\overline{\sigma})$ is generated by global sections, we can find such a $\sigma'$ which has transverse intersections with $\overline{\sigma}$ at points $p_1,\dots,p_{\overline{s}} \in \overline{\Sigma}$. Define $b:\widetilde{\overline{\Sigma}}\rightarrow \overline{\Sigma}$ to be the blowing up of $\overline{\Sigma}$ at the points $p_1,\dots,p_{\overline{s}}$. Let $\widetilde{\overline{\pi}}:\widetilde{\overline{\Sigma}} \rightarrow \overline{\Sigma}$ denote the projection $\overline{\pi}\circ b$. Let $\widetilde{\overline{g}}$ denote $\overline{g}\circ b$. Let $\widetilde{\overline{\sigma}}:B \rightarrow \widetilde{\overline{\Sigma}}$ and $\widetilde{\sigma}:B \rightarrow \widetilde{\overline{\Sigma}}$ denote the proper transforms of $\overline{\sigma}$ and $\sigma'$ respectively. Notice that $\widetilde{\overline{\sigma}}$ and $\widetilde{\sigma}$ are disjoint sections. So the data $${\left}( {\left}(\widetilde{\overline{p}}:\widetilde{\overline{\Sigma}} \rightarrow B, \widetilde{\overline{\sigma}}, \widetilde{\sigma} {\right}), \widetilde{\overline{f}}: \widetilde{\overline{\Sigma}} \rightarrow X {\right})$$ is a family of stable pointed maps, i.e. a $1$-morphism $\widetilde{\overline{\zeta}}_e: B \rightarrow {\overline{{\mathcal}M}_{0,2}({X,e})}$. By the deformation theory in subsection \[subsec-unstable\], we have a short exact sequence: $$\begin{CD} 0 @>>> (\widetilde{\sigma})^* {\mathcal O}_{\widetilde{\overline{\Sigma}}}(\widetilde{\sigma}) @>>> {\left}( \widetilde{\overline{\zeta}}_e {\right})^* T_{{\overline{{\mathcal}M}_{0,2}({X,e})}} @>>> {\left}( \overline{\zeta} {\right})_e^* T_{{\overline{{\mathcal}M}_{0,1}({X,e})}} @>>> 0 \end{CD}$$ Of course we have that $(\widetilde{\sigma})^* {\mathcal O}_{\widetilde{\overline{\Sigma}}}(\widetilde{\sigma})$ is the trivial invertible sheaf ${\mathcal O}_B$. In particular, we have that ${\left}( \widetilde{\overline{\zeta}}_e {\right})^* T_{{\overline{{\mathcal}M}_{0,2}({X,e})}}$ is generated by global sections. Define $\widetilde{\pi}:\widetilde{\Sigma} \rightarrow B$ to be the family of curves obtained by taking the connected sum of $\Sigma$ and $\widetilde{\overline{\Sigma}}$ glued along the sections $\sigma$ and $\tau$ respectively. Here $\widetilde{\pi}$ is the unique morphism such that $\widetilde{\pi}|_{\widetilde{\overline{\Sigma}}} = \widetilde{\overline{\pi}}$ and $\widetilde{\pi}|_\Sigma = \pi$. Define $\widetilde{g}:\widetilde{\Sigma}\rightarrow X$ to be the unique morphism such that $\widetilde{g}|_{\widetilde{\overline{\Sigma}}} = \widetilde{\overline{g}}$ and such that $\widetilde{g}|_{\Sigma} = g$. Define $\widetilde{\sigma}:B \rightarrow \widetilde{\Sigma}$ to be the section from the last paragraph. This gives a family of stable maps: $$\widetilde{\zeta} = {\left}( {\left}( \widetilde{\pi}:\widetilde{\Sigma} \rightarrow B, \widetilde{\sigma}:B \rightarrow \widetilde{\Sigma} {\right}), \widetilde{g}: \widetilde{\Sigma} \rightarrow X {\right}).$$ Define $\tau$ to be the unique genus $0$ stable $A$-graph with two vertices $v_1,v_2$, with one edge joining $v_1$ and $v_2$, with one flag attached to $v_1$, with $\beta(v_1)=e$, and with $\beta(v_2)=1$. In other words, $\tau$ is the dual graph of a stable map with one marked point, with reducible domain consisting of two irreducible components, where the component with the marked point has degree $e$ and where the other component has degree $1$. Then $\widetilde{\zeta}:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e+1})}$ is a $1$-morphism which factors through the Behrend-Manin stack ${\overline{{\mathcal}M}}(X,\tau) \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e+1})}$. By lemma \[lem-glue1\], we have a short exact sequence: $$\begin{CD} 0 @>>> \zeta_1^* T_{\text{ev}} @>>> {\left}( \widetilde{\zeta} {\right})^* T_{{\overline{{\mathcal}M}}(X,\tau)} @>>> {\left}( \widetilde{\overline{\zeta}}_e {\right})^* T_{{\overline{{\mathcal}M}_{0,2}({X,e})}} @>>> 0 \end{CD}$$ As the first and third term in this exact sequence are generated by global sections, also ${\left}( \widetilde{\zeta} {\right})^* T_{{\overline{{\mathcal}M}}(X,\gamma)}$ is generated by global sections. Finally, by lemma \[lem-glue2\] we have a short exact sequence: $$0 \rightarrow {\left}( \widetilde{\zeta} {\right})^* T_{{\overline{{\mathcal}M}}(X,\gamma)} \rightarrow {\left}( \widetilde{\zeta} {\right})^* T_{{\overline{{\mathcal}M}_{0,1}({X,e+1})}} \rightarrow \sigma^* {\mathcal O}_{\Sigma}(\sigma) \otimes_{{\mathcal O}_B} {\left}( \widetilde{\overline{\sigma}} {\right})^* {\mathcal O}_{\widetilde{\overline{\Sigma}}}(\widetilde{\overline{\sigma}}) \rightarrow 0$$ Of course ${\left}( \widetilde{\overline{\sigma}} {\right})^* {\mathcal O}_{\widetilde{\overline{\Sigma}}}(\widetilde{\overline{\sigma}})$ is the trivial invertible sheaf. And $\sigma^* {\mathcal O}_{\Sigma}(\sigma)$ is just ${\mathcal O}_B(s)$. Thus the last term in the sequence is an ample invertible sheaf. In particular, ${\left}( \widetilde{\zeta} {\right})^* T_{{\overline{{\mathcal}M}_{0,1}({X,e+1})}}$ is generated by global sections. So the $1$-morphism $\widetilde{\zeta}: B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e+1})}$ is free. Also, the pullback by $\widetilde{\zeta}$ of the normal sheaf of the unramified $1$-morphism, ${\overline{{\mathcal}M}}(X,\tau) \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e+1})}$ is the last term in the sequence, and so has positive degree. One final remark, when $e>1$, the image of $\widetilde{\zeta}$ intersects the divisor $\Delta_{1,e}$ transversely precisely at the images of the points $p_1,\dots, p_{\overline{s}} \in B$, in particular the degree of the ${\mathbb{Q}}$-Cartier divisor class $\widetilde{\sigma}^*{\mathcal O}_{{\overline{{\mathcal}M}_{0,1}({X,e+1})}}(\Delta_{1,e})$ is positive. In the special case $e=1$, we have that $\Delta_{1,e}=\Delta_{e,1} =\Delta_{1,1}$. In this case ${\overline{{\mathcal}M}}(X,\tau)$ is the normalization of $\Delta_{1,1}$ (at least in a neighborhood of the image of $\widetilde{\sigma}$). So the degree of $\widetilde{\sigma}^*{\mathcal O}_{{\overline{{\mathcal}M}_{-,1}({X,e+1})}}(\Delta_{1,1})$ is the sum of the degree of the pullback of the normal sheaf of ${\overline{{\mathcal}M}}(X,\tau)\rightarrow {\overline{{\mathcal}M}_{0,1}({X,e+1})}$ and the divisor $p_1 + \dots + p_{\overline{s}}$, i.e. $s+ \overline{s}$. So in this case, the degree is again positive. Since $\widetilde{\zeta}:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e+1})}$ is free we can find deformations of $\widetilde{\zeta}$ which are in general position. Now by hypothesis  \[hyp-1\], the locus parametrizing stable maps with at least three irreducible components in their domain has codimension $2$. By hypothesis \[hyp-1.75\] the locus parametrizing stable maps with automorphisms has codimension at least $2$. Therefore we can find a deformation $\overline{\xi}_{e+1}^0:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e+1})}$ of $\widetilde{\zeta}$ such that the image of $\overline{\xi}_{e+1}^0:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e+1})}$ misses the locus parametrizing stable maps with at least three irreducible components in their domain and misses the locus parametrizing stable maps with automorphisms. As well, we can assume that the pullback of $T_{{\overline{{\mathcal}M}_{0,0}({X,e+1})}}$ by $\text{pr}\circ \overline{\xi}_{e+1}^0: B \rightarrow {\overline{{\mathcal}M}_{0,0}({X,e+1})}$ is an ample vector bundle since the pullback of $T_{{\overline{{\mathcal}M}_{0,0}({X,e+1})}}$ by $\text{pr}\circ \widetilde{\zeta}$, i.e. by $\zeta':B \rightarrow {\overline{{\mathcal}M}_{0,0}({X,e+1})}$, is an ample bundle. Let us denote $\overline{\xi}_{e+1}^0$ by $$\overline{\xi}_{e+1}^0 = {\left}( {\left}( \varpi^0:\Xi^0 \rightarrow B, \lambda^0:B \rightarrow \Xi^0 {\right}), g^0:\Xi^0 \rightarrow B {\right}).$$ Define $h^0:B \rightarrow X$ to be $g^0\circ \lambda^0$. By assumption, $\overline{h}:B \rightarrow X$ is very twistable, and $h^0:B \rightarrow X$ is a small deformation of $\overline{h}$. So by proposition  \[prop-twistopen\], $h^0$ is very twistable. Since $\widetilde{\sigma}^*{\mathcal O}_{\widetilde{\Sigma}}(\widetilde{\sigma})$ is trivial, we may also assume that ${\left}( \lambda^0 {\right})^* {\mathcal O}_{\Xi^0}(\lambda^0)$ is trivial. Finally, we may assume that all intersections of the image of $\overline{\xi}_{e+1}^0$ and the divisor $\Delta_{1,e}$ are transverse and occur at general points of $\Delta_{1,e}$. In particular, if $b\in B$ is such a point, we may assume that the corresponding stable map $\overline{\xi}_{e+1}^0(b)$ is of the form $((\Xi^0_b,\lambda^0_b),h^0_b:\Xi^0_b \rightarrow X)$ where $\Xi^0_b$ is a reducible curve $L \cup M$, with $L\cap M$ consisting of one node of $\Xi^0_b$ which is a general point of $X$, with $\lambda^0_q \in L$, and such that $L \rightarrow X$ is an embedding of a line which is twistable. Now we define $\widetilde{\overline{\xi}}^0_{e+1}$ to be the modification of $\overline{\xi}^0_{e+1}$ associated to the point $b\in B$, to $L\subset \Xi^0_b$ and $\lambda^0_q \in L$ (c.f. notation \[not-modif\]). We saw above that $h^0:B \rightarrow X$ is very twistable. And the evaluation map $\widetilde{h}_L:L \rightarrow X$ associated to the restriction $\widetilde{\overline{\xi}}^0_{e+1}: L \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e+1})}$ is just the embedding $L\subset X$, which is twistable. By hypothesis \[hyp-1.5\] and the assumption that $L\cap M$ is a general point of $X$, we see that the hypotheses of lemma \[lem-twisttogether2\] are satisfied. So by lemma \[lem-twisttogether2\] we have that the evaluation map $\widetilde{h}:\widetilde{B}\rightarrow X$ associated to $\widetilde{\overline{\xi}}^0_{e+1}$ is also very twistable. Also, notice that the image of $\widetilde{\overline{\xi}}^0_{e+1}$ is contained in the locus of very stable maps. The $1$-morphism $\widetilde{\overline{\xi}}^0_{e+1}$ satifies the criterion of lemma \[lem-modif\], in fact the criterion of remark \[rmk-modif\], thus we can smooth the node of $\widetilde{B}$. Let $\overline{\xi}^1_{e+1}:B \rightarrow $ denote a small deformation which smooths the node of $\widetilde{B}$. We will denote this by $$\overline{\xi}^1_{e+1} = {\left}( {\left}( \pi^1:\Xi^1\rightarrow B, \sigma^1: B \rightarrow \Xi^1 {\right}), g^1: \Xi^1 \rightarrow X {\right}).$$ The claim is that $\overline{\xi}^1_{e+1}$ is inductable. Denote $h^1 = g^1\circ \sigma^1$. Since $\widetilde{h}$ is very twistable and since $h^1$ is a small deformation of $\widetilde{h}$, it follows by proposition \[prop-twistopen\] that $h^1$ is very twistable. Now the image of $\widetilde{\overline{\xi}}^0_{e+1}$ is not contained in the locus of very stable maps, precisely because of the point $r\in L$ where $L\cap M = \{r\}$: the stable map $\widetilde{\overline{\xi}}^0_{e+1}(r)$ is not very stable. But from the description of the smoothing of $\widetilde{\overline{\xi}}^0_{e+1}$ given in remark \[rmk-modif\], we can find small deformations whose image is contained in the very stable locus. So we may assume the image of $\overline{\xi}^1_{e+1}$ is contained in the locus of very stable maps. It remains only to show that $\overline{\xi}^1_{e+1}$ is positive. That $h^1:B \rightarrow T$ is a stable map follows from the fact that $B$ is smooth and $h^1$ is non-constant: in fact the degree of $h^1(B)$ equals the degree of $\widetilde{h}(\widetilde{B})$, which is just $\text{deg}(h(B)) +1$. This proves ($1$) of definition \[defn-pos\]. To show that $h^1:B\rightarrow X$ is unobstructed, it suffices to prove that $\widetilde{h}:\widetilde{B}\rightarrow X$ is unobstructed. By assumption, $h^0:B \rightarrow X$ is unobstructed (being a small deformation of $h:B \rightarrow X$). And the restriction to $L$ of $T_X$ is generated by global sections (since $L$ is general). Thus $\widetilde{h}:\widetilde{B} \rightarrow X$ is unobstructed. This proves ($2$) of definition \[defn-pos\]. The map $$\text{pr}\circ \widetilde{\overline{\xi}}^0_{e+1}: \widetilde{B} \rightarrow {\overline{{\mathcal}M}_{0,0}({X,e})}$$ is constant on $L\subset \widetilde{B}$, and on $B\subset \widetilde{B}$, it is just $\zeta'$. Thus the image is contained in the unobstructed locus. So we have ($3$) of definition  \[defn-pos\]. The pullback of $T_{{\overline{{\mathcal}M}_{0,0}({X,e})}}$ to $\widetilde{B}$ is trivial on $L$ and equals $(\zeta')^*T_{{\overline{{\mathcal}M}_{0,0}({X,e})}}$ on $B$. Since $(\zeta')^*T_{{\overline{{\mathcal}M}_{0,0}({X,e})}}$ is ample, we conclude from lemma \[lem-DAcrit\] that the pullback to $\widetilde{B}$ of $T_{{\overline{{\mathcal}M}_{0,0}({X,e})}}$ is deformation ample. Since $\overline{\xi}^1_{e+1}$ is a small deformation of $\widetilde{\overline{\xi}}^0_{e+1}$, the pullback of $T_{{\overline{{\mathcal}M}_{0,0}({X,e})}}$ via $\overline{\xi}^1_{e+1}$ is deformation ample by lemma \[lem-opDA\]. This proves ($4$) of definiton  \[defn-pos\]. Now for $\widetilde{\overline{\zeta}}^0_{e+1}$, the pullback of $\widetilde{\sigma}^*{\mathcal O}_{\widetilde{\Sigma}}(\sigma)$ is trivial when restricted to $B\subset \widetilde{B}$, but on $L\subset \widetilde{B}$ it is ${\mathcal O}_L(1)$. So by lemma \[lem-DAcrit\], it is deformation ample. Since $\overline{\zeta}^1_{e+1}$ is a small deformation of $\widetilde{\overline{\xi}}^0_{e+1}$, by lemma \[lem-opDA\] we have that $(\sigma^1)^*{\mathcal O}_{\Xi^1}(\sigma^1)$ is deformation ample. So $\overline{\xi}^1_{e+1}$ is positive. Therefore it is inductable. To define the $1$-morphisms $\overline{\xi}^k_{e+1}:B \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e+1})}$ we repeat the procedure above. Given $\overline{\xi}^k_{e+1}$ and a point $b\in B$ whose image is a general point of $\Delta_{(1,e)}$, say $(\Xi^k_b,\lambda^k_b,g^k_b:\Xi^k_b \rightarrow X)$ with $\Xi^k_b = L \cup M$, $\lambda^k_b \in L$ and $g^k_b:L \rightarrow X$ an embedding of a twistable line, we define $\widetilde{B}^k$ to be the connected sum of $B$ and $L$ with $b\in B$ identified with $\lambda^k_b \in L$. Then we define $$\widetilde{\overline{\xi}}^k_{e+1}:\widetilde{B}^k \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e+1})}$$ to be the modification of $\overline{\xi}^k_{e+1}$ associated to $b\in B$, $L\subset \Xi^k_b$ and $\lambda^k_b \in L$. By the same argument as above, deformations of $\widetilde{\overline{\xi}}^k_{e+1}$ smooth the node of $\widetilde{B}^k$, and a small deformation $\overline{\xi}^{k+1}_{e+1}$ of $\widetilde{\overline{\xi}}^k_{e+1}$ is inductable. It is quite simple to work out the degrees of the pullbacks by $\widetilde{\overline{\xi}}^0_{e+1}$ of the tautological divisors of ${\overline{{\mathcal}M}_{0,1}({{\mathbb{P}}^N,e+1})}$ in terms of the degrees of the pullbacks by $\zeta_1$ and $\overline{\zeta}_e$ of the tautological divisors. This is left to the interested reader. Twistable lines {#sec-twist} =============== In this section, we will prove that if $n+1\geq d^2$, then for a general hypersurface $X_d \subset {\mathbb{P}}^n$ of degree $d$ and a general line $L\subset X$, we have that $L$ is a twistable line on $X$. To prove this we introduce some incidence correspondences. Let $N_d = \binom{n+d}{n}-1$ and let ${\mathbb{P}}^{N_d}$ denote the projective space parametrizing hypersurfaces $X_d\subset {\mathbb{P}}^n$ of degree $d$. Let ${{\mathcal}X} \subset {\mathbb{P}}^{N_d} \times {\mathbb{P}}^n$ denote the universal family of degree $d$ hypersurfaces in ${\mathbb{P}}^n$. Let ${\mathbb{G}}(1,n)$ denote the Grassmannian variety of lines in ${\mathbb{P}}^n$. Let $F({{\mathcal}X}) \subset {\mathbb{P}}^{N_d} \times {\mathbb{G}}(1,n)$ denote the parameter space of pairs $([X],[L])$ where $X\subset {\mathbb{P}}^n$ is a hypersurface of degree $d$, $L\subset {\mathbb{P}}^n$ is a line and $L\subset X$. Observe that the projection $F({{\mathcal}X}) \rightarrow {\mathbb{G}}(1,d)$ is a projective bundle of relative dimension $N_d - (d+1)$. Let $P(t) = (t+1)^2$ denote the Hilbert polynomial of a quadric surface in ${\mathbb{P}}^3$, and let $U \subset \textit{Hilb}^{P(t)}({{\mathbb{P}}^n})$ denote the open subscheme parametrizing subschemes $\Sigma \subset {\mathbb{P}}^n$ which are projectively equivalent to a smooth, quadric surface in ${\mathbb{P}}^3 \subset {\mathbb{P}}^n$. Let $V \subset U \times {\mathbb{G}}(1,n)$ denote the parameter space of pairs $([\Sigma],[L])$ where $\Sigma \subset {\mathbb{P}}^n$ is a smooth quadric surface, where $L\subset {\mathbb{P}}^n$ is a line, and where $L\subset \Sigma$. The projection map $V\rightarrow U$ has a Stein factorization $V \rightarrow \tilde{U} \rightarrow U$ where $\tilde{U} \rightarrow U$ is a finite, étale double cover, and where $V\rightarrow \tilde{U}$ is a ${\mathbb{P}}^1$-bundle. Let $W\subset {\mathbb{P}}^{N_d} \times U \times {\mathbb{G}}(1,n)$ denote the parameter space of triples $([X],[\Sigma],[L])$ where $X\subset {\mathbb{P}}^n$ is a hypersurface of degree $d$, $([\Sigma],[L])$ is a point of $V$ and where $\Sigma \subset X$. The projection map $W \rightarrow V$ is a projective bundle of relative dimension $N_d - (d+1)^2$. Now for a triple $([X],[\Sigma],[L])\in W$, there is a map (well-defined up to nonzero scalar) $d_X:{\mathbb{C}}^{n+1} \rightarrow H^0({\mathbb{P}}^n,{\mathcal O}_{{\mathbb{P}}^n}(d-1))$ which evaluates the partial derivatives of a defining equation of $X$. We may compose this map with the restriction map $H^0({\mathbb{P}}^n,{\mathcal O}_{{\mathbb{P}}^n}(d)) \rightarrow H^0(\Sigma,{\mathcal O}_{\Sigma}(d-1))$. Let this map be denoted by $d_{X,\Sigma}:{\mathbb{C}}^{n+1} \rightarrow H^0(\Sigma,{\mathcal O}_{\Sigma}(d-1))$. More precisely, let $E$ be the trivial vector bundle on $W$ of rank $n+1$, let $G$ be the vector bundle on $U$ whose fiber at a point $\Sigma$ is just $H^0(\Sigma,{\mathcal O}_{\Sigma}(d-1))$, and let $F$ be the vector bundle on $W$ which is $\text{pr}_1^*({\mathcal O}_{{\mathbb{P}}^{N_d}}(1))\otimes \text{pr}_2^*G$. Then there is a map of vector bundles $d:E \rightarrow F$ whose fiber over $([X],[\Sigma],[L])$ is the map $d_{X,\Sigma}$ constructed above. Let $W^o\subset W$ denote the open subscheme (possibly empty) over which $d$ is surjective (i.e. the complement of the support of the cokernel of $d$). \[lem-quadric1\] For any point $([X],[\Sigma],[L])\in W^o$, we have 1. $X$ is smooth along $\Sigma$, 2. $H^i{\left}( \Sigma, N_{\Sigma/X} {\right})$ is zero for $i>0$, 3. $H^i{\left}( \Sigma,N_{\Sigma/X}(-L) {\right})$ is zero for $i>0$, 4. $H^i{\left}( \Sigma,N_{\Sigma/X}\otimes {\mathcal O}_{\Sigma}(-1) {\right})$ is zero for $i>0$, 5. $H^1{\left}( L, N_{L/X} {\right})$ is zero, 6. $H^1{\left}( L, N_{L/X}(-1) {\right})$ is zero, 7. the projection morphism $W \rightarrow {\mathbb{P}}^{N_d}$ given by $([X],[\Sigma],[L]) \mapsto [X]$ is smooth at $([X],[\Sigma],[L])$, 8. the projection morphism $F({{\mathcal}X}) \rightarrow {\mathbb{P}}^{N_d}$ given by $([X],[L]) \mapsto [X]$ is smooth at $([X],[L])$, and 9. the projection morphism $\pi:W\rightarrow F({{\mathcal}X})$ given by $([X],[\Sigma],[L]) \mapsto ([X],[L])$ is smooth at $([X],[\Sigma],[L])$. Since the partial derivatives of a defining equation of $X$ generate $H^0(\Sigma,{\mathcal O}_{\Sigma}(d-1))$, in particular the locus where they all vanish is disjoint from $\Sigma$. By the Jacobian criterion, we conclude that $X$ is smooth along $\Sigma$. For a smooth quadric surface $\Sigma \subset {\mathbb{P}}^3 \subset {\mathbb{P}}^n$, we have a short exact sequence: $$\begin{CD} 0 @>>> N_{\Sigma/{\mathbb{P}}^3} @>>> N_{\Sigma/{\mathbb{P}}^n} @>>> N_{{\mathbb{P}}^3/{\mathbb{P}}^n}|_{\Sigma} @>>> 0 \end{CD}$$ Since $N_{\Sigma/{\mathbb{P}}^3} \cong {\mathcal O}_{\Sigma}(2)$ and since $N_{{\mathbb{P}}^3/{\mathbb{P}}^n}|_{\Sigma} \cong {\mathcal O}_{\Sigma}(1)^{\oplus (n-3)}$, we have a short exact sequence: $$\begin{CD} 0 @>>> {\mathcal O}_{\Sigma}(2) @>>> N_{\Sigma/{\mathbb{P}}^n} @>>> {\mathcal O}_{\Sigma}(1)^{\oplus (n-3)} @>>> 0 \end{CD}$$ From this it is easy to compute that $H^i{\left}( \Sigma,N_{\Sigma/{\mathbb{P}}^n} {\right})$, $H^i{\left}( \Sigma,N_{\Sigma/{\mathbb{P}}^n}\otimes {\mathcal O}_{\Sigma}(-1) {\right})$ and $H^i{\left}( \Sigma,N_{\Sigma/{\mathbb{P}}^n}\otimes {\mathcal O}_{\Sigma}(-L) {\right})$ are all zero for $i>0$. There is a short exact sequence: $$\begin{CD} 0 @>>> N_{\Sigma/X} @>>> N_{\Sigma/{\mathbb{P}}^n} @>>> N_{X/{\mathbb{P}}^n}|_\Sigma @>>> 0 \end{CD}$$ Of course $N_{X/{\mathbb{P}}^n}|_{\Sigma} \cong {\mathcal O}_{\Sigma}(d)$, and for ${{\mathcal}L} = {\mathcal O}_{\Sigma}$, for ${{\mathcal}L} = {\mathcal O}_{\Sigma}(-L)$, and for ${{\mathcal}L} = {\mathcal O}_{\Sigma}(-1)$, we compute that $H^i{\left}( \Sigma, {\mathcal O}_{\Sigma}(d) \otimes {{\mathcal}L} {\right})$ is zero for $i>0$. The bundle $N_{\Sigma/{\mathbb{P}}^n}$ was computed in the last paragraph. For the line bundles ${{\mathcal}L} = {\mathcal O}_\Sigma$, ${{\mathcal}L} = {\mathcal O}_{\Sigma}(-L)$ and ${{\mathcal}L} = {\mathcal O}_{\Sigma}(-1)$, we computed that $H^i(\Sigma, N_{\Sigma/{\mathbb{P}}^n}\otimes {{\mathcal}L})$ is zero for $i>0$. It immediately follows from the long exact sequence in cohomology that $H^2{\left}( \Sigma, N_{\Sigma/X} \otimes {{\mathcal}L} {\right})$ is zero. We also conclude that $H^1(\Sigma, N_{\Sigma/X}\otimes {{\mathcal}L})$ is zero iff the corresponding map $$H^0(\Sigma, N_{\Sigma/{\mathbb{P}}^n}\otimes {{\mathcal}L}) \rightarrow H^0(\Sigma, N_{X/{\mathbb{P}}^n}|_\Sigma)$$ is surjective. In the case that ${{\mathcal}L} = {\mathcal O}_\Sigma(-1)$, the map from the last paragraph factors the map $$H^0({\mathbb{P}}^n,T_{{\mathbb{P}}^n}(-1)) \rightarrow H^0(\Sigma,{\mathcal O}_{\Sigma}(d-1)).$$ But this map is precisely the map $d_{X,\Sigma}:{\mathbb{C}}^{n+1} \rightarrow H^0(\Sigma,{\mathcal O}_{\Sigma}(d-1))$. Since $d_{X,\Sigma}$ is surjective, we conclude that $H^1(\Sigma, N_{\Sigma/X}\otimes {\mathcal O}_{\Sigma}(-1))$ is zero. To see that $H^1(\Sigma,N_{\Sigma/X}\otimes {\mathcal O}_{\Sigma}(-L))$ is zero, observe that we have ${\mathcal O}_{\Sigma}(1) \cong {\mathcal O}_{\Sigma}(L+L')$ where $L'\subset \Sigma$ is any line in the ruling opposite to the ruling of $L$. Thus we have a commutative diagram: $$\begin{CD} H^0( \Sigma , N_{\Sigma/{\mathbb{P}}^n}(-1) ) \otimes_{{\mathbb{C}}} H^0( \Sigma, {\mathcal O}_{\Sigma}(L') ) @>>> H^0( \Sigma, N_{\Sigma/{\mathbb{P}}^n}(-L) \\ @VVV @VVV \\ H^0( \Sigma , N_{X/{\mathbb{P}}^n}|_{\Sigma}(-1) ) \otimes_{{\mathbb{C}}} H^0( \Sigma , {\mathcal O}_{\Sigma}(L') ) @>>> H^0( \Sigma, N_{X/{\mathbb{P}}^n}|_{\Sigma}(-L) ) \\ \end{CD}$$ The top vertical horizontal arrow is surjective by the last paragraph. Moreover, the right vertical arrow is $$\begin{array}{cc} H^0{\left}( \Sigma, {\mathcal O}_{\Sigma} {\left}( (d-1)L + (d-1)L' {\right}) {\right}) \otimes H^0{\left}( \Sigma, {\mathcal O}_{\Sigma}(L') {\right}) \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \ \ \ \ \ \ \ \ \ \ \ \ \rightarrow H^0{\left}( \Sigma, {\mathcal O}_{\Sigma}{\left}( (d-1)L + dL' {\right}) {\right}), \end{array}$$ which is clearly surjective. Therefore we conclude that the bottom horizontal arrow is also surjective, i.e. $H^1(\Sigma, N_{\Sigma/X}(-L) )$ is zero. The proof that $H^1(\Sigma, N_{\Sigma/X})$ is zero is almost identical to the proof that $H^1(\Sigma, N_{\Sigma/X}(-L)$ is zero and is left to the reader. To see that $H^1{\left}( L, N_{L/X}(-1) {\right})$ is zero, first observe we have a short exact sequence: $$\begin{CD} 0 @>>> N_{\Sigma/X}(-1) @>>> N_{\Sigma/X}(-L') @>>> N_{\Sigma/X}|_L(-1) @>>> 0 \end{CD}$$ By our computations and the long exact sequence in cohomology, we conclude that $H^1{\left}( L, N_{\Sigma/X}|_L(-1) {\right})$ is zero. Next observe that we have a short exact sequence: $$\begin{CD} 0 @>>> N_{L/\Sigma}(-1) @>>> N_{L/X}(-1) @>>> N_{\Sigma/X}|_L(-1) @>>> 0 \end{CD}$$ Of course $N_{L/\Sigma} \cong {\mathcal O}_L(1)$, so $H^1{\left}( L,N_{L/\Sigma}(-1) {\right})$ is zero. And we have seen that $H^1{\left}( L,N_{\Sigma/X}|_L(-1) {\right})$ is zero. Therefore by the long exact sequence in cohomology, we conclude that $H^1{\left}(L, N_{L/X}(-1) {\right})$ is zero. By an almost identical argument, we also conclude that $H^1{\left}( L, N_{L/X} {\right})$ is zero. Now by [@K proposition I.2.14.2], the obstruction space for the relative Hilbert scheme $\textit{Hilb}^{P(t)}({{\mathcal}X}/{\mathbb{P}}^{N_d})$ at a point $([X],[\Sigma])$ is contained in $H^1(\Sigma,N_{\Sigma/X})$. Since the obstruction space vanishes, it follows by [@K theorem 2.10] that $\textit{Hilb}^{P(t)}({{\mathcal}X}/{\mathbb{P}}^{N_d}) \rightarrow {\mathbb{P}}^{N_d}$ is smooth at $([X],[\Sigma])$. As we have seen $W \rightarrow \textit{Hilb}^{P(t)}({{\mathcal}X}/{\mathbb{P}}^{N_d})$ is smooth. Therefore the composition $W \rightarrow {\mathbb{P}}^{N_d}$ is smooth along $W^o$. For basically the same reason as above, we conclude that the projection map $F({{\mathcal}X}) \rightarrow {\mathbb{P}}^{N_d}$ is smooth along the image of $\pi:W^o \rightarrow F({{\mathcal}X})$. Since $W^o \rightarrow {\mathbb{P}}^{N_d}$ is smooth, and since $F({{\mathcal}X}) \rightarrow {\mathbb{P}}^{N_d}$ is smooth along the image of $W^o$, to prove that $\pi$ is smooth along $W^o$, it suffices to check that the derivative map $d\pi: T_{W^o/{\mathbb{P}}^{N_d}} \rightarrow \pi^* T_{F({{\mathcal}X})/{\mathbb{P}}^{N_d}}$ is surjective at every point. This exactly reduces to the statement that $H^0 {\left}( \Sigma, N_{\Sigma/X} {\right}) \rightarrow H^0{\left}( L, N_{\Sigma/X}|_L {\right})$ is surjective. Since the cokernel is contained in $H^1{\left}( \Sigma, N_{\Sigma/X}(-L) {\right})$, which is zero, we conclude the map is surjective. Therefore $\pi:W^o \rightarrow F({{\mathcal}X})$ is smooth. Now suppose that $([X],[\Sigma],[L])$ is a point in $W^o$. We associate to this triple a morphism $\zeta: L \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ as follows. Let $\sigma:L \rightarrow \Sigma$ be the inclusion and let $\text{pr}_L: \Sigma \rightarrow L$ be the unique projection such that $\sigma$ is a section of $\text{pr}_L$. Let $g:\Sigma \rightarrow X$ be the inclusion. Then we have a morphism $\zeta:L \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ given by the data: $$\zeta = {\left}(\text{pr}_L:\Sigma \rightarrow L, \sigma:L \rightarrow \Sigma, g:\Sigma \rightarrow X {\right}).$$ \[lem-quadric2\] If $([X],[\Sigma],[L])$ is a triple in $W^o$ and if $X$ is smooth, then the corresponding morphism $\zeta:L \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ is flexible. We need to check the axioms of definition \[defn-twisting\]. Since $g\circ \sigma:L \rightarrow X$ is an embedding, in particular this map is stable, i.e. axiom $(1)$ is satisfied. By part $(5)$ of lemma \[lem-quadric1\], we conclude that ${\overline{{\mathcal}M}_{0,0}({X,1})}$ is unobstructed at $[g\circ \sigma: L \rightarrow X]$, i.e. axiom $(2)$ is satisfied. To check axiom $(3)$, consider the normal bundle ${{\mathcal}N}$ of the regular embedding $(\text{pr}_L, g): \Sigma \rightarrow L \times X$. This fits into a short exact sequence: $$\begin{CD} 0 @>>> \text{pr}_L^* T_L @>>> {{\mathcal}N} @>>> N_{\Sigma/X} @>>> 0 \end{CD}$$ By part $(3)$ of remark \[rmk-twisting\], we need to check that $R^1 {\left}(\text{pr}_L{\right})_* {{\mathcal}N}(-\sigma)$ is zero. It is clear that for each fiber $L'$ of $\text{pr}_L:\Sigma \rightarrow L$, we have that ${{\mathcal}N}(-\sigma)|_{L'}$ is just $N_{L'/X}(-1)$. By part $(6)$ of lemma \[lem-quadric1\], we conclude that $H^1{\left}(L',N_{L'/X}(-1){\right})$ is zero. Therefore we conclude that $R^1 {\left}( \text{pr}_L {\right})_* {{\mathcal}N}(-\sigma)$ is zero. By part $(3)$ of remark \[rmk-twisting\], we have that $\zeta^* T_{\text{ev}}$ is equivalent to ${\left}( \text{pr}_L {\right})_* {{\mathcal}N}(-\sigma)$. Twisting the short exact sequence above by ${\mathcal O}_{\Sigma}(-L)$ and applying the long exact sequence of higher direct images, we see that ${\left}( \text{pr}_L {\right})_* {{\mathcal}N}(-\sigma)$ fits between ${\left}( \text{pr}_L {\right})_* \text{pr}_L^* T_L(-L)$ and ${\left}( \text{pr}_L {\right})_* N_{\Sigma/X}(-L)$ with cokernel $R^1 {\left}( \text{pr}_L {\right})_* \text{pr}_L^* T_L(-L)$. For any fiber $L'$ of $\text{pr}_L$, we have $T_L(-L)|_{L'}$ is isomorphic to ${\mathcal O}_{L'}(-1)$. Therefore ${\left}(\text{pr}_L{\right})_* \text{pr}_L^* T_L(-1)$ and $R^1 {\left}(\text{pr}_L {\right})_* \text{pr}_L^* T_L(-L)$ are both zero, i.e. ${\left}( \text{pr}_L {\right})_* {{\mathcal}N}(-L)$ is isomorphic to ${\left}( \text{pr}_L {\right})_* N_{\Sigma/X}(-L)$. To show that axiom $(4)$ holds, we want to prove that for any point $p'\in L$ with corresponding fiber $L' = \text{pr}_L^{-1}{\left}\{p'{\right}\}$, we have that ${\left}( \text{pr}_L {\right})_* N_{\Sigma/X}(-L)\otimes {\mathcal O}_L(-p')$ has no $H^1$. Observe that since $R^1{\left}( \text{pr}_L {\right})_* \text{pr}_L^* T_L(-L-L')$ and $R^1 {\left}( \text{pr}_L {\right})_* {{\mathcal}N}(-L -L')$ are both zero, it follows from the long exact sequence of higher direct images that also $R^1 {\left}( \text{pr}_L {\right})_* N_{\Sigma/X}(-L-L')$ is zero. Therefore by the Leray spectral sequence, we conclude that $H^1{\left}( \Sigma, N_{\Sigma/X}(-L-L') {\right})$ equals $H^1 {\left}( L, {\left}( \text{pr}_L {\right})_*{\left}( N_{\Sigma/X}(-L) {\right})(-p') {\right})$. But by part $(4)$ of lemma \[lem-quadric1\], this is zero. Thus axiom $(4)$ is satisfied. Finally, observe that $\sigma^* {\mathcal O}_{\Sigma}(\sigma)$ is the trivial line bundle, so axiom $(5)$ is satisfied. By lemma \[lem-quadric2\], for any pair $([X],[L])\in F({{\mathcal}X})$, if we can find a corresponding triple $([X],[\Sigma],[L])\in W^o$, then it follows that $L$ is a twistable line on $X$. Now we come to the main result of this section. \[prop-quadric\] If $n+1 \geq d^2$ and $d\geq 2$, then $W^o\rightarrow F({{\mathcal}X})$ is dominant. Thus for a general pair $([X],[L])\in F({{\mathcal}X})$, we have that $L$ is a twistable line on $X$. By part $(9)$ of lemma \[lem-quadric1\], it suffices to prove that $W^o$ is nonempty. Let $I_d$ be the set of pairs of integers $I_d=\{(i,j) :0\leq i,j \leq d-1, i+j \geq 2\}$. Choose coordinates on ${\mathbb{P}}^n$ of the form $(Y_0,Y_1,Y_2,Y_3) \cup {\left}( X_{i,j} {\right})_{(i,j)\in I_d} \cup (Z_m: m = 1,\dots, n+1-d^2)$. Let $\Sigma \subset {\mathbb{P}}^n$ be the smooth quadric surface with ideal $$I_{\Sigma} = \langle Y_0 Y_3 - Y_1 Y_2 \rangle + \langle X_{i,j} | (i,j) \in I_d \rangle + \langle Z_m | m= 1, \dots, n+1-d^2 \rangle.$$ This is the image of the embedding $f:{\mathbb{P}}^1 \times {\mathbb{P}}^1 \rightarrow {\mathbb{P}}^n$ given by sending a point ${\left}( [U_0:U_1], [V_0:V_1] {\right})\in {\mathbb{P}}^1 \times {\mathbb{P}}^1$ to $${\left}( [U_0:U_1],[V_0:V_1] {\right}) \mapsto [U_0V_0: U_0V_1: U_1V_0: U_1V_1: 0 : \dots :0 ].$$ We make the following convention. Given a pair $(i,j)$ of integers, we set $k=\min(i,j)$, we set $i'=i-k$ and we set $j'=j-k$. Consider the hypersurface $X\subset {\mathbb{P}}^n$ with defining equation $$F = {\left}( Y_0 Y_3 - Y_1 Y_2 {\right}) Y_3^{d-2} + \sum_{(i,j) \in I_d} Y_0^k Y_1^{i'} Y_2^{j'} Y_3 X_{i,j}.$$ It is clear that $\Sigma \subset X$. We claim that the derivative map $dF:{\mathbb{C}}^{n+1} \rightarrow H^0{\left}(\Sigma,{\mathcal O}_{\Sigma}(d-1) {\right})$ is surjective. Observe first that we have $$\begin{aligned} \frac{\partial F}{\partial Y_0} \mapsto U_1^{d-1}V_1^{d-1}, \frac{\partial F}{\partial Y_1} \mapsto U_1^{d-1}V_0V_1^{d-2}, \\ \frac{\partial F}{\partial Y_2} \mapsto U_0U_1^{d-2}V_1^{d-1}, \frac{\partial F}{\partial Y_3} \mapsto U_0U_1^{d-2}V_0V_1^{d-2}.\end{aligned}$$ And observe that for $(i,j)\in I_d$, we have that $$\frac{\partial F}{\partial X_{i,j}} \mapsto U_0^iU_1^{d-1-i}V_0^jV_1^{d-1-j}.$$ Since the partial derivatives of the form $\frac{\partial F}{\partial Y_i}$ give the terms $U_0^iU_1^{d-1-i}V_0^jV_1^{d-1-j}$ with $(i,j) = (0,0), (0,1), (1,0),$ and $(1,1)$, and since these are precisely the pairs $(i,j)$ not contained in $I_d$, we conclude that $dF$ is surjective. Thus, for any line $L\subset \Sigma$, we have that $([X],[\Sigma],[L])$ is in $W^o$. A very positive, very twisting family of lines ============================================== In the last section, we proved that if $n+1 \geq d^2$, and $d\geq 2$, then for a general hypersurface $X_d\subset {\mathbb{P}}^n$ of degree $d$, a general line $L\subset X$ is twistable, in other words hypothesis \[hyp-2\] holds. In this section, we will prove that if $n\geq d^2 + d+2$, and $d\geq 3$ then there exists a morphism $\overline{\zeta}_1:{\mathbb{P}}^1 \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ which is both very twistable and very positive. This provides the base case for the induction argument of section \[sec-induct\]. The arguments in this section are very similar to those of the last section. In that section, the key result was that for $([X],[L])$ general, there is a quadric surface $\Sigma$ with $L\subset \Sigma \subset X$ such that $H^i{\left}(\Sigma,N_{\Sigma/X}(-1){\right})$ is zero for $i>0$. This result in turn reduced to finding a single degree $d$ polynomial $F$ on ${\mathbb{P}}^n$, vanishing on some quadric surface $\Sigma$, and such that $$d_{F,\Sigma}:{\mathbb{C}}^{n+1} \rightarrow H^0{\left}(\Sigma, {\mathcal O}_{\Sigma}(d-1){\right})$$ is surjective. In this section, the role of $L\subset X$ will be replaced by a rational normal curve $C_0\subset X$ of some degree $k\leq n$ (in the end we will only need the case $k=2d-3$). The role of the quadric surface will be replace by a rational normal scroll $\Sigma$ of degree $2k-1$ such that $C_0 \subset \Sigma \subset X$. The cohomology vanishing result of the last section will be replaced by the vanishing of $H^i{\left}(\Sigma, N_{\Sigma/X}(-C_0 - 2L) {\right})$ for $i>0$, where $L$ is any line of ruling of $\Sigma$. The computation in this section will be to find a single degree $d$ polynomial $F$ on ${\mathbb{P}}^n$, vanishing on $\Sigma$, and such that the image of the map, $$d_{F,\Sigma}: {\mathbb{C}}^{n+1} \rightarrow H^0{\left}( \Sigma,{\mathcal O}_{\Sigma}(d-1) {\right}),$$ let’s call it $W\subset H^0{\left}( \Sigma, {\mathcal O}_{\Sigma}(d-1) {\right})$, has the property that the induced map $$W \otimes H^0 {\left}(\Sigma,{\mathcal O}_{\Sigma}{\left}( (k-3)L {\right}) {\right}) \rightarrow H^0 {\left}( \Sigma, {\mathcal O}_{\Sigma}(d-1) \otimes {\mathcal O}_{\Sigma}{\left}( (k-3) L {\right}) {\right})$$ is surjective. A similar polynomial $F$ to that of the last section satisfies this condition. Generating Linear Systems on ${\mathbb F}_1$ -------------------------------------------- In the last section, the relevant surface was the Hirzebruch surface ${\mathbb F}_0 = {\mathbb{P}}^1 \times {\mathbb{P}}^1$ embedded as a quadric surface. In this section, the relevant surface is the Hirzebruch surface ${\mathbb F}_1$ embedded as a rational normal scroll of degree $2k-1$. We will perform our computations using the projective model of ${\mathbb F}_1$: $${\mathbb F}_1 = {\left}\{ ([T_0:T_1],[T_0U:T_1U:V]) \in {\mathbb{P}}^1 \times {\mathbb{P}}^2 | T_0 (T_1U) = T_1 (T_0U) {\right}\}.$$ In the equation above, $T_0U$ and $T_1U$ are simply variables on ${\mathbb{P}}^2$. We denote the projection maps by $\text{pr}_1:{\mathbb F}_1 \rightarrow {\mathbb{P}}^1$ and $\text{pr}_2: {\mathbb F}_1 \rightarrow {\mathbb{P}}^2$. We denote by ${\mathcal O}_{{\mathbb F}_1}(F)$ the line bundle $\text{pr}_1^* {\mathcal O}_{{\mathbb{P}}^1}$ and by ${\mathcal O}_{{\mathbb F}_1}(E+F)$ the line bundle $\text{pr}_2^* {\mathcal O}_{{\mathbb{P}}^2}$. Here ${\mathcal O}_{{\mathbb F}_1}(E)$ is the divisor class of the directrix $E\subset {\mathbb F}_1$. This explains our terminology $T_0U$ and $T_1U$: $U$ is a nonzero element of $H^0{\left}({\mathbb F}_1,{\mathcal O}_{{\mathbb F}_1}(E) {\right})$, and $T_0U$ and $T_1U$ are the products of $U$ with the two global sections $T_0$ and $T_1$ of $H^0{\left}({\mathbb F}_1, {\mathcal O}_{{\mathbb F}_1}(F) {\right})$. We note that the line bundles ${\mathcal O}_{{\mathbb F}_1}(E+F)$ and ${\mathcal O}_{{\mathbb F}_1}(F)$ generate the Picard group of ${\mathbb F}_1$. Thus we adopt the terminology for line bundles on ${\mathbb F}_1$: $${\mathcal O}(a,b) := {\mathcal O}_{{\mathbb F}_1}{\left}( a(E+F) + bF {\right}).$$ Note that $E+F$ and $F$ are each NEF, but not ample. We conclude that these two line bundles generate the NEF cone. Thus every NEF line bundle on ${\mathbb F}_1$ is of the form ${\mathcal O}(a,b)$ for some nonnegative integers $a,b$. Now suppose that ${\mathcal O}(a,b)$ is some NEF line bundle and $W\subset H^0{\left}( {\mathbb F}_1, {\mathcal O}(a,b) {\right})$ is a linear series. \[defn-cgen\] For an integer $c\geq 0$, we say that $W$ is a $c$-*generating linear system*, if the associated map $$\mu_{W,c}: W \otimes H^0{\left}({\mathbb F}_1, {\mathcal O}(0,c){\right}) \rightarrow H^0{\left}( {\mathbb F}_1, {\mathcal O}(a,b+c) {\right})$$ is surjective. The question we want to answer is, when is $W$ a $c$-generating linear system. In particular, what is the minimal necessary dimension of a $c$-generating linear system? To simplify the answer, we write $b-1 = \beta_d(c+1) + \beta_r$ where $\beta_d,\beta_r$ are integers with $0 \leq \beta_r < c+1$, and we write $a+b-1 = \alpha_d(c+1) + \alpha_r$ where $\alpha_d, \alpha_r$ are integers with $0\leq \alpha_r < c+1$. \[lem-comput\] Define the functions $$\begin{aligned} M(a,b,c) = a^2 + {\left}(2b+3(c+1) {\right})a + 2b +4(c+1) + 2, \\ E(a,b,c) = {\left}(\beta_r^2 - (c+1)\beta_r{\right}) - {\left}(\alpha_r^2 - (c-1)\alpha_r{\right}) \end{aligned}$$ The minimal necessary dimension for a $c$-generating linear system $$W\subset H^0{\left}( {\mathbb F}_1, {\mathcal O}(a,b) {\right})$$ is $\text{dim}(W) = \frac{1}{2(c+1)}{\left}( M(a,b,c) + E(a,b,c) {\right})$. This is just a computation. For any nonnegative integers $a',b'$ there is a decreasing filtration on $H^0{\left}( {\mathbb F}_1, {\mathcal O}(a',b') {\right})$ given by $$F^iH^0{\left}( {\mathbb F}_1, {\mathcal O}(a',b'){\right}) = H^0{\left}( {\mathbb F}_1, {\mathcal O}(a',b')(-iE) {\right}).$$ For any linear system $W\subset H^0{\left}( {\mathbb F}_1, {\mathcal O}_(a,b) {\right})$, there is an induced filtration $F^iW = F^i\cap W$. And the multiplication map $\mu_{W,c}$ respects the filtrations on $W$ and on $H^0{\left}( {\mathbb F}_1, {\mathcal O}(a,b+c) {\right})$. If $\mu_{W,c}$ is surjective, then the associated graded pieces $$\text{gr}^i\mu_{W,c}: \text{gr}^iW \otimes H^0{\left}( {\mathbb F}_1, {\mathcal O}(0,c) {\right}) \rightarrow \text{gr}^i H^0{\left}( {\mathbb F}_1, {\mathcal O}(a,b+c) {\right})$$ are all surjective. As the dimension of $W$ is the sum of the dimensions of the pieces $\text{gr}^i W$, we should compute the minimum possible dimension of a vector subspace $W^i \subset \text{gr}^i H^0{\left}( {\mathbb F}_1, {\mathcal O}(a,b) {\right})$ such that $\text{gr}^i\mu_{W^i,c}$ is surjective (where $\text{gr}^i\mu_{W^i,c}$ is the obvious map). Of course the associated graded parts for ${\mathcal O}(a',b')$ are just $$\text{gr}^i H^0{\left}( {\mathbb F}_1, {\mathcal O}(a',b') {\right}) = {\left}\{ \begin{array}{ll} H^0{\left}( E, {\mathcal O}_E(b'+i) {\right}), & 0\leq i \leq a' \\ \{0\}, & i > a' \end{array} {\right}.$$ So we are looking for a subset $W^i \subset H^0{\left}( E, {\mathcal O}_E(b+i) {\right})$ such that the multiplication map $$\text{gr}^i \mu_{W^i,c}: W^i \otimes H^0{\left}( E, {\mathcal O}_E(c) {\right}) \rightarrow H^0{\left}( E, {\mathcal O}_E(b+c+i) {\right})$$ is surjective. Counting the dimensions of the spaces on the left and the right, we have $$\text{dim}(W^i){\left}(c+1{\right}) \geq {\left}(b+c+i+1{\right}),$$ in other words, $$\text{dim}(W^i) \geq {\left}\lfloor\frac{b+i-1}{c+1} {\right}\rfloor +2.$$ On the other hand, we can acheive this bound: simply take $W^i$ to be generated by the set of monomials: $$\{ U^i V^{a-i} T_0^{(b+i)-j(c+1)}T_1^{j(c+1)}| j=1,\dots,r\}\cup \{ U^i V^{a-i} T_1^{b+i}\}$$ where $r = {\left}\lfloor\frac{b+i-1}{c+1} {\right}\rfloor$. Thus we have that the minimum necessary dimension for a $c$-generating linear series is $$\text{dim}(W) = 2a+2 + \sum_{i=0}^a {\left}\lfloor \frac{b+i-1}{c+1} {\right}\rfloor.$$ If we write $b-1 = \beta_d(c+1) + \beta_r$ with $0\leq \beta_r < c+1$ and if we write $a+b-1 = \alpha_d(c+1) + \alpha_r$ with $0 \leq \alpha_r < c+1$, then the sum above is just $$2a+2 + \sum_{i=0}^a {\left}\lfloor \frac{b+i-1}{c+1} {\right}\rfloor = \frac{1}{2(c+1)}{\left}( M(a,b,c) + E(a,b,c) {\right})$$ where $M(a,b,c)$ and $E(a,b,c)$ are as above. For the next result, we introduce the Cox homogeneous coordinate ring: $$S = S({\mathbb F}_1) := {\mathbb{C}}[T_0,T_1,U,V] = \oplus_{(a,b) \in {\mathbb{Z}}^2} H^0{\left}( {\mathbb F}_1, {\mathcal O}(a,b) {\right}).$$ This is a ${\mathbb{Z}}^2$-graded ring, graded by $\text{deg}(T_0) = \text{deg}(T_1) = (0,1)$, $\text{deg}(V) = (1,0)$ and $\text{deg}(U) = (1,-1)$. For any multi-degree $(a,b)\in {\mathbb{Z}}^2$, we have that $S_{(a,b)} = H^0{\left}( {\mathbb F}_1, {\mathcal O}(a,b) {\right})$. We put a graded lexicographical monomial order on $S$ where the grading is by total degree ($\text{deg(a,b)} = a+b$), and where $U>V>T_0>T_1$. In the proof of the lemma above, a special role was played by the linear system $$\begin{aligned} W_0(a,b,c) = \text{span}{\left}\{ U^i V^{a-i} T_0^{(b+i)-j(c+1)}T_1^{j(c+1)}|i=0,\dots, a, j=1,\dots,r(i) {\right}\} \\ + \text{span}{\left}\{U^i V^{a-i} T_1^{b+i}{\right}|i=0,\dots,a\}.\end{aligned}$$ Here $r(i) = {\left}\lfloor\frac{b+i-1}{c+1} {\right}\rfloor$. \[lem-comput2\] Suppose $W\subset H^0{\left}( {\mathbb F}_1, {\mathcal O}(a,b) {\right})$ is a linear system. If the vector space of initial terms $\text{IN}(W)$ contains $W_0(a,b,c)$, then $W$ is a $c$-generating linear system. Clearly we have that the vector space of initial terms of $\text{image}(\mu_{W,c})$ satisfies $$\text{IN}(W)\cdot S_{(0,c)} \subset \text{IN}{\left}(\text{image}(\mu_{W,c}) {\right}).$$ So if $\text{IN}(W)$ contains $W_0(a,b,c)$, then we have that $$W_0(a,b,c)\cdot S_{(0,c)} \subset \text{IN}{\left}( \text{image}(\mu_{W,c}) {\right}).$$ By construction, $W_0(a,b,c)\cdot S_{(0,c)} = S_{(a,b+c)}$. So we have $\text{IN}{\left}( \text{image}(\mu_{W,c}) {\right}) = S_{(a,b+c)}$, and therefore $\text{image}(\mu_{W,c}) = S_{(a,b+c)}$. Therefore $W$ is a $c$-generating linear system. An important special case for us is when $a=d-1, b=(d-1)(k-1)$ and $c=k-3$ for some positive integers $d$ and $k$ (here $d$ will be the degree of the hypersurface $X\subset {\mathbb{P}}^n$, and $k$ will be the degree of the curve $C_0\subset X$). In particular, if $k\geq 2d-3$, then we have $b-1 = (d-1)(k-2) + d-2$ and $a+b-1 = (d-1)(k-2) + 2d-3$. So the formulas above reduce to $$\begin{array}{rcl} M(d-1,(d-1)(k-1),k-3) & = & 2(k-2)(d-1)^2 + \\ 5(k-2)(d-1) + 4(k-2) & + & {\left}(3(d-1)^2 + 2(d-1) -2 {\right}), \\ E(d-1,(d-1)(k-2),k-3) & = & (k-2)(d-1) - \\ {\left}( 3(d-1)^2 + 2(d-1) - 2 {\right}) \end{array}$$ So, if $k\geq 2d-3$, we have $$\frac{1}{2(k-2)}(M+E) = (d-1)^2 + 3(d-1) + 2 = d^2 + d.$$ Cohomology Results ------------------ We introduce some incidence correspondences, analogous to those introduced in section \[sec-twist\]. Let $N_d = \binom{n+d}{n} -1$ and let ${\mathbb{P}}^{N_d}$ denote the projective space parametrizing hypersurfaces $X_d \subset {\mathbb{P}}^n$ of degree $d$. Let ${{\mathcal}X} \subset {\mathbb{P}}^{N_d} \times {\mathbb{P}}^n$ denote the universal family of degree $d$ hypersurfaces in ${\mathbb{P}}^n$. Let $k$ be any integer with $1\leq k \leq \frac{n}{2}$ (later we will only need the case that $k=2d-3$). Let ${{\mathcal}R}^k({\mathbb{P}}^n) \subset \textit{Hilb}^{kt+1}({\mathbb{P}}^n)$ denote the open subscheme parametrizing curves $C_0\subset {\mathbb{P}}^n$ which are projectively equivalent to a degree $k$ rational normal curve $C_0 \subset {\mathbb{P}}^k \subset {\mathbb{P}}^n$. Observe that ${{\mathcal}R}^k{\left}({\mathbb{P}}^n{\right})$ is a homogeneous space of $\text{PGL}_{n+1}$, and therefore is smooth and connected. Let ${{\mathcal}R}^k({{\mathcal}X}) \subset {\mathbb{P}}^{N_d} \times {{\mathcal}R}^k({\mathbb{P}}^n)$ denote the parameter space for pairs $([X],[C_0])$ where $C_0\subset X$. Observe that the projection ${{\mathcal}R}^k({{\mathcal}X}) \rightarrow {{\mathcal}R}^k({\mathbb{P}}^n)$ is a projective bundle of relative dimension $N_d - (kd+1)$. Let $Q(t) = \frac{1}{2}(t+1)((2k-1)t+2)$ denote the Hilbert polynomial of a rational normal scroll of degree $2k-1$ in ${\mathbb{P}}^{2k}$. Let ${{\mathcal}U}\subset \textit{Hilb}^{Q(t)}({{\mathbb{P}}^n})$ denote the open subscheme parametrizing subschemes $\Sigma \subset {\mathbb{P}}^n$ which are projectively equivalent to a rational normal scroll of degree $2k-1$ in ${\mathbb{P}}^{2k}\subset {\mathbb{P}}^n$ which is abstractly isomorphic to ${\mathbb F}_1$. Let ${{\mathcal}V}\subset {{\mathcal}U} \times {{\mathcal}R}^k({\mathbb{P}}^n)$ denote the parameter space of pairs $([\Sigma],[C_0])$ where $C_0\subset \Sigma$ and such that, using the isomorphism of $\Sigma$ and ${\mathbb F}_1$, the line bundle of $C_0$ is ${\mathcal O}(1,0)$. The projection map ${{\mathcal}V} \rightarrow {{\mathcal}U}$ factors as an open subset (with nonempty fibers) of a projective bundle over ${{\mathcal}U}$ of relative dimension $2$ (actually each fiber is isomorphic to the ${\mathbb{A}}^2$ of irreducible curves in the linear system $|{\mathcal O}(1,0)|$). Let ${{\mathcal}W} \subset {\mathbb{P}}^{N_d}\times {{\mathcal}U} \times {{\mathcal}R}^k({\mathbb{P}}^n)$ denote the parameter space for triples $([X],[\Sigma],[C_0])$ where $([\Sigma],[C_0])$ is in ${{\mathcal}V}$ and where $\Sigma \subset X$. The projection map ${{\mathcal}W} \rightarrow {{\mathcal}V}$ is a projective bundle of relative dimension $N_d - Q(d)$. Now for a triple $([X],[\Sigma],[C_0]) \in {{\mathcal}W}$, we define $d_{X,\Sigma}: {\mathbb{C}}^{n+1} \rightarrow H^0{\left}( \Sigma, {\mathcal O}_{\Sigma}(d-1) {\right})$ as in section \[sec-twist\]. More precisely, let ${{\mathcal}E}$ be the trivial vector bundle on ${{\mathcal}W}$ of rank $n+1$, let ${{\mathcal}G}$ be the vector on ${{\mathcal}U}$ whose fiber at a point $\Sigma$ is just $H^0{\left}(\Sigma, {\mathcal O}_{\Sigma}(d-1){\right})$, and let ${{\mathcal}F}$ be the vector bundle on ${{\mathcal}W}$ which is $\text{pr}_1^*({\mathcal O}_{{\mathbb{P}}^{N_d}}(1))\otimes \text{pr}_2^* {{\mathcal}G}$. Then there is a map of vector bundles $d:{{\mathcal}E} \rightarrow {{\mathcal}F}$ whose fiber over $([X],[\Sigma],[C_0])$ is the map $d_{X,\Sigma}$ constructed above. Let ${{\mathcal}W}^o\subset {{\mathcal}W}$ be the open subscheme (possibly empty) parametrizing pairs $([X],[\Sigma],[C_0])$ such that the image of $d_{X,\Sigma}$ is a $(k-3)$-generating linear series in $H^0{\left}( \Sigma, {\mathcal O}_{{\mathbb{P}}^n}(d-3)|_\Sigma {\right})$. \[lem-scroll1\] Let $f:{\mathbb F}_1 \rightarrow \Sigma$ be an isomorphism to a degree $2k-1$ rational normal scroll $\Sigma \subset {\mathbb{P}}^{2k}\subset {\mathbb{P}}^n$. For each pair of integers $a,b \geq 0$, consider the bundle $$N(a,b) = f^*{\left}( N_{\Sigma/{\mathbb{P}}^n(-1}) {\right}) \otimes {\mathcal O}(a,b)$$ and the subbundle $$N'(a,b) = f^*{\left}( N_{\Sigma/{\mathbb{P}}^{2k} } {\right}) \otimes {\mathcal O}(a,b).$$ Then we have the following: 1. $N'(0,0)$ is generated by global sections and satisfies $H^i{\left}( {\mathbb F}_1, N'(0,0) {\right})$ is zero for $i>0$, 2. $N(0,0)$ is generated by global sections and satisfies $H^i{\left}( {\mathbb F}_1, N(0,0) {\right})$ is zero for $i>0$, 3. if ${\mathcal F}$ is any coherent sheaf on ${\mathbb F}_1$ which is generated by global sections and satisfies $H^i{\left}( {\mathbb F}_1, {\mathcal F} {\right})$ is zero for $i>0$, then for every pair of nonnegative integers $(a,b)$ we have that ${\mathcal F}(a,b) := {\mathcal F}\otimes {\mathcal O}(a,b)$ is generated by global sections and satisfes $H^i{\left}( {\mathbb F}_1, {\mathcal F}(a,b) {\right})$ is zero for $i>0$. In particular, we conclude that for any pair of nonnegative integers $(a,b)$, we have that $N(a,b)$ (resp. $N'(a,b)$) is generated by global sections and satisfies $H^i{\left}({\mathbb F}_1, N(a,b){\right})$ is zero for $i>0$ (resp. $H^i{\left}( {\mathbb F}_1, N'(a,b) {\right})$ is zero for $i>0$). Recall that $\text{pr}_1:{\mathbb F}_1 \rightarrow {\mathbb{P}}^1$ is the projection morphism such that $\text{pr}_1^*{\mathcal O}_{{\mathbb{P}}^1}(1) = {\mathcal O}_{{\mathbb F}_1}(f)$. Via $\text{pr}_1$, ${\mathbb F}_1$ is isomorphic as a ${\mathbb{P}}^1$-scheme to the total space of the projective bundle: $${\mathbb F}_1 \cong {\mathbb{P}}{\left}( {\mathcal O}_{{\mathbb{P}}^1}(-k) \oplus {\mathcal O}_{{\mathbb{P}}^1}(-(k-1)) {\right}).$$ Under this isomorphism ${\mathcal O}(1,k-1)$ corresponds to the tautological quotient bundle ${\mathcal O}(1)$ on ${\mathbb{P}}{\left}( {\mathcal O}_{{\mathbb{P}}^1}(-k) \oplus {\mathcal O}_{{\mathbb{P}}^1}(-(k-1)) {\right})$. In other words, up to projective equivalence, the map $f:{\mathbb F}_1 \rightarrow {\mathbb{P}}^{2k}$ corresponds to the complete linear system of ${\mathcal O}(1)$, i.e. $f^*{\mathcal O}_{{\mathbb{P}}^{2k}}(1) \cong {\mathcal O}(1)$. Using this identification, it is easy to see that we have a short exact sequence of vector bundles on ${\mathbb F}_1$: $$0 \rightarrow \text{pr}_1^* T_{{\mathbb{P}}^1} \rightarrow \text{pr}_1^*{\left}( {\mathcal O}_{{\mathbb{P}}^1}(1)^{\oplus (2k-1)} {\right})\otimes f^*{\mathcal O}_{{\mathbb{P}}^{2k}}(1) \rightarrow f^* N_{\Sigma/{\mathbb{P}}^{2k}} \rightarrow 0$$ Twisting by $f^*{\mathcal O}_{{\mathbb{P}}^{2k}}(-1)$, we observe that $N'(0,0)$ is a quotient of $\text{pr}_1^*{\left}({\mathcal O}_{{\mathbb{P}}^1}(1)^{\oplus (2k-1)} {\right})$, and so is generated by global sections. Observe that we have the vanishing $${\left}(\text{pr}_1{\right})_*{\left}(\text{pr}_1^*T_{{\mathbb{P}}^1}\otimes f^*{\mathcal O}_{{\mathbb{P}}^{2k}}(-1) {\right}) = R^1{\left}(\text{pr}_1{\right})_*{\left}(\text{pr}_1^*T_{{\mathbb{P}}^1}\otimes f^*{\mathcal O}_{{\mathbb{P}}^{2k}}(-1) {\right}) = \{0\}.$$ Applying the long exact sequence of higher direct images to our short exact sequence (after twisting by $f^*{\mathcal O}_{{\mathbb{P}}^{2k}}(-1)$), we have that $R^1{\left}( \text{pr}_1 {\right})_* {\left}(f^*N_{\Sigma/{\mathbb{P}}^{2k}}(-1) {\right})$ is zero, and ${\left}( \text{pr}_1 {\right})_* {\left}(f^*N_{\Sigma/{\mathbb{P}}^{2k}}(-1) {\right})$ is ${\mathcal O}_{{\mathbb{P}}^1}(1)^{\oplus (2k-1)}$. From this and the Leray spectral sequence associated to $\text{pr}_1:{\mathbb F}_1 \rightarrow {\mathbb{P}}^1$, we conclude that $H^i{\left}({\mathbb F}_1, N'(0,0) {\right})$ is zero for $i>0$. This proves item $(1)$ of the lemma. Also observe that we have a short exact sequence: $$\begin{CD} 0 @>>> N'(0,0) @>>> N(0,0) @>>> {\mathcal O}(0,0)^{\oplus (n-2k)} @>>> 0 \end{CD}$$ Since $H^i{\left}( {\mathbb F}_1, {\mathcal O}_{{\mathbb F}_1} {\right})$ is zero for $i>0$, we conclude that $N(0,0)$ is generated by global sections and satisfies $H^i{\left}({\mathbb F}_1, N(0,0) {\right})$ is zero for $i>0$. This proves item $(2)$ of the lemma. Now suppose that ${\mathcal F}$ is a coherent sheaf on ${\mathbb F}_1$ such that ${\mathcal F}$ is generated by global sections and such that $H^i{\left}( {\mathbb F}_1, {\mathcal F} {\right})$ is zero for $i>0$. We will prove by double induction on $(a,b)$ that the same is true for ${\mathcal F}(a,b) := {\mathcal F}\otimes {\mathcal O}(a,b)$. First we prove the result when $b=0$. We proceed by induction on $a$. If $a=0$, the result is tautological. Thus suppose that $a>0$ and suppose the result is proved for $a-1$. Let $D\subset {\mathbb F}_1$ be a general member of the linear system $|{\mathcal O}(1,0)|$. Then $D$ is a smooth curve isomorphic to ${\mathbb{P}}^1$. Since $D$ is general, we have a short exact sequence: $$\begin{CD} 0 @>>> {\mathcal F}(a-1,0) @>>> {\mathcal F}(a,0) @>>> {\mathcal F}(a,0)|_D @>>> 0 \end{CD}$$ Now $\mathcal{F}|_D$ is generated by global sections. And ${\mathcal O}_{{\mathbb F}_1}(a(e+f))|_D$ is isomorphic to ${\mathcal O}_{{\mathbb{P}}^1}(a)$. Thus we conclude that also ${\mathcal F}(a,0)|_D$ is generated by global sections. By the induction assumption, $H^1{\left}(({\mathbb F}_1,{\mathcal F}(a-1,0) {\right})$ is zero, we conclude by the long exact sequence of cohomology associated to the short exact sequence above, that all the global sections of ${\mathcal F}(a,0)|_D$ lift to global sections of ${\mathcal F}(a,0)$. Therefore ${\mathcal F}(a,0)$ is generated by global sections. Also, a coherent sheaf on ${\mathbb{P}}^1$ which is generated by global sections has no higher cohomology. Combining this with the induction assumption and using the long exact sequence in cohomology associated to the short exact sequence above, we conclude that $H^i{\left}( {\mathbb F}_1, {\mathcal F}(a,0) {\right})$ is zero for $i>0$. Therefore we conclude by induction that for all $a>0$, ${\mathcal F}(a,0)$ is generated by global sections and $H^i{\left}( {\mathbb F}_1, {\mathcal F}(a,0) {\right})$ is zero for $i>0$. Now we prove the result with $b$ arbitrary. We proceed by induction on $b$. If $b=0$, the result was proved in the last paragraph. Thus suppose that $b>0$ and suppose the result is proved for $b-1$. Let $L \subset {\mathbb F}_1$ be a general fiber of $\text{pr}_1$. Then $L$ is smooth and isomorphic to ${\mathbb{P}}^1$. Since $L$ is general, we have a short exact sequence: $$\begin{CD} 0 @>>> {\mathcal F}(a,b-1) @>>> {\mathcal F}(a,b) @>>> {\mathcal F}(a,b)|_L @>>> 0 \end{CD}$$ Now ${\mathcal O}_{{\mathbb F}_1}(a(e+f)+bf)|_L$ is isomorphic to ${\mathcal O}_{{\mathbb{P}}^1}(a)$. By a similar analysis to that in the last paragraph, we conclude that ${\mathcal F}(a,b)$ is generated by global sections and that $H^i{\left}( {\mathbb F}_1, {\mathcal F}(a,b) {\right})$ is zero for $i>0$. So item $(3)$ is proved by induction on $b$. \[lem-scroll2\] Let $([X],[\Sigma],[C_0]) \in {{\mathcal}W}^o$ be any triple, and let $f:{\mathbb F}_1 \rightarrow \Sigma$ be some fixed isomorphism. Let $N'(a,b)$ and $N(a,b)$ be as in lemma \[lem-scroll1\]. Also let us denote $$N_X(a,b) f^*{\left}( N_{\Sigma/X}(-1) {\right})\otimes {\mathcal O}(a,b)$$ Then we have the following: 1. $X$ is smooth along $\Sigma$ 2. for each pair of nonnegative integers $(a,b)$, we have $H^i{\left}( {\mathbb F}_1, N_X(a,b+k-3) {\right})$ is zero for $i>0$, 3. for any line of ruling $L\subset \Sigma$, $H^1{\left}( L, N_{L/X}(a-1) {\right})$ is zero for $a\geq 0$ any integer, 4. $H^1{\left}( C_0, N_{C_0/X}(a-2) {\right})$ is zero for $a\geq 0$ any integer, 5. the projection morphism ${{\mathcal}W} \rightarrow {\mathbb{P}}^{N_d}$ given by $([X],[\Sigma],[C_0]) \mapsto [X]$ is smooth at $([X],[\Sigma],[C_0])$, 6. for any line $L\subset \Sigma$, the projection morphism $F({{\mathcal}X}) \rightarrow {\mathbb{P}}^{N_d}$ given by $([X],[L]) \mapsto [X]$ is smooth at $([X],[L])$, 7. the projection morphism ${{\mathcal}R}^k({{\mathcal}X}) \rightarrow {\mathbb{P}}^{N_d}$ given by $([X],[C_0]) \mapsto [X]$ is smooth at $([X],[C_0])$, and 8. the projection morphism $\pi: {{\mathcal}W} \rightarrow {{\mathcal}R}^k({{\mathcal}X})$ given by $([X],[\Sigma],[C_0]) \mapsto ([X],[C_0])$ is smooth at $([X],[\Sigma],[C_0])$. Since the partial derivatives of a defining equation of $X$ give a $c$-generating linear series, in particular they generate the sheaf ${\mathcal O}_{\Sigma}(d-1)$. Thus, there is no point of $\Sigma$ at which all the partial derivatives vanish. By the Jacobian criterion, we conclude that $X$ is smooth along $\Sigma$. This proves item $(1)$. For item $2$, we observe that we have a short exact sequence: $$\begin{CD} 0 @>>> N_{\Sigma/X} @>>> N_{\Sigma/{\mathbb{P}}^n} @>>> N_{X/{\mathbb{P}}^n}|_\Sigma @>>> 0 \end{CD}$$ For ease of notation, define $\alpha = a + (d-1)$ and $\beta = b+(d-1)(k-1)$. We have a short exact sequence: $$\begin{CD} 0 @>>> N_X(a,b) @>>> N(a,b) @>>> {\mathcal O}(\alpha,\beta) @>>> 0 \end{CD}$$ When $a,b \geq 0$, it follows by lemma \[lem-scroll1\] that $H^i{\left}( {\mathbb F}_1, N(a,b) {\right})$ is zero for $i\geq 0$. By a simple calculation, we also see that $$H^i{\left}( {\mathbb F}_1,{\mathcal O}(\alpha,\beta) {\right}) = \{0\}, i\geq 0.$$ So we conclude that $H^2{\left}( {\mathbb F}_1,N_X(a,b) {\right})$ is zero, and $H^1{\left}( {\mathbb F}_1, N_X(a,b) {\right})$ is zero iff the following map is surjective: $$H^0{\left}( {\mathbb F}_1, N(a,b) {\right}) \rightarrow H^0{\left}( {\mathbb F}_1, {\mathcal O}(\alpha,\beta) {\right}).$$ We have a commutative diagram: $$\begin{CD} H^0{\left}( {\mathbb F}_1, N(a,b) {\right}) \otimes H^0{\left}( {\mathbb F}_1, {\mathcal O}(a',b') {\right}) @>>> H^0{\left}( {\mathbb F}_1, N(a+a',b+b') {\right}) \\ @VVV @VVV \\ H^0{\left}( {\mathbb F}_1, {\mathcal O}{\left}(\alpha,\beta {\right}) {\right}) \otimes H^0{\left}( {\mathbb F}_1, {\mathcal O}(a',b') {\right}) @>>> H^0{\left}( {\mathbb F}_1, {\mathcal O}{\left}( \alpha+a', \beta+b' {\right}) {\right}) \end{CD}$$ We conclude that if $H^1{\left}( {\mathbb F}_1, N_X(a,b) {\right})$ is zero and if $a',b' \geq 0$, then we also have that $H^1{\left}( {\mathbb F}_1, N_X(a+a',b+b') {\right})$. So to prove item $(2)$, we are reduced to the case $a=0, b=k-3$. But then the commutative diagram above factors as $$\begin{CD} H^0{\left}( {\mathbb F}_1, T_{{\mathbb{P}}^n} {\right}) \otimes H^0{\left}( {\mathbb F}_1, {\mathcal O}(0,k-3) {\right}) @>>> H^0{\left}( {\mathbb F}_1, N(0,k-3) {\right}) \\ @VVV @VVV \\ H^0{\left}( {\mathbb F}_1, {\mathcal O}{\left}(d-1,(d-1)(k-1) {\right}) {\right}) \otimes H^0{\left}( {\mathbb F}_1, {\mathcal O}(0,k-3) {\right}) @>>> H^0{\left}( {\mathbb F}_1, {\mathcal O}{\left}(\alpha,\beta {\right}) {\right}) \end{CD}$$ By definition, the composition $$H^0{\left}( {\mathbb F}_1 T_{{\mathbb{P}}^n} {\right}) \otimes H^0{\left}( {\mathbb F}_1, {\mathcal O}(0,k-3) {\right}) \rightarrow H^0{\left}( {\mathbb F}_1, {\mathcal O}{\left}(\alpha, \beta {\right}) {\right})$$ is surjective iff the triple $([X],[\Sigma],[C_0])$ is in ${\mathcal W}^o$. So if $([X],[\Sigma],[C_0])$ is in ${\mathcal W}^o$, then $H^1{\left}( {\mathbb F}_1, N_X(0,k-3) {\right})$ is zero. Thus item $(2)$ holds. For item $(3)$, observe that we have a short exact sequence: $$\begin{CD} 0 @>>> N_{L/\Sigma}(a-1) @>>> N_{L/X}(a-1) @>>> N_{\Sigma/X}|_L(a-1) @>>> 0 \end{CD}$$ Of course $N_{L/\Sigma} \cong {\mathcal O}_L$, thus $H^1{\left}( L, N_{L/\Sigma}(a-1) {\right})$ is zero for $a\geq 0$. So to prove item $(3)$, it suffices to prove that $H^1{\left}(L, N_{\Sigma/X}|_L(a-1) {\right})$ is zero. Since ${\mathcal O}(a-1,b)|_L \cong {\mathcal O}_L(a-1)$, we have a short exact sequence: $$\begin{CD} 0 @>>> N_X(a,k-3) @>>> N_X(a,k-2) @>>> N_{\Sigma/X}|_L(a-1) @>>> 0 \end{CD}$$ By item $(2)$, for $a\geq 0$ the higher cohomology of the first two terms vanishes. Thus by the long exact sequence in cohomology associated to this short exact sequence, we have that $H^1{\left}( L, N_{\Sigma/X}|_L(a-1) {\right})$ is zero for $a\geq 0$. This proves item $(3)$. Item $(4)$ is almost identical to item $(3)$ and is left as an exercise to the reader. By [@K proposition 2.14.2], the obstruction space for the relative Hilbert scheme $\textit{Hilb}^{Q(t)}({{\mathcal}X}/{\mathbb{P}}^{N_d})$ at a point $([X],[\Sigma])$ is contained in $H^1{\left}(\Sigma, N_{\Sigma/X} {\right})$. For a triple $([X],[\Sigma],[C_0])$ in ${{\mathcal}W}^o$, it follows from item $(2)$ that $$H^1{\left}( \Sigma, N_{\Sigma/X} {\right}) = H^1{\left}( {\mathbb F}_1, N_X(1,k-1) {\right})$$ vanishes. It follows by [@K theorem 2.10] that $\textit{Hilb}^{Q(t)}({{\mathcal}X}/{\mathbb{P}}^{N_d}) \rightarrow {\mathbb{P}}^{N_d}$ is smooth at $([X],[\Sigma])$. Of course the projection ${{\mathcal}W}^o \rightarrow \textit{Hilb}^{Q(t)}({{\mathcal}X}/{\mathbb{P}}^{N_d})$ is an open subset of a projective bundle, and so is smooth. Thus we conclude that the composite morphism ${{\mathcal}W}^o \rightarrow {\mathbb{P}}^{N_d}$ is smooth. This proves item $(5)$. Item $(6)$ is similar to item $(5)$ and follows from item $(3)$ which shows that $H^1{\left}( L, N_{L/X} {\right})$ is zero. Item $(7)$ is similar to item $(5)$ and follows from item $(4)$ which shows that $H^1{\left}( C_0, N_{C_0/X} {\right})$ is zero. Since ${{\mathcal}W}^o \rightarrow {\mathbb{P}}^{N_d}$ is smooth at $([X],[\Sigma],[C_0])$ and since ${{\mathcal}R}^k \rightarrow {\mathbb{P}}^{N_d}$ is smooth at $([X],[C_0])$, to prove that $\pi:{{\mathcal}W}^0 \rightarrow {{\mathcal}R}^k({{\mathcal}X})$ is smooth at $([X],[\Sigma],[C_0])$, it suffices to check that the derivative map $d\pi: T_{{{\mathcal}W}^0/{\mathbb{P}}^{N_d}} \rightarrow \pi^* T_{{{\mathcal}R}^k({{\mathcal}X})/{\mathbb{P}}^{N_d}}$ is surjective at $([X],[\Sigma],[C_0])$. This exactly reduces to the statement that $H^0 {\left}( \Sigma, N_{\Sigma/X} {\right}) \rightarrow H^0 {\left}( C_0, N_{\Sigma/X}|_{C_0} {\right})$ is surjective. Since the cokernel is contained in $H^1{\left}( {\mathbb F}_1, N_X(0,k-1) {\right})$, which is zero by item $(3)$, we conclude the derivative $d\pi$ is surjective. Therefore $\pi:{{\mathcal}W}^o \rightarrow {{\mathcal}R}^k({{\mathcal}X})$ is smooth. This proves item $(8)$. Now suppose that $([X],[\Sigma],[C_0])$ is a point in ${{\mathcal}W}^o$. Let $\sigma:C_0 \rightarrow \Sigma$ be the inclusion and let $\text{pr}_{C_0}: \Sigma \rightarrow C_0$ be the unique projection such that $\sigma$ is a section of $\text{pr}_L$ (in the model of $\Sigma$ as ${\mathbb F}_1$, $\text{pr}_{C_0}$ is simply $\text{pr}_1$). Let $g:\Sigma \rightarrow X$ be the inclusion. Then we have an induced morphism $\zeta:C_0 \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ given by the diagram: $$\begin{CD} \Sigma @> g >> X \\ @VV \text{pr}_{C_0} V \\ C_0 \end{CD}$$ \[lem-scroll3\] If $([X],[\Sigma],[C_0])$ is a triple in ${{\mathcal}W}^o$, then the corresponding morphism $\zeta:C_0 \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$ is very twisting and very positive. First we check the axioms of definition \[defn-verytwisting\]. Since $g\circ \sigma:C_0 \rightarrow X$ is an embedding, in particular this map is stable, i.e. axiom $(1)$ is satisfied. By item $(7)$ of lemma \[lem-scroll2\], we conclude that ${\overline{{\mathcal}M}_{0,0}({X,k})}$ is unobstructed at $[g\circ \sigma: C_0 \rightarrow X]$, i.e. axiom $(2)$ is satisfied. The argument that axiom $(3)$ holds is exactly the same as the argument that axiom $(3)$ holds in the proof of lemma \[lem-quadric2\], where item $(6)$ of lemma \[lem-quadric1\] is replaced by item $(3)$ of lemma \[lem-scroll2\]. As in the proof of lemma \[lem-quadric2\], we have that $\zeta^*T_{\text{ev}}$ is isomorphic to ${\left}( \text{pr}_{C_0} {\right})_* N_X(0,k-1)$. Of course $\zeta^*T_{\text{ev}}$ is ample iff $H^1{\left}( C_0, \zeta^*T_{\text{ev}}(-2) {\right})$ is zero. By the isomorphism above and a Leray spectral sequence argument analogous to the one in the proof of lemma \[lem-quadric2\], this cohomology group equals $H^1{\left}( {\mathbb F}_1, N_X(0,k-3) {\right})$. By item $(2)$ of lemma \[lem-scroll2\], this group is zero. Therefore $\zeta^*T_{\text{ev}}$ is an ample bundle. So axiom $(4)$ is satisfied. Finally, observe that $\sigma^* {\mathcal O}_{\Sigma}(\sigma)$ is ${\mathcal O}_{C_0}(1)$ and so is ample. So axiom $(5)$ is satisfied. Thus $\zeta$ is a very twisting family. Next we consider the axioms in definition \[defn-pos\]. Axioms $(1)$, $(2)$ and $(3)$ follow immediately from axioms $(1)$, $(2)$ and $(3)$ of definition \[defn-twisting\] proved above. To see that axiom $(4)$ holds, observe that we have a short exact sequence: $$\begin{CD} 0 @>>> \zeta^* T_{\text{ev}} @>>> \zeta^* T_{{\overline{{\mathcal}M}_{0,1}({X,1})}} @>>> {\left}( g\circ \sigma{\right})^* T_X @>>> 0 \end{CD}$$ We have seen above that $\zeta^* T_{\text{ev}}$ is ample. Moreover, it follows by item $(4)$ of lemma \[lem-scroll2\] that $N_{C_0/X}$ is an ample vector bundle. Since also $T_{C_0}$ is an ample line bundle, we conclude that $T_X|_{C_0}$ is an ample vector bundle. Since the first and third terms in the short exact sequence above are ample, we conclude that $\zeta^* T_{{\overline{{\mathcal}M}_{0,1}({X,1})}}$ is an ample vector bundle. Since $\zeta^* \text{pr}^* T_{{\overline{{\mathcal}M}_{0,0}({X,1})}}$ is a quotient bundle of $\zeta^* T_{{\overline{{\mathcal}M}_{0,1}({X,1})}}$, we also have that $\zeta^* \text{pr}^* T_{{\overline{{\mathcal}M}_{0,0}({X,1})}}$ is an ample vector bundle. So axiom $(4)$ holds. Finally, we have seen above that $\sigma^*{\mathcal O}_{\Sigma}(\sigma)$ equals ${\mathcal O}_{C_0}(1)$, which is ample. Thus axiom $(5)$ holds. So $\zeta$ is a very positive family. \[prop-scroll\] If $n \geq d^2 + d + 2$ and if $d\geq 3$, for $k=2d-3$, we have that ${{\mathcal}W}^o \rightarrow {{\mathcal}R}^k({{\mathcal}X})$ is dominant, and ${{\mathcal}R}^k({{\mathcal}X}) \rightarrow {\mathbb{P}}^{N_d}$ is dominant. So for $[X]\in {\mathbb{P}}^{N_d}$ general, there exists a very twisting, very positive family $\zeta: C_0 \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$. By item $(8)$ of lemma \[lem-scroll2\], it suffices to prove that ${{\mathcal}W}^o$ is nonempty. We have to find a pair $([X],[\Sigma])$ such that for $a=d-1$, $b=(d-1)(k-1)$ and for $c=k-3$, we have that the image of the derivative map $$d_{X,\Sigma}:H^0{\left}( ({\mathbb{P}}^n,T_{{\mathbb{P}}^n}(-1) {\right}) \rightarrow H^0{\left}( {\mathbb F}_1, {\mathcal O}(a,b) {\right})$$ is a $c$-generating linear system. Recall that $S = {\mathbb{C}}[T_0,T_1,U,V]$ is the ${\mathbb{Z}}^2$-graded Cox homogeneous coordinate ring of ${\mathbb F}_1$. Let $A_d$ denote the set of monomials in the vector subspace $S_{(d-1,(d-1)(k-1))}$ which occur in the linear system $W_0(a,b,c)$, i.e. $$\begin{array}{c} A_d = \\ {\left}\{ U^i V^{d-1-i} T_0^{((d-1)(k-1)+i)-j(k-2)}T_1^{j(k-2)}|i=0,\dots, d-1, j=1,\dots,r(i) {\right}\} \\ \cup {\left}\{U^i V^{d-1-i} T_1^{(d-1)(k-1)+i}{\right}|i=0,\dots,d-1\} \end{array}$$ where $r(i) = d-1+{\left}\lfloor\frac{d-2+i}{k-2} {\right}\rfloor$. Let $B_d$ denote the set of monomials of the form $$\begin{array}{c} B_d = \\ {\left}\{U^{d-1} T_0^{(d-1)k - i(k-2)} T_1^{i(k-2)} | i=0,\dots, d-2 {\right}\} \cup \\ {\left}\{ U^{d-2} V T_0^{(d-1)k-1-i(k-2)} T_1^{i(k-2)} | i=0,\dots, d-2 {\right}\} \cup \\ {\left}\{ U^{d-3} V^2 T_0^{(d-1)k-2-(j+2)(k-2)} T_1^{(j+2)(k-2)} | j=0,\dots, d-4 {\right}\} \cup \\ {\left}\{ U^{d-4} V^3 T_0^{(d-1)k-3-(j+2)(k-2)} T_1^{(j+2)(k-2)} | j=0,\dots, d-4 {\right}\}. \end{array}$$ Let $C_d$ denote the set of monomials $C_d = A_d - B_d$. Now $A_d$ contains $d^2 + d$ monomials, and $B_d$ contains $4d-8$ monomials. Choose homogeneous coordinates on ${\mathbb{P}}^n$ of the form $${\left}\{Y_0,\dots,Y_{2k} {\right}\} \cup {\left}\{ X_M | M \in C_d {\right}\} \cup {\left}\{Z_l | l=1,\dots, n-(d^2+d+2) {\right}\}.$$ Let $f:{\mathbb F}_1 \rightarrow {\mathbb{P}}^{2k} \subset {\mathbb{P}}^n$ be the map defined by sending a point $([T_0:T_1],[T_0U:T_1U:V])$ to the point in ${\mathbb{P}}^n$ with coordinates $X_m = 0, m\in C_d$, with $Z_l=0, l=1,\dots, n-(d^2+d)$, and with $$\begin{aligned} Y_0 = T_0^kU,\dots, Y_i = T_0^{k-i}T_1^iU, \dots, Y_k = T_1^kU, \\ Y_{k+1} = T_0^{k-1}V,\dots, Y_{k+1+j} = T_0^{k-1-j}T_1^j V,\dots, Y_{2k} = T_1^{k-1}V.\end{aligned}$$ This is an embedding whose image $\Sigma = f({\mathbb F}_1)$ is a rational normal scroll of degree $2k-1$. Now the pullback map $H^0{\left}( ({\mathbb{P}}^{2k},{\mathcal O}_{{\mathbb{P}}^{2k}}(1) {\right}) \rightarrow H^0{\left}( {\mathbb F}_1, {\mathcal O}(1,k-1) {\right})$ is surjective by construction. And the natural map $$\text{Sym}^{d-1}H^0 {\left}( {\mathbb F}_1, {\mathcal O}(1,k-1) {\right}) \rightarrow H^0 {\left}( {\mathbb F}_1, {\mathcal O}{\left}(d-1,(d-1)(k-1){\right}) {\right})$$ is surjective. Therefore the pullback map $$H^0{\left}( {\mathbb{P}}^{2k},{\mathcal O}_{{\mathbb{P}}^{2k}}(d-1) {\right}) \rightarrow H^0{\left}( {\mathbb F}_1, {\mathcal O}{\left}(d-1,(d-1)(k-1) {\right}) {\right})$$ is surjective. For each monomial $M\in C_d$, choose a polynomial $G_M(Y_0,\dots,Y_{2k})$ such that $f^*G_M = M$. Consider the hypersurface $X\subset {\mathbb{P}}^n$ with defining equation $$\begin{aligned} F = \sum_{i=0}^{d-2} {\left}(Y_iY_{k+2+i} - Y_{i+1}Y_{k+1+i}{\right}) Y_0^{d-2-i}Y_{k-3}^i + \\ \sum_{j=0}^{d-4}{\left}(Y_{d-1+j}Y_{k+d+1+j} - Y_{d+j}Y_{k+d+j} {\right})Y_0^{d-4-j}Y_{k-3}^jY_{2k-3}Y_{2k+1-d} + \\ \sum_{M\in C_d}G_M(Y_0,\dots,Y_{2k})X_M.\end{aligned}$$ The corresponding derivative map $d_{X,\Sigma}$ acts on the partial derivatives $\frac{\partial F}{\partial Y_i}$ by $$\begin{array}{lcl} \frac{\partial F}{\partial Y_0} & \mapsto & U^{d-2}VT_0^{(d-1)k-2}; \\ \frac{\partial F}{\partial Y_{i+1}} & \mapsto & -U^{d-2} V T_0^{(d-1)k-1-i(k-2)} T_1^{i(k-2)} + \\ & & U^{d-2} V T_0^{(d-1)k-2-(i+1)(k-2)} T_1^{(i+1)(k-2)}, \\ & & i=0,\dots,d-3, \\ \frac{\partial F}{\partial Y_{d-1}} & \mapsto & -U^{d-2} V T_0^{(d-1)k-1-(d-2)(k-2)} T_1^{(d-2)(k-2)} + \\ & & U^{d-4} V^3 T_0^{(d-1)k-4-2(k-2)} T_1^{2(k-2)+1}, \\ \frac{\partial F}{\partial Y_{d+j}} & \mapsto & -U^{d-4} V^3 T_0^{(d-1)k-3-(j+2)(k-2)} T_1^{(j+2)(k-2)} + \\ & & U^{d-4} V^3 T_0^{(d-1)k-4-(j+3)(k-2)} T_1^{(j+3)(k-2)},\\ & & j=0,\dots, d-5, \\ \frac{\partial F}{\partial Y_{k-1}} & \mapsto & -U^{d-4} V^3 T_0^{(d-1)k-3-(d-2)(k-2)} T_1^{(d-2)(k-2)}, \\ \frac{\partial F}{\partial Y_k} & \mapsto & 0, \\ \frac{\partial F}{\partial Y_{k+1}} & \mapsto & -U^{d-1} T_0^{(d-1)k-1}T_1, \\ \frac{\partial F}{\partial Y_{k+2+i}} & \mapsto & U^{d-1} T_0^{(d-1)k - i(k-2)} T_1^{i(k-2)} - \\ & & U^{d-1} T_0^{(d-1)k-1-(i+1)(k-2)} T_1^{(i+1)(k-2)+1},\\ & & i=0,\dots, d-3, \\ \frac{\partial F}{\partial Y_{k+d}} & \mapsto & U^{d-1} T_0^{(d-1)k-(d-2)(k-2)} T_1^{(d-2)(k-2)} - \\ & & U^{d-3} V^2 T_0^{(d-1)k - 3 -2(k-2)} T_1^{2(k+2) + 1}, \\ \frac{\partial F}{\partial Y_{k+d+1+j}} & \mapsto & U^{d-3} V^2 T_0^{(d-1)k-2-(j+2)(k-2)} T_1^{(j+2)(k-2)} - \\ & & U^{d-3} V^2 T_0^{(d-1)k-3-(j+3)(k-2)} T_1^{(j+3)(k-2)+1},\\ & & j=0,\dots, d-5, \\ \frac{\partial F}{\partial Y_{2k}} & \mapsto & U^{d-3} V^2 T_0^{(d-1)k-2-(d-2)(k-2)} T_1^{(d-2)(k-2)}. \end{array}$$ In the list above, every partial derivative is a sum of at most two monomials, i.e. each partial derivative is a binomial, and the first term listed is the initial term with respect to our monomial order. From this list, it is clear that every monomial in $B_d$ occurs as the initial term of some partial derivative. Also, for each $M\in C_d$, the partial derivative $\frac{\partial F}{\partial X_M}$ simply maps to $M$. Thus every monomial in $C_d$ occurs as the initial term of some partial derivative. Thus every monomial in $A_d$ occurs as the initial term of some partial derivative. So the vector space of initial terms of $\text{image}(d_{X,\Sigma})$ contains $W_0(a,b,c)$. Therefore, by lemma \[lem-comput2\], we conclude that $\text{image}(d_{X,\Sigma})$ is a $c$-generating linear system. So, for any irreducible curve $C_0$ in the linear system of $|{\mathcal O}(1,0)|$, we have that $([X],[\Sigma],[C_0])$ is in ${{\mathcal}W}^o$. Proof of the main theorem ========================= In this section we prove theorem \[thm-thm1\]. As explained at the end of section \[sec-results\], if $d< \frac{n+1}{2}$, then for a general hypersurface $X_d\subset {\mathbb{P}}^n$, hypothesis \[hyp-1\], hypothesis \[hyp-1.5\], and hypothesis \[hyp-1.75\] are satisfied. By proposition \[prop-quadric\], if $d\geq 2$ and $d^2 \leq n+1$, then for $X_d\subset {\mathbb{P}}^n$ general we have that hypothesis \[hyp-2\] is satisfied. Finally, if $d\geq 3$ and if $d^2+d+2 \leq n$, then for $X_d\subset {\mathbb{P}}^n$ general there exists a very twisting, very positive family $\zeta:C_0 \rightarrow {\overline{{\mathcal}M}_{0,1}({X,1})}$. Thus $(\zeta,\zeta)$ is an inducting pair. Now by theorem \[thm-induction\], we conclude that for every $e \geq 1$ there exists an inducting pair $(\zeta_1,\overline{\zeta}_e)$. In particular, there exists a very positive $1$-morphism $\overline{\zeta}_e:C \rightarrow {\overline{{\mathcal}M}_{0,1}({X,e})}$. As seen in the proof of theorem \[thm-induction\], we may assume that $C$ is smooth, i.e. $C$ is equal to ${\mathbb{P}}^1$, and that the image of $\text{pr}\circ \overline{\zeta}_e:{\mathbb{P}}^1 \rightarrow {\overline{{\mathcal}M}_{0,0}({X,e})}$ is contained in the smooth part of the fine moduli locus. So, by item $(2)$ of remark \[rmk-pos\], we conclude that $\overline{\zeta}_e$ is a *very free* morphism. And by [@HRS2 proposition 7.4], we have that ${\overline{{\mathcal}M}_{0,0}({X,e})}$ is an irreducible variety. Therefore by [@K theorem IV.3.7], we conclude that ${\overline{{\mathcal}M}_{0,0}({X,e})}$ is rationally connected.

--- abstract: 'A theory describing the operation of a superconducting nanowire quantum interference device (NQUID) is presented. The device consists of a pair of thin-film superconducting leads connected by a pair of topologically parallel ultra-narrow superconducting wires. It exhibits intrinsic electrical resistance, due to thermally-activated dissipative fluctuations of the superconducting order parameter. Attention is given to the dependence of this resistance on the strength of an externally applied magnetic field aligned perpendicular to the leads, for lead dimensions such that there is essentially complete and uniform penetration of the leads by the magnetic field. This regime, in which at least one of the lead dimensions—length or width—lies between the superconducting coherence and penetration lengths, is referred to as the [*mesoscopic*]{} regime. The magnetic field causes a pronounced oscillation of the device resistance, with a period [*not*]{} dominated by the Aharonov-Bohm effect through the area enclosed by the wires and the film edges but, rather, in terms of the geometry of the leads, in contrast to the well-known Little-Parks resistance of thin-walled superconducting cylinders. A detailed theory, encompassing this phenomenology quantitatively, is developed through extensions, to the setting of parallel superconducting wires, of the Ivanchenko-Zil’berman-Ambegaokar-Halperin theory of intrinsic resistive fluctuations in a current-biased Josephson junctions and the Langer-Ambegaokar-McCumber-Halperin theory of intrinsic resistive fluctuations in superconducting wires. In particular, it is demonstrated that via the resistance of the NQUID, the wires act as a probe of spatial variations in the superconducting order parameter along the perimeter of each lead: in essence, a superconducting phase gradiometer.' author: - 'David Pekker, Alexey Bezryadin, David S. Hopkins, and Paul M. Goldbart' title: | Operation of a superconducting nanowire quantum interference device\ with mesoscopic leads --- Introduction ============ The Little-Parks effect concerns the electrical resistance of a thin cylindrically-shaped superconducting film and, specifically, the dependence of this resistance on the magnetic flux threading the cylinder [@LP; @Tinkham]. It is found that the resistance is a periodic function of the magnetic field, with period inversely proportional to the cross-sectional area of the cylinder. Similarly, in a DC SQUID, the critical value of the supercurrent is periodic in magnetic field, with period inversely proportional to the area enclosed by the SQUID ring [@Tinkham]. In this Paper, we consider a mesoscopic analog of a DC SQUID. The analog consists of a device composed of a thin superconducting film patterned into two mesoscopic leads that are connected by a pair of (topologically) parallel, short, weak, superconducting wires. Thus, we refer to the device as an NQUID (superconducting nanowire quantum interference device). The only restriction that we place on the wires of the device is that they be thin enough for the order parameter to be taken as constant over each cross-section of a wire, varying only along the wire length. In principle, this condition of one-dimensionality is satisfied if the wire is much thinner than the superconducting coherence length $\xi$. In practice, it is approximately satisfied provided the wire diameter $d$ is smaller than $4.4\,\xi$  [@LikharevRMP]. For thicker wires, vortices can exist inside the wires, and such wires may not be assumed to be one dimensional. By the term [*mesoscopic*]{} we are characterizing phenomena that occur on length-scales larger than the superconducting coherence length $\xi$ but smaller than the electromagnetic penetration depth $\lambda_\perp$ associated with magnetic fields applied perpendicular to the superconducting film. We shall call a lead mesoscopic if at least one of its two long dimensions is in the mesoscopic regime; the other dimension may be either mesoscopic or macroscopic. Thus, a weak magnetic field applied perpendicular to a mesoscopic lead will penetrate the lead without appreciable attenuation and without driving the lead from the homogeneous superconducting state to the Abrikosov vortex state. This is similar to the regime of operation of superconducting wire networks; see e.g., Ref. [@AB1]. The nanowires connecting the two leads are taken to be topologically parallel (i.e. parallel in the sense of electrical circuitry): these nanowires and edges of the leads define a closed geometrical contour, which will be referred to as the [*Aharonov-Bohm (AB) contour*]{}. In our approach, the nanowires are considered to be links sufficiently weak that any effects of the nanowires on the superconductivity in the leads can be safely ignored. The theory presented here has been developed to explain experiments conducted on DNA-templated NQUIDs [@us_in_science]. These experiments measure the electrical resistivity of a pair of superconducting nanowires suspended between long superconducting strips (see Fig. \[device\]). In them, a current source is used to pass DC current from a contact on the far end of the left lead to one on the far end of the right lead. The voltage between the contacts is measured (and the resistance is hence determined) as a function of the magnetic field applied perpendicular to the plane of the strips. ![\[device\] Schematic depiction of the superconducting phase gradiometer. A current $J$ is passed through the bridges in the presence of a perpendicular magnetic field of strength $B$. ](\figdir/device.eps){width="8cm"} {width="15cm"} In the light of the foregoing remarks, the multiple-connectedness of the device suggests that one should anticipate oscillations with magnetic field, e.g., in the device resistance. Oscillations are indeed observed. But they are distinct from the resistance oscillations observed by Little and Parks and from the critical current oscillations observed in SQUID rings. What distinguishes the resistance oscillations reported in Ref. [@us_in_science] from those found, e.g., by Little and Parks? First, the most notable aspect of these oscillations is the value of their period. In the Little-Parks type of experiment, the period is given by $\Phi_0/ 2 a b$, where $\Phi_0(\equiv h c / 2 e)$ is the superconducting flux quantum, $2 a$ is the bridge separation, and $b$ is the bridge length, i.e., the superconducting flux quantum divided by the area of the AB contour (see Fig. \[coordinates\]). In a high-magnetic-field regime, such periodic behavior is indeed observed experimentally, with the length of the period somewhat shorter but of the same order of magnitude as in the AB effect [@us_in_science]. However, in a low-magnetic-field regime, the observed period is appreciably smaller (in fact by almost two orders of magnitude for our device geometry). Second, because the resistance is caused by thermal phase fluctuations (i.e. phase slips) in very narrow wires, the oscillations are observable over a wide range of temperatures ($\sim 1\,\text{K}$). Third, the Little-Parks resistance is wholly ascribed to a rigid shift of the $R(T)$ curve with magnetic field, as $T_\text{c}$ oscillates. In contrast, in our system we observe a periodic broadening of the transition (instead of the Little-Parks—type rigid shift) with magnetic field. Our theory explains quantitatively this broadening via the modulation of the barrier heights for phase slips of the superconducting order parameter in the nanowires. In the experiment, the sample is cooled in zero magnetic field, and the field is then slowly increased while the resistance is measured. At a sample-dependent field ($\sim 5\,\text{mT}$) the behavior switches sharply from a low-field to a high-field regime. If the high-field regime is not reached before the magnetic field is swept back, the low-field resistance curve is reproduced. However, once the high-field regime has been reached, the sweeping back of the field reveals phase shifts and hysteresis in the $R(B)$ curve. The experiments [@us_in_science] mainly address rectangular leads that have one mesoscopic and one macroscopic dimension. Therefore, we shall concentrate on such strip geometries. We shall, however, also discuss how to extend our approach to generic (mesoscopic) lead shapes. We note in passing that efficient numerical methods, such as the boundary element method (BEM) [@BEM], are available for solving the corresponding Laplace problems. This paper is arranged as follows. In Section \[sec\_phys\] we construct a basic picture for the period of the magnetoresistance oscillations of the two-wire device, which shows how the mesoscopic size of the leads accounts for the anomalously short magnetoresistance period in the low-field regime. In Section \[sec:leads\] we concentrate on the properties of mesoscopic leads with regard to their response to an applied magnetic field, and in Section \[sec:bridges\] we extend the LAMH model to take into account the inter-wire coupling through the leads. Analytical expressions are derived for the short- and long-wire limits, whilst a numerical procedure is described for the general case. The predictions of the model are compared with data from our experiment in Section \[sec:experiment\], and we give some concluding remarks in Section \[sec:conclusion\]. Certain technical components are relegated to the appendix, as is the analysis of example multiwire devices. Origin of magnetoresistance oscillations {#sec_phys} ======================================== Before presenting a detailed development of the theory, we give an intuitive argument to account for the anomalously-short period of the magnetoresistance in the low-magnetic-field regime, mentioned above. Device geometry --------------- The geometry of the devices studied experimentally is shown in Fig. \[coordinates\]. Five devices were successfully fabricated and measured. The dimensions of these devices are listed in Table \[table:period\], along with the short magnetoresistance oscillation period. The perpendicular penetration depth $\lambda_\perp$ for the films used to make the leads is roughly $70\,\mu\text{m}$, and coherence length $\xi$ is roughly $5 \, \text{nm}$. Parametric control of the state of the wires by the leads --------------------------------------------------------- ![\[loop\] (a) Close-up of the two nanowires and the leads. The top (bottom) thick arrow represents the integration contour for determining the phase accumulation $\theta_{1, L \leftarrow R}$ ($\theta_{2, L \leftarrow R}$) in the first (second) wire. The dotted arrow in the left (right) lead indicates a possible choice of integration contour for determining the phase accumulation $\delta_{2 \leftarrow 1, L}$ ($\delta_{2 \leftarrow 1, R}$). These contours may be deformed without affecting the values of the various phase accumulations, as long as no vortices are crossed. (b) Sketch of the corresponding superconducting phase at different points along the AB contour when one vortex is located inside the contour.](\figdir/loop4.eps){width="8cm"} The essential ingredients in our model are (i) leads, in which the applied magnetic field induces supercurrents and hence gradients in the phase of the order parameter, and (ii) the two wires, whose behavior is controlled parametrically by the leads through the boundary conditions imposed by the leads on the phase of the order parameters in the wires. For now, we assume that the wires have sufficiently small cross-sections that the currents through them do not feed back on the order parameter in the leads. (In Section \[sec:blcouple\] we shall discuss when this assumption may be relaxed without altering the oscillation period.) The dissipation results from thermally activated phase slips, which cause the superconducting order parameter to explore a discrete family of local minima of the free energy. (We assume that the barriers separating these minima are sufficiently high to make them well-defined states.) These minima (and the saddle-point configurations connecting them) may be indexed by the net (i.e. forward minus reverse) number of phase slips that have occurred in each wire ($n_1$ and $n_2$, relative to some reference state). More usefully, they can be indexed by $n_s = \min(n_1, n_2)$ (i.e. the net number of phase slips that have occurred in both wires) and $n_v = n_1-n_2$ (i.e. the number of vortices enclosed by the AB contour, which is formed by the wires and the edges of the leads). We note that two configurations with identical $n_v$ but distinct $n_s$ and $n_s'$ have identical order parameters, but differ in energy by $$\int I V \, dt=\frac{\hbar}{2 e} \int I \dot{\Theta} \, dt = \frac{h}{2 e} \, I \, (n_s' - n_s),$$ due to the work done by the current source supplying the current $I$, in which $V$ is the inter-lead voltage, $\Theta$ is the inter-lead phase difference as measured between the two points half-way between the wires, and the Josephson relation $\dot{\Theta}=2 e V/ \hbar$ has been invoked. In our model, we assume that the leads are completely rigid. Therefore the rate of phase change, and thus the voltage, is identical at all points inside one lead. For sufficiently short wires, $n_v$ has a unique value, as there are no stable states with any other number of vortices. Due to the screening currents in the left lead, induced by the applied magnetic field $B$ (and independent of the wires), there is a field-dependent phase $\delta_{2 \leftarrow 1, L}(B)=\int_1^2 d\vec{r}\cdot \vec{\nabla}\varphi(B)$ (computed below) accumulated in passing from the point at which wire 1 (the top wire) contacts the left (L) lead to the point at which wire 2 (the bottom wire) contacts the left lead (see Fig. \[loop\]). Similarly, the field creates a phase accumulation $\delta_{2 \leftarrow 1, R}(B)$ between the contact points in the right (R) lead. As the leads are taken to be geometrically identical, the phase accumulations in them differ in sign only: $\delta_{2 \leftarrow 1, L}(B) = -\delta_{2 \leftarrow 1, R}(B)$. We introduce $\delta(B) = \delta_{2 \leftarrow 1, L}(B)$. In determining the local free-energy minima of the wires, we solve the Ginzburg-Landau equation for the wires for each vortex number $n_v$, imposing the single-valuedness condition on the order parameter, $$\theta_{1, L \leftarrow R}-\theta_{2, L \leftarrow R}+2 \delta(B)=2 \pi n_v. \label{eq:phaseconstraint}$$ This condition will be referred to as the [*phase constraint*]{}. Here, $\theta_{1, L \leftarrow R}=\int_R^L d\vec{r}\cdot \vec{\nabla}\varphi(B)$ is the phase accumulated along wire 1 in passing from the right to the left lead; $\theta_{2, L \leftarrow R}$ is similarly defined for wire 2. Absent any constraints, the lowest energy configuration of the nanowires is the one with no current through the wires. Here, we adopt the gauge in which $\boldsymbol{A}= B y \boldsymbol{e}_x$ for the electromagnetic vector potential, where the coordinates are as shown in Fig. \[coordinates\]. The Ginzburg-Landau expression for the current density in a superconductor is $$\boldsymbol{J} \propto \left(\boldsymbol{\nabla} \varphi(\boldsymbol{r}) - \frac{2 e}{\hbar} \boldsymbol{A}(\boldsymbol{r}) \right). \label{currentD0}$$ For our choice of gauge, the vector potential is always parallel to the nanowires, and therefore the lowest energy state of the nanowires corresponds to a phase accumulation given by the flux through the AB contour, $\theta_{1, L \leftarrow R}=-\theta_{2, L \leftarrow R}=2 \pi B a b/\Phi_0$. As we shall show shortly, for our device geometry (i.e. when the wires are sufficiently short, i.e., $b \ll l$), this phase accumulation may be safely ignored, compared to the phase accumulation $\delta(B)$ associated with screening currents induced in the leads. As the nanowires are assumed to be weak compared to the leads, to satisfy the phase constraint (\[eq:phaseconstraint\]), the phase accumulations in the nanowires will typically deviate from their optimal value, generating a circulating current around the AB contour. As a consequence of LAMH theory, this circulating current results in a decrease of the barrier heights for phase slips, and hence an increase in resistance. The period of the observed oscillations is derived from the fact that whenever the magnetic field satisfies the relation $$2 \pi m = 2 \pi \frac{2 a b B}{\Phi_0} + 2 \delta(B) \label{eq:period}$$ \[where $m$ is an integer and the factor of $2$ accompanying $\delta(B)$ reflects the presence of two leads\], there is no circulating current in the lowest in energy state, resulting in minimal resistance. Furthermore, the family of free energy-minima of the two-wire system (all of which, in thermal equilibrium, are statistically populated according to their energies) is identical to the $B=0$ case. The mapping between configurations at zero and nonzero $B$ fields is established by a shift of the index $n_v \rightarrow n_v - m$. Therefore, as the sets of physical states of the wires are identical whenever the periodicity condition (\[eq:period\]) is satisfied, at such values of $B$ the resistance returns to its $B=0$ value. Simple estimate of the oscillation period ----------------------------------------- In this subsection, we will give a “back of the envelope” estimate for the phase gain $\delta(B)$ in a lead by considering the current and phase profiles in one such lead. According to the Ginzburg-Landau theory, in a mesoscopic superconductor, subjected to a weak magnetic field, the current density is given by . Now consider an isolated strip-shaped lead used in the device. Far from either of the short edges of this lead, $\boldsymbol{A}= B y \boldsymbol{e}_x$ is a London gauge [@London], i.e., along all surfaces of the superconductor $\boldsymbol{A}$ is parallel to them; $\boldsymbol{A}\rightarrow 0$ in the center of the superconductor; and $\boldsymbol{\nabla}\cdot \boldsymbol{A}=0$. In this special case, the London relation [^1] states that the supercurrent density is proportional to the vector potential in the London gauge. Using this relation, we find that the supercurrent density is $\boldsymbol{J} \propto - (2e / \hbar) \, \boldsymbol{A} = -(2e/\hbar) \, B y \boldsymbol{e}_x$, i.e., there is a supercurrent density of magnitude $\propto (2e/\hbar) \, B l$ flowing to the left at the top (long) edge of the strip and to the right at the bottom (long) edge. At the two short ends of the strip, the two supercurrents must be connected, so there is a supercurrent density of magnitude $\sim (2e / \hbar) B l$ flowing down the left (short) edge of the strip and up the right (short) edge (see Fig. \[2dj\]). Near the short ends of the strips, our choice of gauge no longer satisfies the criteria for being a London gauge, and therefore $\boldsymbol{\nabla} \phi$ may be nonzero. As, in our choice of gauge, $\boldsymbol{A}$ points in the $\boldsymbol{e}_x$ direction, the supercurrent on the ends of the strip along $\boldsymbol{e}_y$ must come from the $\nabla_y \phi$ term. Near the center of the short edge $\nabla_y \phi = - 2 \pi c_1 l/\Phi_0 B$. The phase difference between the points $(-L,-a)$ and $(-L,a)$ is therefore given by $$\delta(B)=\int_{-a}^a \nabla_y \phi \, dy = - \frac{2 \pi c_1}{\Phi_0} B \, 2 a l, \label{eq:approx_delta_B}$$ where we have substituted $2\pi/\Phi_0$ for $2e/\hbar$ and $c_1(a/l)$ is a function of order unity, which accounts for how the current flows around the corners. As we shall show, $c_1$ depends only weakly on $a/l$, and is constant in the limit $a \ll l$. Finally, we obtain the magnetoresistance period by substituting into : $$\begin{aligned} \Delta B = \left[ \left(\frac{\Phi_0}{c_1\, 4 a l}\right)^{-1} +\left(\frac{\Phi_0}{2 a b}\right)^{-1} \right]^{-1} \, . \label{p1}\end{aligned}$$ Thus, we see that for certain geometries the period is largely determined not by the flux threading through the geometric area $2 a b$ but by the response of the leads and the corresponding effective area $4 a l$, provided the nanowires are sufficiently short (i.e. $b \ll l$), justifying our assumption of ignoring the phase gradient induced in the nanowires by the magnetic field. In fact, we can also make a prediction for the periodicity of the magnetoresistance at high magnetic fields, i.e., when vortices have penetrated the leads (see Section \[sec:likharev\]). To do this, we should replace $l$ in  by the characteristic inter-vortex spacing $r$. Note that if $r$ is comparable to $b$, we can no longer ignore the flux through the AB contour. Furthermore, if $r \ll b$ then the flux through the AB contour determines periodicity and one recovers the usual Aharonov-Bohm type of phenomenology. Mesoscale superconducting leads {#sec:leads} =============================== In this section and the following one we shall develop a detailed model of the leads and nanowires that constitute the mesoscopic device. Vortex-free and vorticial regimes {#sec:likharev} --------------------------------- Two distinct regimes of magnetic field are expected, depending on whether or not there are trapped (i.e. locally stable) vortices inside the leads. As described by Likharev [@Likharev], a vortex inside a superconducting strip-shaped lead is subject to two forces. First, due to the the currents induced by the magnetic field there is a Magnus force pushing it towards the middle of the strip. Second, there is a force due to image vortices (which are required to enforce the boundary condition that no current flows out of the strip and into the vacuum) pulling the vortex towards the edge. When the two forces balance at the edge of the strip, there is no energy barrier preventing vortex penetration and vortices enter. Likharev has estimated of the corresponding critical magnetic field to be $$H_{\rm s} \approx \frac{\Phi_0}{\pi d} \frac{1}{\xi a(1)},$$ where $d(\equiv2 l)$ is the width of the strip and $a(1) \sim 1$ for strips that are much narrower than the penetration depth (i.e. for $d \ll \lambda$). Likharev has also shown that, once inside a strip, vortices remain stable inside it down to a much lower magnetic field $H_{c1}$, given by $$H_\text{c1}= \frac{\Phi_0}{\pi d} \frac{2}{d} \ln \left( \frac{d}{4\xi} \right).$$ At fields above $H_\text{c1}$ the potential energy of a vortex inside the strip is lower than for one outside (i.e. for a virtual vortex [@ref:topology_note]). Therefore, for magnetic fields in the range $H_\text{c1}<H<H_\text{s}$ vortices would remain trapped inside the strip, but only if at some previous time the field were larger than $H_\text{s}$. This indicates that hysteresis with respect to magnetic field variations should be observed, once $H$ exceeds $H_\text{s}$ and vortices become trapped in the leads. In real samples, in addition to the effects analyzed by Likharev, there are also likely to be locations (e.g. structural defects) that can pin vortices, even for fields smaller than $H_{c1}$, so the reproducibility of the resistance [*vs.*]{} field curve is not generally expected once $H_\text{s}$ has been surpassed. As magnetic field at which vortices first enter the leads is sensitive to the properties of their edges, we expect only rough agreement with Likharev’s theory. For sample 219-4, using Likharev’s formula, we estimate $H_\text{s}=11\,\text{mT}$ (with $\xi=5\,\text{nm}$). The change in regime from fast to slow oscillations is found to occur at $3.1\,\text{mT}$ for that sample [@us_in_science]. It is possible to determine the critical magnetic fields $H_\text{s}$ and $H_\text{c1}$ by the direct imaging of vortices. Although we do not know of such a direct measurement of $H_\text{s}$, $H_\text{c1}$ was determined by field cooling niobium strips, and found to agree in magnitude to Likharev’s estimate [@Martinis2004]. Phase variation along the edge of the lead {#sec_exact} ------------------------------------------ In the previous section it was shown that the periodicity of the magnetoresistance is due to the phase accumulations associated with the currents along the edges of the leads between the nanowires. Thus, we should make a precise calculation of the dependence of these currents on the magnetic field, and this we now do. ### Ginzburg-Landau theory To compute $\delta(B)$, we start with the Ginzburg-Landau equation for a thin film as our description of the mesoscopic superconducting leads: $$\alpha \psi+ \beta |\psi|^2 \psi + \frac{1}{2 m^*} \left(\frac{\hbar}{i} \boldsymbol{\nabla} - \frac{e^*}{c} \boldsymbol{A}\right)^2 \psi = 0. \label{GLfree}$$ Here, $\psi$ is the Ginzburg-Landau order parameter, $e^*$ ($=2e$) is the charge of a Cooper pair and $m^*$ is its mass, and $\alpha$ and $\beta$ may be expressed in terms of the coherence length $\xi$ and critical field $H_\text{c}$ via $\alpha=-\hbar^2/2 m^* \xi^2$ and $\beta=4 \pi \alpha^2/H_\text{c}^2$. The assumptions that the magnetic field is sufficiently weak and that the lead is a narrow strip (compared with the magnetic penetration depth) allow us to take the [*amplitude*]{} of the order parameter in the leads to have the value appropriate to an infinite thin film in the absence of the field. By expressing the order parameter in terms of the (constant) amplitude $\psi_0$ and the (position-dependent) phase $\phi(\boldsymbol{r})$, i.e., $$\psi(\boldsymbol{r})=\psi_0 \, e^{i \phi(\boldsymbol{r})},$$ the Ginzburg-Landau formula for the current density, $$\boldsymbol{J}= \frac{e^* \hbar}{2 m^* i} \big(\psi^{\ast} \boldsymbol{\nabla} \psi -\psi\boldsymbol{\nabla} \psi^{\ast}\big) -\frac{{e^*}^2}{m^* c}\psi^{\ast}\psi \boldsymbol{A}(\boldsymbol{r}), \label{current-old}$$ becomes $$\boldsymbol{J}=\frac{e^*}{m^*}\psi_0^2 \big( \hbar \boldsymbol{\nabla} \phi(\boldsymbol{r}) - \frac{e^*}{c} \boldsymbol{A}(\boldsymbol{r}) \big), \label{currentD}$$ and \[after dividing by $e^{i \phi(\boldsymbol{r})}$\] the real and imaginary parts of the Ginzburg-Landau equation become $$\begin{aligned} 0&=\left[\alpha \, \psi_0 + \beta \, \psi_0^3 + \frac{1}{2 m^*} \psi_0 \left| \hbar \boldsymbol{\nabla} \phi(\boldsymbol{r}) - \frac{e^*}{c} \boldsymbol{A}(\boldsymbol{r})\right|^2\right], \label{eq13a}\\ 0&=\frac{\hbar^2}{2 m^* i} \, \psi_0 \left(\nabla^2 \phi(\boldsymbol{r}) - \frac{e^*}{\hbar c} \boldsymbol{\nabla} \cdot \boldsymbol{A}(\boldsymbol{r}) \right). \label{GLimg}\end{aligned}$$ As long as any spatial inhomogeneity in the gauge-covariant derivative of the phase is weak on the length-scale of the coherence length $\big[$ i.e. $\xi \big|\boldsymbol{\nabla} \phi(\boldsymbol{r}) - \frac{e^*}{\hbar c} \boldsymbol{A}(\boldsymbol{r})\big| \ll 1$ $\big]$, the third term in is much smaller than the first two and may be ignored, fixing the amplitude of the order parameter at its field-free infinite thin film value, [*viz.*]{}, $\bar{\psi_0}\equiv\sqrt{-\alpha/\beta}$. To compute $\phi(\boldsymbol{r})$ we need to solve the imaginary part of the Ginzburg-Landau equation. ### Formulation as a Laplace problem We continue to work in the approximation that the amplitude of the order parameter is fixed at $\bar{\psi_0}$. Starting from , we see that for our choice of gauge, $\boldsymbol{A}= B y \boldsymbol{e}_x$, the phase of the order parameter satisfies the Laplace equation, $\nabla^2 \phi=0$. We also enforce the boundary condition that no current flows out of the superconductor on boundary surface $\Sigma$, whose normal is ${\bf n}$: \[eq:bc\] $$\begin{aligned} &\boldsymbol{n}\cdot \boldsymbol{j} \big|_{\Sigma}=0,\\ &\boldsymbol{j}\propto\Big(\boldsymbol{\nabla} \phi - \frac{2 \pi}{\Phi_0} \boldsymbol{A} \Big).\end{aligned}$$ ### Solving the Laplace problem for the strip geometry To solidify the intuition gained via the physical arguments given in Section \[sec\_phys\], we now determine the phase profile for an isolated superconducting strip in a magnetic field. This will allow us to determine the constant $c_1$ in , and hence obtain a precise formula for the magnetoresistance period. To this end, we solve Laplace’s equation for $\phi$ subject to the boundary conditions (\[eq:bc\]). We specialize to the case of a rectangular strip [^2]. In terms of the coordinates defined in Fig. \[coordinates\], we expand $\phi(x,y)$ as the superposition $$\phi(x,y)=\Theta_{\text{L/R}}+\sum_k \left(A_k \, e^{-k x} + B_k \, e^{k x}\right) \sin(k y), \label{eq_phi}$$ which automatically satisfies Laplace’s equation, although the boundary conditions remain to be satisfied. $\Theta_{\text{L(R)}}$ is the phase at the the point in the left (right) lead located half-way between the wires. In other words $\Theta_L=\phi(-L-b,0)$ and $\Theta_R=\phi(-L,0)$ in the coordinate system indicated in Fig. \[coordinates\]. $\Theta_{\text{L/R}}$ are not determined by the Laplace equation and boundary conditions, but will be determined later by the state of the nanowires. We continue working in the gauge $\boldsymbol{A}=B y \, \boldsymbol{e}_x$. The boundary conditions across the edges at $y=\pm l$ (i.e. the long edges) are $\partial_y \phi(x,y=\pm l)=0$. These conditions are satisfied by enforcing $k_n=\pi (n+\frac{1}{2})/l$, where $n=0,1,2,\ldots$. The boundary conditions across the edges at $x=\pm L$ (i.e. the short edges) are $\partial_x \phi(x=\pm L,y)=h y$ (where $h \equiv 2 \pi B/\Phi_0$). This leads to the coefficients in taking the values $$\begin{aligned} B_k&=-A_k=\frac{h}{k_n^3 l} \frac{(-1)^n}{\cosh(k_n L)} & (n=0,1,\ldots),\end{aligned}$$ and hence to the solution $$\begin{aligned} \phi(x,y)=\sum_{n=0}^\infty \frac{(-1)^n\,2 h}{k_n^3 l \cosh(k_nL)}\, \sin(k_n y)\,\sinh(k_n x).\end{aligned}$$ Figure \[3dplots\] shows the phase profiles in the leads, in the region close to the trench that separates the leads. ![Phase profile in the leads in the vicinity of the trench, generated by numerically summing the series for $\phi$ for a finite-length strip. Arrows indicate phases connected by nanowires. []{data-label="3dplots"}](\figdir/3dcurrents2.eps){width="8cm"} Period of magnetoresistance for leads having a rectangular strip geometry {#subsec:period} ------------------------------------------------------------------------- Using the result for the phase that we have just established, we see that the phase profile on the short edge of the strip at $x=-L$ is given by $$\label{eq:nonlinear} \phi(-L,y)= - \frac{2 h l^2}{\pi^2}\sum_{n=0}^\infty \frac{(-1)^n}{(n+\frac{1}{2})^3} \sin \frac{\pi\left(n+\frac{1}{2}\right) y}{l},$$ where we have taken the limit $L\rightarrow \infty$. We would like to evaluate this sum at the points $(x,y)=(-L,\pm a)$. This can be done numerically. For nanowires that are close to each other (i.e. for $a \ll l$), an approximate value can be found analytically by expanding in a power series in $a$ around $y=0$: $$\begin{split} \phi(-L,a)=\phi(-L,0)+a \, \left. \frac{\partial}{\partial y} \phi(-L,y) \right|_{y=0} \quad\quad \\ +\frac{a^2}{2} \left. \frac{\partial^2}{\partial y^2} \phi(-L,y) \right|_{y=0} + O(a^3). \end{split}$$ The first and third terms are evidently zero, as $\phi$ is an odd function of $y$. The second term can be evaluated by changing the order of summation and differentiation. (Higher-order terms are harder to evaluate, as the changing of the order of summation and differentiation does not work for them.)  Thus, to leading order in $a$ we have $$\begin{aligned} \phi(-L,a)\approx-\frac{8 G}{\pi^2} h l a, \label{lin_fit}\end{aligned}$$ where $G\equiv\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^2}\approx 0.916$ is the Catalan number (see Ref. [@catalan_n]). This linear approximation is plotted, together with the actual phase profile obtained by the numerical evaluation of Eq. (\[eq:nonlinear\]), in Fig. \[fit\_plot\]. Hence, the value of $c_1$ in becomes $c_1=8 G /\pi^2 \approx 0.74$, and becomes $$\begin{aligned} B=\frac{\Phi_0}{2 \pi}h= \frac{\pi^2}{8 G} \frac{\Phi_0}{4 a l}.\end{aligned}$$ To obtain this result we used the relation $\delta(B)/2=\phi(-L,a)$. ![Phase profile on the $x=-L$ (i.e. short) edge of the strip. Numerical summation \[Eq. (\[eq:nonlinear\]) with 100 terms\] for $L=2$ and $L=\infty$ ($l=1)$ as well as the linear form from Eq. (\[lin\_fit\]). Note that the linear fit is good near the origin (e.g. for $a \alt 0.25 l$), and the curves for $L=2$ and $L=\infty$ coincide.[]{data-label="fit_plot"}](\figdir/fit_end.eps) Bridge-lead coupling {#sec:blcouple} -------------------- In order to simplify our analysis we have assumed that the nanowires do not exert any influence on the order parameter in the leads. We examine the justification for this assumption in the setting of the experiment that we are attempting to describe [@us_in_science]. The assumption will be valid if the bending of the phase of the order parameter, in order to accommodate any circulating current around the AB contour, occurs largely in the nanowires. As the phase of the order parameter in the leads satisfies the Laplace equation, which is linear, we can superpose the circulating-current solution with the previously-obtained magnetic-field-induced solution. The boundary conditions on the right lead for the circulating-current solution are $\boldsymbol{n} \cdot \boldsymbol{\nabla} \phi=0$ everywhere, except at the two points where the nanowires are attached to the lead \[i.e. at $(x,y)=(L,\pm a)$\]. Treating the nanowires as point current sources, the boundary condition on the short edge of the right lead is $\partial_x \phi = I (\Phi_0/H_\text{c}^2 s \xi) (\delta(y+a)-\delta(y-a)$), where $I$ is the current circulating in the loop, and $H_\text{c}$, $s$, $\xi$ are the film critical field, thickness, and coherence length. By using the same expansion as before, , we obtain the coefficients of the Fourier-series in the long strip limit: $$\begin{aligned} A_k=I (\Phi_0/H_\text{c}^2 s \xi) \frac{2 \sin(ka)}{k l \exp(k l)}.\end{aligned}$$ Having the coefficients of the Fourier series, we can find the phase difference in the right lead between the two points at which the nanowires connect to the right lead, induced in this lead by the current circulating in the loop: $$\begin{aligned} \delta_{cc}=2 I (\Phi_0/H_\text{c}^2 s \xi) \sum_{n=0}^{k=1/w} \frac{\sin^2(k a)}{k l} \sim \ln(2l/\pi w).\end{aligned}$$ Here, we have introduced a large wave-vector $k$ cut-off at the inverse of the width $w$ of the wire. On the other hand, the current flowing through the wire is $$\frac{\xi H_\text{c}^2}{\Phi_0} w s \frac{\Delta \theta}{b},$$ where $\xi$, $H_\text{c}$, and $s$ are the wire coherence length, critical field, and height (recall that $b$ is the wire length). To support a circulating current that corresponds to a phase accumulation of $\Delta \theta$ along one of the wires, the phase difference between the two nanowires in the lead must be on the order of $$\delta_{cc}=\Delta \theta \frac{w}{b} \frac{\left( H_\text{c}^2 s \xi \right)_\text{wire}} {\left( H_\text{c}^2 s \xi \right)_\text{film}} \ln\left(\frac{2 l}{\pi w}\right).$$ For our experiments [@us_in_science], we estimate that the ratio of $\delta_{cc}$ to $\Delta \theta$ is always less than $20\%$, validating the assumption of weak coupling. Strong nanowires ---------------- We remark that the assumption of weak nanowires [*is not obligatory*]{} for the computation the magnetoresistance [*period*]{}. Dropping this assumption would leave the period of the magnetoresistance oscillations unchanged. To see this, consider $\phi_{11}$, i.e., the phase profile in the leads that corresponds to the lowest energy solution of the Ginzburg-Landau equation at field corresponding to the first resistance minimum \[i.e. at B being the first non-zero solution of \]. For this case, and for short wires, the phase gain along the wires is negligible, whereas the phase gain in the leads is $2 \pi$, even for wires with large critical current. Excited states, with vortices threading the AB contour, can be constructed by the linear superposition of $\phi_{11}$ with $\phi_{0n_v}$, where $\phi_{0n_v}$ is the phase profile with $n_v$ vortices at no applied magnetic field. This construction requires that the nanowires are narrow, but works independently of whether nanowires are strong or weak, in the limit that $H \ll H_\text{c}$. The energy of the lowest energy state always reaches its minimum when the applied magnetic field is such that there is no phase gain (i.e. no current) in the nanowires. By the above construction, it is clear that the resistance of the device at this field is the same as at zero field, and therefore the minimum possible. Therefore, our calculation of the period is valid, independent of whether the nanowires are weak or strong. However, the assumption of weak nanowires is necessary for the computation of magnetoresistance amplitude, which we present in the following section. Parallel superconducting nanowires and intrinsic resistance {#sec:bridges} =========================================================== In this section we consider the intrinsic resistance of the device. We assume that this resistance is due to thermally activated phase slips (TAPS) of the order parameter, and that these occur within the nanowires. Equivalently, these processes may be thought of as thermally activated vortex flow across the nanowires. Specifically, we shall derive analytical results for the asymptotic cases of nanowires that are either short or long, compared to coherence length, i.e. Josephson junctions [@IZ; @AH] or Langer-Ambegaokar-McCumber-Halperin (LAMH) wires [@LA; @MH]; see also Ref. [@little]. We have not been able to find a closed-form expression for the intrinsic resistance in the intermediate-length regime, so we shall consider that case numerically. There are two (limiting) kinds of experiments that may be performed: fixed total current and fixed voltage. In the first kind, a specified current is driven through the device and the time-averaged voltage is measured. Here, this voltage is proportional to the net number of phase slips (in the forward direction) per unit time, which depends on the height of the free-energy barriers for phase slips. Why do we expect minima in the resistance at magnetic fields corresponding to $2\delta=2 m \pi$ and maxima at $2\delta=(2m+1)\pi$ for $m$ integral, at least at vanishingly small total current through both wires? For $2\delta=2 m \pi$ the nanowires are unfrustrated, in the sense that there is no current through either wire in the lowest local minimum of the free energy. On the other hand, for $2\delta=(2m+1)\pi$ the nanowires are maximally frustrated: there is a nonzero circulating current around the AB contour. Quite generally, the heights of the free-energy barriers protecting locally stable states decrease with increasing current through a wire, and thus the frustrated situation is more susceptible to dissipative fluctuations, and hence shows higher resistance. Note, however, that due to the inter-bridge coupling caused by the phase constraint, the resistance of the full device is more subtle than the mere addition of the resistances of two independent, parallel nanowires, both carrying the requisite circulating current. In the second kind of experiment, a fixed voltage is applied across the device and the total current is measured. In this situation, the inter-lead voltage is fixed, and therefore the phase drop along each wire is a fixed function of time. Hence, there is no inter-bridge coupling in the fixed voltage regime. Therefore, the resistance of the device would not exhibit magnetic field dependence. If the voltage is fixed far away from the wires, but not in the immediate vicinity of the wires, so that the phase drop along each wire is not rigidly fixed, then some of the magnetic field dependence of the resistance would be restored. In our experiments on two-wire devices, we believe that the situation lies closer to the fixed current limit than to the fixed voltage limit, and therefore we shall restrict our attention to the former limit. In the fixed-current regime, the relevant independent thermodynamic variable for the device is the total current through the pair of wires, i.e., $I\equiv I_1+I_2$. Therefore, the appropriate free energy to use, in obtaining the barrier heights for phase slips, is the Gibbs free energy $G(I)$, as discussed by McCumber [@M68], rather than the Helmholtz free energy $F(\Theta)$ [^3]. In the Helmholtz free energy the independent variable can be taken to be $\Theta \equiv \Theta_\text{L}-\Theta_\text{R}$, i.e., the phase difference across the center of the “trench,” defined modulo $2 \pi$. $G(I)$ is obtained from $F(\Theta)$ via the appropriate Legendre transformation: $$G(I)=F(\Theta)-\frac{\hbar}{2 e} I \Theta,$$ where the second term represents the work done on the system by the external current source. $F(\Theta)$ is the sum of the Helmholtz free energies for the individual nanowires: $$F(\Theta)=F_1(\theta_1)+F_2(\theta_2),$$ where $F_{1(2)}(\theta_{1(2)})$ is the Ginzburg-Landau free energy for first (second) wire and a simplified notation has been used $\theta_1 \equiv \theta_{1,L\leftarrow R}$ and $\theta_2 \equiv \theta_{2,L \leftarrow R}$. $\theta_1$ and $\theta_2$ are related to each other and to $\Theta$ through the phase constraint . Short nanowires: Josephson junction limit ----------------------------------------- If the nanowires are sufficiently short, they may be treated as Josephson junctions. Unlike the case of long nanowires, described in the following subsection, in this Josephson regime there is no metastability, i.e., the free energy of each junction is a single-valued function of the phase difference, modulo $2 \pi$, across it. The phase constraint then implies that there is a rigid difference between the phases across the two junctions. As a consequence, $n_v$ can be set to zero. The Gibbs free energy in such a configuration is then $$\begin{split} G(I)=-\frac{\hbar}{2 e} \Big(I_{\text{c1}} \cos(\theta_1) + I_{\text{c2}} \cos(\theta_2) + I \Theta \Big), \end{split}$$ where $I_{\text{c1}}$ and $I_{\text{c2}}$ are the critical currents for the junctions. In thermodynamic equilibrium, the Gibbs free energy must be minimized, so the dependent variable $\Theta$ must be chosen such that $\partial G(I)/\partial \Theta=0$. Using $\theta_1=\Theta+\delta$ and $\theta_2=\Theta-\delta$, $G(I)$ may be rewritten in the form $$\begin{split} \tilde{G}(I)=-\frac{\hbar}{2 e}\Big( \sqrt{(I_{\text{c1}}+I_{\text{c2}})^2 \cos^2 \delta+(I_{\text{c1}}-I_{\text{c2}})^2 \sin^2 \delta} \cdot \cos(\vartheta)+I\,\vartheta_1\Big), \end{split} \label{GJJ}$$ where we have shifted the free energy by an additive constant $\left(\frac{\hbar}{2 e}\right) I \tan^{-1}\left[\left(\frac{I_{\text{c1}}-I_{\text{c2}}}{I_{\text{c2}} + I_{\text{c1}}}\right)\tan\delta\right]$, and $\vartheta \equiv\Theta+\tan^{-1} \left[ \left(\frac{I_{\text{c1}} - I_{\text{c2}}}{I_{\text{c2}} + I_{\text{c1}}} \right) \tan\delta\right]$. In this model, the option for having $I_{\text{c1}}\neq I_{\text{c2}}$ is kept open. Equation (\[GJJ\]) shows that, up to an additive constant, the free energy of the two-junction device is identical to that of an effective single-junction device with an effective $I_\text{c}$, which is given by $$\label{eq:r1} I_\text{c}=\sqrt{(I_{\text{c1}}+I_{\text{c2}})^2 \cos^2 \delta+(I_{\text{c1}}-I_{\text{c2}})^2 \sin^2 \delta}.$$ Thus, we may determine the resistance of the two-junction device by applying the well-known results for a single junction, established by Ivanchenko and Zil’berman [@IZ] and by Ambegaokar and Halperin [@AH]: \[eq:r2\] $$\begin{aligned} R&=R_\text{n} \frac{2(1-x^2)^{1/2}}{x} \exp\left(-\gamma(\sqrt{1-x^2} + x \sin^{-1} x)\right) \sinh(\pi \gamma x/2) \label{RJ1}, \\ x&\equiv I/I_\text{c}, \hskip2cm \gamma\equiv\hbar I_\text{c}/e k_\text{B} T, \label{RJ3}\end{aligned}$$ where $R_\text{n}$ is the normal-state resistance of the two-junction device. This formula for $R$ holds when the free-energy barrier is much larger than $k_\text{B} T$, so that the barriers for phase slips are high. References [@IZ; @AH] provide details on how to calculate the resistance in the general case of an over damped junction, which includes that of shallow barriers. Figure \[fig:mr219\] shows the fits to the resistance, computed using , as a function of temperature, magnetic field, and total current for sample 219-4. Observe that both the field- and the temperature-dependence are in good agreement with experimental data. In Section \[sec:expamp\] we make more precise contact between theory and experiment, and explain how the data have been fitted. We also note that, as it should, our Josephson junction model exactly coincides with our extension of the LAMH model in the limit of very short wires and for temperatures for which the barrier-crossing approximation is valid. (a) {width="7.5cm"} (b) {width="7.5cm"} Longer nanowires: LAMH regime ----------------------------- In this section we describe an extension of the LAMH model of resistive fluctuations in a single narrow wire [@LA; @MH], which we shall use to make a quantitative estimate of the voltage across the two-wire device at a fixed total current. In this regime the nanowires are sufficiently long that they behave as LAMH wires. We shall only dwell on two-wire systems, but we note in passing that the model can straightforwardly be extended to more complicated sets of lead interconnections, including periodic, grating-like arrays (see Appendix \[app:multi-wire\]). As the sample is not simply connected, i.e., there is a hole inside the AB contour, it is possible that there are multiple metastable states that can support the total current. These states differ by the number of times the phase winds along paths around the AB contour. The winding number $n_v$ changes whenever a vortex (or an anti-vortex) passes across one of the wires. In the present theory, we include two kinds of processes that lead to the generation of a voltage difference between the the leads; see Fig. \[tunnel\_pic\]. In the first kind of process (Fig. \[tunnel\_pic\]a), two phase slips occur simultaneously: a vortex passes across the top wire and, concurrently, an anti-vortex passes across the bottom wire (in the opposite direction), so that the winding number remains unchanged. In the second kind of process (Fig. \[tunnel\_pic\]b), the phase slips occur sequentially: a vortex (or anti-vortex) enters the AB contour by passing across the top (or bottom) wire, stays inside the contour for some time-interval, and then leaves the AB contour through the bottom (or top) wire [@ref:topology_note]. {width="15cm"} Our goal is to extend LAMH theory to take into account the influence of the wires on each other. In Appendix \[LAMH1\], we review some necessary ingredients associated with the LAMH theory of a single wire. As the wires used in the experiments are relatively short (i.e. 10 to 20 zero-temperature coherence lengths in length), we also take care to correctly treat the wires as being of finite length. Recall that we are considering experiments performed at a fixed total current, and accordingly, in all configurations of the order parameter this current must be shared between the top and bottom wires. We shall refer to this sharing, $$I=I_1+I_2,$$ as the [*total current constraint*]{}. Let us begin by considering a phase-slip event in a device with an isolated wire. While the order parameter in that wire pinches down, the end-to-end phase accumulation must adjust to maintain the prescribed value of the current through the wire. Now consider the two-wire device, and consider a phase slip event in one of the wires. As in the single-wire case, the phase accumulation will adjust, but in so doing it will alter the current flowing through other wire. Thus, in the saddle-point configuration of the two-wire system the current splitting will differ from that in the locally stable initial (and final) state. Taking into account the two kinds of phase-slip processes, and imposing the appropriate constraints (i.e. the total current constraint and the phase constraint), we construct the possible metastable and saddle-point configurations of the order parameter in the two-wire system. Finally, we compute the relevant rates of thermally activated transitions between these metastable states, construct a Markov chain [@markov], and determine the steady-state populations of these states. Thus, we are able to evaluate the time-average of the voltage generated between the leads at fixed current due to these various dissipative fluctuations. We mention that we have not allowed for wires of distinct length or constitution (so that the Ginzburg-Landau parameters describing them are taken to be identical). This is done solely to simplify the analysis; extensions to more general cases would be straightforward but tedious. ### Parallel pair of nanowires The total Gibbs free energy for the two-wire system is given by $$G(I)=F_1(\theta_{1})+F_2(\theta_2) -4 \Ecore \Theta \cdot (J_1+J_2). \label{G_of_I}$$ Here, we have followed MH by rewriting the current-phase term in terms of dimensionless currents in wires $i=1,2$, i.e., $J_i$ defined via $I_i=8 \pi c J_i \Ecore/\Phi_0$. Moreover, $\Ecore \equiv \frac{H_\text{c}^2 \xi \sigma}{8 \pi}$ is the condensate energy density per unit length of wire, and $F_i(\theta_i)$ is the Helmholtz free energy for a single wire along which there is a total phase accumulation of $\theta_i$. The precise form of $F_i(\theta_i)$ depends on whether the wire is in a metastable or saddle-point state. We are concerned with making stationary the total Gibbs free energy at specified total current $I$, subject to the phase constraint, . This can be accomplished by making stationary the Helmholtz free energy on each wire, subject to both the total current constraint and the phase constraint, but allowing $\theta_1$ and $\theta_2$ to vary so as to satisfy these constraints—in effect, adopting the total current $I$ as the independent variable. The stationary points of the Helmholtz free energy for a single wire are reviewed in Appendix \[LAMH1\] as implicit functions of $\theta_i$, i.e., the end-to-end phase accumulation along the wire. The explicit variable used there is $J_i$, which is related to $\theta_i$ via . ### Analytical treatment in the limit of long nanowires In the long-wire limit, we can compute the resistance analytically by making use of the single-wire free energy and end-to-end phase accumulation derived by Langer and Ambegaokar [@LA] (and extended by McCumber [@M68] for the case of the constant-current ensemble). Throughout the present subsection we shall be making an expansion in powers of $1/b$, where $b$ is the length of the wire measured in units of the coherence length, keeping terms only to first order in $1/b$. Thus, one arrives at formulæ for the end-to-end phase accumulations and Helmholtz free energies for single-wire metastable ($\text{m}$) and saddle-point ($\text{sp}$) states [@M68]: $$\begin{aligned} \theta_{\text{m}}(\kappa)&=\kappa b \label{phi0}, \\ \theta_{\text{sp}}(\kappa)&=\kappa b + 2 \tan^{-1} \left(\frac{1-3\kappa^2}{2 \kappa^2}\right)^{1/2}, \\ F_{\text{m}}(\kappa)&=-\Ecore \left( b (1-\kappa^2)^2 \right) \label{F0}, \\ F_{\text{sp}}(\kappa)&=-\Ecore \left(b (1-\kappa^2)^2-\frac{8\sqrt{2}}{3} \sqrt{1-3\kappa^2}\right), \label{Fsp}\end{aligned}$$ where $\kappa$ is defined via $J_i=\kappa_i(1-\kappa_i^2)$. In the small-current limit, one can make the further simplification that $J_i\approx\kappa_i$; henceforth we shall keep terms only up to first order in $\kappa$. To this order, the phase difference along a wire in a saddle-point state becomes $$\theta_{\text{sp}}=\kappa b + \pi-2\sqrt{2} \kappa. \label{phi_sp}$$ Next, we make use of these single-wire LAMH results to find the metastable and saddle-point states of the two-wire system, and use them to compute the corresponding barrier heights and, hence, transition rates. At low temperatures, it is reasonable to expect that only the lowest few metastable states will be appreciably occupied. These metastable states, as well as the saddle-point states between them, correspond to pairs, $\kappa_1$ and $\kappa_2$, one for each wire, that satisfy the total current constraint as well as the phase constraint: $$\begin{aligned} \kappa_1+\kappa_2=J \label{cur5}, \\ \theta_1(\kappa_1)-\theta_2(\kappa_2)=2\pi n_v+2\delta, \label{cons5}\end{aligned}$$ where we need to substitute the appropriate $\theta_{\text{m}/\text{sp}}(\kappa_i)$ from for $\theta_i(\kappa_i)$. In the absence of a magnetic field (i.e. $\delta=0$), the lowest energy state is the one with no circulating current, and the current split evenly between the two wires. This corresponds to the solution of with $n=0$, together with the substitution (\[phi0\]) for $\theta_i(\kappa_i)$ for both wires (i.e. $\theta_1=\kappa_1 b$ and $\theta_2=\kappa_2 b$). Thus we arrive at the solution: $$\begin{aligned} \kappa_1&=J/2, & \theta_1&=b J/2, \\ \kappa_2&=J/2, & \theta_2&=b J/2.\end{aligned}$$ If we ignore the lowest (excited) metastable states then only a parallel phase-slip process is allowed. The saddle point for a parallel phase slip corresponds to a solution of with $n=0$ and the substitution (\[phi\_sp\]) for $\theta_i(\kappa_i)$ for both wires: $$\begin{aligned} \kappa_1&=J/2, & \theta_1&=b J/2 + \pi-2\sqrt{2} J/2,\\ \kappa_2&=J/2, & \theta_2&=b J/2+ \pi-2\sqrt{2} J/2.\end{aligned}$$ The change in the phase difference across the center of the trench, $\Delta \Theta \equiv [\Theta_{\text{sp}}-\Theta_{\text{m}}]$, is $\pi-2\sqrt{2}\kappa$ for a forward phase slip, and $-\pi-2\sqrt{2}\kappa$ for a reverse phase slip. The Gibbs free-energy barrier for the two kinds of phase slips, computed by subtracting the Gibbs free energy for the ground state from that of the saddle-point state, is $$\Delta G=\Ecore \left(\frac{16 \sqrt{2}}{3}\pm 4 J \pi\right).$$ The former free-energy is obtained by substituting into for both wires; the latter one is obtained by substituting into for both wires. We note that the Gibbs free-energy barrier heights for parallel phase slips (in both the forward and reverse directions) are just double those of the LAMH result for a single wire. From the barrier heights, we can work out the generated voltage by appealing to the Josephson relation, $V = (\hbar/2 e) \dot{\Theta}$, and to the fact that each phase slip corresponds to the addition (or subtraction) of $2\pi$ to the phase. Hence, we arrive at the current-voltage relation associated with parallel phase slips at $\delta=0$: $$\begin{aligned} V_{\delta=0 \, \text{, par}}=\frac{\hbar}{e} \, \Omega \, e^{-\beta \Ecore \frac{16 \sqrt{2}}{3}} \sinh \left(I/I_0\right), \label{v:d0:par}\end{aligned}$$ where the prefactor $\Omega$ may be computed using time-dependent Ginzburg-Landau theory or extracted from experiment, and $I_0=4 e/\beta h$. If we take into account the two lowest excited states, which we ignored earlier, then voltage can also be generated via sequential phase slips (in addition to the parallel ones, treated above). To tackle this case, we construct a diagram in which the vertices represent the metastable and saddle-point solutions of , and the edges represent the corresponding free energy barriers; see Fig. \[0\_diag\]. Pairs of metastable-state vertices are connected via two saddle-point-state vertices, corresponding to a phase slip on either the top or the bottom wire. To go from one metastable state to another, the system must follow the edge out of the starting metastable state leading to the desired saddle-point state. We assume that, once the saddle-point state is reached, the top of the barrier has been passed and the order parameter relaxes to the target metastable state. (To make the graph more legible, we have omitted drawing the edge that corresponds to this relaxation process.) To find the Gibbs free-energy difference between a metastable state and a saddle-point state, we need to know the phase difference across the center of the trench. To resolve the ambiguity of $2 \pi$ in the definition of $\Theta$, the phase difference can be found by following the wire with no phase slip. To further improve the legibility of Fig. \[0\_diag\], the free-energy barriers are listed in a separate table to the right. Note, that a phase slip on just one of the wires, being only half of the complete process, can be regarded to a gain in phase of $\pm \pi$ for the purposes of calculating voltage, as indicated in both the graph and the table. Once the table of barrier heights has been computed, we can construct a Markov chain on a directed graph, where the metastable states are the vertices—in effect, an explicit version of our diagram. In general, each pair of neighboring metastable states, $s_{n}$ and $s_{n+1}$, are connected by four directed edges: $$\begin{aligned} s_{n}&\xrightarrow[\text{top}]{\hskip1.1cm} s_{n+1} & s_{n}&\xrightarrow[\text{bottom}]{\hskip1.1cm} s_{n+1}\\ s_{n}&\xleftarrow[\text{top}]{\hskip1.1cm} s_{n+1} & s_{n}&\xleftarrow[\text{bottom}]{\hskip1.1cm} s_{n+1}\end{aligned}$$ where the probability to pass along a particular edge is given by $P(\cdot)=\exp{-\beta \Delta G_{(\cdot)}}$, in which $\Delta G_{(\cdot)}$ may be read off from the table in Fig. \[0\_diag\]. We denote the occupation probability of the $n^\text{th}$ metastable state by $o_n$, where $n$ corresponds to the $n$ in the phase constraint (\[eq:phaseconstraint\]). $o_n$ may be computed in the standard way, by diagonalizing the matrix representing the Markov chain [@markov]. Each move in the Markov chain can be associated with a gain in phase across the device of $\pm \pi$, as specified in Fig. \[0\_diag\]. Thus, we may compute the rate of phase-gain, and hence the voltage: $$V=\frac{\Omega \hbar}{4 e} \sum_{\langle n m \rangle} \frac{o_n}{g_{n,m}} \big(P(s_{n}\xrightarrow[\text{top}]{} s_{m}) -P(s_{n}\xrightarrow[\text{bot}]{} s_{m})\big),$$ where the rate prefactor $\Omega$ is to be determined, $\langle n m \rangle$ indicates that the sum runs over neighboring states only, and $g_{n,m}$ keeps track of the sign of the phase-gain for reverse phase-slips: $$g_{n,m}=\left\{ \begin{array}{rc} 1,& \text{ if }m>n, \\ -1,& \text{ if }m<n. \end{array} \right.$$ For the case $\delta=0$, and keeping the bottom three states only, the voltage generated via sequential phase slips turns out to be $$\begin{aligned} V_{\delta=0 \, \text{, seq}}=\frac{2 \hbar}{e} \,\Omega \, e^{-\beta \Ecore \left(\frac{8 \sqrt{2}}{3}+\frac{\pi^2}{b}\right)} \sinh(I/2 I_0). \label{v:d0:seq}\end{aligned}$$ Having dealt with the case of $\delta=0$ (and hence obtained the value of the resistance at magnetic fields corresponding to resistance minima), we now turn to the case of $\delta=\pi/2$, i.e., resistance maxima. In this half-flux quantum situation, there are two degenerate lowest-energy states, with opposite circulating currents. These states are connected by saddle-point states in which a phase-slip is occurring on either the top or bottom wire. The diagram of the degenerate ground states and the saddle-point states connecting them is shown in Fig. \[piby2\_diag\]. By comparing the diagram with the associated Table, it is easy to see that the free-energy barriers are biased by the current, making clockwise traversals of Fig. \[piby2\_diag\] more probable than counter-clockwise traversals. As there are only two metastable states being considered, and as they are degenerate, it is unnecessary to go through the Markov chain calculation; clearly, the two states each have a population of $1/2$. The voltage being generated by the sequential phase-slip is then given by $$\begin{aligned} V_{\delta=\pi/2 \text{, seq}}=\frac{\hbar}{2 e} \, \Omega \, e^{-\beta \Ecore \left( \frac{8 \sqrt{2}}{3}-\frac{\pi^2}{b}\right)} \sinh(I/2 I_0). \label{v:dpb2:seq}\end{aligned}$$ {width="8cm"} $V_{\delta=\pi/2 \text{, seq}}$ is larger than the sum of $V_{\delta=0 \text{, seq}}$ and $V_{\delta=0 \text{, par}}$, so, as expected, the resistance is highest at magnetic fields corresponding to $\delta=\pi/2$. For very long wires, the perturbation of one wire when a phase slip occurs in the other is very small, and therefore we expect that the dependence of resistance on magnetic field will decrease with wire length. Indeed, for very long wires, the difference in barrier heights to sequential phase slips between the $\delta=0$ and $\delta=\pi/2$ cases disappears (i.e.  and agree when $b \gg 1$). ### Numerical treatment for intermediate-length nanowires Instead of using the long-wire approximation, , we can use the exact functions for the end-to-end phase accumulation along a wire $\theta(J(\kappa))$, and the Helmholtz free energy $F_{\text{m}/\text{sp}}(J(\kappa))$. By dropping the long-wire approximation, as the temperature approaches $T_\text{c}$ and the coherence length decreases the picture correctly passes to the Josephson limit. In this approach, the total current and the phase constraints must be solved numerically, as $\theta(J(\kappa))$ is a relatively complicated function. Figure \[fig:jf\_vs\_t\] provides an illustration of how, for a single wire, the function $J(\theta)$ depends on its length. We shall, however, continue to use the barrier-crossing approximation. Because the barriers get shallower near $T_\text{c}$, our results will become unreliable (and, indeed, incorrect) there. The form of the order parameter that satisfies the Ginzburg-Landau equation inside the wire is expressed in . Therefore, to construct the functions $\theta(J)$ and $F_{\text{m}/\text{sp}}(J)$ \[i.e. \], we need to find $u_0(J)$, i.e., the squared amplitude of the order parameter in the middle of the wire. Hence, we need to ascertain suitable boundary conditions obeyed by the order parameter at the ends of the wire. For thin wires, a reasonable hypothesis is that the amplitude of the order parameter at the ends of the wire matches the amplitude in the leads: $$f(z=\pm b/2)^2=\frac{H_{\text{c}\,\,\text{film}}^2(T) \, \xi_\text{film}^2(T)}{H_{\text{c}\,\,\text{wire}}^2(T) \, \xi_\text{wire}^2(T)}.$$ For wires made out of superconducting material the same as (or weaker than) the leads, this ratio is always larger than unity [^4] . ![Squared amplitude $u(b/2)$ of the order parameter at the end of a wire, as a function of its value $u_0=u(0)$ at the mid-point of the of the wire, computed using the $\text{JacobiSN}$ function \[see \], for the case $b=16$. The black line corresponds to trajectories that do not go through a pole; the gray line corresponds to trajectories that do pass through at least one pole. The intersection of the dashed and black lines represents those trajectories that satisfy the boundary condition $u(\pm b/2)=1$. (The intersection of the dashed-dotted and black lines represent trajectories that start and stop at the same point, i.e., $u(b/2)=u_0$).[]{data-label="u_of_u_0"}](\figdir/u_of_u02d.eps){width="8cm"} Once we have computed the functions $\theta(J)$ and $F_{\text{m}/\text{sp}}(J)$ for both saddle-point and metastable states on a single wire, we can use the phase and total current constraints to build the saddle-point and metastable states for the two-wire device. We proceed as before, by constructing a Markov chain for the state of the device, except that now we include in the graph all metastable states of the device. By diagonalizing the Markov chain, we find the populations of the various metastable states and, hence, the rate of gain of $\Theta$. We plot the typical magnetoresistance curves for various temperatures, obtained numerically, as well as $dV/dI$ [*vs.*]{} $T$ for various magnetic fields and total currents. Notice that the resistance at $\delta=0$ and large total currents can exceed that at $\delta=\pi/2$ with low total-current. ![\[fig:jf\_vs\_t\] Current (in units of the critical current) vs. end-to-end phase accumulation for superconducting wires of various lengths: $0\xi$ (solid line), $1.88\xi$, $5.96\xi$, $14.4\xi$ (dotted line). The transition from LAMH to Josephson junction behavior is evident from the loss of multivaluedness of the current, as the wire length is reduced.](\figdir/j_of_q_plots.tex_gr4d.eps) Connections with experiment {#sec:experiment} =========================== In this section a connection is made between our calculations and our experiments [@us_in_science]. First, the predicted period of the magnetoresistance oscillations is compared to the experimentally obtained one. Then, the experimentally-obtained resistance vs. temperature curves are fitted using our extension of the IZAH Josephson junction model (for shorter wires) and our extension of the LAMH wire model (for longer wires). Device fabrication ------------------ Four different devices were successfully fabricated and measured. The devices were fabricated by suspending DNA molecules across a trench and then sputter coating them with the superconducting alloy of MoGe. The leads were formed in the same sputter-coating step, ensuring seamless contact between leads and the wires. Next, the leads were truncated lithographically to the desired width. In the case of device 930-1, after being measured once, its leads were further narrowed using focused ion beam milling, and the device was remeasured. For further details of the experimental procedure see Ref. [@us_in_science]. Comparison between theory and experiment ---------------------------------------- ### Oscillation period The magnetoresistance periods obtained for four different samples are summarized in Table \[table:period\]. The corresponding theoretical periods were calculated using , based on the geometry of the samples which was obtained via scanning electron microscopy. To test the theoretical model, the leads of one sample, sample 930-1, were narrowed using a focused ion beam mill, and the magnetoresistance of the sample was remeasured. The theoretically predicted periods all coincide quite well with the measured values, except for sample 219-4, which was found to have a “+” shaped notch in one of the leads (which was not accounted for in calculating the period). The notch effectively makes that lead significantly narrower, thus increasing the magnetoresistance period, and this qualitatively accounts for the discrepancy. For all samples, when the leads are driven into the vortex state, the magnetoresistance period becomes much longer, approaching the Aharonov-Bohm value for high fields. This is consistent with the theoretical prediction that the period is then given by , but with $l$ replaced by the field-dependent inter-vortex spacing $r$. Sample $b \, (\text{nm})$ $(\mu\text{T})$ error ---------------- -------------------- -- -------- -- ----------------- --------- -- -- ------- ------ -- ------- ----- ---------- 205-4 $121$ $266$ $11267$ $929$ $21$ $947$ $$ $1.9\%$ 219-4 $137$ $594$ $12062$ $388$ $73$ $456$ $6$ $12.8\%$ 930-1 $141$ $2453$ $14480$ $78$ $41$ $77$ $5$ $-1.2\%$ 930-1 (shaved) $141$ $2453$ $8930$ $127$ $14$ $128$ $3$ $0.9\%$ 205-2 $134$ $4046$ $14521$ $47$ $41$ $48$ $9$ $3.0\%$ ### Oscillation amplitude {#sec:expamp} We have made qualitative and quantitative estimates of the resistance of two-bridge devices in several limiting cases. For devices containing extremely short wires \[$b\approx\xi(T)$\], such as sample 219-4, the superconducting wires cannot support multiple metastable states, and thus they operate essentially in the Josephson junction limit, but with the junction critical current being a function of temperature given by LAMH theory as $I_\text{c}(T)=I_\text{c}(0)(1-T/T_\text{c})^{3/2}$. A summary of fits to the data for this sample, using the Josephson junction limit, is shown in Fig. \[fig:mr219\]. On the other hand, for longer wires it is essential to take into account the multiple metastable states, as is the case for sample 930-1, which has wires of intermediate length. A summary of numerical fits for this sample is shown in Fig. \[fig:mr930\]. In all cases, only the two low total-current magnetoresistance curves were fitted. By using the extracted fit parameters, the high total-current magnetoresistance curves were calculated, with their fit to the data serving as a self-consistency check. As can be seen from the fits, our model is consistent with the data over a wide range of temperatures and resistances. We remark, however, that the coherence length required to fit the data is somewhat larger than expected for MoGe. {width="7.5cm"} 1.5cm {width="7.5cm"} ------- ------------------------- ------------------------ ----------------------------- ---------------------------- ---------------------------------- ---------------------------------- (LHS) $R_1=2882.9 \, \Omega$, $R_2=2941.7\, \Omega$, $\xi_{01}=17.3\,\text{nA}$, $\xi_{02}=8.7\,\text{nA}$, $T_{\text{c}1}=3.147\,\text{K}$, $T_{\text{c}2}=3.716\,\text{K}$. (RHS) $R_1=2912 \, \Omega$, $R_2=2912 \, \Omega$, $\xi_{01}=10\,\text{nA}$, $\xi_{02}=9\,\text{nA}$, $T_{\text{c}1}=3\,\text{K}$, $T_{\text{c}2}=3.65\,\text{K}$. ------- ------------------------- ------------------------ ----------------------------- ---------------------------- ---------------------------------- ---------------------------------- Concluding remarks {#sec:conclusion} ================== The behavior of mesoscale NQUIDs composed of two superconducing leads connected by a pair of superconducting nanowires has been investigated. Magnetoresistance measurements [@us_in_science] have revealed strong oscillations in the resistance as a function of magnetic field, and these were found to have anomalously short periods. The period has been shown to originate in the gradients in the phase of the superconducting order parameter associated with screening currents generated by the applied magnetic field. The periods for five distinct devices were calculated, based on their geometry, and were found to fit very well with the experimental results The amplitude of the magnetoresistance has been estimated via extensions, to the setting of parallel superconducting wires, of the IZAH theory of intrinsic resistive fluctuations in a current-biased Josephson junction for the case of short wires and the LAMH theory of intrinsic resistive fluctuations in superconducting wires for pairs of long wires. In both cases, to make the extensions, it was necessary to take into account the inter-wire coupling mediated through the leads. For sufficiently long wires, it was found that multiple metastable states, corresponding to different winding numbers of the phase of the order parameter around the AB contour, can exist and need to be considered. Accurate fits have been made to the resistance vs. temperature data at various magnetic fields and for several devices by suitably tuning the critical temperatures, zero-temperature coherence lengths, and normal-state resistances of the nanowires. As these device are sensitive to the spatial variations in the phase of the order parameter in the leads, they may have applications as superconducting phase gradiometers. Such applications may include the sensing of the presence in the leads of vortices or of supercurrents flowing perpendicular to lead edges. [*Acknowledgments*]{}: This work was supported by the U.S. Department of Energy, Division of Materials Sciences under Award No. DEFG02-91ER45439, through the Frederick Seitz Materials Research Laboratory at the University of Illinois at Urbana-Champaign. AB and DH would like to also acknowledge support from the Center for Microanalysis of Material DOE Grant No. DEFG02-96ER45439, NSF CAREER Grant No. DMR 01-34770, and the A.P. Sloan Foundation. Multi-wire devices {#app:multi-wire} ================== In this appendix we give an example of how to extend the theory Presented in this Paper to the case of devices comprising more than two wires. In our example, we consider an array of $n$ identical short wires (i.e. wires in the Josephson junction limit) spaced at regular intervals. We continue to work at a fixed total current and to ignore charging effects. The end-to-end phase accumulations along the wires are related to each other as $$\begin{aligned} \begin{split} \theta_2&=\theta_1 + 2 \delta, \\ \theta_3&=\theta_1 + 4 \delta, \\ &\vdots \\ \theta_n&=\theta_1 + 2 (n-1) \delta, \end{split}\end{aligned}$$ i.e., $\theta_n-\theta_1=2(n-1)\delta$ (for $n=2,\ldots,N$), where $\delta$ is the phase accumulation in one of the leads between each pair of adjacent wires. The Gibbs free energy of the multi-wire subsystem is given by $$G(I,\theta_1)=-\frac{h}{2 e} \left( I_\text{c} \sum_{m=1}^n \cos \big( \theta_1 + 2(m-1) \delta \big) + I \theta_1 \right),$$ where $I$ is the total current and we are assuming that the wires have identical critical currents. As for the two-junction case, this junction array, is equivalent to a single effective junction. Figure \[fig:multi-wire\] shows the critical current of this effective junction as a function of $\delta$ for devices comprising 2, 5, and 15 wires. The magnetoresistance of such a device then follows from IZAH theory, i.e., . ![Effective single junction critical current for a multi-junction array, as a function of $\delta$. The critical current has been rescaled so that $J_\text{c}(\delta=0)=1$. Note the similarity with a multi-slit interference pattern.[]{data-label="fig:multi-wire"}](\figdir/JJ_multi_plot.eps){width="8cm"} Physical Scales =============== It is convenient to express the results of the long-wire model, Eqs. (\[v:d0:par\],\[v:d0:seq\],\[v:dpb2:seq\]), in terms of macroscopic physical parameters. Following Tinkham and Lau [@TL], we express the condensation energy scale per coherence length of wire as $$\Ecore=0.22 \, k_\text{B} T_\text{c} \, (1-t)^{3/2} \, \frac{R_\text{q}}{R_\text{N}}\frac{b}{\xi(T=0)}, \label{RTEcore}$$ where $t\equiv T/T_\text{c}$, $R_\text{N}$ is the normal-state resistance of the device, and $R_\text{q} \equiv h/4e^2 \approx 6.5\,\text{k}\Omega$ is the quantum of resistance. The LAMH prefactor for sequential phase slips then becomes $$\Omega=\frac{b \sqrt{1-t}}{\xi(T=0)} \left(\frac{8 \sqrt{2} \, \Ecore}{3\, k_\text{B} T_\text{c}}\right)^{1/2} \frac{8 k_\text{B} (T_\text{c}-T)}{\pi \hbar},$$ and for parallel phase slips becomes $$\Omega=\left(\frac{b \sqrt{1-t}}{\xi(T=0)}\right)^2 \left(\frac{16 \sqrt{2} \, \Ecore}{3\, k_\text{B} T_\text{c}}\right)^{1/2} \frac{8 k_\text{B} (T_\text{c}-T)}{\pi \hbar}.$$ The remaining parameters in the model are $R_N$, $T_\text{c}$ and $\xi(T=0)$. The normal-state resistance and the critical temperature may be obtained from the $R$ vs. $T$ curve. The coherence length may be obtained by comparing $\Ecore$ obtained from the critical current at low temperature, via $$I_\text{c}=\frac{2}{3 \sqrt{3}} \frac{16 \pi \Ecore}{\Phi_0},$$ with $\Ecore$ obtained via . In experiment, it is expected that the two wires are not identical. The long-wire model can be easily extended to this case. The number of parameters to be fitted would then expand to include the normal-state resistance for each wire (only one of which is free, as the pair are constrained by the normal-state resistance of the entire device, which can be extracted from the $R$ vs. $T$ curve), a zero-temperature coherence length for each wire, and a critical temperatures for each wire. LAMH theory for a single bridge {#LAMH1} =============================== In this appendix we reproduce useful formulas from LA [@LA], and rewrite them in a way that is convenient for further calculations, especially for numerical implementation. As in the case of single-wire LAMH theory, one starts with the Ginzburg-Landau free energy $$F=\int_{-b/2}^{b/2} \alpha |\psi|^2+\frac{\beta}{2} |\psi|^4 + \frac{\hbar^2}{2 m} |\nabla \psi|^2 \, dz.$$ The relationships between the parameters of the Ginzburg-Landau free energy ($\alpha$ and $\beta$), coherence length $\xi$, the condensation energy per unit coherence length $\Ecore$, the critical field $H_\text{c}$ and the cross-sectional area of the wire $\sigma$ are given by $\frac{\alpha^2}{\beta}=\frac{H_\text{c}^2 \sigma}{8 \pi}=\Ecore/\xi$ and $\xi^2=\frac{\hbar^2}{2 m |\alpha|}$. Following McCumber [@M68], it is convenient to work in terms of the dimensionless units obtained using the transformations: $|\psi|^2 \rightarrow \frac{\alpha}{\beta} |\psi|^2$, $z \rightarrow \sqrt{\frac{2 m |\alpha|}{\hbar^2}} z$, and $b \rightarrow b/\xi=\sqrt{\frac{2 m |\alpha|}{\hbar^2}} b$. In terms of these units, the free energy becomes $$\begin{aligned} F= 2 \Ecore \int_{-b/2}^{b/2} \left(\frac{1}{2}(1-|\psi|^2)^2 +|\nabla \psi|^2 \right) dz.\end{aligned}$$ The Ginzburg-Landau equation is obtained by varying the free energy: $$\delta F = 0 \,\,\, \Rightarrow \,\,\, -\psi + |\psi|^2 \psi - \nabla^2 \psi = 0.$$ By writing $\psi=f e^{i \phi}$ and taking the real and imaginary parts of the Ginzburg-Landau equation one obtains $$\begin{aligned} -f+f^3+(\phi')^2 f&=f'' \label{feq}, \\ 2 \phi'f'+\phi''f&=0 \label{cl1}.\end{aligned}$$ From , one finds the current conservation law: $$f^2 \phi' = J \label{cl},$$ where $J$ is identified with the dimensionless current $\frac{1}{2 i}(\psi^* \nabla \psi-\psi \nabla \psi^*)$. The physical current (in stat-amps) is given by $I=J c H_\text{c}^2 \sigma \xi/\Phi_0$. Expressing $\phi'$ in terms of J, becomes $$\begin{aligned} f''&=-f+f^3+\frac{J^2}{f^3}=-\frac{d}{df}U(f),\end{aligned}$$ where the effective potential $U(f)$ is given by $$\begin{aligned} U(f)=\frac{J^2}{2 f^2} +\frac{f^2}{2}-\frac{f^4}{4}. \label{pot}\end{aligned}$$ Following LA, Eq. (\[pot\]) can usefully be regarded as the equation of motion for a particle with position $f(z)$, where $z$ plays the role of time, moving in the potential $U(f)$ [@LA]. Before proceeding to find the solution of this equation, we pause to consider the type of trajectories that are possible. Later, it will be demonstrated that at the edge of the wire $f(\pm b) \geq 1$, so the particle starts to the right of the hump; see Fig. \[U\_f\]. If the total energy of the particle is less than the height of the hump, the particle will be reflected by the hump. If, however, the particle starts with more energy than the height of the hump, it will pass over the hump and be reflected by the $J^2/2 f^2$ dominated part of $U(f)$. ![“Mechanical potential” $U[u=f^2]$ at an intermediate value of the dimensionless current, plotted as a function of amplitude squared to make comparison with Fig. \[u\_of\_u\_0\] more convenient.[]{data-label="U_f"}](\figdir/U_u2c.eps) The equation of motion can be solved via the first integral (i.e. multiplying both sides by $f$ and integrating with respect to $f$): $$E=\frac{(f')^2}{2}+U(f)\,\,\,\Rightarrow\,\,\,f'=\sqrt{2(E-U(f))},$$ where $E$ is a constant of integration (i.e. the energy of the particle in the mechanical analogy), which gives $$\begin{aligned} z &= \int_{f_0}^f \frac{df}{\sqrt{2(E-U(f))}}= \int_{f_0}^f \frac{f df}{\sqrt{2 f^2 E-J^2-f^4+f^6/2}}.\end{aligned}$$ It is convenient to apply “initial” conditions at the middle of the wire, where $f(z=0)=f_0$, and to integrate towards the edges. We require that the particle come back to its starting point after a “time” $b$, i.e. at the edges of the wire the amplitude of the order parameter must match the boundary condition. Therefore, the middle of the wire must be the turning-point for the particle, i.e., at $z=0$ we have $E=U(f_0)$. What follows next is a series of manipulations via which one can express solution for $f(z)$ in terms of special functions. Step 1: substitution: $f^2\rightarrow u$ $$\begin{aligned} z&= \frac{1}{2} \int_{u_0}^u \frac{du}{\sqrt{2 E u-J^2-u^2+u^3/2}}\end{aligned}$$ Step 2: substitution: $u\rightarrow u_0+\epsilon$ $$\begin{aligned} 2 z &= \int_{0}^{u-u_0} \frac{d\epsilon}{ \big[\epsilon \big(\underbrace{\left(\frac{J^2}{u_0}-u_0+u_0^2\right)}_\alpha+\underbrace{\left(\frac{3}{2} u_0-1\right)}_\beta \epsilon+\epsilon^2/2\big)\big]^{1/2}}\end{aligned}$$ $$\begin{aligned} 2z&=\int_{0}^{u-u_0} \frac{d\epsilon}{(\epsilon(\epsilon+\beta\epsilon+\epsilon^2/2))^{1/2}}\\ &=\int_{0}^{u-u_0} \frac{\sqrt{2} d\epsilon}{(\epsilon(\epsilon+\underbrace{\beta +\sqrt{\beta^2-2\alpha}}_{-u_1})(\epsilon+\underbrace{\beta -\sqrt{\beta^2-2\alpha}}_{-u_2}))^{1/2}}\\ &=\int_{0}^{u-u_0} \frac{\sqrt{2} d\epsilon}{(\epsilon (\epsilon-u_1)(\epsilon-u_2))^{1/2}}\end{aligned}$$ Step 3: substitution: $\epsilon\rightarrow u_1 z^2$ $$\begin{aligned} 2 z&=\frac{2\sqrt{2}}{\sqrt{u_2}}\int_{0}^{\sqrt{\frac{u-u_0}{u_1}}} \frac{d \omega}{((\omega^2-1)(\frac{u_1}{u_2}\omega^2-1))^{1/2}}\\ &=\frac{2\sqrt{2}}{\sqrt{u_2}} \text{EllipticF}\big[\text{ArcSin}\big[\sqrt{\frac{u-u_0}{u_1}}\big],\frac{u_1}{u_2}\big] \label{z_of_f}\end{aligned}$$ The following definitions have been used: $$\begin{aligned} \alpha[u_0] &\equiv J^2/u_0-u_0+u_0^2, & \beta[u_0] &\equiv \frac{3}{2} u_0-1,\\ u_1[\alpha,\beta] &\equiv-\beta-\sqrt{\beta^2-2\alpha}, & u_2[\alpha,\beta] &\equiv-\beta+\sqrt{\beta^2-2\alpha}.\end{aligned}$$ By inverting relation (\[z\_of\_f\]) one obtains an explicit equation for the amplitude of the order parameter as a function of position along the wire (see Fig. \[u\_of\_x\]): $$\begin{aligned} f^2(z) &= u_0 + u_1 \sin^2 \big[\text{JacobiAmplitude}\big[z \sqrt{\frac{u_2}{2}},\frac{u_1}{u_2}\big]\big]\\ &=u_0+u_1 \text{JacobiSn}^2\big[z \sqrt{\frac{u_2}{2}},\frac{u_1}{u_2}\big] \label{f_of_z}\end{aligned}$$ ![Squared amplitude $u$ of the order parameter as a function of position along the wire for the two types of solution: metastable and saddle point. []{data-label="u_of_x"}](\figdir/u_of_x2c.eps){width="7cm"} The end-to-end phase difference along the wire may be found by using the current conservation law. Thus one obtains $$\theta=\int_{-b/2}^{b/2} \frac{J}{f^2(z)} dz = 2 J \int_0^{b/2} \frac{dz}{ u_0+u_1 \text{JacobiSn}^2\big[z \sqrt{\frac{u_2}{2}},\frac{u_1}{u_2}\big]}. \label{etoe}$$ The Helmholtz free energy can be found by substituting the expressions for $f(z)$ and $\phi'(z)$ into the expression for the free energy. One then obtains $$F=4 \Ecore \int_0^{b/2} dz (\frac{1}{2}-2 f^2+f^4+J^2/u_0+u_0-u_0^2/2), \label{fexact}$$ where $E$ was expressed in terms of $u_0$. provide expressions for $\theta$ and $F$ which are true regardless of the length of the wire, and therefore may be used as a starting point for computing the Gibbs free-energy of the various metastable states subject to the total-current and the phase constraints. [99]{} W. A. Little and R. D. Parks, Phys. Rev. Lett. [**9**]{}, 9 (1962); W. A. Little and R. D. Parks, Phys. Rev. [**133**]{}, A 97 (1964). M. Tinkham, [*Introduction to Superconductivity*]{}, McGraw-Hill, New York (1996). K. K. Likharev, Rev. Mod. Phys. [**51**]{} 101 (1979). A. Bezryadin and B. Pannetier, J. Low Temp. Phys. [**98**]{}, 251 (1995); C. C. Abilio, L. Amico, R. Fazio, and B. Pannetier, J. Low Temp. Phys. [**118**]{} 23 (2000). D. S. Hopkins, D. Pekker, P. M. Goldbart and A. Bezryadin, Science [**308**]{}, 1762 (2005). See, e.g., C. A. Brebbia, [*Boundary Element Method for Engineers*]{}, Pentech Press, London (1978). F. London and H. London, Proc. Roy. Soc. (London) [**A109**]{}, 71 (1935); see also Ref. [@Tinkham], pp. 4-6. K. K. Likharev, Sov. Radiophys. [**14**]{}, 722 (1973); see also J. R. Clem, Bull. Am. Phys. Soc. [**43**]{}, 411 (1972); G. M. Maksimova, Phys. Solid State [**40**]{}, 1610 (1998). Here and elsewhere we speak of [*vortices*]{} and [*anti-vortices*]{} entering or leaving the leads or the loop made by the wires. Of course, outside the superconducting regions there can be no vortices or anti-vortices. Nevertheless, we use this language to connote the temporary reduction of the amplitude of the superconducting order parameter during a dissipative fluctuation, and its global consequences for the phase of the order parameter. G. Stan, S. B. Field and J. M. Martinis, Phys. Rev. Lett. [**92**]{}, 097003 (2004). See, e.g., I.S. Gradshtein and I. M. Ryzhik, [*Table of Integrals, Series, and Products*]{}, Academic Press, New York (1965), p. 1036. Yu. M. Ivanchenko and L. A. Zil’berman, JETP Lett. [**8**]{}, 113 (1968); Yu. M. Ivanchenko and L. A. Zil’berman, JETP [**28**]{}, 1272 (1969); V. Ambegaokar and B. I. Halperin, Phys. Rev. Lett. [**22**]{}, 1364 (1969). J. S. Langer and V. Ambegaokar, Phys. Rev. [**164**]{}, 498 (1967). D. E. McCumber and B. I. Halperin, Phys. Rev. B [**1**]{}, 1054 (1970). W. A. Little, Phys. Rev. [**156**]{}, 396 (1967). D. E. McCumber, Phys. Rev. [**172**]{}, 427 (1968). A Markov chain is a device for computing the properties of stochastic processes where each step depends only on the step preceeding it. See, e.g., J. Zinn-Justin, [*Quantum Field Theory and Critical Phenomena*]{}, third ed. (1999), p. 85. M. Tinkham and C. N. Lau, Appl. Phys. Lett. [**80**]{}, 2946 (2002). [^1]: Consider the case in which $\boldsymbol{A}$ is a London gauge everywhere (with our choice of gauge, $\boldsymbol{A}= B y \boldsymbol{e}_x$, this is the case for an infinitely long strip). By using the requirement that $\boldsymbol{\nabla}\cdot \boldsymbol{A}=0$, together with , we see that $\phi$ satisfies the Laplace equation. We further insist that no current flows out of the superconductor, i.e., along all surfaces the supercurrent density, , is always parallel to the surface. Together with the requirement that along all surfaces $\boldsymbol{A}$ is parallel to them, this implies the boundary condition that $\boldsymbol{n} \cdot \boldsymbol{\nabla} \phi=0$. Next, it can be shown that this boundary condition implies that $\phi$ must be a constant function of position in order to satisfy the Laplace equation, and therefore  simplifies to read $\boldsymbol{J} = -(c/8 \pi \lambda_\text{eff}^2) \boldsymbol{A}$, which is known as the London relation. [^2]: This specialization is not necessary, but it is convenient and adequately illustrative [^3]: Recall that the Helmholtz free energy is obtained by minimizing the Ginzburg-Landau free energy functional with respect to the order parameter function $\psi(\boldsymbol{r})$, subject to the phase accumulation constraint $\int_\text{L}^\text{R} d\boldsymbol{r} \cdot \boldsymbol{\nabla} \phi = \theta$. [^4]: In finding $u_0$ there is a minor numerical difficulty. As the amplitude of the order parameter is expressed via the $\text{JacobiSn}$ function, and $\text{JacobiSn}[z \sqrt{u_2/2},u_1/u_2]$ is a doubly periodic function in the first variable, it is not obvious whether $\pm (b/2) \sqrt{u_2/2}$ lies in the first period, as can be seen from Fig. \[u\_of\_u\_0\]. As the trajectory must be simply periodic, $z \sqrt{u_2/2}$ must intersect either a zero or a pole in the first unit quarter cell of the $\text{JacobiSn}$ function. Now, we are only interested in trajectories that escape to $f\rightarrow \infty$ \[as $f(\pm b/2)$ is assumed to be greater than or equal to unity\], so a pole must be intersected. (However, being outside the first period is unphysical, as it means that somewhere along the wire $f=\infty$.) There are exactly two poles in the first unit quarter cell. They are located at $2 v_1+v_2$ and $v_2$, where $v_1 \equiv {\rm K}(u_1/u_2)$ and $v_2 \equiv i {\rm K}(1-u_1/u_2)$, in which ${\rm K}(\cdot)$ is the complete elliptic integral. So, instead of checking whether $\pm (b/2) \sqrt{u_2/2}$ is outside the unit quarter cell, we can just determine which pole $z \sqrt{u_2/2}$ intersects and then see if $\pm (b/2) \sqrt{u_2/2}$ lies beyond that pole or not.

--- author: - | Marcel Filoche$^{1,2\ast}$, Svitlana Mayboroda$^{3}$\ \ \ \ \ \ \ \ \ bibliography: - 'scibib.bib' title: The Hidden Landscape of Localization --- Wave localization occurs in all types of vibrating systems, in acoustics, mechanics, optics, or quantum physics. It arises either in systems of irregular geometry (weak localization) or in disordered systems (Anderson localization). We present here a general theory that explains how the system geometry and the wave operator interplay to give rise to a “landscape" that splits the system into weakly coupled subregions, and how these regions shape the spatial distribution of the vibrational eigenmodes. This theory holds in any dimension, for any domain shape, and for all operators deriving from an energy form. It encompasses both weak and Anderson localizations in the same mathematical frame and shows, in particular, that Anderson localization can be understood as a special case of weak localization in a very rough landscape. #### Introduction {#Intro .unnumbered} Every object in nature vibrates. The vibrations can be acoustic waves in a medium, mechanical deformations of a rigid plate, electromagnetic waves in a cavity or quantum states such as the electronic states in a crystal. In all these cases, the linear behavior of the system can be reduced to the knowledge of the vibrational modes, i.e., the eigenfunctions and eigenvalues of the spatial differential operator associated to the wave equation. A stunning property exhibited by these eigenfunctions in irregular or disordered systems is known as localization: in some cases, the mode amplitude is small everywhere except in a very limited subregion of the domain, even though there is no clearly visible obstacle preventing the vibration to propagate in the rest of the domain, away from its main existence subregion. It has been acknowledged that there are several types of localization, each exhibiting a specific behavior in space or in frequency. First, when caused by the irregular or complex geometry of the vibrating domain, localization is classified as *weak*. It is characterized by a slow decay of the mode amplitude away from its main existence subregion (much slower than exponential) [@Baranger1993; @Felix2007; @Filoche2009; @Heilman2010]. Secondly, localization can arise due to a disorder quenched in the system [@Anderson1958; @Richardella2010]. This phenomenon, called *Anderson localization*, has fascinated scientists since its discovery in 1958 and spurred a wealth of literature [@Lagendijk2009]. In that case, the mode amplitude decays exponentially away from its main existence subregion and the localization is classified as *strong*. Finally, a third and quite different type of localization occurs at high frequencies in specific domains that possess stable orbits such as whispering gallery or bouncing ball modes [@Heller1984]. Confined modes known as *scar* modes then appear asymptotically localized near these orbits. Until today, there has been no theory able to explain how the geometry of the domain or the nature of the disorder is related to the localization of vibrations, to predict in which subregions one can expect localized eigenmodes to appear, and in which frequency range. Moreover, the question of whether the different types of localization are linked remains open. Consider, for instance, the system depicted in Fig. \[fig:cplate\_u\] (left). It has a non trivial shape, possesses 2 inner blocked points (in the left upper region), one crack on the right upper boundary and a bottleneck in its lower region. This can represent either a flexible membrane of complex shape (the spatial differential operator is then the Laplacian), or a rigid thin plate (the operator being the bi-Laplacian). From the knowledge of this geometry only, how can one determine whether and where to expect localization in this structure? In the present paper, we show that one theory can answer these questions. It unifies weak and Anderson localization, and reveals inside each system a hidden landscape that determines the localization subregions, the strength of the confinement, and its frequency dependency. #### The landscape mapping {#landscape .unnumbered} In mathematical terms, a vibrating system is governed by the wave equation associated to a suitable elliptic differential operator $L$. The latter is determined by the nature of vibration and the medium. For instance, the Laplacian $L=-\Delta$ describes the vibration of a soft membrane in 2D, acoustic waves, the quantum states inside a box or cavity; variable coefficient second order operators $L=-{\rm div} \left( A(x)\,\nabla \right)$ pertain to the aforementioned phenomena in inhomogeneous media; and the bi-Laplacian $\Delta^2$ addresses thin plate vibrations in 2D. A study of the vibrational properties of the system can be reduced to the investigation of the eigenmodes of $L$ defined by $$\label{eq11.0} L~\varphi~=~\lambda~\varphi\,\,\mbox{ in } \,\,\Omega,\quad \varphi|_{\partial\Omega}=0$$ We demonstrate here that [*all*]{} eigenmodes are controlled by the same function which has a decisive impact on localization properties. To be specific, for every $\lambda$ and $\varphi$ as above $$\label{eq9} |\varphi(\vec{x})|\leq \lambda \int_\Omega |G(\vec{x},\vec{y})|\,dy,\quad \forall\,\vec{x}\in\Omega,$$ where the mode $\varphi$ is normalized so that $\sup_\Omega |\varphi|=1$, and $G(\vec{x},\vec{y})$ is the Green’s function solving $L~G(\vec{x},\vec{y})=\delta_{\vec{x}}(\vec{y})$ with zero data on the boundary. (For the sake of brevity, we assume Dirichlet boundary data. Other types of boundary conditions will be addressed in forthcoming publications). In other words, there exists a function $u(\vec{x})=\int_\Omega |G(\vec{x},\vec{y})|\,dy$, *independent* of the eigenmode $\varphi$ such that $$\label{eq10} {|\varphi(\vec{x})|}\leq \lambda \,u(\vec{x})$$ Recall that the Green’s function is positive for the Laplacian and, more generally, for all second order differential operators. In that case $u$ admits a remarkably simple definition. It is the solution to the Dirichlet problem $$\label{eq11} L~u~=~1 \,\,\mbox{ in } \,\,\Omega,\quad u|_{\partial\Omega}=0$$ For a higher order operator the boundary condition $u|_{\partial\Omega}=0$ in and refers to the natural Dirichlet data, including vanishing derivatives of appropriate orders. We defer rigorous mathematical justification of – to [@supporting]. In physical terms, $u$ can be interpreted, for instance, as the steady-state deformation of a membrane under a uniform load. Through inequality , the *landscape* $u$ compels the eigenmodes to be small along its lines of local minima (called [*valleys*]{} throughout the paper). As we will show, the network of these valleys, a priori invisible when looking at the domain, but clearly identifiable on the graph of $u$, operates as a driving force that determines the confinement properties for both weak and Anderson localization. Let us illustrate this with an example. Figure \[fig:cplate\_u\] displays two maps of $u$ computed in the same complex geometry for the Laplacian (center) and the bi-Laplacian (right), respectively. The streamlines (lines of the gradient) have been plotted in thin black in order to clearly pinpoint the valleys, which are highlighted as thick red lines. The two cases expose dramatically different patterns. For the Laplacian, one can observe two valleys, and only one of them splits the domain into two disjoint subregions. In the bi-Laplacian case, five valleys form a network yielding a partition of the domain into four disjoint subregions. ![Left: Geometry of a complex domain, with a bottleneck (in the lower part), two inner blocked points, and an inward crack (on the right upper side). The question is: is there localization in this structure, which modes will be localized and where? Center and right: 2D representations of the landscape $u$ for the Laplacian (center) and the bi-Laplacian (right) in the same domain. The colors correspond to the height of the landscape. The (thin black) streamlines help to detect the valley lines, which are then highlighted as thick red curves. These valleys delimit 2 subregions of localization for the Laplacian and 4 subregions for the bi-Laplacian.[]{data-label="fig:cplate_u"}](cplate_geom.jpg "fig:"){width="4.25cm"} ![Left: Geometry of a complex domain, with a bottleneck (in the lower part), two inner blocked points, and an inward crack (on the right upper side). The question is: is there localization in this structure, which modes will be localized and where? Center and right: 2D representations of the landscape $u$ for the Laplacian (center) and the bi-Laplacian (right) in the same domain. The colors correspond to the height of the landscape. The (thin black) streamlines help to detect the valley lines, which are then highlighted as thick red curves. These valleys delimit 2 subregions of localization for the Laplacian and 4 subregions for the bi-Laplacian.[]{data-label="fig:cplate_u"}](cplate_lap_u.jpg "fig:"){width="5.4cm"} ![Left: Geometry of a complex domain, with a bottleneck (in the lower part), two inner blocked points, and an inward crack (on the right upper side). The question is: is there localization in this structure, which modes will be localized and where? Center and right: 2D representations of the landscape $u$ for the Laplacian (center) and the bi-Laplacian (right) in the same domain. The colors correspond to the height of the landscape. The (thin black) streamlines help to detect the valley lines, which are then highlighted as thick red curves. These valleys delimit 2 subregions of localization for the Laplacian and 4 subregions for the bi-Laplacian.[]{data-label="fig:cplate_u"}](cplate_bil_u.jpg "fig:"){width="5.4cm"} We now show rigorously that for any domain and any differential operator the network of valleys built as above triggers wave localization by effectively separating the original domain into weakly coupled vibrating regions. #### The formation of localized modes {#formation .unnumbered} Consider a subregion $\Omega_1$ of the original domain $\Omega$ carved out by the valley lines. It can be thought of, for instance, as any of the four subregions in Figure \[fig:cplate\_u\], right. By construction, $u$ is relatively small (locally minimal) along the boundary of $\Omega_1$. Thus, for low eigenvalues $\lambda$, inequality provides a severe constraint on the mode amplitude $\varphi$ along the boundary $\partial\Omega_1$. As a result, any eigenmode $\varphi$ of the entire domain can locally be viewed as a solution to the problem $$\begin{aligned} \label{eqED1} L~\varphi &= \lambda~\varphi \qquad \rm{in} \quad \Omega_1,\\ \varphi = 0 \qquad \rm{on} \quad \partial \Omega_1\cap \partial\Omega , \qquad & \rm{and} \qquad \varphi = {\varepsilon}\qquad \rm{on} \quad \partial \Omega_1\setminus \partial\Omega,\label{eqED1.0}\end{aligned}$$ where ${\varepsilon}(\vec{x})$ is a quantity smaller than $\lambda u(\vec{x})$ on the boundary of $\Omega_1$. Observe that boundary value problem – is, in fact, akin to the eigenvalue problem in the subregion $\Omega_1$ alone: the differential equation inside the subregion is identical, but on the boundary $\varphi$ is small in rather than just being zero as an eigenvalue problem on $\Omega_1$ would normally warrant. Using properties of the resolvent $(L-\lambda I)^{-1}$ and spectral decomposition, one can establish the following estimate which plays an essential role in understanding the origin of localization: $$\label{eq19} \|\varphi\|_{L^2(\Omega_1)} \leq \left(1+\frac{\lambda}{d_{\Omega_1}(\lambda)}\right) \|{\varepsilon}\|$$ See [@supporting] for the proof. Here, $d_{\Omega_1}(\lambda)$ is the distance from $\lambda$ to the spectrum of the operator $L$ in the subregion $\Omega_1$ (defined as: $\displaystyle d_{\Omega_1}(\lambda) = \min_{\lambda_{k,\Omega_1}} \left\{\left|\lambda-\lambda_{k,\Omega_1}\right|\right\}$, the minimum being taken over all eigenvalues $\left(\lambda_{k,\Omega_1}\right)$ of $L$ in $\Omega_1$), and $\|{\varepsilon}\|$ is the $L^2$-norm of the solution to $Lv=0$ in $\Omega_1$ with data ${\varepsilon}$ on $\partial\Omega_1$ (in the sense of ). In particular, $\|{\varepsilon}\|$ becomes arbitrarily small as ${\varepsilon}$ in Eq.  vanishes. The presence of $d_{\Omega_1}(\lambda)$ in the denominator of the right-hand side of Eq.  assures that whenever $\lambda$ is [*far*]{} from any eigenvalue of $L$ in $\Omega_1$ in relative value, the norm of $\varphi$ in the entire subregion, $\|\varphi\|_{L^2(\Omega_1)}$, has to be smaller than $2\|{\varepsilon}\|$. Consequently, such a mode $\varphi$ is expelled from $\Omega_1$ and must “live" in its complement, exhibiting weak localization. Conversely, $\varphi$ can only be substantial in the subregion $\Omega_1$ when $\lambda$ almost coincides with one of local eigenvalues of the operator $L$ in $\Omega_1$. Moreover, in that case Eq.  yields the conclusion that $\varphi$ itself almost coincides with the corresponding eigenmode of the subregion $\Omega_1$. Thus, we obtain a rigorous scheme elucidating the formation of weak localization. In any subregion delimited by the valleys of $u$, an eigenmode of $\Omega$ has only two possible choices: (1) either its amplitude is very small throughout this subregion, or (2) this mode mimics (both in frequency and in shape) one of the subregion’s own eigenmodes. Consequently, a low frequency eigenmode can cross the boundary between two adjacent subregions only if they possess two similar local eigenvalues. More generally, a [*fully delocalized*]{} eigenmode can only emerge as a [*collection*]{} of local eigenmodes of all subdomains when they all share a common eigenvalue. ![two localized modes of the Laplace operator (left) and four localized modes of the bi-Laplace operator (right). In both cases, the valley network (displayed in red) obtained in Figure \[fig:cplate\_u\] accurately predicts the number and the location of the localization subregions.[]{data-label="fig:cplate_modes"}](cplate_lap_mode01.jpg "fig:"){width="3.6cm"} ![two localized modes of the Laplace operator (left) and four localized modes of the bi-Laplace operator (right). In both cases, the valley network (displayed in red) obtained in Figure \[fig:cplate\_u\] accurately predicts the number and the location of the localization subregions.[]{data-label="fig:cplate_modes"}](cplate_bil_mode01.jpg "fig:"){width="3.6cm"} ![two localized modes of the Laplace operator (left) and four localized modes of the bi-Laplace operator (right). In both cases, the valley network (displayed in red) obtained in Figure \[fig:cplate\_u\] accurately predicts the number and the location of the localization subregions.[]{data-label="fig:cplate_modes"}](cplate_bil_mode02.jpg "fig:"){width="3.6cm"}\ ![two localized modes of the Laplace operator (left) and four localized modes of the bi-Laplace operator (right). In both cases, the valley network (displayed in red) obtained in Figure \[fig:cplate\_u\] accurately predicts the number and the location of the localization subregions.[]{data-label="fig:cplate_modes"}](cplate_lap_mode05.jpg "fig:"){width="3.6cm"} ![two localized modes of the Laplace operator (left) and four localized modes of the bi-Laplace operator (right). In both cases, the valley network (displayed in red) obtained in Figure \[fig:cplate\_u\] accurately predicts the number and the location of the localization subregions.[]{data-label="fig:cplate_modes"}](cplate_bil_mode04.jpg "fig:"){width="3.6cm"} ![two localized modes of the Laplace operator (left) and four localized modes of the bi-Laplace operator (right). In both cases, the valley network (displayed in red) obtained in Figure \[fig:cplate\_u\] accurately predicts the number and the location of the localization subregions.[]{data-label="fig:cplate_modes"}](cplate_bil_mode06.jpg "fig:"){width="3.6cm"} The role played by the valleys of $u$ in localization is even more astonishing when exploring the examples. Let us revisit the domain in Figure \[fig:cplate\_u\]. Figure \[fig:cplate\_modes\], left, displays two localized modes (number 1 and 5) of the Laplacian plotted together with the valley lines computed in Figure \[fig:cplate\_u\], center. Not only these modes obey the pattern predicted by the landscape $u$, but there exists no eigenmode confined to a smaller subregion. In Figure \[fig:cplate\_modes\], right, modes 1, 2, 4, and 6 of the bi-Laplacian are displayed in the same domain together with the valley lines from Figure \[fig:cplate\_u\], right. We observe here a very different and much larger variety of localization behaviors. However, once again, the location, the shape, and the number of the localization subregions exactly match the partition of the domain generated by the valley network of the landscape of $u$ computed in Figure \[fig:cplate\_u\]. #### The effective valley network and Weyl’s law {#effective .unnumbered} Not only the location of the subregions is revealed by the network of valleys, but the height and shape of the valleys gives access to statistical information on the evolution of the localization properties in [*all*]{} frequency ranges. While geometrically the network of valleys is determined by $u$ and does not depend on a particular eigenmode, the [*strength of the confinement*]{} of an eigenmode dictated by Eq.  diminishes as $\lambda$ grows. Indeed, given the normalization chosen in Eq. , Eq.  represents an effective constraint only at those points $\{\vec{x}\}$ where $\lambda~u(\vec{x})$ is smaller than 1. In other words, an eigenmode can actually [*see*]{} only the [*portion*]{} of the initial landscape which satisfies $u<1/\lambda$. We refer to the latter as the *effective* valley network. By definition, at relatively low frequencies the effective network is identical to the full one. However, as $\lambda$ increases, the effective network progressively disappears. Subregions that were initially disjoint begin to merge to form larger subregions. Above a critical value of $\lambda$, what is left of the effective network allows one subregion to percolate throughout the system: starting from that value of $\lambda$, completely new fully delocalized modes can appear. Note that the precise quantitative information on evolution of the effective network with the growth of $\lambda$ is already encoded in the original mapping of $u$. In this vein, it would be extremely useful to estimate the underlying rate of growth of the eigenvalues. Unfortunately, in the full generality of our set-up the Weyl’s law is not available yet. Roughly speaking, it predicts that for any given $\Lambda>0$ the number of the eigenvalues of $L$ below $\Lambda$ is asymptotically $\displaystyle \Lambda^{d/2m}$, where $d$ stands for the dimension and $2m$ is the order of the differential operator $L$. This estimate is widely used in practice and can be employed in the present context to determine the range of frequencies with high response to the impact of $u$, that is, the range of thoroughly localized eigenmodes. In this context, one has to stress that in a domain of self-similar boundary, smaller copies of a valley line would appear at all scales, triggering mode localization for an infinite number of eigenvalues. In that case, even though weak localization essentially affects low frequency eigenmodes, one would always find localized modes in the high frequency limit governed by Weyl’s law. #### From weak to strong: Anderson localization {#anderson .unnumbered} Since Anderson’s work in 1958 [@Anderson1958], strong localization is known to arise due to the presence of structural disorder in a system. For instance, in a gas of non interacting electrons in a crystal, this phenomenon can induce a transition between metallic and insulating behavior as the electronic states become highly localized [@Punnoose2005; @Richardella2010]. More recently, disorder-induced localization has been demonstrated to be a very general phenomenon also observed in acoustics [@Zhang1999], in microwaves [@Laurent2007], or in optics [@Sapienza2010; @Riboli2011]. ![Top left: random potential $V(\vec{x})$. The 2D square domain is divided in 20 $\times$ 20 small squares. In each square, the potential $V$ is assigned an independent random value uniformly distributed between 0 and $V_{max}$ (here 8000). Top right: 3D view of the landscape of $u$, obtained by solving $L u = 1$, where $L = - \Delta + V(\vec{x})$. Bottom: 2D color representation of the map of $u$, together with the streamlines. The thicker lines correspond to the deepest valleys and the moderately thick lines to the more shallow ones. This landscape draws an intricate network of interconnected valleys. One can conjecture that in the limit of a Brownian potential, this network becomes scale invariant and exhibit fractal properties.[]{data-label="fig:random_u"}](random_V.jpg "fig:"){width="7cm"} ![Top left: random potential $V(\vec{x})$. The 2D square domain is divided in 20 $\times$ 20 small squares. In each square, the potential $V$ is assigned an independent random value uniformly distributed between 0 and $V_{max}$ (here 8000). Top right: 3D view of the landscape of $u$, obtained by solving $L u = 1$, where $L = - \Delta + V(\vec{x})$. Bottom: 2D color representation of the map of $u$, together with the streamlines. The thicker lines correspond to the deepest valleys and the moderately thick lines to the more shallow ones. This landscape draws an intricate network of interconnected valleys. One can conjecture that in the limit of a Brownian potential, this network becomes scale invariant and exhibit fractal properties.[]{data-label="fig:random_u"}](random_u3D.jpg "fig:"){width="7cm"} ![Top left: random potential $V(\vec{x})$. The 2D square domain is divided in 20 $\times$ 20 small squares. In each square, the potential $V$ is assigned an independent random value uniformly distributed between 0 and $V_{max}$ (here 8000). Top right: 3D view of the landscape of $u$, obtained by solving $L u = 1$, where $L = - \Delta + V(\vec{x})$. Bottom: 2D color representation of the map of $u$, together with the streamlines. The thicker lines correspond to the deepest valleys and the moderately thick lines to the more shallow ones. This landscape draws an intricate network of interconnected valleys. One can conjecture that in the limit of a Brownian potential, this network becomes scale invariant and exhibit fractal properties.[]{data-label="fig:random_u"}](random_u.jpg "fig:"){width="8cm"} We present here a *fundamentally novel approach* to Anderson localization, resting upon the theory developed in the previous sections. To this end, we compute eigenmodes of the Schrödinger operator with a random potential modeled as follows. The original domain is chosen to be a simple unit square. It is divided into $400=20\times 20$ smaller squares. On each of these smaller squares, the potential $V$ is constant, its value being determined at random uniformly between 0 and $V_{\max}$ (here $V_{\max}=8000$, see Figure \[fig:random\_u\], top left, the arbitrary energy units are taken such that $\hbar^2/2m=1$). The corresponding Schrödinger operator $H = -\Delta + V$ is therefore a second order elliptic operator with variable coefficients falling under the scope of our theory. The valley landscape is further obtained by solving $H~u = 1$ in the entire square, with Dirichlet boundary conditions on 4 sides (see Figure \[fig:random\_u\], top right). One can observe a complex relief, characterized by a network of interconnected valleys of varying depths. Using the streamlines as guidelines, the valley lines are drawn in Figure \[fig:random\_u\], bottom. The thick lines represent the deepest valleys, while the moderately thick lines correspond to the shallower ones. This intricate network reveals a complex partition of the domain into a large number of subregions that was impossible to guess by just looking at the random potential at hand (cf. Figure \[fig:random\_u\], top left). In the continuous limit of a Gaussian random potential, one can conjecture that this network would show similar patterns at all possible valley depths, and hence, exhibit fractal properties. Figure \[fig:random\_modes\] displays 2D color representations of the amplitude for a number of modes (or quantum states), at lower and higher frequencies. For each mode, the *effective* network of valley lines is plotted on top of the amplitude. Since in the present context the eigenvalue is equal to the energy, the effective network can be viewed as a subset of the initial landscape of valleys subject to constraint $u< 1/E$, $E$ being the mode energy. It is striking to observe how all modes are clearly shaped by the valley lines. The fundamental and the first excited states (modes 1 to 8) are localized completely to one of the subregions defined by the network of valley lines. At higher energies (mode 45, 48, 70), the effective valley network starts to shrink, opening breaches in the shallowest valley lines. One can see that these modes are still localized, but now exactly in the much larger subregions defined by the remaining effective network. However, there still exist some small subregions in which one can find localized modes weakly coupled to the rest of the domain (modes 47 and 71). At even higher energies (modes 97-99), the effective valley network is mostly disconnected, allowing a subregion to percolate throughout the entire domain: delocalized states appear. For sake of simplicity, simulations have been carried out in 2D. In 3D, the valleys are not lines but surfaces of minimal value of $u$. The entire valley network has the shape of a foam which separates the domain into a large number of subregions. When the energy increases, the effective valley network evolves by opening gaps in the walls separating adjacent subregions. The localization length then increases accordingly as the average size of a subregion. ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode01.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode02.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode03.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode04.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode05.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode06.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode07.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode08.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode45.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode46.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode47.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode48.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode70.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode71.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode72.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode73.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode97.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode98.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode99.jpg "fig:"){width="3.3cm"} ![Spatial distribution of several quantum states in the random potential of Fig. \[fig:random\_u\]. On top of each state is drawn the effective valley network corresponding to the state energy. One can clearly observe that all modes are localized exactly in one of the subregions delimited by the effective valley network.[]{data-label="fig:random_modes"}](rand_mode100.jpg "fig:"){width="3.3cm"} One can examine in detail the strength of the localization by plotting the mode amplitude on a logarithmic scale (Figure \[fig:log\_modes\]). By doing so, one can notice that the level curves are on average equally spaced, which corresponds to an exponential decay away from the existence subregion. Even more precisely, we once again observe our landscape at work: the amplitude of the mode appears to stay within the same order of magnitude inside each subregion, the decay essentially occurs when crossing a boundary between two adjacent subregions (this boundary corresponds to a valley line). This is particularly clear for both modes in Figure \[fig:log\_modes\] in which their principal existence subregion appears dark red, all nearest neighboring subregions appear light red, the second nearest neighboring subregions essentially orange, etc. Therefore, each subregion is weakly coupled to its neighbors, the mode decaying by a somewhat constant factor each time it crosses a valley line away from the center subregion of existence. Hence, when zoomed towards any subregion, mode localization is of the weak type. However, due to the intricacy of the valley network, the succession of regularly spaced valley lines in the effective network yields an exponential decay of the mode away from its center subregion, over distances much larger than the typical size of a subregion. As a consequence, the mode appears to be exponentially confined: strong localization emerges from successive weak localizations. ![Logarithmic plot (in log10) of the amplitude for two excited states in the potential shown in Fig. \[fig:random\_u\]. One can observe here in detail how strong localization emerges from weak localization. Both modes are located in one of the subregions delimited by the valleys ways. Moreover, their decay is shaped by neighboring subregions. Firstly, the mode amplitude is more or less uniform within any subregion. Secondly, when going away from the main subregion, crossing a valley line corresponds to a decrease by one or two orders of magnitude. This leads to an exponential decay for distance larger than the typical size a subregion.[]{data-label="fig:log_modes"}](rand_log_mode05.jpg "fig:"){width="5cm"} ![Logarithmic plot (in log10) of the amplitude for two excited states in the potential shown in Fig. \[fig:random\_u\]. One can observe here in detail how strong localization emerges from weak localization. Both modes are located in one of the subregions delimited by the valleys ways. Moreover, their decay is shaped by neighboring subregions. Firstly, the mode amplitude is more or less uniform within any subregion. Secondly, when going away from the main subregion, crossing a valley line corresponds to a decrease by one or two orders of magnitude. This leads to an exponential decay for distance larger than the typical size a subregion.[]{data-label="fig:log_modes"}](rand_log_mode08.jpg "fig:"){width="5cm"} Therefore, the exact same concepts and the exact same theory both account for localization in a domain of irregular geometry and localization induced by the presence of disorder. Moreover the notion of effective network extremely accurately predicts the shape of the localization subregions at any energy and therefore allows us to understand the apparition of delocalized states above a critical energy: this critical value is the smallest energy for which the effective network opens all valley lines and allows one subregion to percolate throughout the system. Remarkably, computing the landscape of $u$ and the effective valley network at any energy does not require any knowledge on the quantum states of the system, but yet gives access to accurate information on the confinement of the states within any energy range. One also should add that, although the landscape $u$ is obtained by simply solving one linear system, it nevertheless depends in a complex way upon the quenched disorder introduced in the potential $V$. In particular, different types of randomness would lead to different localization properties. More generally, the question of localization in any disordered system can now be mathematically reformulated in the simple following way: How does the valley network of the landscape of $u$ (its connectivity, sizes, and height), depend on the properties of the disorder in $V$? Localization also directly affects the transport properties in a random medium [@Wang2011]: at zero temperature, the system is insulating if all the quantum states occupied by the electrons are localized. This condition can be reformulated in terms of the landscape: [*for the system to be insulating at T=0, its effective valley network computed at the Fermi level (in other words, the network of valley lines for which $u < 1/E_F$) must allow one subregion to percolate through the entire system*]{}. In 2D, if a subregion crosses the entire domain then its effective valley network has to be disconnected. The percolation transition of the subregion thus exactly corresponds to the percolation transition of the effective valley network itself. This theory also allows us to understand how temperature can modify the conductance of the system. On one hand, its increases the amplitude of disorder that enters the Schrödinger equation. This tends to localize the quantum states and to reduce the conductance. On the other hand, it explores excited states at higher energies, which are less localized, and it couples the states, allowing electrons to hop between states. The existence of a transition at finite temperature in a macroscopic system therefore depends on the outcome of the competition between these effects. Finally one should recall that this theory is formally valid in any dimension. In particular, it means that the localization or delocalization properties of a system of $N$ correlated electrons can also be analyzed, in theory, through the knowledge of the statistical properties of the valley network for the landscape $\displaystyle u = \left(-\sum_{i=1}^N \Delta_i + V\right)^{-1} \mathbf{1}$ in a $3N$-dimensional space. #### Conclusion {#conclusion .unnumbered} What emerges here is a new picture of localization. There are, in fact, not *three* but only *two* types of localization: *low frequency* localization, described by the landscape theory developed in this paper, and *scar* or *high frequency* localization, predicted by the stable orbits of the domain. Our findings demonstrate that low frequency localization is a universal phenomenon, observed for any type of vibration governed by a spatial differential operator $L$ that derives from an energy form. The geometry of the domain and the properties of the operator interplay to create a landscape $u$ which entirely determines the localization properties of the system. First, its network of valley lines (in 2D) or surfaces (in 3D) creates a partition of the initial domain into disjoint subregions which shape the spatial distributions of the vibrational modes and identify precisely the regions confining vibrations. Second, the depth of these valleys determines the strength of the confinement within each subregion. The localization of a given mode of eigenvalue $\lambda$ is controlled by the effective valley network defined as the subset of the entire original valley network subject to the condition $u<1/\lambda$. Consequently, the relative number of localized modes decreases at high frequencies. After partitioning by the landscape, a complex vibrational system can be understood as a collection of weakly coupled subregions whose coupling increases with frequency. The theory holds for systems of irregular geometry as well as for disordered ones. In this framework, *Anderson localization* arises as a specific form of weak localization, “strengthened" by the extremely rough landscape generated by a random potential. More generally, the macroscopic properties materials or systems in which localization plays an essential role, can now be reformulated from the geometrical and analytical characteristics of the effective valley network. This theory of localization opens a number of new problems. Let us mention a few: In the case of domain with fractal boundary, can one relate the asymptotic distribution of eigenmodes with the scaling properties of the valley network? Can one deduce the thermodynamical behavior of non interacting bosons or fermions in a disordered system from the knowledge of its valley network? What are the statistical properties of the landscape of $N$ interacting particles? Finally, one should stress that the effective valley network is promising to become a new tool of primary importance for designing systems with specific vibrational properties. To this end, future studies should investigate in detail the relationship between the geometry of a system (irregular or fractal), the characteristics of the wave operator (order, non-homogeneity, possibly stochastic), and the properties of one key mapping, the resulting valley network. Part of this work was completed during the visit of the second author to the ENS Cachan. Both authors were partially supported by the ENS Cachan through the Farman program. The first author is also partially supported by the ANR Program “Silent Wall" ANR-06-MAPR-00-18. The second author is partially supported by the Alfred P. Sloan Fellowship, the NSF CAREER Award DMS 1056004, and the US NSF Grant DMS 0758500. The authors also wish to thank Guy David for fruitful discussions. #### SUPPORTING ONLINE MATERIAL {#supporting-online-material .unnumbered} #### Materials and methods {#materials-and-methods .unnumbered} #### Preliminaries {#s1 .unnumbered} Let $\Omega$ be an arbitrary bounded open set in ${{\mathbb{R}}}^n$ and let $L$ be any elliptic differential operator associated to a symmetric positive bilinear form $B$ (an energy integral). Essentially all elliptic operators governing wave propagation, whether in acoustics, mechanics, electromagnetism, or quantum physics, are associated with an energy and fall into this category. The most prominent examples include: $$\begin{aligned} \label{eq0.1} \mbox{the Laplacian} \quad &L=-\Delta, \quad &B[u,v]=\int_\Omega\nabla u\,\nabla v \,dx, \\[4pt] \label{eq0.2} \mbox{the Hamiltonian} \quad &L=-\Delta+V(x), \quad 0\leq V(x)\leq C, \quad &B[u,v]=\int_\Omega\nabla u\,\nabla v + V u v \,dx, \\ \label{eq0.3} \mbox{the bilaplacian} \quad &L=\Delta^2=-\Delta(-\Delta), \quad &B[u,v]=\int_\Omega\Delta u\,\Delta v\,dx,\end{aligned}$$ and finally, any second order divergence form elliptic operator $$\label{eq0} L=-{\rm div}\,A(x) \nabla, \qquad B[u,v]=\int_{\Omega} A(x) \nabla u \nabla v\,dx,$$ where $A$ is an elliptic real symmetric $n\times n$ matrix with bounded measurable coefficients, that is, $$\label{eq0.5} A(x)=\{a_{ij}(x)\}_{i,j=1}^n, \,x\in\Omega, \qquad a_{ij}\in L^\infty(\Omega), \qquad \sum_{i,j=1}^{n} a_{ij}(x) \xi_i\xi_j\geq \lambda |\xi|^2,\,\,\forall\,\xi\in{{\mathbb{R}}}^n,$$ for some $\lambda>0$, and $a_{ij}=a_{ji}, \,\,\forall i,j=1,...,n.$ In general, $L$ is a differential operator of order $2m$, $m\in {{\mathbb{N}}}$, defined in the weak sense: $$\label{eq0.4} \int_\Omega Lu \,v\,dx:= B[u,v],\qquad \mbox{for every}\qquad u,v\in \ring H^m(\Omega),$$ where $B$ is a bounded positive bilinear form and $\ring H^m(\Omega)$ is the Sobolev space of functions given by the completion of $C_0^\infty(\Omega)$ in the norm $$\label{eq2} \|u\|_{\ring H^m(\Omega)}:=\|\nabla^m u\|_{L^2(\Omega)},$$ and $\nabla^m u$ denotes the vector of all partial derivatives of $u$ of order $m$. Recall that the Lax-Milgram Lemma ascertains that for every $f\in (\ring H^m(\Omega))^*=:H^{-m}(\Omega)$ the boundary value problem $$\label{eq3} Lu=f, \quad u\in \ring H^m(\Omega),$$ has a unique solution understood in the weak sense. [**Remark**]{}. The weak solution formalism is necessary to treat the Dirichlet boundary problem on an [*arbitrary*]{} bounded domain. When the boundary, the data, and the coefficients of the equation are sufficiently smooth, the weak solution coincides with the [*classical solution*]{} and can be written for a second order operator as $$\label{eq3.1} Lu=f, \qquad u|_{{{\partial\Omega}}}=0,$$ where $u|_{{{\partial\Omega}}}$ is the usual pointwise limit at the boundary. For a $2m$-th order operator the derivatives up to the order $m-1$ must vanish as well, e.g., $$\label{eq3.2} \Delta^2 u=f, \qquad u|_{{{\partial\Omega}}}=0,\qquad \partial_\nu u|_{{{\partial\Omega}}}=0,$$ where $\partial_\nu$ stays for normal derivative at the boundary. In any context, the condition $u\in \ring H^m(\Omega)$ automatically prescribes zero Dirichlet boundary data. On rough domains the definitions akin to , might not make sense, i.e., a pointwise boundary limit might not exist (the solution might be discontinuous at the boundary), and then the Dirichlet data can only be interpreted in the sense of . The classification of domains in which all solutions to the Laplace’s equation are continuous up to the boundary is available due to the celebrated 1924 Wiener criterion [@Wiener]. Over the years, Wiener test has been extended to a variety of operators. We shall not concentrate on this issue, let us just mention the results covering all divergence form second order elliptic operators [@LSW], and the bilaplacian in dimension three [@Mayboroda2009]. Here we shall impose no additional restriction on ${{\partial\Omega}}$ or on the coefficients and work in the general context of weak solutions. 0.08 in For later reference, we also define the Green function of $L$, as conventionally, by $$\label{eq4.1} L_xG(x,y)=\delta_y(x), \quad \mbox{for all} \,x,y\in\Omega, \quad G(\cdot,y)\in \ring H^{m}(\Omega) \quad \mbox{for all} \,y\in\Omega,$$ in the sense of , so that $$\label{eq4.2} \int_{{{\mathbb{R}}}^n}L_xG(x,y) v(x)\,dx=v(y), \quad y\in \Omega,$$ for every $v\in \ring H^{m}(\Omega)$. It is not difficult to show that for a self-adjoint elliptic operator the Green function is symmetric, i.e., $G(x,y)=G(y,x)$, $x,y\in\Omega$. #### Control of the eigenfunctions by the solution to the Dirichlet problem: the landscape {#control-of-the-eigenfunctions-by-the-solution-to-the-dirichlet-problem-the-landscape .unnumbered} Let us now turn to the discussion of the eigenfunctions of $L$. Unless otherwise stated, we assume that $L$ is an elliptic operator in the weak sense described above and that the underlying bilinear form is symmetric, i.e., that $L$ is self-adjoint. The Fredholm theory provides a framework to consider the eigenvalue problem: $$\label{eq8} L\varphi=\lambda \varphi, \quad \varphi\in \ring H^{m}(\Omega),$$ where $\lambda\in{{\mathbb{R}}}$. If there exists a non-trivial solution to , interpreted, as before, in the weak sense then the corresponding $\lambda\in{{\mathbb{R}}}$ is called an eigenvalue and $\varphi\in \ring H^{m}(\Omega)$ is an eigenvector. \[p1\]Let $\Omega$ be an arbitrary bounded open set, $L$ be a self-adjoint elliptic operator on $\Omega$, and assume that $\varphi\in \ring H^{m}(\Omega)$ is an eigenfunction of $L$ and $\lambda$ is the corresponding eigenvalue, i.e., is satisfied. Then $$\label{eq10_sup} \frac{|\varphi(x)|}{\|\varphi\|_{L^{\infty}(\Omega)}}\leq \lambda u(x), \quad\mbox{for all}\,x\in\Omega,$$ provided that $\varphi \in L^\infty(\Omega)$, with $$\label{eq9_sup} u(x)= \int_\Omega |G(x,y)|\,dy, \qquad x\in\Omega.$$ If, in addition, the Green function is non-negative (in the sense of distributions), then $u$ is the solution of the boundary problem $$\label{eq11_sup} Lu=1, \quad u\in \ring H^m(\Omega).$$ [**Remark.**]{} The Green function is positive and eigenfunctions are bounded for the Laplacian , the Hamiltonian , all second order elliptic operators in all dimensions due to the strong maximum principle (see, e.g., [@Gilbarg2001], Section 8.7). Hence, for all such operators , are valid. The situation for the higher order PDEs is more subtle. In fact, even for the bilaplacian the positivity in general fails, and then one has to operate directly with . 0.08in [[*Proof*]{}.]{}By and (with the roles of $x$ and $y$ interchanged) and self-adjointness of $L$, for every $x\in \Omega$ $$\label{eq12} \varphi(x) = \int_\Omega\varphi(y)\, L_y G(x,y)\,dy= \int_\Omega L_y \varphi(y)\, G(x,y)\,dy= \int_\Omega \lambda \,\varphi(y)\, G(x,y)\,dy,$$ and hence, $$\label{eq13} |\varphi(x)|\leq \lambda \,\|\varphi\|_{L^\infty(\Omega)}\int_{\Omega}|G(x,y)|\,dy, \quad x\in \Omega,$$ as desired. Moreover, if the Green function is positive, $$\label{eq14} \int_{\Omega}|G(x,y)|\,dy=\int_{\Omega}G(x,y)\cdot 1\,dy, \quad x\in \Omega,$$ which is by definition a solution of .[$\Box$ 0.08in]{} Vaguely speaking, the inequality provides the “landscape of localization", as the map of $u$ in – draws lines separating subdomains which will “host" localized eigenmodes. We now discuss in details the situation on these subdomains. #### Analysis of localized modes on the subdomains {#analysis-of-localized-modes-on-the-subdomains .unnumbered} The gist of the forthcoming discussion is that, roughly speaking, a mode of $\Omega$ localized to a subdomain $D\subset \Omega$ must be fairly close to an eigenmode of this subdomain, and an eigenvalue of $\Omega$ for which localization takes place, must be close to some eigenvalue of $D$. To this end, let $\varphi$ be one of the eigenmodes of $\Omega$, which exhibits localization to $D$, a subdomain of $\Omega$. This means, in particular, that the boundary values of $\varphi$ on $\partial D$ are small. In fact, in the present context the correct way to interpret “smallness" of $\varphi$ on the boundary of $D$ is in terms of the smallness of an $L$-harmonic function, with the same data as $\varphi$ on $\partial D$. To be rigorous, let us define ${\varepsilon}={\varepsilon}_{\varphi}>0$ as $$\begin{aligned} \label{eq14.1}\nonumber && \mbox{${\varepsilon}=\|v\|_{L^2(D)}$, where $v\in H^m(D)$ is such that} \\ && \mbox{$w:=\varphi-v\in \ring H^m(D)$ (that is, the boundary values of $\varphi$ and $v$ on $\partial D$ coincide),} \\ && \mbox{and $Lv=0$ on $D$ in the sense of distributions.}\nonumber\end{aligned}$$ Then the following Proposition holds. \[pRes\] Let $\Omega$ be an arbitrary bounded open set, $L$ be a self-adjoint elliptic operator on $\Omega$, and $\varphi\in \ring H^{m}(\Omega)$ be an eigenmode of $L$. Suppose further that $D$ is a subset of $\Omega$ and denote by ${\varepsilon}$ the norm of the boundary data of $\varphi$ on $\partial D$ in the sense of . Denote by $\lambda$ the eigenvalue corresponding to $\varphi$. Then either $\lambda$ is an eigenvalue of $L$ in $D$ or $$\label{eq19_sup} \|\varphi\|_{L^2(D)} \leq \left(1+\max_{\lambda_k(D)} \left\{\left|1-\frac{\lambda_k(D)}{\lambda}\right|^{-1}\right\}\right) {\varepsilon},$$ where the maximum is taken over all eigenvalues of $L$ in $D$. [[*Proof*]{}.]{}First of all, note that implies $$\label{eq15} (L-\lambda)w=\lambda v \quad\mbox{on } D,$$ as usually, in the sense of distributions. If $\lambda$ is an eigenvalue of $D$, there is nothing to prove. If $\lambda$ is not an eigenvalue of $D$, we claim that $$\label{eq16} \|w\|_{L^2(D)} \leq \max_{\lambda_k(D)} \left\{\frac{1}{|\lambda-\lambda_k(D)|}\right\} \|\lambda v\|_{L^2(D)}.$$ Indeed, in our setup, the eigenvalues of $L$ are real, positive, at most countable, and moreover, there exists an orthonormal basis of $L^2(D)$ formed by the eigenfunctions of $L$ on $D$, $\{\psi_{k, D}\}_{k}$. In particular, for every $f\in \ring H^m(D)\subset L^2(D)$ we can write $$\label{eq16.1} f=\sum _{k} c_k(f) \psi_k, \qquad c_k(f)=\int_D f\,\psi_k\,dx,$$ with the convergence in $L^2(D)$, and $\|f\|_{L^2(D)}=\left(\sum _{k}c_k(f)^2\right)^{1/2}$. Moreover, such a series $\sum _{k} c_k(f) \psi_k$ converges in $\ring H^m(D)$ as well and $\{\psi_{k, D}\}_k$ form an orthogonal basis of $\ring H^m(D)$. These considerations follow from ellipticity and self-adjointness of $L$ in a standard way using the machinery of functional analysis (see, e.g., [@Evans2010], pp. 355–358 treating the case of the second order operator of the type ). Therefore, for every $\lambda$ not belonging to the spectrum of $L$ on $D$ and $w\in \ring H^m(D)$ with $(L-\lambda) w \in L^2(D)$ (cf. ) we have $$\label{eq16.1.0} \|(L-\lambda)w\|_{L^2(D)} = \left\|\sum_k c_k((L-\lambda)w)\psi_k\right \|_{L^2(D)},$$ where $$\begin{aligned} \label{eq16.1.0.1} c_k((L-\lambda)w)&=&\int_D(L-\lambda)w \,\psi_k\,dx= \int_Dw \,(L-\lambda) \psi_k\,dx\\[4pt] &=& (\lambda_k(D)-\lambda) \int_Dw \, \psi_k\,dx=(\lambda_k(D)-\lambda) c_k(w).\end{aligned}$$ Hence, $$\begin{aligned} \label{eq16.2}\nonumber \|(L-\lambda)w\|_{L^2(D)} &=& \left\|\sum_k (\lambda_k(D)-\lambda) c_k(w)\psi_k\right \|_{L^2(D)}=\left(\sum_k (\lambda_k(D)-\lambda)^2 c_k(w)^2\right)^{1/2}\\[4pt]\nonumber &\geq& \min_{\lambda_k(D)} |\lambda_k(D)-\lambda| \left(\sum_k c_k(w)^2\right)^{1/2}= \min_{\lambda_k(D)} |\lambda_k(D)-\lambda| \,\|w\|_{L^2(D)}, \end{aligned}$$ which leads to inequality . Going further, yields $$\label{eq17} \|w\|_{L^2(D)} \leq \max_{\lambda_k(D)} \left\{\left|1-\frac{\lambda_k(D)}{\lambda}\right|^{-1}\right\} \|v\|_{L^2(D)}\leq \max_{\lambda_k(D)} \left\{\left|1-\frac{\lambda_k(D)}{\lambda}\right|^{-1}\right\} {\varepsilon},$$ and therefore, $$\label{eq18} \|\varphi\|_{L^2(D)} \leq \left(1+\max_{\lambda_k(D)} \left\{\left|1-\frac{\lambda_k(D)}{\lambda}\right|^{-1}\right\}\right) {\varepsilon},$$ as desired.

--- abstract: 'We show existence of solutions to the least gradient problem on the plane for boundary data in $BV(\partial\Omega)$. We also provide an example of a function $f \in L^1(\partial\Omega) \backslash (C(\partial\Omega) \cup BV(\partial\Omega))$, for which the solution exists. We also show non-uniqueness of solutions even for smooth boundary data in the anisotropic case for a nonsmooth anisotropy. We additionally prove a regularity result valid also in higher dimensions.' address: 'W. Górny: Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland.' author: - Wojciech Górny title: 'Planar least gradient problem: existence, regularity and anisotropic case' --- Introduction ============ Many papers, including [@SWZ], [@MNT], [@MRL], [@GRS] describe the least gradient problem, i.e. a problem of minimalization $$\min \{ \int_\Omega |Du|, \quad u \in BV(\Omega), \quad u|_{\partial\Omega} = f \},$$ where we may impose certain conditions on $\Omega$, $f$ and use different approaches to the boundary condition. In [@SWZ] $f$ is assumed to be continuous and the boundary condition is in the sense of traces. They also impose a set of geometrical conditions on $\Omega$, which are satisfied by strictly convex sets; in fact, in dimension two they are equivalent to strict convexity. The authors of [@MNT] also add a positive weight. Another approach is presented in [@MRL], where boundary datum belongs to $L^1(\partial\Omega)$, but the boundary condition is understood in a weaker sense. Throughout this paper $\Omega \subset \mathbb{R}^N$ shall be an open, bounded, strictly convex set with Lipschitz (or $C^1$) boundary. The boundary datum $f$ will belong to $L^1(\partial \Omega)$ or $BV(\partial\Omega)$. We consider the following minimalization problem called the least gradient problem (for brevity denoted by LGP): $$\label{zagadnienie} \inf \{ \int_\Omega |Du|, \quad u \in BV(\Omega), \quad Tu = f \},$$ where $T$ denotes the trace operator $T: BV(\Omega) \rightarrow L^1(\partial\Omega)$. Even existence of solutions in this sense is not obvious, as the functional $$F(u) = \twopartdef{\int_\Omega |Du|}{ u \in BV(\Omega) \text{ and } Tu = f;}{+\infty}{\text{otherwise}}$$ is not lower semicontinuous with respect to $L^1$ convergence. In fact, in [@ST] the authors have given an example of a function $f$ without a solution to corresponding least gradient problem. It was a characteristic function of a certain fat Cantor set. Let us note that it does not lie in $BV(\partial\Omega)$. There are two possible ways to deal with Problem . The first is the relaxation of the functional $F$. Such reformulation and its relationship with the original statement is considered in [@MRL] and [@Maz]. Another way is to consider when Problem has a solution in the classical sense and what is its regularity. This paper uses the latter approach. The main result of the present paper is giving a sufficient condition for existence of solutions of the least gradient problem on the plane. It is given in the following theorem, which will be later proved as Theorem \[tw:istnienie\]: Let $\Omega \subset \mathbb{R}^2$ be an open, bounded, strictly convex set with $C^1$ boundary. Then for every $f \in BV(\partial\Omega)$ there exists a solution of LGP for $f$. Obviously, this condition is not necessary; the construction given in [@SWZ] does not require the boundary data to have finite total variation. We also provide an example of a function $f \in L^1(\Omega) \backslash (C(\partial\Omega) \cup BV(\partial\Omega))$, for which the solution exists, see Example \[ex:cantor\]. Another result included in this article provides a certain regularity property. Theorem \[tw:rozklad\] asserts existence of a decomposition of a function of least gradient into a continuous and a locally constant function. It is not a property shared by all BV functions, see [@AFP Example 4.1]. Let $\Omega \subset \mathbb{R}^N$, where $N \leq 7$, be an open, bounded, strictly convex set with Lipschitz boundary. Suppose $u \in BV(\Omega)$ is a function of least gradient. Then there exist functions $u_c, u_j \in BV(\Omega)$ such that $u = u_c + u_j$ and $(Du)_c = Du_c$ and $(Du)_j = Du_j$, i.e. one can represent $u$ as a sum of a continuous function and a piecewise constant function. They are of least gradient in $\Omega$. Moreover this decomposition is unique up to an additive constant. The final chapter takes on the subject of anisotropy. As it was proved in [@JMN], for an anisotropic norm $\phi$ on $\mathbb{R}^N$ smooth with respect to the Euclidean norm there is a unique solution to the anisotropic LGP. I consider $p-$norms on the plane for $p \in [1, \infty]$ to show that for $p = 1, \infty$, i.e. where the anisotropy is not smooth, the solutions need not be unique even for smooth boundary data (see Examples \[ex:l1\] and \[ex:linfty\]), whereas for $1 < p < \infty$, when the anisotropy is smooth, Theorem \[tw:anizotropia\] asserts that the only connected minimal surface with respect to the p-norm is a line segment, similarly to the isotropic solution. Let $\Omega \subset \mathbb{R}^2$ be an open convex set. Let the anisotropy be given by the function $\phi(x,Du) = \| Du \|_p$, where $1 < p < \infty$. Let $E$ be a $\phi-$minimal set with respect to $\Omega$, i.e. $\chi_E$ is a function of $\phi-$least gradient in $\Omega$. Then every connected component of $\partial E$ is a line segment. Preliminaries ============= Least gradient functions ------------------------ Now we shall briefly recall basic facts about least gradient functions. What we need most in this paper is the Miranda stability theorem and the relationship between functions of least gradient and minimal surfaces. For more information, see [@Giu]. We say that $u \in BV(\Omega)$ is a function of least gradient, if for every compactly supported $($equivalently: with trace zero$)$ $v \in BV(\Omega)$ we have $$\int_\Omega |Du| \leq \int_\Omega |D(u + v)|.$$ We say that $u \in BV(\Omega)$ is a solution of the least gradient problem in the sense of traces $($solution of LGP$)$ for given $f \in L^1(\Omega)$, if $Tu = f$ and for every $v \in BV(\Omega)$ such that $Tv = 0$ we have $$\int_\Omega |Du| \leq \int_\Omega |D(u + v)|.$$ To underline the difference between the two notions, we recall a stability theorem by Miranda: $($[@Mir Theorem 3]$)$ Let $\Omega \subset \mathbb{R}^N$ be open. Suppose $\{ f_n \}$ is a sequence of least gradient functions in $\Omega$ convergent in $L^1_{loc}(\Omega)$ to $f$. Then $f$ is of least gradient in $\Omega$. An identical result for solutions of least gradient problem is impossible, as the trace operator is not continuous in $L^1$ topology. We need an additional assumption regarding traces. A correct formulation would be: \[stabilnosc\] Suppose $f, f_n \in L^1(\partial\Omega)$. Let $u_n$ be a solution of LGP for $f_n$, i.e. $Tu_n = f_n$. Let $f_n \rightarrow f$ in $L^1(\partial\Omega)$ and $u_n \rightarrow u$ in $L^1(\Omega)$. Assume that also $Tu = f$. Then $u$ is a solution of LGP for $f$. To deal with regularity of solutions of LGP, it is convenient to consider superlevel sets of $u$, i.e. sets of the form $\partial \{ u > t \}$ for $t \in \mathbb{R}$. It follows the the two subsequent results: \[lem:jednoznacznoscnadpoziomic\] Suppose $u_1, u_2 \in L^1(\Omega)$. Then $u_1 = u_2$ a.e. iff for every $t \in \mathbb{R}$ the superlevel sets of $u_1$ and $u_2$ are equal, i.e. $\{ u_1 > t \} = \{ u_2 > t \}$ up to a set of measure zero. \[twierdzeniezbgg\] $($[@BGG Theorem 1]$)$\ Suppose $\Omega \subset \mathbb{R}^N$ is open. Let $f$ be a function of least gradient in $\Omega$. Then the set $\partial \{ f > t \}$ is minimal in $\Omega$, i.e. $\chi_{\{ f > t \}}$ is of least gradient for every $t \in \mathbb{R}$. It follows from [@Giu Chapter 10] that in low dimensions $(N \leq 7)$ the boundary $\partial E$ of a minimal set $E$ is an analytical hypersurface $($after modification of $E$ on a set of measure zero$)$. Thus, as we modify each superlevel set of $u$ by a set of measure zero, from Lemma \[lem:jednoznacznoscnadpoziomic\] we deduce that the class of $u$ in $L^1(\Omega)$ does not change. After a change of representative we get that the boundary of each superlevel set of $u$ is a sum of analytical minimal surfaces; thus, we may from now on assume that we deal with such a representative. Also, several proofs are significantly simplified if we remember that in dimension two there is only one minimal surface: an interval. Sternberg-Williams-Ziemer construction -------------------------------------- In [@SWZ] the authors have shown existence and uniqueness of solutions of LGP for continuous boundary data and strictly convex $\Omega$ (or, to be more precise, the authors assume that $\partial \Omega$ has non-negative mean curvature and is not locally area-minimizing). The proof of existence is constructive and we shall briefly recall it. The main idea is reversing Theorem \[twierdzeniezbgg\] and constructing almost all level sets of the solution. According to the Lemma \[lem:jednoznacznoscnadpoziomic\] this uniquely determines the solution. We fix the boundary data $g \in C(\partial \Omega)$. By Tietze theorem it has an extension $G \in C(\mathbb{R}^n \backslash \Omega)$. We may also demand that $G \in BV(\mathbb{R}^n \backslash \overline{\Omega})$. Let $L_t = (\mathbb{R}^n \backslash \Omega) \cap \{ G \geq t \}$. Since $G \in BV(\mathbb{R}^n \backslash \overline{\Omega})$, then for a.e. $t \in \mathbb{R}$ we have $P(L_t, \mathbb{R}^n \backslash \overline{\Omega}) < \infty$. Let $E_t$ be a set solving the following problems: $$\label{sternbergminimalnadlugosc} \min \{ P(E, \mathbb{R}^n): E \backslash \overline{\Omega} = L_t \backslash \overline{\Omega} \},$$ $$\max \{ |E|: E \text{ is a minimizer of \eqref{sternbergminimalnadlugosc}} \}.$$ Let us note that both of these problems have solutions; let $m \geq 0$ be the infimum in the first problem. Let $E_n$ be a sequence of sets such that $P(E_n, \Omega) \rightarrow m$. By compactness of unit ball in $BV(\Omega)$ and lower semicontinuity of the total variation we obtain $\chi_{E_{n_k}} \rightarrow \chi_E$, where $$m \leq P(E, \Omega) \leq P(E_n, \Omega) \rightarrow m.$$ Take $M \leq |\Omega|$ be the supremum in the second problem. Take a sequence o sets $E_n$ such that $|E_n| \rightarrow M$. Then on some subsequence $\chi_{E_{n_k}} \rightarrow \chi_E$, and thus $$M \geq |E| \geq |E_n| - |E_n \triangle E| = |E_n| - \|\chi_{E_n} - \chi_E \|_1 \rightarrow M - 0.$$ Then we can show existence of a set $T$ of full measure such that for every $t \in T$ we have $\partial E_t \cap \partial \Omega \subset g^{-1}(t)$ and for every $t,s \in T$, $s < t$ the inclusion $E_t \subset \subset E_s$ holds. It enables us to treat $E_t$ as superlevel sets of a certain function; we define it by the following formula: $$u(x) = \sup \{t \in T: x \in \overline{E_t \cap \Omega} \}.$$ It turns out that $u \in C(\overline{\Omega}) \cap BV(\Omega)$ and $u$ is a solution to LGP for $g$. Moreover $| \{ u \geq t \} \triangle (\overline{E_t \cap \Omega})| = 0$ for a.e. $t$. Uniqueness proof is based on a maximum principle. In the existence proof in chapter $4$ we are going to use a particularly simple case of the construction. Suppose $\Omega \subset \mathbb{R}^2$ and that $f \in C^1(\partial\Omega)$. Firstly, let us notice that we only have to construct the set $E_t$ for almost all $t$. Secondly, we recall that in dimension $2$ the only minimal surfaces are intervals; thus, to find the set $E_t$, let us fix $t$ and look at the preimage $g^{-1}(t)$. We connect its points with intervals with sum of their lengths as small as possible. It can cause problems, for example if we take $t$ to be a global maximum of the function; thus, let us take $t$ to be a regular value (by Sard theorem almost all values are regular), so the preimage $f^{-1}(t)$ is a manifold. In dimension $2$ this means that the preimage contains finitely many points, because $f$ is Lipschitz and $\partial\Omega$ is compact. As the derivative at every point $p \in f^{-1}(t)$ is nonzero, there is at least one interval in $\partial E_t$ ending in $p$. As is established later in Proposition \[slabazasadamaksimum2\], by minimality of $\partial E_t$ there can be at most one, so there is exactly one interval in $\partial E_t$ ending in every $p \in f^{-1}(t)$. A typical example for the construction, attributed to John Brothers, is to let $\Omega = B(0,1)$ and take the boundary data to be (in polar coordinates, for fixed $r = 1$) the function $f: [0, 2\pi) \rightarrow \mathbb{R}$ given by the formula $f(\theta) = \cos(2 \theta)$; see [@MRL Example 2.7] or [@SZ Example 3.6]. BV on a one-dimensional compact manifold ---------------------------------------- In the general case one may attempt to define BV spaces on compact manifolds using partition of unity; such approach is presented in [@AGM]. It is not necessary for us; it suffices to consider one-dimensional case. Let us consider $\Omega \subset \mathbb{R}^2$ open, bounded with $C^1$ boundary. We may define on $\partial\Omega$ the Hausdorff measure, integrable functions $($which are appoximatively continuous a.e.$)$. We recall (see [@EG Chapter 5.10]) that the one-dimensional $BV$ space on the interval $(a,b) \subset \mathbb{R}$ may be described in the following way: $$f \in BV((a,b)) \Leftrightarrow \sum |f(x_{i})-f(x_{i-1})| \leq M < \infty$$ for every $a < x_0 < ... < x_n < b$, where $x_i$ are points of approximate continuity of $f$. The smallest such constant $M$ turns out to be the usual total variation of $f$. We may extend this definition to the case where we have a one-dimensional manifold diffeomorphic to an open interval if it is properly parametrized, i.e. all tangent vectors have length one. Repeating the proof from [@EG] we get that this definition coincides with the divergence definition. Then we extend it to the case of a one-dimensional compact connected manifold in the following way: We say that $f \in BV(\partial\Omega)$, if after removing from $\partial\Omega$ a point $p$ of approximate continuity of $f$ we have $f \in BV(\partial\Omega \backslash \{ p \})$. The norm is defined to be $$\| f \|_{BV(\partial\Omega)} = \| f \|_1 + \| f \|_{BV(\partial\Omega \backslash \{ p \})}.$$ This definition does not depend on the choice of $p$, as in dimension one the total variation on disjoint intervals is additive, thus for different points $p_1, p_2$ we get that $$\| f \|_{BV(\partial\Omega \backslash \{ p_1 \})} = \| f \|_{BV((p_1,p_2))} + \| f \|_{BV((p_2,p_1))} = \| f \|_{BV(\partial\Omega \backslash \{ p_2 \})},$$ where $(p_1,p_2)$ is an oriented arc from $p_1$ to $p_2$. Thus all local properties of $BV(\partial\Omega)$ hold; we shall recall the most important one for our considerations: \[stw:bvdim1\] Let $E \subset \partial\Omega$ be a set of finite perimeter, i.e. $\chi_E \in BV(\partial\Omega)$. Then, if we take its representative to be the set of points of density $1$, then $\partial E = \partial^{*} E = \{ p_1, ..., p_{2n} \}$ and $P(E, \partial\Omega) = 2n$. Here $\partial^{*} E$ denotes the reduced boundary of $E$, i.e. the set where a measure-theoretical normal vector exists; see [@EG Chapter 5.7]. However, some global properties need not hold. For example, the decomposition theorem $f = f_{ac} + f_j + f_s$ does not hold; consider $\Omega = B(0,1)$, $f = \arg (z)$. The main reason is that $\pi_1(\partial\Omega) \neq 0$. Regularity of least gradient functions ====================================== In this section we are going to prove several regularity results about functions of least gradient, valid up to dimension $7$. We start with a weak form of the maximum principle and later prove a result on decomposition of a least gradient function into a continuous and jump-type part; this decomposition holds not only at the level derivatives, but also at the level of functions. We will extensively use the blow-up theorem, see [@EG Section 5.7.2]. \[tw:blowup\] For each $x \in \partial^{*} E$ define the set $E^r = \{ y \in \mathbb{R}^N: r(y-x) + x \in E \}$ and the hyperplane $H^{-}(x) = \{ y \in \mathbb{R}^N: \nu_E (x) \cdotp (y - x) \leq 0 \}$. Then $$\chi_{E^r} \rightarrow \chi_{H^{-}(x)}$$ in $L^1_{loc}(\mathbb{R}^N)$ as $r \rightarrow 0$. It turns out that on the plane Theorem \[twierdzeniezbgg\] may be improved to an analogue of the maximum principle for linear equations; geometrically speaking, the linear weak maximum principle states that every level set touches the boundary. \[slabazasadamaksimum\] $($weak maximum principle on the plane$)$\ Let $\Omega \subset \mathbb{R}^2$ be an open bounded set with Lipschitz boundary and suppose $u \in BV(\Omega)$ is a function of least gradient. Then for every $t \in \mathbb{R}$ the set $\partial \{ u > t \}$ is empty or it is a sum of intervals, pairwise disjoint in $\Omega$, such that every interval connects two points of $\partial \Omega$. By the argument from [@Giu Chapter 10] for every $t \in \mathbb{R}$ the set $\partial \{ u > t \}$ is a sum of intervals and $\partial \{ u > t \} = \partial^* \{ u > t \}$. Obviously $\partial \{ u > t \}$ is closed in $\Omega$. Suppose one of those intervals ends in $x \in \Omega$. Then the normal vector at $x$ is not well defined (the statement of the Theorem \[tw:blowup\] does not hold), so $x \notin \partial^* \{ u > t \}$. Thus $x \notin \partial \{ u > t \}$, contradiction. Similarly suppose two such intervals intersect in $x \in \Omega$. Then the measure-theoretic normal vector at $x$ has length smaller then $1$, depending on the angle between the two intervals. Thus $x \notin \partial^* \{ u > t \}$, contradiction. If we additionally assume that $\Omega$ is convex, then the union is disjoint also on $\partial\Omega$: \[slabazasadamaksimum2\] Let $\Omega \subset \mathbb{R}^2$ be an open, bounded, convex set with Lipschitz boundary and suppose $u \in BV(\Omega)$ is a function of least gradient. Then for every $t \in \mathbb{R}$ the set $\partial \{ u > t \}$ is empty or it is a sum of intervals, pairwise disjoint in $\overline{\Omega}$, such that every interval connects two points of $\partial \Omega$. Suppose that at least two intervals in $\partial E_t$ end in $x \in \partial\Omega$: $\overline{xy}$ and $\overline{xz}$. We have two possibilities: there are countably many intervals in $\partial E_t$, which end in $x$, with the other end lying in the arc $\overline{yz} \subset \partial\Omega$ which does not contain $x$; or there are finitely many. The first case is excluded by the monotonicity formula for minimal surfaces, see for example [@Sim Theorem 17.6, Remark 37.9], as from Theorem \[twierdzeniezbgg\] $E$ is a minimal set and only finitely many connected components of the boundary of a minimal set may intersect any compact subset of $\Omega$. In the second case we may without loss of generality assume that $\overline{xy}$ and $\overline{xz}$ are adjacent. Consider the function $\chi_{E_t}$. In the area enclosed by the intervals $\overline{xy}, \overline{xz}$ and the arc $\overline{yz} \subset \partial\Omega$ not containing $x$ we have $\chi_{E_t} = 1$ and $\chi_{E_t} = 0$ on the two sides of the triangle (or the opposite situation, which we handle similarly). Then $\chi_{E_t}$ is not a function of least gradient: the function $\widetilde{\chi_{E_t}} = \chi_{E_t} - \chi_{\Delta xyz}$ has strictly smaller total variation due to the triangle inequality. This contradicts Theorem \[twierdzeniezbgg\]. The result above is sharp. As the following example shows, we may not relax the assumption of convexity of $\Omega$. Denote by $\varphi$ the angular coordinate in the polar coordinates on the plane. Let $\Omega = B(0,1) \backslash (\{ \frac{\pi}{4} \leq \varphi \leq \frac{3\pi}{4} \} \cup \{ 0 \}) \subset \mathbb{R}^2$, i.e. the unit ball with one quarter removed. Take the boundary data $f \in L^1(\partial \Omega)$ to be $$f(x,y) = \twopartdef{1}{y \geq 0}{0}{y < 0.}$$ Then the solution to the least gradient problem is the function (defined inside $\Omega$) $$u(x,y) = \twopartdef{1}{y \geq 0}{0}{y < 0,}$$ in particular $\partial \{ u \geq 1 \}$ consists of two horizontal line segments whose closures intersect at the point $(0,0) \in \partial\Omega$. Note that in this example the set $\Omega$ is star-shaped, but it is not convex. In higher dimensions, we are going to need a result from [@SWZ] concerning minimal surfaces: \[stw:sternbergpowmin\] $($[@SWZ Theorem 2.2])\ Suppose $E_1 \subset E_2$ and let $\partial E_1, \partial E_2$ are area-minimizing in a open set $U$. Further, suppose $x \in \partial E_1 \cap \partial E_2 \cap U$. Then $\partial E_1$ and $\partial E_2$ agree in some neighbourhood of $x$. $($weak maximum principle$)$\ Let $\Omega \subset \mathbb{R}^N$, where $N \leq 7$ and suppose $u \in BV(\Omega)$ is a function of least gradient. Then for every $t \in \mathbb{R}$ the set $\partial \{ u > t \}$ is empty or it is a sum of minimal surfaces $S_{t,i}$, pairwise disjoint in $\Omega$, which satisfy $\partial S_{t,i} \subset \partial \Omega$. Let us notice, that with only subtle changes the previous proof works also in the case $N \leq 7$, i.e. when boundaries of superlevel sets are minimal surfaces. From [@Giu Chapter 10] it follows that for $t \in \mathbb{R}$ the set $\partial \{ u > t \}$ is a sum of minimal surfaces $S_{t,i}$ and $\partial \{ u > t \} = \partial^* \{ u > t \}$. Obviously $\partial \{ u > t \}$ $($boundary in topology of $\Omega)$ is closed in $\Omega$, so $\partial S_{t,i} \cap \Omega = \emptyset$ $($boundary in topology of $\partial \{ u > t \})$; suppose otherwise. Let $x \in \partial S_{t,i} \cap \Omega$. Then in $x$ the blow-up theorem does not hold, so $x \notin \partial \{ u > t \}$, contradiction. Now suppose that $S_{t,i}$ and $S_{t,j}$ are not disjoint in $\Omega$. Then from the Proposition \[stw:sternbergpowmin\] applied to $E_1 = E_2 = \{ u > t \}$ we get $S_{t,i} = S_{t,j}$. Let $E_1 \subset E_2$ and suppose that $E_1$ and $E_2$ are sets of locally bounded perimeter and let $x \in \partial^{*} E_1 \cap \partial^{*} E_2$. Then $\nu_{E_1}(x) = \nu_{E_2}(x)$. We are going to use the blow-up theorem (Theorem \[tw:blowup\]). First notice that the inclusion $E_1 \subset E_2$ implies $$E_1^r = \{ y \in \mathbb{R}^N: r(y-x) + x \in E_1 \} \subset \{ y \in \mathbb{R}^N: r(y-x) + x \in E_2 \} = E_2^r.$$ We keep the same notation as in Theorem \[tw:blowup\] and use it to obtain $$\chi_{H_1^-(x)} \leftarrow \chi_{E_1^r} \leq \chi_{E_1^r} \rightarrow \chi_{H_2^-(x)},$$ where the convergence holds in $L^1_{loc}$ topology. Thus $H_1^-(x) = H_2^-(x)$, so $\nu_{E_2}(x) = \nu_{E_2}(x)$. \[stw:zbiorskokow\] For $u \in BV(\Omega)$ the structure of its jump set is as follows: $$J_u = \bigcup_{s,t \in \mathbb{Q}; s \neq t} (\partial^{*} \{ u > s \} \cap \partial^{*} \{ u > t \}).$$ Let $x \in J_u$. By definition of $J_u$ the normal vector at $x$ is well defined. The same applies to the trace values from both sides: let us denote them by $u^{-} (x) < u^{+} (x)$. But then there exist $s,t \in \mathbb{Q}$ such that $u^{-} (x) < s < t < u^{+} (x)$, so $x \in \partial^{*} \{ u > s \} \cap \partial^{*} \{ u > t \}$. On the other hand, let $x \in \partial^{*} \{ u > s \} \cap \partial^{*} \{ u > t \}$. From the previous proposition the normal vectors coincide, so the normal at $x$ does not depend on $t$ and we may define traces from both sides as $$u^{+}(x) = \sup \{ t: x \in \partial^{*} \{ u > s \} \cap \partial^{*} \{ u > t \} \};$$ $$u^{-}(x) = \inf \{ t: x \in \partial^{*} \{ u > s \} \cap \partial^{*} \{ u > t \} \}.$$ More precisely, the trace is uniquely determined up to a measure zero set by the mean integral property from [@EG Theorem 5.3.2]. But it holds for all $x \in \partial^{*} \{ u > s \} \cap \partial^{*} \{ u > t \}$; from the weak maximum principle this set divides $\Omega$ into two disjoint parts, $\Omega^+$ and $\Omega^-$. Let $\Omega^+$ be the part with greater values of $u$ in the neighbourhood of the cut. If $u^+(x) < \sup \{ t: x \in \partial^{*} \{ u > s \} \cap \partial^{*} \{ u > t \} \} = s$, then for sufficiently small neighbourhoods of $x$ we would have $u \geq s$, so $\fint_{B(x,r) \cap \Omega^+} |u^+(x) - u(y)| \geq |u^+(x) - s| > 0$, contradiction. The other cases are analogous. Suppose $u \in BV(\Omega)$ is a least gradient function. Then $J_u = \bigcup_{k=1}^{\infty} S_k$, where $S_k$ are pairwise disjoint minimal surfaces. In addition, the trace of $u$ from both sides is constant along $S_k$; in particular the jump of $u$ is constant along $S_k$. We follow the characterisation of $J_u$ from the Proposition \[stw:zbiorskokow\]. For every $t$ the set $\partial^{*} \{ u > t \}$ is a minimal surface. Proposition \[stw:sternbergpowmin\] ensures that if $\partial^{*} \{ u > s \} \cap \partial^{*} \{ u > t \} \neq \emptyset$, then their intersecting connected components $S_{s,i}$, $S_{t,j}$ coincide. In particular, the trace from both sides defined as above is constant along $S_{t,j}$. Thus connected components of $J_u$ coincide with connected components of $\partial^{*} \{ u > t \}$ for some $t$, so by weak maximum principle they are minimal surfaces non-intersecting in $\Omega$ with boundary in $\partial \Omega$. As the area of each such surface is positive, there is at most countably many of them. \[tw:rozklad\] Let $\Omega \subset \mathbb{R}^N$, where $N \leq 7$, be an open, bounded, strictly convex set with Lipschitz boundary. Suppose $u \in BV(\Omega)$ is a function of least gradient. Then there exist functions $u_c, u_j \in BV(\Omega)$ such that $u = u_c + u_j$ and $(Du)_c = Du_c$ and $(Du)_j = Du_j$, i.e. one can represent $u$ as a sum of a continuous function and a piecewise constant function. They are of least gradient in $\Omega$. Moreover this decomposition is unique up to an additive constant. 1\. From the previous theorem $J_u = \bigcup_{k=1}^{\infty} S_k$, where $S_k$ are pairwise disjoint minimal surfaces with boundary in $\partial\Omega$. The jump along each of them has a constant value $a_k$. They divide $\Omega$ into open, pairwise disjoint sets $U_i$. 2\. We define $u_j$ in the following way: let us call any of the obtained sets $U_0$. Let us draw a graph such that the sets $U_i$ are its vertices. $U_i$ and $U_j$ are connected by an edge iff $\partial U_i \cap U_j = S_k$, i.e. when they have a common part of their boundaries. To such an edge we ascribe a weight $a_k$. Example of such construction is presented on the picture above. As $S_k$ are disjoint in $\Omega$ and do not touch the boundary, then such a graph is a tree, i.e. it is connected and there is exactly one path connecting two given vertices. Thus, we define $u_j$ by the formula $$u_j(x) = \sum_{\text{path connecting } U_0 \text{ with } U_i} a_k, \text{ when } x \in U_j.$$ Such a function is well defined, as our graph has no cycles. It also does not depend on the choice of $U_0$ up to an additive constant (if we chose some $U_1$ instead, the function would change by a summand $\sum_{\text{path connecting } U_0 \text{ with } U_1} a_k)$. We see that $u_j \in L^1(\Omega)$ and that it is piecewise constant.\ 3. We notice that $Du_j = (Du)_j$, as $u_j$ is constant on each $U_i$, $J_{u_j} = J_u$ and the jumps along connected components of $J_u$ have the same magnitude. Thus we define $u_c = u - u_j$. We see that $(Du_c)_j = 0$.\ 4. The $u_c$, $u_j$ defined above are functions of least gradient. Suppose that $u_j$ is not a a function of least gradient, i.e. there exists $v \in BV(\Omega)$ such that $\int_\Omega |Dv| < \int_\Omega |Du_j|$ and $Tu_j = Tv$. Then we would get $$\int_\Omega |Du| \leq \int_\Omega |D(u_c + v)| \leq \int_\Omega |Du_c| + \int_\Omega |Dv| < \int_\Omega |Du_c| + \int_\Omega |Du_j| = \int_\Omega |Du|,$$ where the first inequality follows from $u$ being a function of least gradient, and the last equality from measures $Du_c$ and $Du_j$ being mutually singular. The proof for $u_c$ is analogous. 5\. The function $u_c$ is continuous. As $u_c$ is of least gradient, then if it isn’t continuous at $x \in \Omega$, then a certain set of the form $\partial \{ u_c > t \}$ passes through $x$; otherwise $u_c$ would be constant in the neighbourhood of $x$. But in that case $u_c$ has a jump along the whole connected component of $\partial \{ u_c > t \}$ containing $x$, which is impossible as $(Du_c)_j = 0$. 6\. What is left is to prove uniqueness of such a decomposition. Let $u = u_c^1 + u_j^1 = u_c^2 + u_j^2$. Changing the order of summands we obtain $$u_c^1 - u_c^2 = u_j^2 - u_j^1,$$ but the distributional derivative of the left hand side is a continuous measure, and the distributional derivative of the right hand side is supported on the set of zero measure with respect to $\mathcal{H}^{n-1}$, so both of them are zero measures. But the condition $Dv = 0$ implies $v = \text{const}$, so the functions $u_c^1$, $u_c^2$ differ by an additive constant. In this decomposition $u_c$ isn’t necessarily continuous up to the boundary. Let us use the complex numbers notation for the plane. We take $\Omega = B(1,1)$. Let the boundary values be given by the formula $f(z) = \arg(z)$. Then $u = u_c = \arg(z) \in BV(\Omega) \cap C^{\infty}(\Omega)$, but $u$ isn’t continuous at $0 \in \partial\Omega$. Existence of solutions on the plane =================================== We shall prove existence of solutions on the plane for boundary data in $BV(\partial\Omega)$. We are going to use approximations of the solution in strict topology. Proposition \[podciagzbiezny\] will ensure us that existence of convergent sequences of approximations in $L^1$ topology is not a problem; Theorem \[tw:scislazb\] will upgrade it to strict convergence. The Miranda stability theorem (Theorem \[stabilnosc\]) ends the proof. Later, we shall see an example of a discontinuous function $f$ of infinite total variation such that the solution to the LGP exists. \[podciagzbiezny\] Suppose $f_n \rightarrow f$ in $L^1(\partial\Omega)$. $u_n$ are solutions of LGP for $f_n$. Then $u_n$ has a convergent subsequence, i.e. $u_{n_k} \rightarrow u$ in $L^1(\Omega)$. As the trace operator is a surjection, by the Open Mapping Theorem it is open. Let us fix $\widetilde{f} \in BV(\Omega)$ such that $T\widetilde{f} = f$ and a sequence of positive numbers $\varepsilon_n \rightarrow 0$. Then by continuity and openness of $T$ the image of a ball $B(\widetilde{f}, \varepsilon_n)$ contains a smaller ball $B(T\widetilde{f}, \delta_n)$ for another sequence of positive numbers $\delta_n \rightarrow 0$. As $f_n \rightarrow f$ in $L^1(\partial\Omega)$, there exists a subsequence $f_{n_k}$ such that $f_{n_k} \in B(f, \delta_n) = B(T\widetilde{f}, \delta_n)$, so the set $T^{-1}(f_{n_k})$ is non-empty; there exists a preimage of $f_{n_k}$ by $T$ in $B(\widetilde{f}, \varepsilon_n)$. Let us call it $\widetilde{f_n}$. Obviously $\widetilde{f_n} \rightarrow \widetilde{f}$ in $BV(\Omega)$. Thus, after possibly passing to a subsequence, there exist functions $\widetilde{f_n}, \widetilde{f}$ such that $\widetilde{f_n} \rightarrow \widetilde{f}$ in $BV(\Omega)$ and $T\widetilde{f_n} = f_n, T\widetilde{f} = f$. Now we may proceed as in [@HKLS Proposition 3.3]. Let us estimate from above the norm of $\|u_n - \widetilde{f_n}\|_{BV}$: $$\begin{gathered} \|u_n - \widetilde{f_n}\|_{BV} = \|u_n - \widetilde{f_n}\|_1 + \int_\Omega |D(u_n - \widetilde{f_n})| \leq (C + 1) \int_\Omega |D(u_n - \widetilde{f_n})| \leq \\ \leq (C + 1) (\int_\Omega |Du_n| + \int_\Omega |D\widetilde{f_n}|) \leq 2(C + 1) \int_\Omega |D\widetilde{f_n}| \leq M < \infty\end{gathered}$$ where the inequalities follow from Poincaré inequality $($as $u_n - f_n$ has trace zero$)$, triangle inequality and the fact that $u_n$ is solution of LGP for $f_n$. The common bound follows from convergence of $\widetilde{f_n}$. Thus, by compactness of the unit ball of $BV(\Omega)$ in $L^1(\Omega)$ we get a convergent subsequence $u_{n_k} - \widetilde{f_{n_k}} \rightarrow v$ in $L^1(\Omega)$. But $\widetilde{f_n} \rightarrow \widetilde{f}$ in $BV(\Omega)$, so as well in $L^1(\Omega)$; thus $u_{n_k} \rightarrow v + \widetilde{f} = u$ in $L^1(\Omega)$. We are going to need three lemmas. The first two are straightforward and their proofs can be found as a step in the proof of co-area formula, see [@EG Section 5.5]. The third one is a convenient version of Fatou lemma. \[lem:zb1\] Let $f_n \rightarrow f$ in $L^1(\Omega)$. Then there exists a subsequence $f_{n_k}$ such that $\chi_{\{ f_{n_k} \geq t \}} \rightarrow \chi_{\{ f \geq t \}}$ in $L^1(\Omega)$ for a.e. $t$. \[lem:zb2\] Suppose $\chi_{\{ f_{n} \geq t \}} \rightarrow \chi_{\{ f \geq t \}}$ in $L^1(\Omega)$ for a.e. $t$. Then $f_n \rightarrow f$ in $L^1_{loc}(\Omega)$. If additionally $f, f_n$ form a bounded family in $L^{\infty}(\Omega)$, then this covergence holds also in $L^1(\Omega)$. \[lem:zb3\] Suppose that $g, g_n \geq 0$. If additionally $g \leq \liminf g_n$ a.e. and $\lim \int_\Omega g_n \, dx = \int_\Omega g \, dx < \infty$, then $g_n \rightarrow g$ in $L^1(\Omega)$. Let $f_+ = \max(f, 0)$ and $f_- = \max(-f, 0)$. Let us note that $$\int_\Omega |g - g_n| = \int_\Omega (g - g_n)_+ + \int_\Omega (g - g_n)_-$$ and $$0 \leftarrow \int_\Omega (g - g_n) = \int_\Omega (g - g_n)_+ - \int_\Omega (g - g_n)_-,$$ so it suffices to prove that $\int_\Omega (g - g_n)_+ \rightarrow 0$ to show that $g_n \rightarrow g$ in $L^1(\Omega)$. Now let us see what happens to (well defined) upper limit of the sequence $\int_\Omega (g - g_n)_+$: $$0 \leq \limsup \int_\Omega (g - g_n)_+ \leq \int_\Omega \limsup (g - g_n)_+ = \int_\Omega \limsup \max(g - g_n, 0) =$$ $$= \int_\Omega \max(g + \limsup (- g_n), 0) = \int_\Omega \max(g - \liminf g_n), 0) = \int_\Omega 0 = 0.$$ where inequality follows from the (inverse) Fatou lemma: by definition $0 \leq (g - g_n)_+ \leq g$, and $g$ is integrable, so we can apply the Fatou lemma. To prove equalities we use the fact that $\limsup (- g_n) = - \liminf g_n$ and the assumption that $g \leq \liminf g_n$ a.e. Thus $\int_\Omega (g - g_n)_+ \rightarrow 0$, so $g_n \rightarrow g$ in $L^1(\Omega)$. \[tw:scislazb\] Let $\Omega \subset \mathbb{R}^2$ be an open, bounded, strictly convex set with $C^1$ boundary and suppose $f \in BV(\partial\Omega)$. Let $f_n \rightarrow f$ strictly in $BV(\partial\Omega)$, where $f_n$ are smooth. Denote the unique solution of LGP for $f_n$ by $u_n$. Then on some subsequence $u_{n_k}$ we have strict convergence in $BV(\Omega)$ to a function $u \in BV(\Omega)$. In particular $Tu = f$. 1\. As we have $f_n \rightarrow f$ strictly in $BV(\partial\Omega)$, we by definition also convergence in $L^1(\partial\Omega)$. Thus, by Lemma \[lem:zb1\], after possibly passing to a subsequence we have convergence $\chi_{\{ f_n \geq t \}} \rightarrow \chi_{\{ f \geq t \}}$ for a.e. $t$. 2\. By co-area formula $$\int_{\partial\Omega} |Df_n| = \int_{\mathbb{R}} P(E_t^n, \partial\Omega) \, dt \rightarrow \int_{\mathbb{R}} P(E_t, \partial\Omega) dt = \int_{\partial \Omega} |Df|,$$ and lower semicontinuity of the total variation gives us $P(E_t, \partial\Omega) \leq \liminf P(E_t^n, \partial\Omega) < \infty$ for a.e. $t$. We observe that the conditions in Lemma \[lem:zb3\] are fulfilled and we obtain convergence $P(E_t^n, \partial\Omega) \rightarrow P(E_t, \partial\Omega)$ in $L^1(\mathbb{R})$, so after possibly passing to a subsequence we have pointwise convergence for a.e. $t$. Consequently $\chi_{\{ f_n \geq t \}} \rightarrow \chi_{\{ f \geq t \}}$ strictly in $BV(\partial\Omega)$. 3\. As $\partial\Omega \in C^1$ and $f_n \in C^1(\partial\Omega)$, then by Sard theorem the set $\mathcal{T}$ of such $t$, which are regular values for all $f_n$, is of full measure. Recalling the Sternberg-Williams-Ziemer construction we get that for every $t \in \mathcal{T}$ every point of $\partial E_t^n \cap \partial\Omega$ is an end of at least one interval; according to Proposition \[slabazasadamaksimum2\] it is an end of exactly one interval. 4\. From now on it is necessary that we are in dimension $N = 2$. Let $t \in \mathcal{T}$. As $\partial\Omega$ is one-dimensional, then $P(E_t^n, \partial\Omega) \in \mathbb{N}$ and $D \chi_{\{ f_n \geq t \}}$ is a sum $\sum_{i = 1}^M \pm \delta_{x_i}$. Furthermore, by Proposition \[stw:bvdim1\] there exists a representative of the set $E_t^n$, which is a sum of closed arcs between consecutive points $x_i$. By Lemma \[lem:jednoznacznoscnadpoziomic\] we can change all representatives of the sets $E_t^n$ not changing $f_n$ itself. We do the same for $E_t$. As $f_n$ are smooth functions, such form of $E_t^n$ follows directly from their smoothness; this needs not be the case for $E_t$. 5\. As $\chi_{\{ f_n \geq t \}} \rightarrow \chi_{\{ f \geq t \}}$ strictly, then for sufficiently large $n$ $P(E_t^n, \partial\Omega) = P(E_t, \partial\Omega)$. What is more, their derivatives converge in weak\* topology; but we have an exact representation of those derivative. This gives us convergence $x_i^n \rightarrow x_i$ for every $i$. 6\. We apply the Sternberg-Williams-Ziemer construction to the sequence $f_n$. The set $\partial E_t^n$ is a sum of intervals, disjoint in $\Omega$, connecting certain pairs of points among $x_i^n$. By definition of $\mathcal{T}$ every point of $\partial E_t^n \cap \partial\Omega$ is an end of exactly one interval. This gives us convergence $\chi_{\{ u_n \geq t \}} \rightarrow \chi_{\{ u \geq t \}}$ w $L^1(\Omega)$ for a.e. $t$. Because of continuity of the metric in $\mathbb{R}^2$ we get $P(E_t^n, \Omega) = \sum \| x_i^n - x_j^n \| \rightarrow \sum \| x_i - x_j \| = P(E_t, \Omega)$. 7\. Let us see that $P(E_t^n, \Omega) \leq P(\Omega, \mathbb{R}^N)$. Indeed, $\partial E_t^n$ is a sum of intervals, disjoint in $\Omega$, connecting certain pairs of points among $x_i^n$. If we choose a different connection between them, for example by drawing a full convex polygon with vertices in $x_i^n$, by minimality of $\partial E_t^n$ the polygon has a larger perimeter. If we use arcs on $\partial\Omega$ instead, the perimeter would be even larger, as intervals are minimal surfaces in $\mathbb{R}^2$. 8\. Since the functions $\chi_{\{ u_n \geq t \}}$ converge in $L^1(\Omega)$ for a.e. $t$ to $\chi_{\{ u \geq t \}}$, then by Lemma \[lem:zb2\] we have convergence $u_n \rightarrow u$ in $L^1(\Omega)$. Furthermore in step $6$ we proved convergence $P(E_t^n, \Omega) \rightarrow P(E_t, \Omega)$ for a.e. $t$, so by dominated convergence theorem (by step $7$ this sequence is bounded) we have convergence $P(E_t^n, \Omega) \rightarrow P(E_t, \Omega)$ in $L^1(\mathbb{R})$. By co-area formula $\int_\Omega |Du_n| \rightarrow \int_\Omega |Du|$, which gives that $u_n \rightarrow u$ strictly in $BV(\Omega)$. \[tw:istnienie\] Let $\Omega \subset \mathbb{R}^2$ be an open, bounded, strictly convex set with $C^1$ boundary. Then for every $f \in BV(\partial\Omega)$ there exists a solution of LGP for $f$. For each $f \in BV(\partial\Omega)$ we can find a sequence $f_n$ of class $C^{\infty}(\partial\Omega)$ strictly convergent to $f$. Let $u_n$ be solutions of LGP for $f_n$. Then after possibly passing to subsequence we have that $u_n \rightarrow u$ strictly in $BV(\Omega)$; but the trace is continuous in the strict topology, so $Tu = f$. Thus by Miranda stability theorem (Theorem \[stabilnosc\]) we get that $u$ is a solution of LGP for $f$. Take $\Omega = B(0,1)$. As we know from [@ST], when $f$ is a characteristic function of a certain fat Cantor set, then the least gradient problem has no solution. Thus, we would expect that if we approximated the boundary function and constructed solutions of LGP for the approximation, then the trace of the limit would be incorrect. To settle this, let $f_n$ be a function of the $n-$th stage of the Cantor set construction. Then $u_n \rightarrow 0$ in $L^1(\Omega)$: Let $f_0(\theta) = \chi_{[0,1]}$. We construct $f_1$ by removing from the middle of $[0,1]$ an interval of length $2^{-2}$, i.e. $f_1 = \chi_{[0,3/8] \cup [5/8,1]}$. In the second stage we remove from the middle of both intervals an interval of length $2^{-4}$ and obtain $f_2 = \chi_{[0,5/32] \cup [7/32, 3/8] \cup [5/8,25/32] \cup [27/32, 1]}$. During the $n-$th stage of construction we remove an interval of length $2^{-2n}$ from the middle of all existing $2^{n-1}$ intervals. Let us see what is the length of all such intervals. Let $a_n$ be the length of an interval at the $n-$th stage of construction. Then $a_{n} = \frac{a_{n-1}}{2} - \frac{1}{2^{2n+1}}$. As $a_0 = 1$, we obtain a direct formula $a_n = \frac{2^n + 1}{2^{2n+1}}$. Now we take the fat Cantor set to be on the circle, i.e. the interval \[0,1\] corresponds to angles measured in radians. On the rest of the circle we set the function $f$ to be $0$. Let us compare at every stage of construction the sum of lengths of the red intervals and the green ones. After trygonometric considerations we have to check the following inequality: $$\label{ineq:cantor} \sqrt{1 - \cos(a_n)} + \sqrt{1 - \cos(\frac{1}{2^{2n}})} > 2 \sqrt{1 - \cos(a_{n+1})}.$$ Substitute $x = 2^{-n}$ and use the direct formula for $a_n$. It changes to the inequality $$g(x) = \sqrt{1 - \cos(\frac{x(x+1)}{2})} + \sqrt{1 - \cos(x^2)} - 2 \sqrt{1 - \cos(\frac{x(x+2)}{8})} > 0.$$ But $g$ satisfies $g(0) = 0$ and its derivative is positive on $(0,1)$, so $g > 0$ on $(0,1)$, thus the inequality holds for all $n$. Thus, as on every stage of construction the sides of the trapezoid are shorter than the bases. It means that the solution of LGP for $f_n$ takes value $0$ on the trapezoid (as we minimize $P(E_t, B(0,1))$ for $t \in (0,1)$). In the next stage of construction the value on this trapezoid will remain zero and we will make the same reasoning on two adjacent smaller trapezoids. From the construction of Cantor set the sequence $u_n$ is nonincreasing and for every point $x$ inside the circle at a sufficiently large stage of construction we would have $u_n(x) = 0$. Thus $u_n \rightarrow 0$ a.e.; but it is bounded from above by $1$, so the convergence holds also in $L^1(\Omega)$. \[ex:cantor\] Let us make a slight change to the previous example: consider another fat Cantor set. More precisely, take a set almost of full measure such that the inequality holds in the opposite direction; it is possible due to the triangle inequality. Thus at every stage of construction it is more efficient (minimizing lengths of level sets) to remove $2^{n-1}$ curvilinear triangles from the set $\{ u_n = 1 \}$ than to repeat the above construction, i.e. add trapezoids to the set $\{ u_n = 0 \}$. Thus at every stage of construction the set $\{ u_n = 1 \}$ will be a sum of trapezoids mentioned before, so the trace of $u$ equals $f$. Also $u_n \rightarrow u$ in $L^1(\Omega)$, as it converges a.e. Thus we obtained that there exists a solution to LGP for a certain discontinuous $f \notin BV(\partial\Omega)$. Anisotropic case ================ This section is devoted to the anisotropic least gradient problem. We discuss $l^p$ norms on the plane for $p \in [1, \infty]$. We prove a non-uniqueness result for $p = 1, \infty$ and discuss how the solutions look like for $p \in (1, \infty)$. We shall use the notation introduced in [@Maz]. A continuous function $\phi: \overline{\Omega} \times \mathbb{R}^n \rightarrow [0, \infty)$ is called a metric integrand, if it satisfies the following conditions:\ \ $(1)$ $\phi$ is convex with respect to the second variable for a.e. $x \in \overline{\Omega}$;\ $(2)$ $\phi$ is homogeneous with respect to the second variable, i.e. $$\forall x \in \overline{\Omega}, \quad \forall \xi \in \mathbb{R}^n, \quad \forall t \in \mathbb{R} \quad \phi(x, t \xi) = |t| \phi(x, \xi);$$ $(3)$ $\phi$ is bounded in $\Omega$, i.e. $$\exists \Gamma > 0 \quad \forall x \in \overline{\Omega}, \quad \forall \xi \in \mathbb{R}^n \quad 0 \leq \phi(x, \xi) \leq \Gamma |\xi|.$$ \ $(4)$ $\phi$ is elliptic in $\Omega$, i.e. $$\exists \lambda > 0 \quad \forall x \in \overline{\Omega}, \quad \forall \xi \in \mathbb{R}^n \quad \lambda |\xi| \leq \phi(x, \xi).$$ These conditions are sufficient for most of the cases considered in scientific work: they are satisfied for the classical LGP, i.e. $(\phi(x, \xi) = |\xi|)$, as well as for the $l_p$ norms, $p \in [1, \infty]$ and for weighted LGP considered in [@JMN]: a function $\phi(x, \xi) = g(x) |\xi|$, where $g \geq c > 0$. The polar function of $\phi$ is $\phi^0: \overline{\Omega} \times \mathbb{R}^N \rightarrow [0, \infty)$ defined as $$\phi^0 (x, \xi^*) = \sup \, \{ \langle \xi^*, \xi \rangle : \xi \in \mathbb{R}^N, \phi(x, \xi) \leq 1 \}.$$ Let $$\mathcal{K}_\phi(\Omega) = \{ \mathbf{z} \in X_\infty(\Omega) : \phi^0(x,\mathbf{z}(x)) \leq 1 \text{ for a.e. } x \in \Omega, \quad [\mathbf{z}, \nu] = 0 \}.$$ For a given function $u \in L^1(\Omega)$ we define its $\phi-$total variation in $\Omega$ by the formula (another notation used in the literature is $\int_\Omega \phi(x, Du)$): $$\int_\Omega |Du|_\phi = \sup \, \{ \int_\Omega u \, \mathrm{div} \, \mathbf{z} \, dx : \mathbf{z} \in \mathcal{K}_\phi(\Omega) \}.$$ If $\int_\Omega |Du|_\phi < \infty$, we say that $u \in BV_\phi(\Omega)$. If $\phi$ is a metric integrand, by properties $(3)$ and $(4)$ we have that $\lambda \int_\Omega |Du| \leq \int_\Omega |Du|_\phi \leq \Gamma \int_\Omega |Du|$, so $BV_\phi(\Omega) = BV(\Omega)$. We also know ([@AB Chapter 3]) that when $\phi$ is continuous and elliptic in $\Omega$, then in the definition of $\mathcal{K}_\phi(\Omega)$ we can replace the condition $[\mathbf{z}, \nu] = 0$ with a demand that $\mathbf{z} \in C_c^1(\Omega)$, so we recover the classical definition. \[lem:scisaniz\] When $\phi$ is continuous and elliptic in $\Omega$, then similarly to the classical case ([@AB Chapter 4]) we recover lower semicontinuity of the $\phi-$total variation, the notion of $\phi-$perimeter of a set and the co-area formula. We also recover the approximation by $C^\infty$ functions in the strict topology, even in the strong form proved by Giusti in [@Giu Corollaries 1.17, 2.10]: let $v \in BV_\phi(\Omega)$, $Tv = f$. Then there exists a sequence of $C^\infty$ functions $v_n$ such that $v_n \rightarrow v$ strictly in $BV_\phi(\Omega)$ such that $Tv = f$. For an explicit use we shall need the following integral representation ([@AB], [@JMN]): \[stw:repcalkowa\] Let $\varphi: \overline{\Omega} \times \mathbb{R}^N \rightarrow \mathbb{R}$ be a metric integrand. Then we have an integral representation: $$\int_\Omega |Du|_\phi = \int_\Omega \phi(x, \nu^u(x)) \, |Du|,$$ where $\nu^u$ is the Radon-Nikodym derivative $\nu^u = \frac{d Du}{d |Du|}$. In particular, if $E \subset \Omega$ and $\partial E$ is sufficiently smooth (at least $C^1$), then we have a representation $$P_\phi(E, \Omega) = \int_\Omega \phi(x, \nu_E) \, d \mathcal{H}^{n-1},$$ where $\nu_E$ is the external normal to $E$. For $p \in [1, \infty)$ we define the $p-$th norm of a vector on the plane by the formula $\| (x, y) \|_p = (|x|^p + |y|^p)^{1/p}$. For $p = \infty$ it is defined as $\| (x, y) \|_\infty = \sup(|x|, |y|)$. Let us note that $\| \cdotp \|_1 \geq \| \cdotp \|_2 \geq \| \cdotp \|_\infty$ and that the case $p = 2$ is isotropic. We aim to prove that for nonsmooth anisotropy the solutions need not be unique (and in general are not unique); to achieve this goal, we will study how do minimal surfaces with respect to the $p-$th norm look like. At first let us see an example that the solution is unique: \[stw:uniqueness\] Let $\Omega \subset \mathbb{R}^2$ be an open, bounded, strictly convex set. Take $\phi(x, Du) = \| Du \|_1$. Let $f \in C(\partial \Omega)$. Denote by $u$ the solution to isotropic LGP for $f$. Then, if the boundaries of superlevel sets of $u$ are parallel to the axes of the coordinate system, then $u$ is a unique solution of the anisotropic LGP with respect to the $l^1$ norm. Let $v \in BV(\Omega)$, $Tv = f$. Then $$\int_\Omega |Dv|_1 \geq \int_\Omega |Dv|_2 \geq \int_\Omega |Du|_2.$$ By uniqueness of solution to Euclidean LGP the second inequality is strict, if only $u \neq v$. As the boundaries of superlevel sets of $u$ are parallel to the axes of the coordinate system, we have $\int_\Omega |Du|_1 = \int_\Omega |Du|_2$; it follows that $u$ is a unique solution to the anisotropic LGP. \[ex:uniqueness\] Let $\Omega = B(0,1)$. Take $\phi(x, Du) = \| Du \|_1$. Let $f(\theta) = \cos(2 \theta)$. We construct the isotropic solution $u$ using Sternberg-Williams-Ziemer construction. We notice, as the picture below shows, that the boundaries of superlevel sets of $u$ are parallel to the axes of the coordinate system. By Proposition \[stw:uniqueness\] the solution to the anisotropic LGP is unique. \[stw:niejednoznacznosc\] Let $\Omega \subset \mathbb{R}^2$ be an open, bounded, strictly convex set. Take $\phi(x, Du) = \| Du \|_1$. Let $f \in C(\partial \Omega)$. Denote by $u$ the solution to isotropic LGP for $f$. Then, if for some $t$ the boundaries of superlevel sets of $u$ are not parallel to the axes of the coordinate system, then the solution to the anisotropic LGP with respect to the $l^1$ norm is not unique. 1\. Take $v \in C^1(\Omega)$ with trace $f$. Then the co-area formula reads $$\int_\Omega |Dv|_1 = \int_{\mathbb{R}} P_1 (E_t, \Omega) dt,$$ in particular $v$ is a solution to anisotropic LGP iff $P_1(E_t, \Omega)$ is minimal for a.e. $t$. As $v$ is smooth, $v |_{\partial\Omega} = f$, then by Sard theorem for a.e. $t$ the set $\{ v = t \}$ is a smooth manifold; as such, it is an at most countable sum of smooth curves disjoint in $\Omega$. 2\. We want to find the lower bound for $\int_\Omega |Dv|_1$. We shall find it for a larger class of functions: continuous functions, for which the sets $\{ v = t \}$ are at most countable sums of smooth curves disjoint in $\Omega$. We have to extend our class of functions, as we need to be able to eliminate closed curves from the disjoint sum: if there were any closed curves, then by setting $v = t$ in the open set enclosed by such curves we obtain a function with strictly smaller total variation, but not necessarily smooth. Thus we may assume that $\partial \{ v \geq t \}$ is a disjoint sum of open curves. Let us note that they must end in points $p \in f^{-1}(t) \subset \partial\Omega$. 3\. According to the co-area formula, it is sufficient to construct superlevel sets of $v$ such that $P_1(E_t, \Omega)$ is minimal; then $\int_\Omega |Dv|_1$ would be minimal as well. Let us suppose additionally that $\partial E_t$ does not contain any vertical intervals, i.e. we may represent a level set from the point $(x,y)$ to $(z,t)$ as a graph of a $C^1$ function $g$. Let us note that at the point $((s, g(s)))$ the Radon-Nikodym derivative $\nu^{\chi_{E_t}}$ is perpendicular to the level set, so it is a vector $(- \sin \theta, \cos \theta)$, where $g'(s) = \tan \theta$. Thus $\phi(x, \nu^{\chi_{E_t}}) = |\sin \theta| + |\cos \theta|$. As $|D \chi_{E_t}| = \mathcal{H}^{n-1}|_{\partial E_t}$, then, using the representation introduced by Proposition \[stw:repcalkowa\], we have to minimize the integral (we may assume that $x < z$): $$P(E_t, \Omega) = \int_\Omega \phi(x, \nu^{\chi_{E_t}}) |D \chi_{E_t}| = \int_{\partial E_t} (|\sin \theta| + |\cos \theta|) d\mathcal{H}^{n-1} =$$ $$= \int_{x}^{z} (|\sin \theta| + |\cos \theta|) \sqrt{1 + (\tan \theta)^2} dx = \int_{x}^{z} (|\sin \theta| + |\cos \theta|) \frac{1}{|\cos \theta|} dx =$$ $$= \int_{x}^{z} (1 + |\tan \theta|) dx = |z - x| + \int_{x}^{z} |g'| dx \geq |z - x| + |t - y|,$$ where the inequality becomes equality iff $g$ is monotone (remember we assumed it to be $C^1$). Thus there are multiple functions minimizing this integral. 4\. Now we allow $\partial E_t$ to contain vertical intervals. The difference is purely technical, as we have to divide our integral into two parts. Let us suppose that the (orientated) length of $i-$th vertical interval equals $\lambda_i$, then we have $$\int_{\text{graph part of } \partial E_t} (|\sin \theta| + |\cos \theta|) dl + \int_{\text{vertical part of } \partial E_t} (1 + 0) dl = \int_{x}^{z} (1 + |g'|) dx + \sum_{i = 1}^{\infty} |\lambda_i| =$$ $$= \int_{x}^{z} |g'| dx + |z - x| + \sum_{i = 1}^{\infty} |\lambda_i| \geq |t - y - \sum_{i = 1}^{\infty} \lambda_i| + |z - x| + \sum_{i = 1}^{\infty} |\lambda_i| \geq |z - x| + |t - y|,$$ where the inequality becomes equality iff $g$ is monotone (remember we assumed it to be $C^1$) and all the vertical intervals are orientated in the same direction as $g'$. Thus there are multiple functions minimizing this integral. We have proved that in a class containing all smooth functions the problem of minimizing perimeter of a set $E_t$ doesn’t have a unique solution. 5\. Let us denote by $u$ the solution to the Euclidean LGP. Let us notice that intervals are graphs of monotone functions, so an interval mimimizes the above integral; thus, by co-area formula, the value of $\int_\Omega |Dv|_1$ is bounded from below by $$\int_\Omega |Dv|_1 = \int_{\mathbb{R}} P_1(E_t, \Omega) \geq \int_{\mathbb{R}} P_1(\{ u > t \}, \Omega),$$ so by Remark \[lem:scisaniz\] such inequality holds for all $v \in BV_1(\Omega)$ such that $Tv = f$. In particular, the Euclidean solution is also a solution to the anisotropic LGP. But if we choose $v$ such that its level sets $\{ v = t \}$ be monotone for almost all $t$, then its total variation is exactly the same (it is possible due to the non-parallelism assumption). Thus the solution to this anisotropic LGP is not unique. \[ex:l1\] Let $\Omega = B(0,1)$. Take $\phi(x, Du) = \| Du \|_1$. Let $f \in C^{\infty}(\partial\Omega)$ be given as $f = \cos(2 \theta - \pi/2)$. Then the solution to the anisotropic LGP is not unique. At first, let us see that the Euclidean solution is a rotation of the function $u$ from Example \[ex:uniqueness\], so we may apply the procedure from Proposition \[stw:niejednoznacznosc\]. We observe that for fixed $t \in (0,1)$ its preimage contains points of the form $A_1 = (a,b)$, $A_2 = (b, a)$, $A_3 = (-a, -b)$, $A_4 = (-b, -a)$; then, applying the above calculation to the function $f$, we see that the two possible connections, $A_1 A_2, A_3 A_4$ and $A_1 A_4, A_2 A_3$ have perimeter lengths $4 |a - b|$ and $4 |a + b|$ respectively; we choose the former as the level set $E_t$. Similar calculation holds for $t \in (-1,0)$. But if we choose $v$ such that its level sets $\{ v = t \}$ be monotone for almost all $t$, then their perimeter (and, by co-area formula, its total variation) stays exactly the same. Thus the solution to this anisotropic LGP is not unique; an example of a non-Euclidean solution is presented on the picture below. \[ex:linfty\] Now let $p = \infty$. If we make a similar calculation, we obtain that the perimeter of a level set connecting points $(x,y)$ with $(z,t)$ equals $$\int_{\partial E_t} \max(|\sin \theta|, |\cos \theta|) d\mathcal{H}^{n-1} = \int_{x}^{z} \max(|\sin \theta|, |\cos \theta|) \sqrt{1 + (\tan \theta)^2} dx =$$ $$= \int_{x}^{z} \max(|\sin \theta|, |\cos \theta|) \frac{1}{|\cos \theta|} dx = \int_{x}^{z} \max(1, |\tan \theta|) dx = \int_{x}^{z} \max(1, |g'|) dx \geq |z - x|,$$ where the inequality becomes equality iff $|g'| \leq 1$; in other words, the angle between the level set and the $x$ coordinate axis is not greater than $\frac{\pi}{4}$. Thus, if we take the function $f(\theta) = \cos(2 \theta)$, the solution is not unique; we apply this result for $t \in (-1,0)$ and then apply it again for $t \in (0,1)$ considering the level set as a function of $y$. A solution different than the Euclidean one is presented on the picture below. Nevertheless, it may still happen that the solution is unique: it is the case if we take such $f$ that the Euclidean solution has all level sets at an angle $\frac{\pi}{4}$ to the coordinate axes. For example we can take $f(\theta) = \cos(2 \theta - \frac{\pi}{2})$. Now let $1 < p < \infty$. By [@JMN Theorems 1.1, 1.2] for continuous boundary data the anisotropic LGP has a unique solution, because the norm $\| \cdotp \|_p$ is a smooth function of the Euclidean norm outside $(0,0)$. We will show that connected components of boundaries of superlevel sets of functions of $\phi-$least gradient are line segments, similarly to for the isotropic norm $\| \cdotp \|_2$; in fact, due to an anisotropic analogue of Theorem \[twierdzeniezbgg\] proved in [@Maz Theorem 3.19], it is enough to show that the boundaries of minimal sets are line segments. \[tw:anizotropia\] Let $\Omega \subset \mathbb{R}^2$ be an open convex set. Let the anisotropy be given by the function $\phi(x,Du) = \| Du \|_p$, where $1 < p < \infty$. Let $E$ be a $\phi-$minimal set with respect to $\Omega$, i.e. $\chi_E$ is a function of $\phi-$least gradient in $\Omega$. Then every connected component of $\partial E$ is a line segment. Let $(x,y),(z,t)$ be two points on the same connected component of $\partial E$. We have to minimize an integral analogous to the previous one (notation stays the same): $$L(x, g, g') = \int_{\partial E_t} (|\sin \theta|^p + |\cos \theta|^p)^{\frac{1}{p}} d\mathcal{H}^{n-1} = \int_{x}^{z} (|\sin \theta|^p + |\cos \theta|^p)^{\frac{1}{p}} \sqrt{1 + (\tan \theta)^2} dx =$$ $$= \int_{x}^{z} (|\sin \theta|^p + |\cos \theta|^p)^{\frac{1}{p}} \frac{1}{|\cos \theta|} dx = \int_{x}^{z} (1 + |\tan \theta|^p)^{\frac{1}{p}} dx = \int_{x}^{z} (1 + |g'|^p)^{\frac{1}{p}} dx.$$ The Euler$-$Lagrange equation for the functional $L$ takes form $$0 = \frac{\partial L}{\partial g} = \frac{d}{dx} (\frac{\partial L}{\partial g'}) = \frac{d}{dx} (\mathrm{sgn}(g') (g')^{p-1} (1 + |g'|^p)^{\frac{1}{p} - 1})$$ $$\mathrm{sgn}(g') (g')^{p-1} (1 + |g'|^p)^{\frac{1}{p} - 1} = \text{ const}.$$ Taking absolute value and raising both sides to power $\frac{p}{p-1}$ we obtain $$\frac{|g'|^p}{1 + |g'|^p} = \text{ const} = C,$$ thus $g' =$ const. 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--- abstract: 'We consider algebras that can be realized as PBW deformations of (Artin-Schelter) regular algebras. This is equivalent to the homogenization of the algebra being regular. It is shown that the homogenization, when it is a geometric algebra, contains a component whose points are in 1-1 correspondence with the simple modules of the deformation. We classify all PBW deformations of 2-dimensional regular algebras and give examples of 3-dimensional deformations. Other properties, such as the skew Calabi-Yau property and closure under tensor products, are considered.' author: - Jason Gaddis bibliography: - 'biblio.bib' title: 'PBW deformations of Artin-Schelter regular algebras' --- =1 Introduction ============ In [@artsch], Artin and Schelter introduced a class of 3-dimensional graded algebras which may be regarded as noncommutative versions of the polynomial ring in three variables. These algebras, which posses suitably nice growth and homological properties, are now called (Artin-Schelter) regular algebras. In [@atv], Artin, Tate, and Van den Bergh showed that these algebras have geometric interpretations related to point schemes in projective space. The classification of 3-dimensional regular algebras was completed in the two aforementioned articles and the classification of higher dimensional regular algebras is an active open problem. We consider algebras on the periphery of this classification. Indeed, there are many non-graded algebras with similar properties to regular algebras. Examples include the quantum Weyl algebras and the enveloping algebra of the two-dimensional solvable Lie algebra. If $A$ is an algebra such that its homogenization, $H(A)$, is regular, we say $A$ is *essentially regular*. This is equivalent to the associated graded ring, $\operatorname{gr}(A)$, being regular and equivalent to $A$ being a *PBW deformation* of $\operatorname{gr}(A)$ (Proposition \[grprop\]). If $x_0$ is the homogenizing element, then one can pass certain properties between $A$ and $H(A)$ (Propositions \[cntr\] and \[hprops\]) via the isomorphism $H(A)[x_0{^{-1}}] {\cong}A[x_0^{\pm 1}]$. In Section \[cy\], we review the definition of a skew Calabi-Yau algebra and show that all essentially regular algebras are skew Calabi-Yau. All 3-dimensional regular algebras are geometric algebras, that is, they are canonically identified with a pair $(E,\sigma)$ where $E \subset \mathbb{P}(V^*)$ is a scheme and $\sigma \in \operatorname{Aut}(E)$. If $A$ is essentially regular and $H(A)$ is geometric, then there exists a subscheme $E_1 \subset E$ whose points are in 1-1 correspondence with the 1-dimensional simple modules of $H(A)$ (Proposition \[dim1\]). In case $A$ is 2-dimensional and not PI, these are all of the finite-dimensional simple modules of $H(A)$ (Proposition \[fdim\]). We generalize this result in Proposition \[Itors\] and make progress towards a conjecture by Walton that all finite-dimensional simple modules of a non-PI deformed Sklyanin algebra are 1-dimensional [@walton]. Relying on work done in [@twogen], we classify all 2-dimensional essentially regular algebras (Corollary \[eclass\]) and compile some known examples of 3-dimensional essentially regular algebras (Examples \[ex1\]-\[ex3\]). Finally, in Section \[fivedim\], we show that the property of being essentially regular is closed under tensor products and use this to give examples of 5-dimensional regular algebras. Definitions and initial properties {#props} ================================== Let ${K}$ be an algebraically closed, characteristic zero field. All algebras are ${K}$-algebras and all isomorphisms should be read as ‘isomorphisms of ${K}$-algebras’. Suppose $A$ is defined as a factor algebra of the free algebra on $n$ generators by $m$ polynomial relations, i.e., $$\begin{aligned} \label{form} A = {K}\langle x_1,\hdots,x_n \mid f_1,\hdots,f_m \rangle.\end{aligned}$$ Throughout, we assume the generators $x_i$ all have degree 1. Given a (noncommutative) polynomial $f \in {K}\langle x_1,\hdots,x_n\rangle$ with $\alpha=\deg(f)$, write $$\begin{aligned} \label{fexpl} f = \sum_{\gamma \in \Gamma} c_\gamma x_{i_1}^{\alpha_{\gamma_1}} \cdots x_{i_\ell}^{\alpha_{\gamma_\ell}}, c_\gamma \in {K}, \alpha_{\gamma_i} \in {K}, \sum_{i=1}^\ell \alpha_{\gamma_i} \leq \alpha,\end{aligned}$$ where $\Gamma$ ranges over distinct monomials in the free algebra and all but finitely many of the $c_\gamma$ are zero. The homogenization of $f$ is then $$\begin{aligned} \hat{f} = \sum_{\gamma \in \Gamma} c_\gamma x_{i_1}^{\alpha_{\gamma_1}} \cdots x_{i_\ell}^{\alpha_{\gamma_\ell}} x_0^{\alpha_{\gamma_0}},\end{aligned}$$ where $x_0$ is a new, central indeterminate and $\alpha_{\gamma_0}$ is chosen such that $\sum_{i=0}^\ell \alpha_{\gamma_i} = \alpha$. Then $\hat{f}$ is homogeneous. \[hom-def\]Let $A$ be of form (\[form\]). The **homogenization** $H(A)$ of $A$ is the ${K}$-algebra on the $n+1$ generators $x_0,x_1,\hdots,x_n$ subject to the homogenized relations $\hat{f}_i$ as well as the additional relations $x_0x_i-x_ix_0$ for all $i \in \{1,\hdots,n\}$. A *filtration* ${\mathcal{F}}$ on an algebra $A$ is a collection of vector spaces $\{{\mathcal{F}}_n(A)\}$ such that $${\mathcal{F}}_n(A) \subset {\mathcal{F}}_{n+1}(A), {\mathcal{F}}_n(A) \cdot {\mathcal{F}}_m(A) \subset {\mathcal{F}}_{n+m}(A) \text{ and } \bigcup {\mathcal{F}}_n(A) = A.$$ The filtration ${\mathcal{F}}$ is said to be *connected* if ${\mathcal{F}}_0(A)={K}$ and ${\mathcal{F}}_\ell(A) = 0$ for all $\ell < 0$. The *associated graded algebra* of $A$ is $\operatorname{gr}_{\mathcal{F}}(A) := \bigoplus_{i \geq 0} {\mathcal{F}}_i(A)/{\mathcal{F}}_{i-1}(A)$. The algebra $\operatorname{gr}_{\mathcal{F}}(A)$ is said to be connected if the filtration ${\mathcal{F}}$ is connected. Associated to the pair $(A,{\mathcal{F}})$ is also the *Rees ring* of $A$, $$R_{\mathcal{F}}(A) := \bigoplus_{n \geq 0} {\mathcal{F}}_n(A) x_0^n.$$ For an algebra defined by generators and relations, as above, there is a standard connected filtration wherein ${\mathcal{F}}_\ell(A)$ is the span of all monomials of degree at most $\ell$. Since this filtration is the only one we consider, we drop the subscript on $\operatorname{gr}(A)$ and $R(A)$. One can always recover $A$ and $\operatorname{gr}(A)$ from $H(A)$ via $A {\cong}H(A)/(x_0-1)H$ and $\operatorname{gr}(A) {\cong}H(A)/x_0H$, respectively. An algebra is said to be *graded* if $\operatorname{gr}(A) = A$. In this case, we write $A_m$ for the vector space spanned by homogeneous elements of degree $m$. If $A$ is graded, then the *global dimension* of $A$, $\operatorname{gld}(A)$, is the projective dimension of the trivial module ${K}_A = A/A_+$, where $A_+$ is the augmentation ideal generated by all degree one elements. Let $V$ be a ${K}$-algebra generating set for $A$ and $V^n$ the set of degree $n$ monomials in $A$. The Gelfand-Kirilov (GK) dimension of $A$ is defined as $${\text{GK.dim}}(A) := \limsup_{n \rightarrow \infty} \log_n (\dim V^n).$$ The algebra $A$ is said to be *AS-Gorenstein* if $\operatorname{Ext}_A^i({K}_A,A){\cong}\delta_{i,d} {_A}{K}$ where $\delta_{i,d}$ is the Kronecker delta and $d=\operatorname{gld}(A)$. A connected graded algebra is said to be (Artin-Schelter) **regular** of dimension $d$ if $H$ has finite global dimension $d$, finite GK dimension, and is AS-Gorenstein. We say an algebra $A$ is **essentially regular** of dimension $d$ if $H(A)$ is regular of dimension $d+1$. In the case that $x_0$ is not a zero divisor, we have $H(A) {\cong}R(A)$ ([@wu], Proposition 2.6) and $H(A)$ becomes a regular central extension of $\operatorname{gr}(A)$ (see [@casshel], [@bruyn]). However, this need not always be the case, which the next example illustrates, and so we choose to use $H(A)$ instead of $R(A)$ in the above definition. Let $A={K}\langle x_1,x_2 \mid x_1^2-x_2\rangle$. Then $A {\cong}{K}[x]$. However, the algebra $H(A)$ is not regular. Indeed, $$x_1x_2x_0 = x_1x_1^2 = x_1^2x_1 = x_2x_0x_1 = x_2x_1x_0 \Rightarrow (x_1x_2-x_2x_1)x_0=0.$$ Thus, either $H(A)$ is not a domain or else $H(A)$ is commutative. The latter cannot hold because $H(A)/x_0H(A) {\cong}{K}\langle x_1,x_2 \mid x_1^2\rangle$ is not commutative. By [@atv2], Theorem 3.9, all regular algebras of dimension at most four are domains. Hence, in this case, $H(A) {\ncong}R(A)$. If $A$ is essentially regular of dimension at most three, then $H(A)$ is domain. We will assume, hereafter, that $x_0$ is not a zero divisor in $H(A)$ when $A$ is essentially regular. The dimension of an essentially regular algebra is not the same as its global dimension in all cases. The first Weyl algebra, ${A_1({K})}= {K}\langle x,y \mid yx-xy+1 \rangle$, is dimension two essentially regular but has global dimension one. It would be interesting to know whether there is a lower bound on the global dimension of an essentially regular algebra based on that of its homogenization. The following lemma is useful in passing properties between $A$ and $H(A)$. \[loclem\]Suppose $x_0$ is not a zero divisor. Then $H(A)[x_0{^{-1}}] {\cong}A[x_0^{\pm 1}]$. Let $f$ be a defining relation for $A$ and $\hat{f}$ the homogenized relation in $H$. Let $\alpha=\deg(f)$ and write $f$ as in (\[fexpl\]). Then $$0 = x_0^{-\alpha}\hat{f} = \sum_{\gamma \in \Gamma} c_\gamma (x_0{^{-1}}x_{i_1})^{\alpha_{\gamma_1}} \cdots (x_0{^{-1}}x_{i_\ell})^{\alpha_{\gamma_\ell}}.$$ If we let $X_0=x_0$ and $X_i=x_0{^{-1}}x_i$ for $i > 0$, then the $\{X_i\}_{i \geq 0}$ generate $A[x_0^{\pm 1}]$ in $H[x_0{^{-1}}]$. Conversely, in $A[x_0^{\pm 1}]$ we have $$0 = x_0^\alpha f = \sum_{\gamma \in \Gamma} c_\gamma (x_0x_{i_1})^{\alpha_{\gamma_1}} \cdots (x_0x_{i_\ell})^{\alpha_{\gamma_\ell}}.$$ If we let $X_0=x_0$ and $X_i=x_0x_i$ for $i > 0$, then the $\{X_i\}_{i \geq 0}$ generate $H[x_0{^{-1}}]$ in $A[x_0^{\pm 1}]$. Let ${\mathcal{Z}}(A)$ denote the center of $A$. One would expect a natural equivalence between the center of a homogenization and the homogenization of a center. The next proposition formalizes that idea. \[cntr\]Suppose $x_0$ is not a zero divisor. By identify generators, we have $${\mathcal{Z}}(H(A)) = H({\mathcal{Z}}(A)).$$ By [@rowen], Propositions 1.2.20 (ii) and 1.10.13, along with Lemma \[loclem\], $$H({\mathcal{Z}}(A))[x_0{^{-1}}] {\cong}{\mathcal{Z}}(A)[x_0^{\pm 1}] = {\mathcal{Z}}(A[x_0^{\pm 1}]) {\cong}{\mathcal{Z}}(H(A)[x_0{^{-1}}]) = {\mathcal{Z}}(H(A))[x_0{^{-1}}].$$ Thus, $H({\mathcal{Z}}(A))[x_0{^{-1}}] {\cong}{\mathcal{Z}}(H(A))[x_0{^{-1}}]$. It remains to be shown that the subalgebras $H({\mathcal{Z}}(A))$ and ${\mathcal{Z}}(H(A))$ are isomorphic and, moreover, the elements can be identified by generators. Let $\hat{f} \in H({\mathcal{Z}}(A))$, then $\hat{f} \in H({\mathcal{Z}}(A))[x_0{^{-1}}]$ and, by Lemma \[loclem\], $f \in {\mathcal{Z}}(A)[x_0^{\pm 1}]$. Since only positive powers of $x_0$ appear in $f$, then $f \in {\mathcal{Z}}(A)[x_0]$. Therefore, $$f \in {\mathcal{Z}}(A)[x_0^{\pm 1}] = {\mathcal{Z}}(A[x_0^{\pm 1}]),$$ and so $\hat{f} \in {\mathcal{Z}}(H(A)[x_0{^{-1}}])={\mathcal{Z}}(H(A))[x_0{^{-1}}]$. Again, since only positive powers of $x_0$ appear in $\hat{f}$, then $\hat{f} \in {\mathcal{Z}}(H(A))$. The converse is similar. We now consider properties that pass between an essentially regular algebra and its homogenization. \[hprops\]Let $A$ be essentially regular and $H=H(A)$. 1. $A$ is prime if and only if $H$ is prime; 2. $A$ is PI if and only if $H$ is PI; 3. $A$ is noetherian if $H$ is noetherian; 4. $H$ is noetherian if $\operatorname{gr}(A)$ is noetherian; 5. $H$ is not primitive. \(1) is well-known since $x_0$ is central and not a zero divisor. (2) is a consequence of Proposition \[cntr\]. (3) is clear because $A$ is a factor algebra of $H$ and (4) follows from [@atv], Lemma 8.2. The algebra $H$ is affine over the uncountable, algebraically closed field ${K}$, so (5) follows from [@kirkkuz], Proposition 3.2. Let $F$ be the set of relations defining $A$ and let $R$ be the relations $F$ filtered by degree. There is a canonical surjection $\operatorname{gr}(A) \rightarrow B := {K}\langle x_1,\hdots x_n \mid R\rangle$. We say $A$ is a *Poincaré-Birkhoff-Witt (PBW) deformation* of $\operatorname{gr}(A)$ if that map is an isomorphism. The next proposition shows that essentially regular algebras are equivalent to PBW deformations of regular algebras. \[grprop\]An algebra $A$ is essentially regular if and only if $\operatorname{gr}(A)$ is regular. Moreover, if $A$ is essentially regular, then it is a PBW deformation of $\operatorname{gr}(A)$. Let $B=\operatorname{gr}(A)$ and $H=H(A)$. Since $x_0\in H$ is central and not a zero divisor, then by the Rees Lemma ([@rotman], Theorem 8.34), $\operatorname{Ext}_B^n({K}_B,B) {\cong}\operatorname{Ext}_{H}^{n+1}({K}_H,H)$. Hence, $B$ is AS-Gorenstein if and only if $H$ is. Moreover, since $B$ (resp. $H$) is graded, then $\operatorname{gld}(B)=\operatorname{pd}({K}_B)$ (resp. $\operatorname{gld}(H)=\operatorname{pd}({K}_H)$). By the Rees Lemma, $\operatorname{gld}(B)=d$ if and only if $\operatorname{gld}(H)=d+1$. The sequence $0 \rightarrow x_0H \rightarrow H \rightarrow B \rightarrow 0$ is exact, so ${\text{GK.dim}}(B) \leq {\text{GK.dim}}(H)-1 < \infty$ when $H$ is regular. Conversely, if $B$ is regular, then ${\text{GK.dim}}(A)={\text{GK.dim}}(B) < \infty$. Localization at the central regular element $x_0$ in $H$ and in $A[x_0]$ preserves GK dimension ([@mcrob], Proposition 8.2.13). This, combined with Lemma \[loclem\], gives, $${\text{GK.dim}}(H) = {\text{GK.dim}}(H[x_0{^{-1}}]) = {\text{GK.dim}}(A[x_0^{\pm 1}]) = {\text{GK.dim}}(A)+1 < \infty.$$ That $A$ is a PBW deformation now follows from [@cassidy], Theorem 1.3. \[essAS\]If $A$ is regular, then $A$ is essentially regular. If $A$ is a noetherian essentially regular algebra, then $A$ has finite global and GK dimension. By [@mcrob], Corollary 6.18, and because $\operatorname{gr}(A)$ is regular, $\operatorname{gld}A \leq \operatorname{gld}\operatorname{gr}(A) < \infty$. The statement on GK dimension follows from the proof of Proposition \[grprop\]. \[polyext\]Let $\xi$ be a central indeterminate over an algebra $A$. The polynomial ring $A[\xi]$ is essentially regular of dimension $d$ if and only if $A$ is essentially regular of dimension $d-1$. We need only observe that $H(A[\xi])=H(A)[\xi]$ and that regularity is preserved under polynomial extensions. In the next section we observe that essential regularity is preserved under certain skew polynomial extensions. We end this section with a classification of examples of essentially regular algebras in dimension two and several examples of those of dimension three. If $A$ is essentially regular of dimension two, then $H=H(A)$ is dimension three regular. Hence, $H$ either has three generators subject to three quadratic relations, or else it has two generators subject to two cubic relations. Since $H$ is a homogenization, then the commutation relations of $x_0$ give two quadratic relations, so there must be some presentation in the first form. Since $A {\cong}H/(x_0-1)H$, then the commutation relations drop off and we are left with one quadratic relation. Thus, to determine the 2-dimensional essentially regular algebras, we begin with a list of all algebras defined by two generators subject to a single quadratic relation. An easy consequence is that a subset of this list contains the 2-dimensional essentially regular algebras. \[classification\] Suppose $A {\cong}{K}\langle x,y \mid f \rangle$ where $f$ is a polynomial of degree two. Then $A$ is isomorphic to one of the following algebras: $$\begin{aligned} &{\mathcal{O}_q({K}^2)}, f=xy-qyx ~ (q \in {K}^\times), & ~ & {A_1^q({K})}, f=xy-qyx+1 ~ (q \in {K}^\times), \\ &{\mathcal{J}}, f=yx-xy+y^2, & ~ & {\mathcal{J}_1}, f=yx-xy+y^2+1, \\ &{\mathfrak{U}}, f=yx-xy+y, & ~ & {{K}[x]}, f= x^2-y, \\ &{R_{x^2}}, f=x^2, & ~ & {R_{x^2-1}}, f=x^2-1,\\ &{R_{yx}}, f=yx, & ~ & {\mathcal{S}}, f=yx-1.\end{aligned}$$ Furthermore, the above algebras are pairwise non-isomorphic, except $${\mathcal{O}_q({K}^2)}{\cong}\mathcal{O}_{q{^{-1}}}({K}^2) \text{ and } {A_1^q({K})}{\cong}A_1^{q{^{-1}}}({K}).$$ \[eclass\]The dimension two essentially regular algebras are $${\mathcal{O}_q({K}^2)}, {A_1^q({K})}, {\mathcal{J}}, {\mathcal{J}_1}, {\mathfrak{U}}.$$ The algebras ${\mathcal{O}_q({K}^2)}$ and ${\mathcal{J}}$ are 2-dimensional regular [@artsch]. On the other hand, ${R_{x^2}}$ and ${R_{yx}}$ are not domains and therefore not regular [@atv2]. Therefore, ${A_1^q({K})}$, ${\mathcal{J}_1}$ and ${\mathfrak{U}}$ are essentially regular of dimension two whereas ${{K}[x]}$, ${R_{x^2-1}}$ and ${\mathcal{S}}$ are not by Proposition \[grprop\]. In the following, we collect a few examples of 3-dimensional essentially regular algebras. We have already observed that if $H$ is 3-dimensional regular, then $H$ is 3-dimensional essentially regular (Corollary \[essAS\]) and if $A$ is 2-dimensional regular, then $A[\xi]$ is 3-dimensional essentially regular (Corollary \[polyext\]). ([@lbvdb])\[ex1\] Let $L$ be the Lie algebra over ${K}$ generated by $\{x_1,x_2,x_3\}$ subject to the relations $[x_i,x_j] = \sum_{k=1}^n \alpha_{ij,k} x_k$. The enveloping algebra $U(L)$ is 3-dimensional essentially regular. A special case of this is when $L=\mathfrak{sl}_2(\mathbb{C})$ (see [@lesmith]). ([@cassidy])\[ex2\] Essentially regular algebras need not be skew polynomial rings. The down-up algebra $A(\alpha,\beta,\gamma)$ for $\alpha,\beta,\gamma \in {K}$ is defined as the ${K}$-algebra on generators $d,u$ subject to the relations $d^2u = \alpha dud + \beta ud^2 + \gamma d$, $du^2 = \alpha udu + \beta u^2 d + \gamma u$. The algebra $A(\alpha,\beta,\gamma)$ is 3-dimensional essentially regular if and only if $\beta \neq 0$. More generally, PBW deformations of generic cubic regular algebras were computed by Fl[ø]{}ystad and Vatne in [@flovat]. \[ex3\]Let $A$ be an algebra with generators $x_0,x_1,x_2$ and let $r_i=x_i x_j - q x_j x_i + s(x_k)$, $\deg s \leq 2$, $q \in {K}^\times$, where $i=0,1,2$ and $j\equiv i+1 \mod 3, k \equiv i+2 \mod 3$. Since $\operatorname{gr}(A)$ is a Sklyanin algebra, then $A$ is 3-dimensional essentially regular by Proposition \[grprop\]. Skew Calabi-Yau algebras {#cy} ======================== Closely related to regular algebras is the more recent notion of a Calabi-Yau and skew Calabi-Yau algebra. We define (skew) Calabi-Yau algebras and show that essentially regular algebras have this property. This is then used to show that the property of essential regularity is preserved under certain skew polynomial extensions. The *enveloping algebra* of $A$ is defined as $A^e:=A {\otimes}A^{op}$. If $M$ is both a left and right $A$-module, then $M$ is an $A^e$ module with the action given by $(a {\otimes}b)\cdot x = axb$ for all $x \in M$, $a,b \in A$. Correspondingly, given automorphisms $\sigma,\tau \in \operatorname{Aut}(A)$, we can define the twisted $A^e$-module ${^\sigma}M^\tau$ via the rule $(a {\otimes}b)\cdot x = \sigma(a)x\tau(b)$ for all $x \in M$, $a,b \in A$. When $\sigma$ is the identity, we omit it. An algebra $A$ is said to be **homologically smooth** if it has a finite resolution by finitely generated projectives as an $A^e$-module. The length of this resolution is known as the Hochschild dimension of $A$. The Hochschild dimension of $A$ is known to coincide with the global dimension of $A$ ([@berger], Remark 2.8). An algebra $A$ is said to be **skew Calabi-Yau** of dimension $d$ if it is homologically smooth and there exists an automorphism $\tau \in \operatorname{Aut}(A)$ such that there are isomorphisms $$\operatorname{Ext}_{A^e}^i(A,A^e) {\cong}\begin{cases}0 & \text{ if } i\neq d \\ A^\tau & \text{ if } i = d.\end{cases}$$ If $\tau$ is the identity, then $A$ is said to be **Calabi-Yau**. The condition on $\operatorname{Ext}$ in the above definition is sometimes referred to as the *rigid Gorenstein* condition [@brzha]. Regular algebras are important examples of (skew) Calabi-Yau algebras. This idea was formalized recently in [@rrz]. \[rrz\]An algebra is connected graded Calabi-Yau if and only if it is regular. On the other hand, (skew) Calabi-Yau algebras need not be graded. For example, the enveloping algebra of any finite-dimensional Lie algebra $\mathfrak{g}$ is Calabi-Yau if $\operatorname{trace}(\operatorname{ad}_\mathfrak{g}(x))$ for all $x \in \mathfrak{g}$ ([@he], Lemma 4.1). PBW deformations of Calabi-Yau algebras were studied by Berger and Taillefer [@berger]. Their main result is that PBW deformations of Calabi-Yau algebras defined by quivers and potentials are again Calabi-Yau. More recently, Wu and Zhu proved that a PBW deformation of a noetherian Koszul Calabi-Yau algebra is Calabi-Yau if and only if its Rees ring is [@wu]. We generalize their result partially below. \[cyprop\]If $A$ is essentially regular, then $A$ is skew Calabi-Yau. By Proposition \[grprop\], $A$ is a PBW deformation of the regular algebra $\operatorname{gr}(A)$. By [@yekzha], $A$ has a rigid dualizing complex $R=A^\sigma[n]$ for some integer $n$ and some $\sigma \in \operatorname{Aut}(A)$. This is precisely the condition for $A$ to be rigid Gorenstein. Since $\operatorname{gr}(A)$ is regular (Proposition \[grprop\]), then $\operatorname{gr}(A)$ is Calabi-Yau by Theorem \[rrz\]. Thus, $\operatorname{gr}(A)$ is homologically smooth and so, by [@mcrob], Theorem 7.6.17, $A$ is homologically smooth. If $A$ is a PBW deformation of a graded skew Calabi-Yau algebra. Then $A$ is skew Calabi-Yau. By hypothesis, $\operatorname{gr}(A)$ is graded skew Calabi-Yau. Hence, $\operatorname{gr}(A)$ is regular by Theorem \[rrz\] and so $A$ is essentially regular. Hence, $A$ is skew Calabi-Yau by Theorem \[cyprop\]. Let $R$ be a ring, $\sigma \in Aut(R)$ and $\delta$ a $\sigma$-derivation, that is, $\delta:R \rightarrow R$ satisfies the twisted Leibniz rule, $\delta(rs) = \sigma(r)\delta(s) + \delta(r)s$ for all $r,s \in R$. The *skew polynomial ring* (or Ore extension) $R[\xi;\sigma,\delta]$ is defined via the commutation rule $\xi r = \alpha(r)\xi + \delta(r)$ for all $r \in R$. Suppose $A$ is of the form . If $\sigma \in \operatorname{Aut}(A)$ with $\deg(\sigma(x_i)) = 1$, then $\sigma$ lifts to an automorphism $\hat{\sigma} \in \operatorname{Aut}(H(A))$ defined by $\hat{\sigma}(x_0)=x_0$ and $\hat{\sigma}(x_i)=\widehat{\sigma(x_i)}$ for $i > 0$. To see this, let $g$ be a defining relation for $A$ and $\hat{g}$ the corresponding relation in $H(A)$. For a generator $x_i$ of $A$, $\sigma(x_i)=y_{i,1} + y_{0,1}$ for some $y_{i,1} \in A_1$ and $y_{0,1} \in A_0={K}$. Thus, $\hat{\sigma}(x_i)=y_{i,1}+y_{0,1}x_0$. We must show that this rule implies $\hat{\sigma}(g) = \widehat{\sigma(g)}$. Suppose $\deg(g)=d$ and write $g = \sum_{i=0}^d g_i$ with $\deg(g_i)=i$. Then $\sigma(g_i) = \sum_{j=0}^i g_{i,j}$ where $\deg(g_{i,j})=j$. Thus, $$\widehat{\sigma(g)} = \widehat{\sum_{i=0}^d \sigma(g_i)} = \widehat{\sum_{i=0}^d \sum_{j=0}^i g_{i,j}} = \sum_{i=0}^d \sum_{j=0}^i g_{i,j} x_0^{d-j}.$$ Now $\hat{g} = \sum_{i=0}^d g_i x_0^{d-i}$ and a similar computation shows $$\hat{\sigma}(\hat{g}) = \sum_{i=0}^d \hat{\sigma}(g_i) x_0^{d-i} = \sum_{i=0}^d \sum_{j=0}^i (g_{i,j} x_0^{i-j}) x_0^{d-i} = \sum_{i=0}^d \sum_{j=0}^i g_{i,j} x_0^{d-j}.$$ Similarly, if $\delta$ is a $\sigma$-derivation of $A$ with $\delta(x_i)\leq 2$, then $\hat{\delta}$ is $\hat{\sigma}$-derivation of $H(A)$ with $\hat{\delta}(x_0)=0$ and $\hat{\delta}(x_i)=\widehat{\delta(x_i)}$ for $i>0$. \[hsp\]Let $A$, $\sigma$, and $\delta$ be as above. Then $H(A[\xi;\sigma,\delta]) = H(A)[\xi;\hat{\sigma},\hat{\delta}]$. Let $f_1,\hdots,f_m$ be the defining relations for $A$. The defining relations for $A[\xi;\sigma,\delta]$ are then $f_1,\hdots,f_m$ along with $e_1,\hdots,e_n$ where $e_i=x_i\xi-\sigma(\xi) x_i - g_i$. The defining relations for $H(A[\xi;\sigma,\delta])$ are then $\hat{f}_i,\hdots,\hat{f}_m$ along with $\hat{e}_i = x_i\xi-\hat{\sigma}(\xi) x_i - \hat{g}_i$. By Lemma \[hsp\], these are precisely the defining relations for $H(A)[\xi;\hat{\sigma},\hat{\delta}]$. Let $A$ be essentially regular. If $\sigma$ and $\delta$ are as above, then $A[\xi;\sigma,\delta]$ is essentially regular. Let $R = H(A)[\xi;\hat{\sigma},\hat{\delta}]$. By Lemma \[hsp\], it suffices to prove that $R$ is regular. Since $H(A)$ is regular, then it is Calabi-Yau. By [@lww], Theorem 3.3, skew polynomial extensions of Calabi-Yau algebras are Calabi-Yau and so $R$ is Calabi-Yau. Moreover, $\hat{\sigma}$ and $\hat{\delta}$ preserve the grading on $H(A)$ and so $R$ is graded. Thus, by Theorem \[rrz\], $R$ is regular. If $\delta=0$ (so $\hat{\delta}=0$), then $\xi$ is a normal element in $H(A)[\xi;\hat{\sigma}]$. Thus, by the Rees Lemma and [@mcrob], Proposition 7.2.2, it follows that $A[\xi;\sigma]$ essentially regular implies $A$ is essentially regular. It is not clear if this holds in the case $\delta \neq 0$. Geometry of Homogenized Algebras {#geom} ================================ In [@atv], Artin, Tate, and Van den Bergh showed that every dimension two and dimension three regular algebra surjects onto a *twisted homogeneous coordinate ring*. We begin this section by defining a twisted homogeneous coordinate ring following the exposition in [@keeler]. We then go on to define the related concept of a *geometric algebra*, which was originally called an *algebra defined by geometric data* by Vancliff and Van Rompay [@vanvan]. While one would not expect such a construction for deformations of regular algebras, one might hope to recover information about the deformed algebra from the geometry associated to the homogenization of a deformation. We show that, in certain cases, this geometry allows us to classify all finite-dimensional simple modules of a deformed regular algebra. Let $E$ be a projective scheme and $\sigma \in \operatorname{Aut}(E)$. Set ${\mathcal{L}}_0={\mathcal{O}}_E$, $\mathcal{L}_1 = \mathcal{L}$, and $${\mathcal{L}}_d = {\mathcal{L}}{\otimes}_{{\mathcal{O}}_E} {\mathcal{L}}^\sigma {\otimes}_{{\mathcal{O}}_E} \cdots {\otimes}_{{\mathcal{O}}_E} {\mathcal{L}}^{\sigma^{d-1}} \text{ for } d \geq 2.$$ Define the (graded) vector spaces ${\mathcal{B}}_m = H^0(E,\mathcal{L}_m)$. Taking global sections of the natural isomorphism ${\mathcal{L}}_d {\otimes}_{{\mathcal{O}}_E} \mathcal{L}_e^{\sigma^d} {\cong}{\mathcal{L}}_{d+e}$ gives a multiplication defined by $a \cdot b = a\sigma^m(b) \in {\mathcal{B}}_{m+n}$ for $a \in {\mathcal{B}}_m$, $b \in {\mathcal{B}}_n$. Thus, if $\sigma = {\mathrm{id}}_E$, then this construction defines the the (commutative) homogeneous coordinate ring of $E$. The **twisted homogeneous coordinate ring** of $E$ with respect to $\mathcal{L}$ and $\sigma$ is the $\mathbb{N}$-graded ring ${\mathcal{B}}={\mathcal{B}}(E,{\mathcal{L}},\sigma):=\bigoplus_{d \geq 0} H^0(E,\mathcal{L}_d)$ with multiplication defined as above. Artin and Stafford have shown that every domain of GK dimension two is isomorphic to a twisted homogeneous coordinate ring [@artstaf]. Hence, if $H$ is 3-dimensional regular and $g \in H_3$ is a normal element which is not a zero-divisor, then $H/gH$ must be isomorphic to some ${\mathcal{B}}$. By ([@atv], Theorem 6.8), every 3-dimensional regular algebra contains such an element, though it may be zero. To define a geometric algebra, we make a slight change of notation to conform to convention. In addition, we specialize to the case of quadratic algebras. These algebras were originally defined by Vancliff and Van Rompay. They were renamed *geometric algebras* by Mori [@mori] and we use that definition here. The free algebra ${K}\langle x_0,x_1,\hdots,x_n\rangle$ is equivalent to the tensor algebra $T(V)$ on the vector space $V=\{x_0,\hdots,x_1\}$. If $H$ is quadratic and homogeneous, then we write $H=T(V)/(R)$ where $R$ is the set of defining polynomials of $H$. Any defining polynomial may be regarded as a bilinear form $f:V {\otimes}_k V \rightarrow {K}$. Write $f = \sum \alpha_{ij} x_i {\otimes}x_j$, $\alpha_{ij} \in {K}$. If $p,q \in \mathbb{P}(V^*)$, written as $p=(a_0:a_1:\cdots:a_n)$ and $q=(b_0:b_1:\cdots:b_n)$, then $f(p,q) = \sum \alpha_{ij} a_ib_j$. Define the *vanishing set* of $R$ to be $${\mathcal{V}}(R) = \{ (p,q) \in \mathbb{P}(V^*) \times \mathbb{P}(V^*) \mid f(p,q)=0 \text{ for all } f \in R\}.$$ A homogeneous quadratic algebra $H=T(V)/(R)$ is called **geometric** if there exists a scheme $E \subset \mathbb{P}(V^*)$ and $\sigma \in \operatorname{Aut}E$ such that $$\begin{aligned} \textbf{G1} ~ &{\mathcal{V}}(R) = \{(p,\sigma(p)) \in \mathbb{P}(V^*) \times \mathbb{P}(V^*) \mid p \in E\}, \\ \textbf{G2} ~ &R = \{f \in V {\otimes}_k V \mid f(p,\sigma(p)) = 0 \text{ for all } p \in E\}.\end{aligned}$$ The pair $(E,\sigma)$ is called the **geometric data** corresponding to $H$. All regular algebras of dimension $d \leq 3$ are geometric. The classification of quadratic regular algebras given in [@atv] shows that either $E=\mathbb{P}^2$, or else $E$ is a cubic divisor in $\mathbb{P}^2$. The projective scheme $E$ is referred to as the point scheme of $H$. It is not true that every 4-dimensional regular algebra is geometric [@rompay]. However, it seems that they are in the generic case. \[geo\]If $A$ is a PBW deformation of a quadratic geometric algebra, then $H(A)$ is geometric. Let $V=\{x_1,\hdots,x_n\}$ and $W=\{x_0\} \cup V$. Write $$H(A)=T(W)/(R) \text{ and } \operatorname{gr}(A)=T(V)/(S).$$ Choose $p,q \in {\mathcal{V}}(R)$ and write $$p=(a_0:a_1:\cdots:a_n), ~~q=(b_0:b_1:\cdots:b_n).$$ Let $e_i=x_0x_i-x_ix_0$, $i=1,\hdots,n$, be the commutation relations of $x_0$ in $H$. Suppose $a_0=0$, then $e_i(p,q)=0$ implies $a_ib_0=0$ for $i=1,\hdots,n$. Since $q$ is not identically zero, then $b_0=0$. Reversing the argument, we see that $a_0=0$ if and only if $b_0=0$. Let $E_0 \subset {\mathcal{V}}(R)$ be those points with the first coordinate zero and define $\sigma{\mid_{E_0}}$ to be the automorphism corresponding to $\operatorname{gr}(A)$. Then $E_0$ is $\sigma$-invariant and the restriction $\left(E_0,\sigma{\mid_{E_0}}\right)$ is the geometric pair for $\operatorname{gr}(A)$. Let $E_1 \subset E$ be those points with first coordinate nonzero. If $a_0 \neq 0$, then $b_0 \neq 0$ and so there is no loss in letting $a_0=b_0=1$. Hence, $e_i(p,q)=0$ implies $a_i=b_i$ and so we define $\sigma{\mid_{E_1}} = {\mathrm{id}}_{E_1}$. We define the scheme $E=E_0 \cup E_1 \subset \mathbb{P}(V^*)$ where $E_0$ corresponds to the point scheme of $\operatorname{gr}(A)$ and $E_1$ corresponds to the diagonal of ${\mathcal{V}}(R)$, that is, $p \in E_1$ if $(p,p) \in {\mathcal{V}}(R)$. Define the automorphism $\sigma$ where $\sigma{\mid_{E_0}}$ is the automorphism corresponding to $\operatorname{gr}(A)$ and $\sigma{\mid_{E_1}}={\mathrm{id}}_{E_1}$. Thus, $E_1$ and $E_2$ are $\sigma$-invariant and ${\mathcal{V}}(R)$ is the graph of $E$. It is left to check that **G2** holds. Let $F$ be $R$ reduced to $E_0$. If $f \in F$, then $f(p,p)=0$ for all $p \in E_1$, so $f$ corresponds to a relation in commutative affine space. Define $C={K}[x_1,\hdots,x_n]/(F)$. If $g \in V {\otimes}_k V$ such that $g(p,\sigma(p))=0$ for all $p\in E_1$, then $\hat{g}:=g{\mid_{E_1}} \in {\mathcal{V}}(F)$. By the Nullstellensatz, $\hat{g}^n \in F$ for some $n$. On the other hand, if $p \in E_0$, then either $g$ is a commutation relation or else $\operatorname{gr}(g) \in S$. Thus, $g$ is quadratic and so $n=1$. Hence, $g \in R$. If $H$ is geometric, then $H_1 \approx {\mathcal{B}}_1$ and so there is a surjection $\mu:H \rightarrow {\mathcal{B}}$. When $H$ is noetherian, $I=\ker\mu$ is finitely generated by homogeneous elements and so there is hope of pulling information about $H$ back from ${\mathcal{B}}$. Let $M$ be a finite-dimensional simple module of $H$. Then $M$ is either $I$-torsion or it is $I$-torsionfree. Those of the first type may be regarded as modules over $H/I {\cong}{\mathcal{B}}$. Those of the second type are not as tractable, though results from [@atv2] give us a complete picture in the case that $H$ is regular of dimension 3. Our goal is to generalize the following example to homogenizations of 2-dimensional essentially regular algebras that are not PI. If $H={H({\mathcal{J}_1})}$, then $E_1 = \{(1:a:\pm i)\}$. The finite-dimensional simple modules of ${\mathcal{J}_1}$ are exactly of the form ${\mathcal{J}_1}/((x_1-a){\mathcal{J}_1}+ (x_2 \pm i){\mathcal{J}_1})$. They are all non-isomorphic. The following conjecture is closely related to these results. Let $S$ be a PBW deformation of a Sklyanin algebra that is not PI. Then all finite-dimensional simple modules of $S$ are 1-dimensional. While we cannot answer this conjecture in its entirety, we make progress towards the affirmative by showing that the modules which are torsion over the canonical map $H(S) \rightarrow {\mathcal{B}}$ are 1-dimensional. Let $A$ be essentially regular and $H=H(A)$ geometric with geometric pair $(E,\sigma)$. If $f$ is a defining relation of $H$ and $p=(p_0: p_1: \cdots : p_n) \in E_1$, then $f(p,\sigma(p))=0$ implies $\sigma(p)=p$. Thus, we write $f(p,p)=0$ or, more simply, $f(p)=0$. This is equivalent to defining the module, $$M_p = H/((x_0-1)H + (x_1-p_1)H + \cdots + (x_n-p_n)H ).$$ Since $H$ acts on $M_p$ via scalars, then $M$ is 1-dimensional. Since $x_0-1 \in \operatorname{Ann}(M_p)$, then $M_p$ may be identified with the $A$-module $A/((x_1-p_1)A + \cdots + (x_n-p_n)A )$. By an abuse notation, we also call this $A$-module $M_p$. Conversely, if $M=\{v\}$ is a 1-dimensional (simple) $A$-module, then $A$ acts on $M$ via scalars, say $x_i.v = c_iv$, $c_i \in {K}$, $i=1,\hdots,n$. By setting $x_0.v=v$, $M$ becomes an $H$-module. This action must satisfy the defining relations of $H$ and so setting $p_i=c_i$ gives $f(p)=0$. We have now proved the following. \[dim1\]Let $A$ be essentially regular. The $A$-module $M$ is $1$-dimensional if and only if $M {\cong}M_p$ for some $p \in E_1$. Moreover, if $M_p$ and $M_q$ are 1-dimensional simple modules of $A$, then $M_p{\cong}M_q$ if and only if $p=q$. For $A$ essentially regular, we believe that certain conditions will imply that these are all of the finite-dimensional simple modules. In the following, we will show that this is the case when $A$ is $2$-dimensional essentially regular and not PI. \[simps\]If $A$ is essentially regular and $M$ is a finite-dimensional simple module of $A$, then $\operatorname{Ann}(M) \neq 0$. Let $H=H(A)$. Write $M=M_A$ (resp. $M=M_H$) when $M$ is regarded as an $A$-module (resp. $H$-module). If $N_H \subset M_H$ as an $H$-module, then $N_A \subset M_A$, so $N_H=0$ or $N_H=M_H$. Thus, $M_H$ is a simple module and, moreover, $\dim_A(M_A)=\dim_H(M_H)$. By [@walton], Lemma 3.1, if $M_H$ is a finite-dimensional simple module and $P$ is the largest graded ideal contained in $\operatorname{Ann}(M_H)$, then ${\text{GK.dim}}(H/P)$ is 0 or 1. If $P=0$, then ${\text{GK.dim}}(H/P)={\text{GK.dim}}(H)>1$ when $H$ is regular of dimension greater than 1. Hence, if $\dim(M_H)>1$, then $\operatorname{Ann}(M_H)\neq 0$. Since $x_0.m=m$, then $x_0-1 \in \operatorname{Ann}(M_H)$, but $x_0-1$ is not a homogeneous element so $x_0-1 \notin P$. Let $r \in P \subset \operatorname{Ann}(M_H)$ with $r\neq 0$. If $r \in {K}[x_0]$ with $r \neq x_0-1$, then $1 \in P$ so $\operatorname{Ann}(M_H)=H$. Hence, $r \notin {K}[x_0]$ and so $r \not\equiv 0 \mod (x_0-1)$. Thus, $\operatorname{Ann}(M_A)\neq 0$. The following result is well-known. Suppose $A={\mathcal{J}}$ or $A={\mathcal{O}_q({K}^2)}$ with $q \in {K}^\times$ a nonroot of unity. Then every finite-dimensional simple module of $A$ is 1-dimensional. (Sketch) Let $M$ be such a module and let $xy$ and $y$ be the standard generators of $A$. By Lemma 5.9, $\operatorname{Ann}(M)$ is a nonzero prime ideal. In the case $A={\mathcal{O}_q({K}^2)}$, one checks that $C=\{x^iy^j \mid i,j \in \mathbb{N}\}$ is an Ore set and $AC{^{-1}}$ is a simple ring, so that every prime ideals contains either $x$ or $y$. Now $A/xA {\cong}{K}[y]$ and $A/yA {\cong}{K}[x]$ and the result follows. For $A={\mathcal{J}}$, one repeats with $C=\{y^i \mid i \in \mathbb{N}\}$ so that every nonzero prime ideal contains $y$. The following result applies to homogenizations of 2-dimensional essentially regular algebras. However, in light of Proposition \[geo\], it seems reasonable that it may apply to certain higher dimensional algebras as well. \[fdim\]Let $A$ be an essentially regular algebra of dimension two that is not PI. If $M$ is a finite-dimensional simple $A$-module, then $M$ is 1-dimensional. Let $g \in H=H(A)$ be the canonical element such that $H/gH {\cong}{\mathcal{B}}= B(E,\mathcal{L},\sigma)$ and let $Q=\operatorname{Ann}M$. Because $|\sigma|=\infty$, the set of $g$-torsionfree simple modules of $H$ is empty ([@atv2], Theorem 7.3). Hence, we may assume $M$ is $g$-torsion and therefore $M$ corresponds to a finite-dimensional simple module of ${\mathcal{B}}$. Since $H$ is a homogenization, then $g=g_0g_1$ where $g_i \notin {K}$ for $i=1,2$. It is clear that $x_0 \mid g$ so set $g_0=x_0$. Hence, $g_0 \in Q$ or $g_1 \in Q$ because $Q$ is prime. If $g_0$ and $g_1$ are irreducible, then the point scheme $E$ decomposes as $E=E_0 \cup E_1$. Thus, $M$ corresponds to a finite-dimensional simple module of ${\mathcal{B}}(E_0,\mathcal{L},\sigma{\mid_{E_0}})$ or ${\mathcal{B}}(E_1,{\mathcal{L}},\sigma{\mid_{E_1}})$. In the first case, we have that ${\mathcal{B}}(E_0,{\mathcal{L}},\sigma{\mid_{E_0}})$ is isomorphic to the twisted homogeneous coordinate ring of ${\mathcal{O}_q({K}^2)}$ or ${\mathcal{J}}$. Since $\sigma{\mid_{E_1}} = {\mathrm{id}}$, then ${\mathcal{B}}(E_1,{\mathcal{L}},\sigma{\mid_{E_1}})$ is commutative. Hence, $H/Q$ is commutative and $Q$ contains $x_0-1$ so $M$ is a 1-dimensional simple module of $A$. If $g$ divides into three linear factors $g_i$, $i=1,2,3$, then ${\mathcal{B}}/g_i{\mathcal{B}}$ is isomorphic to the twisted homogeneous coordinate ring of ${\mathcal{O}_q({K}^2)}$ or ${\mathcal{J}}$ for each $i$. As a corollary, we recover a well-known result regarding the Weyl algebra. The first Weyl algebra ${A_1({K})}$ has no finite-dimensional simple modules. If $p \in E_1$, then $p=(1,a,b)$. The defining relation $f=x_1x_2-x_2x_1-1$ gives $f(p,p) = ab-ba-1 = 1 \neq 0$. Hence, the point scheme $E_1$ is empty. More generally, suppose $A$ is a PBW deformation of a noetherian geometric algebra and $H=H(A)$. By Proposition \[geo\], $H$ is geometric. Let $(E,\sigma)$ be the geometric data associated to $H$ and let $I$ be the kernel of the canonical map $H \rightarrow {\mathcal{B}}(E,{\mathcal{L}},\sigma)$. Let $E_1$ be the fixed points of $E$ and $E_0 = E\backslash E_1$. We say $F \subset E_0$ is *reducible* if there exists disjoint and nonempty subschemes $F',F'' \subset F$ such that $F=F' \cup F''$ and $\sigma(F') \subset F'$, $\sigma(F'') \subset F''$. We say $F$ is *reduced* if it is not reducible. \[Itors\]With the above notation. If $M$ is a finite-dimensional simple module of $H$ that is $I$-torsion, then $M$ is either a module over $\operatorname{gr}(A)$ or else $M$ is 1-dimensional. Let $M$ be an $I$-torsion simple module of $H$, so we may regard $M$ as a simple module of ${\mathcal{B}}$. Let $Q=\operatorname{Ann}(M)$ and so $Q \neq 0$ by Lemma \[simps\]. If $P$ is the largest homogeneous prime ideal contained in $Q$, then $P$ corresponds to a reduced closed subscheme of $F \subset E$ and ${\mathcal{B}}/P {\cong}{\mathcal{B}}(F,\mathcal{O}_F(1),\sigma{\mid_F})$ ([@dmtwist], Lemma 3.3). These subschemes are well-understood in this case, and so either $F$ corresponds to a subscheme in the twisted homogeneous coordinate ring associated to $\operatorname{gr}(A)$, or else $F$ is fixed pointwise by $\sigma{\mid_F}$, in which case ${\mathcal{B}}/P$ is commutative. If $S$ is a deformed Sklyanin algebra that is not PI, then the only finite-dimensional simple module over $\operatorname{gr}(S)$ is the trivial one ([@walton], Theorem 1.3). By [@bruyn], there are exactly 8 fixed points in $E$. Hence, all $I$-torsion, finite-dimensional simple modules are 1-dimensional. The algebra $U(\mathfrak{sl}_2({K}))$ is essentially regular of dimension three, is not PI, but does have finite-dimensional simple modules of every dimension $n$. There are other examples of essentially regular algebras exhibiting the same behavior (see [@redman], [@benkroby]). This leads to the following conjecture. Let $A$ be essentially regular of dimension three that is not PI. Then either all finite-dimensional simple modules are 1-dimensional or else $A$ has finite dimensional simple modules of arbitrarily large dimension. We end this section with a brief foray into the PI case. Suppose $A$ is prime PI and essentially regular. By Proposition \[hprops\], $H=H(A)$ is also prime PI. Moreover, if we let $Q_A$ be the quotient division ring of $A$ and $Q_H$ that of $H$, then $$\operatorname{PI-deg}(H)=\operatorname{PI-deg}(Q_H) = \operatorname{PI-deg}(Q_A(x_0)) = \operatorname{PI-deg}(A[x_0]) = \operatorname{PI-deg}(A).$$ If $A$ is 2-dimensional essentially regular and PI, then $A={A_1^q({K})}$ or $A={\mathcal{O}_q({K}^2)}$ with $q$ a primitive $n$th root of unity. In each case, $\operatorname{PI-deg}(H(A))=n$. One can also show that $n=|\sigma|$ where $\sigma$ is the automorphism of the geometric pair corresponding to $H(A)$. The proof of Proposition \[fdim\] implies that the $g$-torsion finite-dimensional simple modules of $A$ correspond to the finite-dimensional simple modules of ${\mathcal{O}_q({K}^2)}$. The $g$-torsionfree simple modules of $H$ are in 1-1 correspondence with those of $H[g{^{-1}}]$. Let $\Lambda_0$ be its degree 0 component. Since $H$ contains a central homogeneous element of degree 1, then $\operatorname{PI-deg}(\Lambda_0) = \operatorname{PI-deg}(H) = n$ ([@leb], page 149). Thus, by [@walton], Theorem 3.5, the $g$-torsionfree simple modules of $H$ all have dimension $n$. A 5-dimensional family of regular algebras {#fivedim} ========================================== Suppose $A$ and $B$ are regular. In terms of generators and relations, the algebra $C=A {\otimes}B$ is easy to describe. Let $\{x_i\}$ be the generators for $A$ and $\{y_i\}$ those for $B$. Let $\{f_i\}$ be the relations for $A$ and $\{g_i\}$ those for $B$. Associate $x_i \in A$ with $x_i {\otimes}1 \in A {\otimes}B$, and similarly for the $y_i$. Then $A {\otimes}B$ is the algebra on generators $\{x_i,y_i\}$ satisfying the relations $\{f_i,g_i\}$ along with the relations $x_iy_j-y_jx_i=0$ for all $i,j$. A similar description holds when $A$ and $B$ are essentially regular. By comparing global dimension, one sees that $H(A {\otimes}B) {\ncong}H(A) {\otimes}H(B)$. However, a related identity will be used to prove the following proposition. Let $A$ and $B$ be essentially regular algebras. Then $A {\otimes}B$ is essentially regular. We must show that $H(A {\otimes}B)$ is regular given that $H(A)$ and $H(B)$ are. Suppose $z_0$ is the homogenizing element in $H(A {\otimes}B)$ and $x_0$, $y_0$ those in $H(A)$ and $H(B)$, respectively. By Proposition \[grprop\] and [@maowu], Proposition 3.5, it suffices to prove the following: $$\begin{aligned} \label{teniso} H(A {\otimes}B)/z_0H(A {\otimes}B) {\cong}H(A)/x_0H(A) {\otimes}H(B)/y_0H(B).\end{aligned}$$ This is clear from the defining relations for the given algebras. \[tencor\]Let $A$ and $B$ be $2$-dimensional essentially regular. Then $H(A {\otimes}B)$ is regular of dimension five. Using the techniques developed above, we hope to understand the module structure of algebras of the form $H(A {\otimes}B)$. Let $A=B={\mathcal{J}}$ with generating sets $\{x_1,x_2\}$ and $\{y_1,y_2\}$, respectively. Let $\hat{x}_i=x_i {\otimes}1$ and $\hat{y}_i=1 {\otimes}y_i$ for $i=1,2$. Then $C = A {\otimes}B$ is generated by $\{\hat{x}_1,\hat{x}_2,\hat{y}_1,\hat{y}_2\}$ and the defining relations are $$\begin{aligned} f &= \hat{x}_1\hat{x}_2-\hat{x}_2\hat{x}_1+\hat{x}_1^2, \\ g &= \hat{y}_1\hat{y}_2-\hat{y}_2\hat{y}_1+\hat{y}_1^2, \\ h_{ij} &= \hat{x}_i\hat{y}_j-\hat{y}_j\hat{x}_i \text{ for } i,j \in \{1,2\}.\end{aligned}$$ Let $E^A,E^B,E^C$ be the point schemes of $A,B$ and $C$, respectively. We claim that $E^C {\cong}E^A \cup E^B$. Let $p=(a_1:a_2:a_3:a_4) \in \mathbb{P}^3$. Then $p \in E^C$ if there exists $q=(b_1:b_2:b_3:b_4) \in \mathbb{P}^3$ such that $(p,q)$ is a zero for the above defining relations. The relation $f_1$ gives $\frac{a_1}{a_2}=\frac{b_1}{b_1+b_2}$ and $f_2$ gives $\frac{a_3}{a_4}=\frac{b_3}{b_3+b_4}$. Substituting into the the additional relations gives $a_3=a_4=0$ or else $a_1=a_2=0$. In the first case the points correspond to $E_A$ and otherwise to $E_B$. The following proposition generalizes the above example. Suppose $A$ and $B$ are regular and $C=A {\otimes}B$ is not commutative. Then $E^C = E^A \cup E^B$. Let $\{x_1,\hdots,x_n,y_1,\hdots,y_m\}$ be the generators of $C$ subject to relations $\{f_i$, $g_i$, $h_{ij}\}$ such that the subalgebra generated by the $x_i$ (resp. $y_i$) subject to the relations $f_i$ (resp. $g_i$) is isomorphic to $A$ (resp. $B$). Identify $A$ and $B$ with their respective images in $C$. Let $h_{ij}=x_iy_j-y_jx_i$ for $1 \leq i \leq n,1 \leq j \leq m$. Let $E$ be the point scheme of $C$ and $\sigma$ the corresponding automorphism. Choose $p = (a_1:\cdots:a_n:b_1:\cdots:b_m) \in E_A \times E_B$ and $q=\sigma(p)=(c_1:\cdots:c_n:d_1:\cdots:d_m)$. We claim either $a_i=0$ for all $i$ or else $b_i=0$ for all $i$. Let $\sigma_A=\sigma{\mid_A}$ and $\sigma_B=\sigma{\mid_B}$. Suppose there exists $l,k$ such that $a_l\neq 0$ and $b_k \neq 0$. There is no loss in letting $a_l=1$. Hence, $0=h_{lj}(p,q)=d_j-b_jc_l$. If $c_l=0$, then $d_j=0$ for all $j$. Hence, $\sigma_B (b_1:\hdots:b_m)=0$, a contradiction, so $c_l\neq 0$. Then $b_j=d_j$ for all $j$. Thus, either $a_i=0$ for all $i$ or else $\sigma_B$ is constant, so $B$ is commutative. An identical argument shows that either $b_i=0$ for all $i$ or else $\sigma_A$ is constant, so $A$ is commutative. If $A$ and $B$ are commutative, then so is $C$. If $A$ and $B$ are essentially regular, then the point scheme of $H(A {\otimes}B)$ has two components, $E_0$ and $E_1$. The component $E_0$ corresponds to that of $H(A {\otimes}B)/z_0H(A {\otimes}B)$ (see (\[teniso\])). An argument similar to that from the previous proposition shows that $E_1$ corresponds to $E_1^A \cup E_1^B$. Consequently, if $M$ is a 1-dimensional simple module of $A {\otimes}B$, then $M$ is isomorphic to a 1-dimensional simple module of $A$ or $B$. Acknowledgements {#acknowledgements .unnumbered} ================ The author would like to thank his advisor, Allen Bell, for his assistance throughout this project. Additionally, the author thanks to Dan Rogalski for helpful conversations at the MSRI Summer Graduate Workshop on Noncommuatative Algebraic Geometry and correspondence afterwards.

--- abstract: 'We show that the recently considered McGreevy-Swingle model for Majorana fermions in the presence of a ’t Hooft-Polyakov magnetic monopole arises when the Jackiw-Rebbi model is constrained to be conjugation self dual.' author: - 'S.-H. Ho' title: 'Constrained Jackiw-Rebbi model gives McGreevy-Swingle model ' --- The Dirac equation in a topological background has been studied in various dimensions, such as the background of a kink in one spatial dimension, a vortex in two spatial dimensions, a ’t Hooft-Polyakov magnetic monopole and a dyon in three dimensions [@Jackiw:1975fn][@Jackiw:1981ee], and there exist normalizable Dirac zero modes in all cases. The zero modes of Majorana fermions, however, are only found in the cases of a kink in one dimension [@Jackiw:1975fn] and a vortex background in two dimensions [@Jackiw:1981ee]. A well separated pair of Majorana zero modes can define a quibit since it is a degenerate two-state system, whose state is stored nonlocally [@Hasan:2010xy]. Due to this feature and obeying non-Abelian statistics, Majorana zero modes caught a lot of attention in physics because of its potential application on quantum computing [@Nayak:2008zza]. Recently, McGreevy and Swingle considered a three spatial dimension model for Weyl fermions coupled to a ’t Hooft-Polyakov monopole and a scalar field in the $SU(2)$ adjoint representation [@McGreevy:2011if] [@Teo:2009qv]. In [@McGreevy:2011if], they solved the zero mode Dirac equation explicitly and found the exact solutions for the Majorana zero modes. In this brief report, we show that the Jackiw-Rebbi model with a Dirac fermion in the fundamental representation of $SU(2)$ gauge group, once the conjugation condition is imposed on the Dirac field, reproducs the single Wyel fermion case of McGreevy-Swingle model. This indicates that the Majorana feature of the model is not only shown in the zero mode, but in the whole field. However, the quantum version of the theory is problematic because of the Witten anomaly [@McGreevy:2011if][@Witten:1982fp]. Let us start from the Lagrangian density (3.1) in [@Jackiw:1975fn]: \[eq1\] && = |\_a i \^ (D\_)\_[ab]{} \_b -gG |\_a T\^A\_[ab]{} \_b \^A ,\ (D\_)\_[ab]{} && =\_ \_[ab]{}-i g A\_\^A T\^A\_[ab]{} where $\psi_a$ is a four-component Dirac spinor and a two-component $SU(2)$ isospinor, $A_{\mu}^A$ is the vector potential, $T^A$ is the $SU(2)$ generator, $g_{\mu\nu}=diag (+1, -1, -1, -1) $ and $G$ is a positive dimensionless coupling constant. Here $(a,b)$ are isospin indices while spin indices are suppressed. (We work in the chiral representation for the gamma matrices and the fundamental representation for the $SU(2)$ matrices. Thus we use gamma matrix conventions of [@McGreevy:2011if] rather than [@Jackiw:1975fn].) \[eq2\] \^ && = ( [cc]{}0 & \^\ |\^ & 0 ),\ T\^[A]{} && =,  A=1,2,3 Here $\sigma^{\mu}=(1, \vec{\sigma})$ and $\bar{\sigma}^{\mu}=(1, -\vec{\sigma})$. From (\[eq1\]) we can derive the Dirac equation \[eq3\] (i\^ (\_ \_[ab]{}-i A\_\^A \^A\_[ab]{}) - \^A\_[ab]{} \^A ) \_b =0 or equivalently \[eq4\] &&H\_[ab]{} \_b \_b = i \_t \_a= E \_a,\ && =-i ,\ &&=\^0 =( [cc]{}- & 0\ 0 & ),    =\^0 =( [cc]{}0 & 1\ 1 & 0 ). The conjugated field \[eq5\] \_a\^c ( [cc]{}0 & i \^2\ i \^2 & 0 ) i \^2\_[ab]{} \^\*\_b C\_[ab]{} \^\*\_b satisfies the equation \[eq6\] H\_[ab]{}\_b\^c = -E \_a\^c owing to \[eq7\] (C H C\^[-1]{})\_[ab]{} = - (H\^\*)\_[ab]{} . Now we impose the conjugation constraint on the Dirac spinor $\Psi_a = \left(\begin{array}{c}\xi_a \\\eta_a\end{array}\right) $, \[eq8\] \_a\^c && = \_a,\ \_a = i \^2 i \^2\_[ab]{} \_b\^\* &&,   \_a =i \^2 i \^2\_[ab]{} \_b\^\* . Replacing the unconstrained Dirac spinor $\psi_a$ by the constrained $\Psi_a$ we can rewrite (\[eq1\]) in terms of the two component field $\xi_a$: \[eq9\] && = \^\_a ( [cc]{}i |\^ ( \_ \_[ab]{} - A\_\^A \^A\_[ab]{} ) & - \^A \^A\_[ab]{}\ - \^A \^A\_[ab]{} & i \^ ( \_ \_[ab]{} - A\_\^A \^A\_[ab]{} ) ) \_b\ &&= 2 \^\_a i |\^ ( \_ \_[ab]{} - A\_\^A \^A\_[ab]{} ) \_b - \_a\^[T]{} ( i\^2 \^A)\_[ab]{} \^A i \^2 \_b - \^\_a (i \^A \^2)\_[ab]{} \^A i \^2 \^\*\_b\ This is Equation (2.1) in [@McGreevy:2011if]. The single zero mode $\psi_a^0$ is present both in the unconstrained JR model and the constrained McGS model since its mode function satisfies $\psi_a^0 = C_{ab} \psi_b^{0*}$. A similar story has been told in two spatial dimensions: an unconstrained Dirac equation with conjugation properties like (\[eq5\]) and (\[eq6\]) describes graphene; when a conjugation constraint is imposed, the equation reduces to two components and describes Majorana fermions [@Chamon:2010ks]. We have shown that the McGS model emerges when the energy reflection conjugation is imposed on JR model with a Dirac fermion in $SU(2)$ fundamental representation. Here we also note that the Majorana fermion in $SU(2)$ adjoint representation in JR model cannot be achieved since it seems to be no way to impose the conjugation constraint in isovector fermion case [^1]. We thank R. Jackiw for suggesting this calculation and for discussion with J. McGreevy and B. Swingle. This work is supported by the National Science Council of R.O.C. under Grant number: NSC98-2917-I-564-122. [99]{} R. Jackiw and C. Rebbi, Phys. Rev.  D [**13**]{}, 3398 (1976). R. Jackiw, P. Rossi, Nucl. Phys.  [**B190**]{}, 681 (1981). M. Z. Hasan, C. L. Kane, Rev. Mod. Phys.  [**82**]{}, 3045 (2010). C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. Das Sarma, Rev. Mod. Phys.  [**80**]{}, 1083 (2008). J. McGreevy and B. Swingle, Phys. Rev.  D [**84**]{}, 065019 (2011). For most recent studies of Majorana zero mode in three spatial dimension which motivated the study in [@McGreevy:2011if], see: J. C. Y. Teo, C. L. Kane, Phys. Rev.  Lett [**104**]{}, 046401 (2010), M. Freedman, M. B. Hastings, C. Nayak, X. -L. Qi, K. Walker, Z. Wang, Phys. Rev.  B [**83**]{}, 115132 (2011), and M. Freedman, M. B. Hastings, C. Nayak, X. -L. Qi, \[arXiv:1107.2731\]. E. Witten, Phys. Lett.  [**B117**]{}, 324-328 (1982). C. Chamon, R. Jackiw, Y. Nishida, S. Y. Pi and L. Santos, Phys. Rev.  B [**81**]{}, 224515 (2010) [^1]: In the case of isovector fermion, the corresponding conjugated field is defined by $\Psi^c_a \equiv \left(\begin{array}{cc}0 & i \sigma^2 \\i \sigma^2 & 0\end{array}\right) \Psi^*_a$. Once we impose the conjugation constraint $\Psi^c=\Psi$ we only have the trivial solution $\Psi=0$.

--- abstract: 'Graph link prediction is an important task in cyber-security: relationships between entities within a computer network, such as users interacting with computers, or system libraries and the corresponding processes that use them, can provide key insights into adversary behaviour. Poisson matrix factorisation (PMF) is a popular model for link prediction in large networks, particularly useful for its scalability. In this article, PMF is extended to include scenarios that are commonly encountered in cyber-security applications. Specifically, an extension is proposed to explicitly handle binary adjacency matrices and include known covariates associated with the graph nodes. A seasonal PMF model is also presented to handle dynamic networks. To allow the methods to scale to large graphs, variational methods are discussed for performing fast inference. The results show an improved performance over the standard PMF model and other common link prediction techniques.' address: | $\color{blue}\dagger$ – Department of Mathematics, Imperial College London\ $\color{blue}\star$ – Advanced Research in Cyber-Systems, Los Alamos National Laboratory author: - '$\color{blue}{}^\dagger$' - '$\color{blue}{}^\star$' - '$\color{blue}{}^\dagger$' bibliography: - 'biblio.bib' title: Graph link prediction in computer networks using Poisson matrix factorisation --- , ,\ \ anomaly detection, dynamic networks, new link prediction, Poisson matrix factorisation, statistical cyber-security, variational inference Introduction {#intro} ============ In recent years, there has been a significant increase in investment from both government and industry in improving cyber-security using statistical and machine learning techniques on a wide range of data collected from computer networks [@HeardAdams18; @Jeske18]. One significant research challenge associated with these networks is [*link prediction*]{}, defined as the problem of predicting the presence of an edge between two nodes in a network graph, based on observed edges and attributes of the nodes [@liben]. Adversaries attacking a computer network often affect relationships (links) between nodes within these networks, such as users authenticating to computers, or clients connecting to servers. New links (previously unobserved relationships) are of particular interest, as many attack behaviours such as lateral movement [@neil], phishing, and data retrieval, can create [*new links*]{} between network entities [@Metelli19]. In practical cyber applications, it is necessary to use relatively simple and scalable statistical methods, given the size and inherently dynamic nature of these networks. Away from cyber applications, the link prediction problem has been an active field of research [see, for example, @dunlavy; @lu; @menon], being similar, especially in its static formulation, to recommender systems [@adomavicius]. Link prediction problems have been successfully tackled using probabilistic matrix factorisation methods, especially classical Gaussian matrix factorisation [@prob_mat_fact], and are currently widely used in the technology industry [see, for example, @agarwal_aoas; @Khanna13; @Paquet13; @Johnson14]. Poisson matrix factorisation (PMF) [@canny; @dunson; @cemgil; @gopalan] emerged as a suitable model in the link prediction framework, for its flexibility and scalability. The methodological contribution of this article is to present extensions of the PMF model, suitably adapted to scenarios which are commonly encountered in cyber-security applications. Traditionally, Poisson matrix factorisation methods are used on adjacency matrices of natural numbers representing, for example, ratings of movies provided by different users. In cyber-security applications, the counts associated with network edges are complicated by repeated observations, polling at regular intervals, and the intrinsic burstiness of the events [@Heard14]. As a result, the standard PMF model whereby counts associated with links are assumed to follow a Poisson distribution cannot be applied directly. Instead, indicator functions are applied and an extension for PMF on binary adjacency matrices is discussed. Next, a framework for including categorical covariates within the PMF model is introduced, which also allows for modelling of new nodes appearing within a network. Finally, extensions of the PMF model to incorporate seasonal dynamics are presented. The rest of the article is organised as follows: Section \[sec:data\] presents the computer network data which are to be analysed. Section \[background\] formally introduces Poisson matrix factorisation for network link prediction, and Section \[pmf\_covs\] discusses the proposed PMF model for binary matrices and labelled nodes. A seasonal extension is provided in Section \[seasonal\_pmf\]. Finally, results of the analysis are presented in Section \[results\_section\]. LANL computer network data {#sec:data} ========================== The methodologies in this article have been developed to provide insight into authentication data extracted from the publicly released “Unified Host and Network Dataset” from Los Alamos National Laboratory (LANL) [@Turcotte18]. The data contain authentication logs collected over a 90-day period from most computers in the Los Alamos National Laboratory enterprise network running a Microsoft Windows operating system. An example record from the data is {"UserName": "User865586", "EventID": 4624, "LogHost": "Comp256596", "LogonID": "0x5aa8bd4", "DomainName": "Domain001", "LogonTypeDescription": "Network", "Source": "Comp782342", "AuthenticationPackage": "Kerberos", "Time": 87264, "LogonType": 3}. From each authentication record, the following fields are extracted for analysis: the user credential that initiates the event ([UserName]{}), the computer where the authentication originates ([Source]{}), and the computer the credential is authenticating to (most often [LogHost]{}). Two bipartite graphs are generated from the data: first, the network users and the computers from which they authenticate, denoted *User – Source*; second, the same users and the computers or servers they are connecting to, denoted *User – Destination*. As generic notation, let $\mathbb G=(U,V,E)$ represent one of these bipartite graphs, where $U=\{u_1,u_2,\ldots\}$ is the set of [*users*]{} and $V=\{v_1,v_2,\ldots\}$ a set of [*computers*]{} (sometimes referred to as [*hosts*]{}). The set $E\subseteq U\times V$ represents the observed edges, such that $(u,v)\in E$ if user $u\in U$ connected to host $v\in V$ in a given time interval. A finite set of edges $E$ can be represented as a rectangular ${\vert{U}\vert}\times{\vert{V}\vert}$ binary adjacency matrix ${\mathbf}A$, where $A_{ij} =\mathds 1_E\{(u_i,v_j)\}$. For each user and computer, a list of categorical covariates were also obtained; these included *job titles* and *locations* for the users, and *subnets* and *types* for the computers. In total, there are $K=\numprint{1064}$ factor levels available for the user credentials and $H=735$ factor levels for the computers. One objective of this article is to present methodology for incorporating such covariates within Poisson matrix factorisation. As mentioned in Section \[intro\], for cyber-security applications it would be valuable to accurately predict and assess the significance of new links. Importantly, many new links are formed each day as part of normal operating behaviour of a computer network; to demonstrate this, Figure \[new\_links\_plot\] shows the the total number of edges and the percentage of those that are new for the *User – Source* and *User – Destination* graphs. Even though the relative percentage is small, this would still provide many more alerts than could be practically acted upon each day. ![Number of links per day (top), and proportion of those that are new (bottom), after $20$ days of observation of the LANL computer network. [ **Solid red**]{} curve: *User – Source*. [ **Dashed blue**]{} curve: *User – Destination*.[]{data-label="new_links_plot"}](Pictures/wls_new_per_day_REGULAR.pdf){width="\textwidth"} Background on Poisson matrix factorisation {#background} ========================================== Let ${\mathbf}N\in\mathbb N^{{\vert{U}\vert}\times{\vert{V}\vert}}$ be a matrix of non-negative integers $N_{ij}$. For recommender system applications, $N_{ij}$ could represent information about how a user $i$ rated an item $j$, or a count of the times they have clicked on or purchased the item. The hierarchical Poisson factorisation model [@gopalan] models $N_{ij}$ using a Poisson link function with rate given by the inner product between user-specific latent features $\bm\alpha_i\in\mathbb R^R_+$ and host-specific latent features $\bm\beta_j\in\mathbb R^R_+$, for a positive integer $R\geq 1$: $$N_{ij}\sim\mathrm{Pois}(\bm\alpha_i^\top\bm\beta_j)=\mathrm{Pois}\left({\sum}_{r=1}^R \alpha_{ir}\beta_{jr}\right). \label{pmf_equation}$$ The specification of the model is completed in a Bayesian framework using gamma hierarchical priors on the latent parameters: $$\begin{aligned} &\alpha_{ir} \sim\Gamma(a^{(\alpha)},\zeta_i^{(\alpha)}), \ i=1,\dots,{\vert{U}\vert},\ r=1,\dots,R, \\ &\beta_{jr}\sim\Gamma(a^{(\beta)},\zeta_j^{(\beta)}), \ j=1,\dots,{\vert{V}\vert},\ r=1,\dots,R, \\ & \zeta_i^{(\alpha)}\sim\Gamma(b^{(\alpha)},c^{(\alpha)}),\ \zeta_j^{(\beta)}\sim\Gamma(b^{(\beta)},c^{(\beta)}), \label{pmf_standard}\end{aligned}$$ where each of the gamma distribution parameters $a^{(\alpha)}, b^{(\alpha)}, c^{(\alpha)}, a^{(\beta)}, b^{(\beta)}, c^{(\beta)}$ are positive real numbers which must be specified. An advantage of PMF over competing models [for example, those of @hoff; @prob_mat_fact] is that the likelihood only depends on the number of observed links, meaning evaluating the likelihood is $\mathcal O(\mathrm{nnz}({\mathbf}N))$, where $\mathrm{nnz}(\cdot)$ is the number of non-zero elements in the matrix, compared to $\mathcal O({\vert{U}\vert}\times{\vert{V}\vert})$ for most statistical network models. Networks observed in real-world applications tend to be extremely sparse, $\mathrm{nnz}({\mathbf}N)\ll {\vert{U}\vert}\times{\vert{V}\vert}$, which makes PMF scalable to very large graphs. The Poisson matrix factorisation model has been used as a building block for multiple extensions. For example, [@chaney] developed *social Poisson factorisation* to include latent *social influences* in personalised recommendations. [@gopalan_cpf] developed *collaborative topic Poisson factorisation*, which adds a document topic offset to the standard PMF model to provide content-based recommendations and thereby tackle the challenge of recommending new items, referred to in the literature as cold starts. These ideas of combining collaborative filtering and content-based filtering are further developed in [@Zhang15], [@singh] and [@da_silva], where social influences are added as constraints in the latter. Note that these methods allow for item-specific covariate information to be incorporated into the model but not explicitly user specific covariate information. The approach outlined in this article allows for both user and item (host)-specific covariate information to be included. Finally, all of these methods model binary adjacency matrices (often referred to as “implicit data”) using the Poisson link function for convenience, despite the incorrect range that is implied. Dynamical extensions to PMF have also been studied. [@charlin] use Kalman filter updates to dynamically correct the rates of the Poisson distributions. [@schein; @schein2] propose a temporal version of PMF using the two main tensor factorisation algorithms: canonical polyadic and Tucker decompositions. [@Hosseini18] combine the PMF model with the Poisson process to produce dynamic recommendations. In general, despite the extensive treatments of PMF in a dynamical context, seasonality has not been explicitly accounted for. This article further aims to fill this gap and propose a viable seasonal PMF model. PMF with labelled nodes and binary adjacency matrices {#pmf_covs} ===================================================== Suppose that there are $K$ covariates associated with each user and $H$ covariates for each host. Let the value of the covariate $k$ for user $i$ be denoted as $x_{ik}$. Similarly, let the value of the covariate $h$ for host $j$ be $y_{jh}$. In cyber-security applications, and more generically in the network literature, the main interest is on categorical covariates, which indicate memberships of known groupings or clusters of nodes. For the remaining of this article, the covariates will be assumed to be binary indicators representing one-hot encodings of categorical variables. Several approaches for including nodal covariates in recommender systems using non-probabilistic matrix factorisation methods such as the Singular Value Decomposition (SVD) have been discussed in the literature [for some examples, see @Nguyen13; @Fithian18; @Dai19]. To model binary links, it is assumed that the count $N_{ij}$ is a latent random variable, and the binary indicator variable $A_{ij}=\mathds 1_{\mathbb N_+}(N_{ij})$ is a censored Poisson draw with a corresponding Bernoulli distribution. This type of link has been referred to in the literature as the Bernoulli-Poisson (BerPo) link [@acharya; @zhou]. The full extended model is $$\begin{aligned} A_{ij}\vert N_{ij} &=\ \mathds 1_{\mathbb N_+}(N_{ij}),\\ \ N_{ij}\vert\bm\alpha_i,\bm\beta_j,\bm\Phi &\sim \mathrm{Pois}\left(\bm\alpha_i^\top\bm\beta_j+{\boldsymbol}1_K^\top(\bm\Phi\odot{\boldsymbol}x_i{\boldsymbol}y_j^\top){\boldsymbol}1_H\right), \\ &= \mathrm{Pois}\left({\sum}_{r=1}^R\alpha_{ir}\beta_{jr}+{\sum}_{k=1}^K{\sum}_{h=1}^H\phi_{kh}x_{ik}y_{jh}\right)\label{nij}\end{aligned}$$ where ${\boldsymbol}1_n$ is a vector of $n$ ones, $\odot$ is the Hadamard element-wise product, and ${\boldsymbol}x_i=\{x_{ik}\}$, ${\boldsymbol}y_j=\{y_{jh}\}$ are $K$ and $H$-dimensional binary vectors of covariates. The $R$-dimensional latent features $\bm\alpha_i$ and $\bm\beta_j$, appear in the traditional PMF model, given in and $\bm\Phi=\{\phi_{kh}\}\in\mathbb R^{K\times H}_+$ is a matrix of interaction terms for each combination of the covariates. Under model , $${\mathbb P}(A_{ij}=1)=1- \exp\left(-{\sum}_{r=1}^R\alpha_{ir}\beta_{jr}-{\sum}_{k=1}^K{\sum}_{h=1}^H\phi_{kh}x_{ik}y_{jh}\right).\label{eq:paij}$$ To provide intuition for these extra terms, assume for the cyber-security application that a binary covariate for job title [manager]{} is provided for the users, and that a binary covariate for the location [research lab]{} for the hosts. If user $i$ is a manager and host $j$ is located in a research lab, then $\phi_{kh}$ expresses a correction to the rate $\bm\alpha_i^\top\bm\beta_j$ for a manager connecting to a machine in a research lab. The covariate term is inspired by the bilinear mixed-effects models for network data in [@hoff_bilinear]. The same hierarchical priors are used for $\bm\alpha_i$ and $\bm\beta_j$ and the following prior distribution completes the specification of the model: $$\begin{aligned} \phi_{kh}\vert\zeta^{(\phi)} &\sim\Gamma(a^{(\phi)},\zeta^{(\phi)}),\ k=1,\dots,K,\ h=1,\dots,H,\\ \zeta^{(\phi)}&\sim\Gamma(b^{(\phi)},c^{(\phi)}). \notag\end{aligned}$$ Note that this model provides a natural way for handling what the literature commonly refers to as cold starts, where new users or hosts appear in the network. Provided that covariate-level information is known about new entities, then the estimates for $\bm \Phi$ can be used to make predictions about links where $\bm \alpha_i$ and $\bm \beta_j$ for new user $i$ or new host $j$ could be initialised from the prior or some other global statistic based on other users and hosts. Bayesian inference {#bayes_inference} ------------------ Given an observed matrix ${\mathbf}A$, inferential interest is on the marginal posterior distributions of the parameters $\bm\alpha_i$ and $\bm\beta_j$ for all the users and hosts, and the regression parameters $\bm\Phi$ for the covariates, since these govern the predictive distribution for the edges observed in the future. A common approach for performing inference is adopted, where additional latent variables are introduced. Given the (assumed) unobserved count $N_{ij}$, a further set of latent counts $Z_{ijl},\ l=1,\dots,R+KH$, are used to represent the contribution of each component $l$ to the total latent count, such that $N_{ij}=\sum_{l} Z_{ijl}$. For $l\leq R$, $Z_{ijl}\sim\mathrm{Pois}(\alpha_{il}\beta_{jl})$. Otherwise, $l$ refers to a $(k,h)$ covariate pair, and $Z_{ijl}\sim\mathrm{Pois}(\phi_{kh})$. This construction ensures that $N_{ij}$ has precisely the Poisson distribution specified in . Inference using Gibbs sampling is straightforward, as the full conditionals all have closed form expressions, but sampling-based methods do not scale well with network size. Instead, a variational inference procedure is discussed. Variational inference schemes have already been successfully used in the literature for probabilistic matrix factorisation models with a number of different link functions [@Nakajima10; @Seeger12; @Lobato14]. Variational inference {#inference} --------------------- Variational inference [see, for example, @blei] is an optimisation based technique for approximating intractable distributions, such as the joint posterior density $p(\bm\alpha,\bm\beta,\bm\Phi,\bm\zeta,{\mathbf}N,{\mathbf}Z\vert{\mathbf}A)$, with a proxy $q(\bm\alpha,\bm\beta,\bm\Phi,\bm\zeta,{\mathbf}N,{\mathbf}Z)$ from a given distributional family $\mathcal{Q}$, and then finding the member $q^\star\in\mathcal{Q}$ that minimises the Kullback-Leibler (KL) divergence to the true posterior. Usually the KL-divergence cannot be explicitly computed, and therefore an equivalent objective, called the [*evidence lower bound*]{} (ELBO), is maximised instead: $$\mathrm{ELBO}(q) = \mathbb E^q[\log p(\bm\alpha,\bm\beta,\bm\Phi,\bm\zeta,{\mathbf}N,{\mathbf}Z,{\mathbf}A)]- \mathbb E^q[\log q(\bm\alpha,\bm\beta,\bm\Phi,\bm\zeta,{\mathbf}N,{\mathbf}Z)], \label{elbo}$$ where the expectations are taken with respect to $q(\cdot)$. The proxy distribution $q(\cdot)$ is usually chosen to be much simpler form than the posterior distribution so that maximising the ELBO is tractable. As in [@gopalan] the [*mean-field variational family*]{} is used, where the latent variables in the posterior are considered to be independent and governed by their own distribution, so that: $$\begin{gathered} q(\bm\alpha,\bm\beta,\bm\zeta,\bm\Phi,{\mathbf}N,{\mathbf}Z) = {\prod}_{i,r} q(\alpha_{ir}\vert\lambda_{ir}^{(\alpha)},\mu_{ir}^{(\alpha)})\times{\prod}_{j,r}q(\beta_{jr}\vert\lambda_{jr}^{(\beta)},\mu_{jr}^{(\beta)}) \\ \times{\prod}_{k,h}q(\phi_{kh}\vert\lambda_{kh}^{(\phi)},\mu_{kh}^{(\phi)})\times{\prod}_i q(\zeta_i^{(\alpha)}\vert\nu_i^{(\alpha)},\xi_i^{(\alpha)}) \times{\prod}_j q(\zeta_j^{(\beta)}\vert\nu_j^{(\beta)},\xi_j^{(\beta)}) \\ \times q(\zeta^{(\phi)}\vert\nu^{(\phi)},\xi^{(\phi)})\times{\prod}_{i,j}q(N_{ij},{\mathbf}Z_{ij}\vert\theta_{ij},\bm\chi_{ij}). \label{mean_field}\end{gathered}$$ The objective function is optimised using [*coordinate ascent mean field variational inference*]{} (CAVI), whereby each density or variational factor is optimised while holding the others fixed [see @bishop; @blei for details]. Using this algorithm the optimal form of each variational factor is: $$q^\star(v_j)\propto\exp\left\{\mathbb E^q_{-j}\left[\log p(v_{j}\vert{\boldsymbol}v_{-j},{\mathbf}A)\right]\right\}, \label{cavi_update}$$ where $v_j$ is an element of a *partition* of the full set of parameters ${\boldsymbol}v$, and the expectation is taken with respect to the variational densities that are currently held fixed for ${\boldsymbol}v_{-j}$, defined as ${\boldsymbol}v$ excluding the parameters in the subset $v_j$. Convergence of the CAVI algorithm is determined by monitoring the change in the ELBO over subsequent iterations. Since the prior distributions are chosen to be conjugate, the full conditionals in are all available analytically. Conditional on $N_{ij}$, ${\mathbf}Z_{ij}=(Z_{ij1},\dots,Z_{ij(R+KH)})$ has a multinomial distribution, $${\mathbf}Z_{ij}\vert N_{ij},\bm\alpha_i,\bm\beta_j,\bm\Phi \sim\mathrm{Mult}\left(N_{ij},\bm\pi_{ij}\right),$$ where $\bm\pi_{ij}$ is the probability vector proportional to $$(\alpha_{i1}\beta_{j1},\dots,\alpha_{iR}\beta_{jR},\phi_{11}x_{i1}y_{j1},\dots,\phi_{KH}x_{iK}y_{jH}).$$ Therefore, setting $\psi_{ij}=\bm\alpha_i^\top\bm\beta_j+{\boldsymbol}1_K^\top(\bm\Phi\odot{\boldsymbol}x_i{\boldsymbol}y_j^\top){\boldsymbol}1_H$, $$\begin{aligned} p(N_{ij},{\mathbf}Z_{ij}\vert\bm\alpha_i,\bm\beta_j,\bm\Phi,{\mathbf}A) =\left\{\begin{array}{ll} \mathrm{Pois}_+(\psi_{ij})\mathrm{Mult}(N_{ij},\bm\pi_{ij}) & A_{ij}>0, \\ \delta_0(N_{ij})\delta_{{\boldsymbol}0}({\mathbf}Z_{ij}) & A_{ij}=0, \end{array}\right. \label{block_gibbs}\end{aligned}$$ where $\mathrm{Pois}_+(\cdot)$ denotes the zero-truncated Poisson distribution. The user and host latent features complete conditionals are gamma, where $$\begin{gathered} \alpha_{ir}\vert\bm\beta,\zeta_i^{(\alpha)},{\mathbf}Z \sim\Gamma\left(a^{(\alpha)}+{\sum}_{j=1}^{{\vert{V}\vert}} Z_{ijr},\zeta_i^{(\alpha)}+{\sum}_{j=1}^{{\vert{V}\vert}}\beta_{jr}\right), \\ \beta_{jr}\vert\bm\alpha,\zeta_j^{(\beta)},{\mathbf}Z \sim\Gamma\left(a^{(\beta)}+{\sum}_{i=1}^{{\vert{U}\vert}} Z_{ijr}, \zeta_j^{(\beta)} + {\sum}_{i=1}^{{\vert{U}\vert}} \alpha_{ir}\right),\end{gathered}$$ and finally, $$\begin{gathered} \zeta_i^{(\alpha)}\vert\bm\alpha_i\sim\Gamma\left(b^{(\alpha)}+Ra^{(\alpha)},c^{(\alpha)}+{\sum}_{r=1}^R \alpha_{ir} \right),\\ \zeta_j^{(\beta)}\vert\bm\beta_j\sim\Gamma\left(b^{(\beta)}+Ra^{(\beta)},c^{(\beta)}+{\sum}_{r=1}^R \beta_{jr} \right). \label{zeta_eta}\end{gathered}$$ Similarly, $$\begin{gathered} \phi_{kh}\vert\zeta^{(\phi)},{\mathbf}Z\sim \Gamma\left(a^{(\phi)}+{\sum}_{i=1}^{{\vert{U}\vert}}{\sum}_{j=1}^{{\vert{V}\vert}} {Z_{ijl}},\ \zeta^{(\phi)}+{\sum}_{i=1}^{{\vert{U}\vert}} x_{ik}{\sum}_{j=1}^{{\vert{V}\vert}} y_{jh}\right), \label{phis} \notag\\ \zeta^{(\phi)}\vert{\boldsymbol}\Phi\sim\Gamma\left(b^{(\phi)}+KHa^{(\phi)},\ c^{(\phi)}+{\sum}_{k=1}^K {\sum}_{h=1}^H\phi_{kh}\right),\end{gathered}$$ where $l$ is the index corresponding to the covariate pair $(k,h)$. Since all the conditionals are exponential families, each $q(v_j)$ obtained from is from the same exponential family [@blei]. Hence, with the exception of $q(N_{ij},{\mathbf}Z_{ij}\vert\theta_{ij},\bm\chi_{ij})$, the proxy distributions in are all gamma; for example, $q(\alpha_{ir}\vert\lambda_{ir}^{(\alpha)},\mu_{ir}^{(\alpha)})=\Gamma(\lambda_{ir}^{(\alpha)},\mu_{ir}^{(\alpha)})$. The update equations for the variational parameters $\{\bm\lambda, \bm\mu, \bm\nu, \bm\xi, \bm\theta, \bm\chi\}$ can be obtained using , which is effectively the expected parameter of the full conditional with respect to $q$. For further details on obtaining the update equations for the Poisson and multinomial parameters $\bm\theta$ and $\bm\chi$, see Appendix \[appendix\_vi\]. The full variational inference algorithm is detailed in Algorithm \[algo\_vi\]. Note that each update equation only depends upon the elements of the matrix where $A_{ij} > 0$, providing computational efficiency for large sparse matrices. initialise $\bm\lambda,\bm\mu$ and $\bm\xi$ from the prior,\ set $\nu_i^{(\alpha)}=b^{(\alpha)}+Ra^{(\alpha)},\ \nu_j^{(\beta)}=b^{(\beta)}+Ra^{(\beta)},\ \nu^{(\phi)}=b^{(\phi)}+KHa^{(\phi)}$,\ calculate $\tilde x_k=\sum_{i=1}^{{\vert{U}\vert}}x_{ik},\ k=1,\dots,K$ and $\tilde y_h=\sum_{j=1}^{{\vert{V}\vert}}y_{jh}$,\ Link prediction --------------- Given the optimised values of the parameters of the variational approximation $q^\star(\cdot)$ to the posterior, a Monte Carlo posterior model estimate of $\mathbb P(A_{ij}=1)$ can be obtained by averaging over $M$ samples from $q^\star(\alpha_{ir}\vert\lambda_{ir}^{(\alpha)},\mu_{ir}^{(\alpha)}),\ q^\star(\beta_{jr}\vert\lambda_{jr}^{(\beta)},\mu_{jr}^{(\beta)})$, and $q^\star(\phi_{kh}\vert\lambda_{kh}^{(\phi)},\mu_{kh}^{(\phi)})$: $$\hat{\mathbb P}(A_{ij}=1)=1-\frac{1}{M}{\sum}_{m=1}^M \exp\left(-{\sum}_{r=1}^R\alpha_{ir}^{(m)}\beta_{jr}^{(m)}-{\sum}_{h,k}^{K,H}\phi_{kh}^{(m)}x_{ik}y_{jh}\right). \label{full_pval}$$ Alternatively, a computationally fast way to approximate ${\mathbb P}(A_{ij}=1)$ plugs in the parameters of the estimated variational distributions: $$\tilde{\mathbb P}(A_{ij}=1|\hat\alpha_{ir}, \hat\beta_{jr}, \hat\phi_{kh})=1-\exp\left(-{\sum}_{r=1}^R\hat\alpha_{ir}\hat{\beta}_{jr}-{\sum}_{h,k}^{K,H}\hat\phi_{kh}x_{ik}y_{jh}\right), \label{biased_pval}$$ where, for example, $\hat\alpha_{ir}=\lambda_{ir}^{(\alpha)}/\mu_{ir}^{(\alpha)}$, the mean of the gamma proxy distribution. Note that clearly gives a biased estimate, and by Jensen’s inequality $\hat{\mathbb P}(A_{ij}=1) \leq \tilde{\mathbb P}(A_{ij}=1)$ in expectation, but it carries a much lower computational burden. The approximation in has been successfully used for link prediction and network anomaly detection purposes in [@turcotte]. Dynamic networks: seasonal PMF {#seasonal_pmf} ============================== The previous sections have been concerned with making inference from a single adjacency matrix ${\mathbf}A$. Now, consider observing a sequence of adjacency matrices ${\mathbf}A_1,\dots,{\mathbf}A_T$, representing snapshots of the same network over time. Further, suppose this time series of adjacency matrices has seasonal dynamics with some known fixed seasonal period, $P$; for example, $P$ could be one day, one week or one year. To recognise time dependence, a third index $t$ is required, such that $A_{ijt}$ denotes the $(i,j)$-th element of the matrix ${\mathbf}A_t$, $t=1,\ldots,T$. As in Section \[pmf\_covs\], there are assumed to be underlying counts $N_{ijt}$ which are treated as latent variables, and the sequence of observed adjacency matrices is obtained by $A_{ijt}=\mathds 1_{\mathbb N_+}(N_{ijt})$. To account for seasonal repetition in connectivity patterns, the model proposed for the latent counts is: $$\begin{aligned} \label{seasonal_model} N_{ijt}\sim\ &\mathrm{Pois}\left({\sum}_{r=1}^R\alpha_{ir}\gamma_{i{t^\prime}r}\beta_{jr}\delta_{j{t^\prime}r} + {\sum}_{k=1}^{K}{\sum}_{h=1}^H \phi_{kh}x_{ik}y_{jh} \right) \\ =\ &\mathrm{Pois}\left(({\boldsymbol}\alpha_i\odot{\boldsymbol}\gamma_{i{t^\prime}})^\top({\boldsymbol}\beta_j\odot{\boldsymbol}\delta_{j{t^\prime}}) + {\boldsymbol}1_K^\top(\bm\Phi\odot{\boldsymbol}x_i{\boldsymbol}y_j^\top){\boldsymbol}1_H\right), \end{aligned}$$ where, for example, ${t^\prime}=1+(t\bmod P)$. In general, more complicated functions for ${t^\prime}$ might be required, as in Section \[seasonal\_sec\]. The priors on $\alpha_{ir}$ and $\beta_{jr}$ are those given in ; these parameters represent a baseline level of activity, which is constant over time. The two additional parameters $\gamma_{i{t^\prime}r}$ and $\delta_{j{t^\prime}r}$ represent corrections to these rates for seasonal segment ${t^\prime}\in \{1,\ldots,P\}$. Note that for some applications, it may be anticipated that there is a seasonal adjustment to the rate for the interaction terms of the covariates, in which case temporal adjustments could be also added to $\bm\Phi$. For identifiability, it is necessary to impose constraints on the seasonal adjustments so that, for example, for all $i,j,r$, $\gamma_{i1r}=\delta_{j1r}=1$. For ${t^\prime}>1$, the following hierarchical priors are placed on $\gamma_{i{t^\prime}r}$ and $\delta_{j{t^\prime}r}$: $$\begin{gathered} \gamma_{i{t^\prime}r}\sim\Gamma(a^{(\gamma)},\zeta_{{t^\prime}}^{(\gamma)}),\ \zeta_{{t^\prime}}^{(\gamma)}\sim\Gamma(b^{(\gamma)},c^{(\gamma)}),\\ \delta_{j{t^\prime}r}\sim\Gamma(a^{(\delta)},\zeta_{{t^\prime}}^{(\delta)}),\ \zeta_{{t^\prime}}^{(\delta)}\sim\Gamma(b^{(\delta)},c^{(\delta)}).\end{gathered}$$ Inference for the seasonal model can be performed following the same principles of Section \[inference\]; full details are given in Appendix \[inference\_seasonal\]. Results {#results_section} ======= The extensions to the PMF model detailed in Sections \[pmf\_covs\] and \[seasonal\_pmf\] are now used to analyse the LANL authentication data described in Section \[sec:data\]. For this analysis, the data are split into a training set corresponding to the first 56 days of activity, and a test set corresponding to days 57 through 82. During the latter time period, LANL conducted a *red-team* exercise, where the security team test the robustness of the network by attempting to compromise other network hosts; labels of known compromised authentication events will be used for validation. Summary statistics about the data are provided in Table \[summ\_table\], where “cold starts” refer to links originating from new users and hosts in the test data. Figure \[adj\_plots\] shows binary heat map plots of the adjacency matrices obtained from the training period for each the two graphs, *User – Source* and *User – Destination*. In all analyses, variational inference is used to estimate the parameters based on the the training data, with a threshold for convergence being $10^{-5}$ for relative difference between two consecutive values of the ELBO . The prior hyperparameters are set to $a^{*}=b^{*}=1$ and $c^{*}=0.1$, although the algorithm is fairly robust to the choice of these parameters. The number of latent features $R=20$ and was chosen using the criterion of the elbow in the scree-plot of singular values [@Zhu]. =1.2mm [c c c c c]{} & &\ & & & &\ & & & &\ & & & &\ & & & &\ & & & &\ & & & &\ [.495]{} ![Training set adjacency matrices for the two graphs (spy-plot). Nodes are sorted by in-degree and out-degree.[]{data-label="adj_plots"}](Pictures/adj_usersip.pdf "fig:"){width="\textwidth"} \[adjsip\] [.495]{} ![Training set adjacency matrices for the two graphs (spy-plot). Nodes are sorted by in-degree and out-degree.[]{data-label="adj_plots"}](Pictures/adj_userdip.pdf "fig:"){width="\textwidth"} \[adjdip\] Including covariates -------------------- First, results are presented for the extended PMF (EPMF) model with covariates, discussed in Section \[pmf\_covs\]. The training and test binary adjacency matrices are created from all the data obtained over the respective periods, although initially all links relating to new entities (users or computers) in the test period are removed. Performance is evaluated by the receiver operating characteristic (ROC) curve and the corresponding area under the curve (AUC). The AUC is used as a measure of quality of classification and will allow for the predictive power of the different models to be ranked. Due to the large computational effort of scoring all entries in the adjacency matrix, the AUC is estimated by subsampling the negative class at random from the zeros in the adjacency matrix formed from the test data; the sample sizes is chosen to be three times the size of the number of edges in the test set. Experimentation showed that such a balance leads to reliable estimates of the AUC. The AUC scores are summarised in Table \[auc\_table\]. For evaluating the AUC scores for new links (edges in the test set not present in the training set), the negative class was also restricted to entries in the training adjacency matrix for which $A_{ij}=0$. =1.2mm [c c c c c]{} & &\ & & & &\ & & & &\ & & & &\ ![ROC curves for standard PMF and extended PMF on *User – Source*, $R=20$.[]{data-label="roc"}](Pictures/tikz/roc_curve.pdf){width=".8\textwidth"} Table \[auc\_table\] shows that the AUC for *User – Destination* does not change significantly when the extended model is used; however, for *User – Source*, the extended PMF model offers a significant improvement. The difference in the results between the two networks can be explained by the contrasting structures of the adjacency matrices. The density of the graph for *User – Destination* is $0.184\%$ and for *User – Source*, $0.031\%$. However, despite *User – Destination* having a higher density, the links are concentrated on a small number of dominant nodes, as can be seen in Figure \[adjdip\]. Therefore, the prediction task is relatively easy: the probability of a link is roughly approximated by a function of the degree of the node, and adding additional information is not particularly beneficial. For *User – Source*, as can be seen by Figure \[adjsip\], the links are more evenly distributed between the nodes, and the prediction task is more difficult. Hence, in this setting, including additional information about known groupings is crucial to improve the predictive capability of the model. The ROC curves for the *User – Source* graph are shown in Figure \[roc\]. Cold starts ----------- As discussed in Section \[pmf\_covs\], the extended PMF model allows for prediction of new entities or nodes in the network (cold starts). To assess performance on links in the test set involving new users or hosts, the estimates of the covariate coefficients $\hat\phi_{kh} = \lambda_{kh}^{(\phi)}/\mu_{kh}^{(\phi)}$ from the training period are used. The latent feature values are set equal to the mean of all users and hosts observed in the training set. For comparison against a baseline model, the regular PMF model is used where the latent features are set as above; this has the effect of comparing against the global mean of latent features. Cold starts can be divided between new *users* and new *hosts*, and the AUC scores for prediction for each case are presented in Table \[cold\_starts\]. To calculate the AUC, the negative class is randomly sampled from the rows and columns corresponding to the new users and hosts, respectively. Again, there are only minor performance gains for *User – Destination*, and the regular PMF model using the global average of the latent features provides surprisingly good results. As discussed above, this can be explained by the prediction task being much simpler, and well approximated by a simple degree-based model. In contrast, for the *User – Source* graph the extended PMF model shows very good predictive performance for cold starts. =1.2mm [c c c c c]{} & &\ & & & &\ & & & &\ & & & &\ Red-team -------- The motivation for this work is the detection of cyber attacks; to assess performance from an anomaly detection standpoint, the event labels from the red-team attack are used as a binary classification problem. Figure \[roc\_redteam\] plots the ROC curves and AUC scores from the standard and extended PMF models, and improvements in detection capability are obtained using EPMF. Similarly to the previous cases, the predictive performance gain is most notable for *User – Source*. ![ROC curves for prediction of red-team events for standard PMF and extended PMF.[]{data-label="roc_redteam"}](Pictures/tikz/roc_redteam.pdf){width=".8\textwidth"} Rival methods ------------- The results in Table \[auc\_table\] are compared with other common link prediction methods in Table \[auc\_compare\]: tSVD [@dhillon], tKatz [@dunlavy], and a degree-based model where the probability of a link is approximated as $\mathbb P(A_{ij}=1)=1-\exp(-d^\mathrm{out}_id^\mathrm{in}_j)$ where $d^\mathrm{out}_i$ and $d^\mathrm{in}_j$ are the out-degree and in-degree of each node. Overall, when compared to the results in Table \[auc\_table\], the PMF models achieve impressive improvements over competing matrix factorisation techniques. =1.2mm [c c c c c]{} & &\ & & & &\ & & & &\ & & & &\ & & & &\ Seasonal modelling {#seasonal_sec} ------------------ To investigate dynamic modelling, binary adjacency matrices ${\mathbf}A_1,\dots,{\mathbf}A_{82}$ are constructed for each day across the train and test periods. The seasonal PMF model with the inclusion of covariates (SEPMF) is then compared against EPMF; for EPMF, the adjacency matrices are assumed to be independent realisations randomly generated from a fixed set of latent features. Due to a “9 day-80 hour” work schedule operated at LANL, whereby employees can elect to take vacation every other Friday, the seasonal period is assumed to be comprised of four segments: weekdays (Monday - Thursday), weekends (Saturday and Sunday), and two separate segments for alternating Friday’s. For each model, binary classification is performed using the model predictive scores calculated across the entire period. For the positive class, scores are calculated for all user-host pairs $(i,j)$ such that $A_{ijt}=1$ for at least one $t$ in the test set; for the negative class, a random sample of $(i,j)$ pairs such that $A_{ijt}=0$ for all $t$ in the test set are obtained, with sample size equal to three times the total number of observed links. Table \[auc\_seasonal\] presents the resulting AUC scores. For both networks, the seasonal model does not globally outperform the extended PMF model for [*all links*]{}. However, improvements are obtained for prediction of the *new links*. One explanation for the weaker overall performance could be the reduced training sample size implied for the seasonal model: EPMF in a dynamic setting assumes that the all daily graphs have been sampled from the same process, whereas if the seasonal model is used then the daily graphs are only informative for the corresponding seasonal segments. In addition, as briefly mentioned in Section \[intro\], elements of the data exhibit strong polling patterns, often due to computers automatically authenticating on users’ behalves [@Turcotte18]; some of the links that exhibit polling will not exhibit seasonal patterns, as the human behaviour has not been separated from the automated behaviour. =1.2mm [c c c c c]{} & &\ & & & &\ & & & &\ & & & &\ On the other hand, improvements in the estimation of new links, despite the reduced training sample size, demonstrates that it can be beneficial to understand the temporal dynamics of the network for these cases. Considering the context of the application, it might be perfectly normal for a user to authenticate to a computer during the week; however, that same authentication would be extremely unusual on the weekends when the user is not present at work. Without the seasonal model this behavioural difference in would be missed. Since the red-team activity only took place during working hours, this consideration is not tested by the labelled red-team data. Conclusion ========== In this article, extensions of the standard Poisson matrix factorisation model have been proposed, motivated by applications to computer network data, in particular the LANL enterprise computer network. The extensions are threefold: handling binary matrices, including covariates for users and hosts in the PMF framework, and accounting for seasonal effects. The counts $N_{ijt}$ have been treated as censored, and it has been assumed that only the binary indicator $A_{ijt}=\mathds 1_{\mathbb N_+}(N_{ijt})$ is observed. Starting from the hierarchical Poisson matrix factorisation model of [@gopalan], which only includes the latent features $\bm\alpha_i$ and $\bm\beta_j$, covariates have been included through the matrix of coefficients $\bm\Phi$. Seasonal adjustments for the coefficients are obtained through the variables $\bm\gamma_{i{t^\prime}}$ and $\bm\delta_{j{t^\prime}}$. A variational inference algorithm is given, suitably adapted for the Bernoulli-Poisson link. The results show improvements over competing models for link prediction purposes on the real computer network data. Including covariates provides significant improvement in predictive performance and allows for prediction of new nodes within the network. The seasonal model enables time-varying anomaly scores and offers marginal improvements for predicting new links, which are of primary interest in cyber-security applications. Acknowledgements {#acknowledgements .unnumbered} ================ The authors acknowledge funding from the Los Alamos National Laboratory, EPSRC and the Heilbronn Institute for Mathematical Research. Research presented in this article was supported by the Laboratory Directed Research and Development program of Los Alamos National Laboratory (New Mexico, USA) under project number 20180607ECR. Variational inference in PMF models =================================== Variational inference in the binary PMF model {#appendix_vi} --------------------------------------------- As all of the factors in the variational approximation given in take the same distributional form of the complete conditionals, $$\begin{aligned} q(N_{ij},\ &{\mathbf}Z_{ij}\vert\theta_{ij},\bm\chi_{ij}) = \left\{\begin{array}{ll} \mathrm{Pois}_+(\theta_{ij})\mathrm{Mult}(N_{ij},\bm\chi_{ij}) & A_{ij}>0, \\ \delta_0(N_{ij})\delta_{{\boldsymbol}0}({\mathbf}Z_{ij}) & A_{ij}=0. \end{array}\right. \label{factorisation}\end{aligned}$$ Let $\psi_{ij}$ denote the rate $\sum_{r=1}^R\alpha_{ir}\beta_{jr}+\sum_{k=1}^K\sum_{h=1}^H\phi_{kh}x_{ik}y_{jh}$ of the Poisson distribution for $N_{ij}$, and $\psi_{ijl},\ l=1,\dots,R+KH$, represent the individual elements in the sum. To get the update equations for $\bm\theta$ and $\bm\chi$, following , for $A_{ij}>0$, $$\mathbb E^q_{-N_{ij},{\mathbf}Z_{ij}}\left\{ \log p(N_{ij},{\mathbf}Z_{ij}\vert \bm\alpha_i,\bm\beta_j,\bm\Phi) \right\} = \sum_l \left\{ Z_{ijl}\mathbb E^q_{-N_{ij},{\mathbf}Z_{ij}}\left(\log\psi_{ijl}\right) -\log(Z_{ijl}!)\right\} + k, \label{expect_nij}$$ where $k$ is a constant with respect to $N_{ij}$ and ${\mathbf}Z_{ij}$. Hence: $$q^\star(N_{ij},{\mathbf}Z_{ij}) \propto {\prod}_{l=1}^{R+KH} \exp\left\{ \mathbb E^q_{-N_{ij},{\mathbf}Z_{ij}}(\log\psi_{ijl}) \right\}^{Z_{ijl}} \Big/ Z_{ijl}!,$$ with domain of ${\mathbf}Z_{ij}$ constrained to have $\sum_l Z_{ijl}>0$. Multiplying and dividing the expression by $N_{ij}!$ and $[\sum_l \exp\{ \mathbb E^q_{-N_{ij},{\mathbf}Z_{ij}}(\log\psi_{ijl})\}]^{N_{ijl}}$ gives a distribution which has the same form of . Therefore the rate $\theta_{ij}$ of the zero truncated Poisson is updated using $\sum_l \exp\{ \mathbb E^q_{-N_{ij},{\mathbf}Z_{ij}}(\log\psi_{ijl})\}$, see step 5 in Algorithm \[algo\_vi\] for the resulting final expression. The update for the vector of probabilities $\bm\chi_{ij}$ is given by a slight extension of the standard result for variational inference in the PMF model [@gopalan] to include the covariate terms, see step 6 of Algorithm \[algo\_vi\]. The remaining updates are essentially analogous to standard PMF [@gopalan]. Inference in the seasonal model {#inference_seasonal} ------------------------------- The inferential procedure for the seasonal model follows the same guidelines used for the non-seasonal model. Given the unobserved count $N_{ijt}$, latent variables $Z_{ijtl}$ are added, representing the contribution of the component $l$ to the total count $N_{ijt}$: $N_{ijt}=\sum_{l} Z_{ijtl}$. The full conditional for $N_{ijt}$ and $Z_{ijt}$ follows , except the rate for the Poisson and probability vectors for the multinomial will now depend on the seasonal parameters $\gamma_{i{t^\prime}r}$, $\delta_{j{t^\prime}r}$, and $\omega_{kh{t^\prime}}$. Letting $p$ denote a seasonal segment in $\{1,\ldots,P\}$ the full conditionals for the rate parameters are: $$\begin{gathered} \alpha_{ir}\vert{\mathbf}Z,\bm\beta,\bm\gamma,\bm\delta,\zeta_i^{(\alpha)} \overset{d}{\sim}\Gamma\left(a^{(\alpha)}+\sum_{j=1}^{{\vert{V}\vert}}\sum_{t=1}^T Z_{ijtr}, \zeta_i^{(\alpha)}+\sum_{t=1}^T\gamma_{i{t^\prime}r}\sum_{j=1}^{{\vert{V}\vert}}\beta_{jr}\delta_{j{t^\prime}r} \right), \\ \gamma_{ipr}\vert{\mathbf}Z,\bm\alpha,\bm\beta,\bm\delta,\zeta_p^{(\gamma)} \overset{d}{\sim}\Gamma\left(a^{(\gamma)}+\sum_{j=1}^{{\vert{V}\vert}}\sum_{t:{t^\prime}=p} Z_{ijtr}, \zeta_p^{(\gamma)}+\alpha_{ir}\sum_{j=1}^{{\vert{V}\vert}} \beta_{jr}\sum_{t:{t^\prime}=p} \delta_{j{t^\prime}r} \right), \notag \\ \phi_{kh}\vert{\mathbf}Z, \zeta^{(\phi)} \overset{d}{\sim}\Gamma\left(a^{(\phi)}+\sum_{i=1}^{{\vert{U}\vert}}\sum_{j=1}^{{\vert{V}\vert}}\sum_{t=1}^T Z_{ijtl}, \zeta^{(\phi)} +T\tilde x_k\tilde y_h \right), \end{gathered}$$ where $\tilde x_k=\sum_{i=1}^{{\vert{U}\vert}} x_{ik}$ and $\tilde y_h=\sum_{j=1}^{{\vert{V}\vert}} y_{jh}$. Similar results are available for $\beta_{jr}$ and $\delta_{jpr}$. Also: $$\begin{gathered} \zeta_p^{(\gamma)}\vert\bm\gamma\overset{d}{\sim}\Gamma\left(b^{(\gamma)}+{\vert{U}\vert}Ra^{(\gamma)},c^{(\gamma)}+\sum_{i=1}^{{\vert{U}\vert}}\sum_{r=1}^R \gamma_{ipr} \right),\end{gathered}$$ and similarly for $\zeta_p^{(\delta)}$. For $\zeta_i^{(\alpha)}$ and $\zeta_j^{(\beta)}$, the conditional distribution is equivalent to . The mean-field variational family is again used implying a factorisation similar to , so that $$\begin{aligned} q(&\bm\alpha,\bm\beta,\bm\Phi,\bm\gamma,\bm\delta, \bm\zeta,{\mathbf}N,{\mathbf}Z) = {\prod}_{i,j,t}q(N_{ijt},{\mathbf}Z_{ijt}\vert\theta_{ijt},\bm\chi_{ijt}) \times {\prod}_{i,r} q(\alpha_{ir}\vert\lambda_{ir}^{(\alpha)},\mu_{ir}^{(\alpha)}) \\ & \times{\prod}_{j,r}q(\beta_{jr}\vert\lambda_{jr}^{(\beta)},\mu_{jr}^{(\beta)})\times{\prod}_{k,h}q(\phi_{kh}\vert\lambda_{kh}^{(\phi)},\mu_{kh}^{(\phi)}) \times{\prod}_iq(\zeta_i^{(\alpha)}\vert\nu_i^{(\alpha)},\xi_i^{(\alpha)}) \\ & \times{\prod}_jq(\zeta_j^{(\beta)}\vert\nu_j^{(\beta)},\xi_j^{(\beta)}) \times q(\zeta^{(\phi)}\vert\nu^{(\phi)},\xi^{(\phi)}) \times{\prod}_{i,q,r}q(\gamma_{ipr}\vert\lambda_{ipr}^{(\gamma)},\mu_{ipr}^{(\gamma)}) \\ &\times{\prod}_{j,q,r}q(\delta_{jpr}\vert\lambda_{jpr}^{(\delta)},\mu_{jpr}^{(\delta)}) \times {\prod}_p q(\zeta_p^{(\gamma)}\vert\nu_p^{(\gamma)},\xi_p^{(\gamma)})q(\zeta_p^{(\delta)}\vert\nu_p^{(\delta)},\xi_p^{(\delta)}).\end{aligned}$$ As in Section \[inference\], each $q(\cdot)$ has the same form of the full conditional distributions for the corresponding parameter or group of parameters. Again the variational parameters are updated using CAVI and a similar algorithm is obtained to that detailed in Algorithm \[algo\_vi\], where steps 7, 8, 9 and 10 are modified to include the time dependent parameters. It follows that for the user-specific parameters the update equations take the form: $$\begin{gathered} \lambda_{ir}^{(\alpha)} = a^{(\alpha)}+\sum_{j=1}^{{\vert{V}\vert}}\sum_{t=1}^T \frac{A_{ijt}\theta_{ijt}\chi_{ijtr}}{1-e^{-\theta_{ijt}}},\ \mu_{ir}^{(\alpha)} = \frac{\nu_i^{(\alpha)}}{\xi_i^{(\alpha)}}+\sum_{t=1}^T\frac{\lambda_{i{t^\prime}r}^{(\gamma)}}{\mu_{i{t^\prime}r}^{(\gamma)}}\sum_{j=1}^{{\vert{V}\vert}}\frac{\lambda_{jr}^{(\beta)}}{\mu_{jr}^{(\beta)}}\frac{\lambda_{j{t^\prime}r}^{(\delta)}}{\mu_{j{t^\prime}r}^{(\delta)}}, \notag \\ \lambda_{ipr}^{(\gamma)}=a^{(\gamma)}+\sum_{j=1}^{{\vert{V}\vert}}\sum_{t:{t^\prime}=p} \frac{A_{ijt}\theta_{ijt}\chi_{ijtr}}{1-e^{-\theta_{ijt}}},\ \mu_{ipr}^{(\gamma)} = \frac{\nu_p^{(\gamma)}}{\xi_p^{(\gamma)}}+\frac{\lambda_{ir}^{(\alpha)}}{\mu_{ir}^{(\alpha)}}\sum_{j=1}^{{\vert{V}\vert}} \frac{\lambda_{jr}^{(\beta)}}{\mu_{jr}^{(\beta)}}\sum_{t:{t^\prime}=p} \frac{\lambda_{j{t^\prime}r}^{(\delta)}}{\mu_{j{t^\prime}r}^{(\delta)}},\end{gathered}$$ and similar results can be obtained for the host-specific parameters $\lambda_{ir}^{(\beta)}$, $\mu_{jr}^{(\beta)}$, $\lambda_{jpr}^{(\delta)}$ and $\mu_{jpr}^{(\delta)}$. The updates for $\nu_i^{(\alpha)},\ \xi_i^{(\alpha)},\ \nu_j^{(\beta)}$ and $\xi_j^{(\beta)}$ are identical to steps 7 and 8 in Algorithm \[algo\_vi\]. For the covariates $$\begin{gathered} \lambda_{kh}^{(\phi)}=a^{(\phi)}+\sum_{i=1}^{{\vert{U}\vert}}\sum_{j=1}^{{\vert{V}\vert}}\sum_{t=1}^T \frac{A_{ijt}\theta_{ijt}\chi_{ijtl}}{1-e^{-\theta_{ijt}}},\ \mu_{kh}^{(\phi)} = \frac{\nu^{(\phi)}}{\xi^{(\phi)}}+\tilde x_k\tilde y_hT. \end{gathered}$$ The updates for $\nu^{(\phi)}$ and $\xi^{(\phi)}$ are the same as step 9 in Algorithm \[algo\_vi\]. Finally, for the time dependent hyperparameters: $$\begin{aligned} \nu_p^{(\gamma)} = b^{(\gamma)} + {\vert{U}\vert}Ra^{(\gamma)},\ \xi_p^{(\gamma)} = c^{(\gamma)} + \sum_{i=1}^{{\vert{U}\vert}}\sum_{r=1}^R \frac{\lambda_{ipr}^{(\gamma)}}{\mu_{ipr}^{(\gamma)}},\end{aligned}$$ and similarly for $\nu_p^{(\delta)}$ and $\xi_p^{(\delta)}$. The updates for $\theta_{ijt}$ and $\bm\chi_{ijt}$ are similar to Appendix \[appendix\_vi\]. The required expectation $\mathbb E^q_{-N_{ijt},{\mathbf}Z_{ijt}}\{\log p(N_{ijt},{\mathbf}Z_{ijt}\vert \bm\alpha_i,\bm\beta_j,\bm\gamma_{i{t^\prime}},\bm\delta_{j{t^\prime}},\bm\Phi\}$ can be expanded similarly to , and the update equations for $\theta_{ijt}$ and $\chi_{ijtr}$ can be derived similarly to Appendix \[appendix\_vi\]: $$\begin{aligned} \theta_{ijt} =&\ \sum_{r=1}^R \exp\left\{\Psi(\lambda_{ir}^{(\alpha)})-\log(\mu_{ir}^{(\alpha)})+\Psi(\lambda_{jr}^{(\beta)})-\log(\mu_{jr}^{(\beta)})\right. \\ &\hspace{2.3cm} \left. +\ \Psi(\lambda_{i{t^\prime}r}^{(\gamma)})-\log(\mu_{i{t^\prime}r}^{(\gamma)})+\Psi(\lambda_{j{t^\prime}r}^{(\delta)})-\log(\mu_{j{t^\prime}r}^{(\delta)})\right\} \\ &+\sum_{k=1}^K\sum_{h=1}^H x_{ik}y_{jh}\exp\left\{\Psi(\lambda_{kh}^{(\phi)})-\log(\mu_{kh}^{(\phi)}) \right\}, \\[1em] \chi_{ijtl}\propto&\left\{ \begin{array}{ll} \exp\left\{\Psi(\lambda_{il}^{(\alpha)})-\log(\mu_{il}^{(\alpha)})+\Psi(\lambda_{jl}^{(\beta)})-\log(\mu_{jl}^{(\beta)})\right. & \\ \hspace{.8cm}\left.\ +\ \Psi(\lambda_{i{t^\prime}l}^{(\gamma)})-\log(\mu_{i{t^\prime}l}^{(\gamma)})+\Psi(\lambda_{j{t^\prime}l}^{(\delta)})-\log(\mu_{j{t^\prime}l}^{(\delta)})\right\} &l\leq R, \\ x_{ik}y_{jh}\exp\left\{\Psi(\lambda_{kh}^{(\phi)})-\log(\mu_{kh}^{(\phi)}) \right\} & l>R. \end{array}\right. \end{aligned}$$ <span style="font-variant:small-caps;">Francesco Sanna Passino, Nicholas A. Heard\ Department of Mathematics\ Imperial College London\ 180 Queen’s Gate\ SW7 2AZ London (United Kingdom)</span>\ *E-mail:* [`francesco.sanna-passino16@imperial.ac.uk`]{}\ <span style="font-variant:small-caps;">Melissa J.M. Turcotte\ Advanced Research in Cyber-Systems\ Los Alamos National Laboratory\ Bikini Atoll Rd, SM 30\ Los Alamos, NM 87545 (USA)</span>\ *E-mail:* [`mturcotte@lanl.gov`]{}

--- abstract: 'The non-radiative scattering amplitudes for electron-positron annihilation into quark and lepton pairs in the TeV energy range are calculated in the double-logarithmic approximation. The expressions for the amplitudes are obtained using infrared evolution equations with different cut-offs for virtual photons and for $W$ and $Z$ bosons, and compared with previous results obtained with an universal cut-off.' author: - 'A. Barroso' - 'B.I. Ermolaev' - 'M. Greco' - 'S.M. Oliveira' - 'S.I. Troyan' title: 'Electroweak $2 \rightarrow 2$ amplitudes for electron-positron annihilation at TeV energies' --- Introduction ============ Next future linear $e^+e^-$ colliders will be operating in a energy domain which is much higher than the electroweak bosons masses, so that the full knowledge of the scattering amplitudes for $e^+e^-$ annihilation into quark and lepton pairs will be needed. The forward-backward asymmetry for $e^+e^-$ annihilation into leptons or hadrons produced at energies much greater than the $W$ and $Z$ boson masses has been recently considered in Ref. [@egt], where the electroweak radiative corrections were calculated to all orders in the double-logarithmic approximation (DLA). It was shown that the effect of the electroweak DL radiative corrections on the value of the forward-backward asymmetry is quite sizable and grows rapidly with the energy. As usual, the asymmetry is defined as the difference between the forward and the backward scattering amplitudes over the sum of them. These amplitudes were calculated in Ref. [@egt] in DLA, by introducing and solving the Infrared Evolution Equations (IREE). This method is a very simple and the most efficient instrument for performing all-orders double-logarithmic calculations (see Ref. [@flmm] and Refs. therein). In particular, when it was applied in Ref. [@flmm] to calculate the electroweak Sudakov (infrared-divergent) logarithms, it led easily to the proof of the exponentiation of the Sudakov logarithms. At that moment this was in contradiction to the non-exponentiation claimed in Ref. [@pciaf] and obtained by other means. This contradiction provoked a large discussion about the exponentiation. The exponentiation was confirmed eventually by the two-loop calculations in Refs. [@me]-[@dp] and by summing up the higher loop DL contributions in Refs. [@kp] and [@m]. These Sudakov logarithms provide the whole set of DL contributions to the $2 \to 2$ amplitudes only when the process is considered in the hard kinematic region where all the Mandelstam variables $s,t,u$ are of the same order. On the other hand, when the kinematics of the $2 \to 2$ processes is of the Regge type, besides the Sudakov logarithms, another kind of DL contributions arises, coming from ladder Feynman graphs. Accounting for those (infrared stable) contributions it leads, instead of simple exponentials, to much more complicated expressions for the scattering amplitudes. This was first shown in Ref. [@ggfl], where in the framework of pure QED, the scattering amplitudes for the forward and backward $e^+e^- \to \mu^+\mu^-$ annihilation were calculated in the Regge kinematics. One example of high-energy electroweak processes in the Regge kinematics was considered in Ref. [@flmm], where the backward scattering amplitude was calculated, for the annihilation of a lepton pair with same helicities into another pair of leptons. More general calculations of the forward and backward electroweak scattering amplitudes were done in Ref [@egt]. However, both calculations in Refs. [@egt] and  [@flmm] were done under the assumption that the transverse momenta $k_{i \perp}$ of the virtual photons and virtual $W,Z$ -bosons were much greater than the masses of the weak bosons. In other words, the same infrared cut-off $M$ in the transverse momentum space, was used for all virtual electroweak bosons, i.e., $$\label{1cut} k_{i\perp} \gg M \geq M_W \approx M_Z .$$ Obviously, while $M$ is the natural infrared cut-off for the logarithmic contributions involving $W,Z$ bosons, the cut-off for the photons can be chosen independently. in accord with the experimental resolution in a given observed process. Indeed the assumption (\[1cut\]), although simplifying the calculations a lot, is unnecessary and an approach that involves different cut-offs for photons and $W,Z$ weak bosons would be more interesting and suitable for phenomenological applications. This technique involving different cut-offs for photons and for $W,Z$ bosons was applied in Ref. [@flmm], for calculating the double-logarithmic contributions of soft virtual electroweak bosons (the Sudakov electroweak logarithms) but not for the scattering amplitudes in the regions of Regge kinematics. In the present paper we generalize the results of Refs. [@flmm] and  [@egt], and obtain new double-logarithmic expressions for the $2 \to 2$ - electroweak amplitudes in the forward and backward kinematics. These expressions involve therefore different infrared cut-offs for virtual photons and virtual weak bosons. Throughout the paper we assume that the photon cut-off, $\mu$, and the $W,Z$ boson cut-off, $M$, satisfy the relations $$\label{cuts} M \geq M_{W,Z},~~~\mu \geq m_f$$ where $m_f$ is the largest mass of the quarks or leptons involved in the process. Notice that the values of $M$ and $\mu$ could be widely different. Let us remind that in order to study a scattering amplitude $A(s,t)$ in the Regge kinematics $s \gg -t$ (where $s$ and $t$ are the standard Mandelstam variables), it is convenient to represent $A(s,t)$ in the following form: $A(s,t) = A^{(+)}(s,t)+ A^{(-)}(s,t)$, with $ A^{(\pm)}(s,t) = (1/2)[A(s,t) \pm A(-s,t)]$ called the positive (negative) signature amplitudes. We shall consider only amplitudes with the positive signatures. The IREE for the negative signature electroweak amplitudes can be obtained in a similar way, see e.g. Ref. [@egt] for more details. The paper is organized as follows: in Sect. 2 we define the kinematics and express the scattering amplitude for the $e^+e^-$ annihilation in terms of invariant amplitudes. In Sect. 3, we construct the evolution equations for the invariant amplitudes for the case when in the center mass (cm) frame, the scattering angles are very small. First, we obtain the IREE equations in the integral form and then we transform them in the simpler, differential form. These differential equations are solved in Sect. 4 and explicit expressions for the invariant amplitudes involving the Mellin integrals are obtained. In Sect. 5, we consider the case of large scattering angles, or when the Mandelstam variables s, t and u are all large. Sect. 6 deals with the expansion of the invariant amplitudes into the perturbative series in order to extract the first-loop and the second-loop contributions. Then we compare these contributions to the analogous terms obtained when one universal cut-off is used and study their difference. The effect of high-order contributions in the two approaches is further studied in Sect. 7 where the asymptotic expressions of the amplitudes are compared. Finally, Sect. 8 contains our concluding remarks. Invariant amplitudes for the annihilation processes {#ELAST} =================================================== Let us consider a general process where the lepton $l^k(p_1)$ and its anti-particle $\bar{l}_i(p_2)$ annihilate into a quark or a lepton $q ^{k'}(p'_1)$ and its anti-particle $\bar{q}_{i'}({p'}_2)$ (see Fig. 1): $$\label{annih} l^k(p_1)\bar{l}_i(p_2) \to q^{k'}({p'}_1) \bar{q}_{i'}({p'}_2) ~.$$ ![Scattering amplitude of the annihilation of Eq. 3.[]{data-label="fig.T1"}](Ampfig1.epsi) For this process, the most complicate case occurs when both the initial and the final particles (anti-particles) are left-handed (right-handed). The scattering amplitudes for other helicities can be obtained easily from the formulae derived for this case. As there is no technical difference when considering the annihilation into quarks or leptons, we present parallel results for the annihilation into a quark-antiquark or a lepton-antilepton pair. According to our assumption, the initial lepton belongs to the weak isodoublet $(\nu,~e)$. The final lepton belongs to another doublet, e.g. $(\nu_{\mu},~\mu)$, and the final quarks are also from a doublet, e.g. $(u, ~d)$. The antilepton and the antiquark belong to the charge conjugate doublets. Obviously, the scattering amplitude $A$ for the annihilation can be written as follows: $$\label{A} A = q^{k'}({p'}_1) \bar{q}_{i'}({p'}_2) A^{i i'}_{k' k}l^k(p_1)\bar{l}_i(p_2) ~ ,$$ where the $SU(2)$ matrix amplitude $A^{i i'}_{k' k}$ has to be calculated. We will consider it in DLA. The DL contributions to $A^{i i'}_{k' k}$ are different according to the kinematics of the process. The kinematics is defined by appropriate relations among the Mandelstam variables $s,~t,~u$, $$\label{studef} s = (p_1 + p_2)^2,~t = (p_1 - {p'}_1)^2, ~u = (p_1 - {p'}_2)^2 ~.$$ Throughout this paper we assume that $\sqrt{s} \gg M_{W,Z}$. The kinematical regime defined as $$\label{hard} -t \sim -u \sim s$$ is called the hard kinematics and corresponds to large cm scattering angles $\theta \equiv \theta_{ \mathbf{p_1\,{p'}_1}} \sim 1$ . Radiative corrections to the annihilation in this kinematics yield DL contributions. There are also two other Regge-type kinematical regimes where DL contributions appear. First, there is the configuration where $$\label{tkin} s \sim -u \gg -t ~.$$ We call it the $t$ -kinematics. According to the terminology introduced in [@ggfl], it is the forward kinematics (with respect to the charge flow) for $e^+e^- \to \mu^+ \mu^-$ and $e^+e^- \to d \bar{d}$. At the same time, it corresponds to the backward kinematics for $e^+e^- \to u \bar{u}$. In this kinematics, $\theta \ll 1$. Second, there is the opposite kinematics where $\theta \sim\pi$ and therefore $$\label{ukin} s \sim -t \gg -u ~.$$ We define the configuration (\[ukin\]) as the $u$-kinematics. It corresponds to the forward scattering for $e^+e^- \to u \bar{u}$ and the backward scattering for $e^+e^- \to \mu^+ \mu^-$,  $e^+e^- \to d \bar{d}$. To simplify the calculations, it is convenient to introduce the projection operators $(P_j)^{i i'}_{k' k} ~(j = 1,2,3,4)$, so that $A^{i i'}_{k' k}$ can be written in the following form: $$\label{invampl} A^{i i'}_{k' k} = \frac{\bar{u}(-{p'}_2) \gamma_{\mu} u({p'}_1) \bar{u}(-p_2) \gamma^{\mu} u(p_1)}{s} |\Big[ (P_j)_{kk'}^{ii'} A_j + (P_{j + 1})_{kk'}^{ii'} A_{j + 1}\Big] ~,$$ where $j = 1$ for the $t$-kinematics and $j = 3$ for the $u$-kinematics. The representation (\[invampl\]) reduces the calculation of the matrix amplitude $A^{i i'}_{k' k}$ to the calculation of the invariant amplitudes $A_j$. The explicit expressions for the operators $(P_c)^{i i'}_{k' k} ~(c = 1,..,4)$ can be taken from Ref. [@egt]: $$\begin{aligned} \label{pr} (P_1)_{kk'}^{ii'} = \frac12\delta_k^{i'} \delta^i_{k'}, &&(P_2)_{kk'}^{ii'} = 2 (t_c)_k^{i}(t_c)^{i'}_{k'}, \\ \nonumber (P_3)_{kk'}^{ii'} = \frac{1}{2} \left[\delta^i_k \delta^{i'}_{k'}- \delta^{i'}_{k} \delta^i_{k'}\right]~, &&(P_4)_{kk'}^{ii'} = \frac{1}{2} \left[\delta^i_k \delta_{k'}^{i'} + \delta^{i'}_{k} \delta^i_{k'}\right] ~.\end{aligned}$$ According to the results of Ref. [@egt], the forward $(A_F)$ and backward $(A_B)$ amplitudes of the $e^+e^-$ annihilation into quarks are expressed through invariant amplitudes $A_j$ as follows: $$\begin{aligned} \label{amplquark} A_F(e^+e^- \to u\bar{u}) = (A_3 + A_4)/2, ~A_B(e^+e^- \to u\bar{u}) = A_4 , \\ \nonumber ~A_F(e^+e^- \to d\bar{d}) = (A_1 + A_2)/2, ~A_B(e^+e^- \to d\bar{d}) = A_2~.\end{aligned}$$ and the annihilation into leptons is expressed through the leptonic invariant amplitudes very similarly: $$\begin{aligned} \label{ampllept} A_F(e^+e^- \to \mu^+\mu^-) = (A_1 + A_2)/2, ~A_B(e^+e^- \to \mu^+\mu^-) = A_2 ~, \\ \nonumber A_F(e^+e^- \to \nu_{\mu} \bar{\nu_{\mu}}) = (A_3 + A_4)/2, ~~A_B(e^+e^- \to \nu_{\mu} \bar{\nu_{\mu}}) = A_4 ~.\end{aligned}$$ We have used the general notation $A_j$ for the invariant amplitudes in Eqs. (\[amplquark\], \[ampllept\]) and we will keep using this notation until Sect. 4. In order to calculate the amplitudes $A_j$ to all orders in the electroweak couplings in the DLA, we construct and solve some infrared evolution equations (IREE). These equations describe the evolution of $A_j,~(j = 1,2,3,4)$ with respect to an infrared cut-off. We introduce two such cut-offs, $\mu$ and $M$. We presume that $M \approx M_Z \approx M_W$ and use this cut-off to regulate the DL contributions involving soft (almost on-shell) virtual $W,Z$ -bosons. In order to regulate the IR divergences arising from soft photons we use the cut-off $\mu$ and we assume that $\mu \approx m_q \ll M$ where $m_q$ is the maximal quark mass involved. Both cut-offs are introduced in the transverse momentum space (with respect to the plane formed by momenta of the initial leptons) so that the transverse momenta $k_i$ of virtual photons obey $$\label{mu} k_{i\perp} > \mu ~,$$ while the momenta $k_i$ of virtual $W,Z$ -bosons obey $$\label{M} k_{i\perp} > M ~.$$ Let us first consider $A_j$ in the collinear kinematics where, in the cm frame, the produced quarks or leptons move very close to the $e^+e^-$ -beams. In order to fix such kinematics, we implement Eq. (\[tkin\]) by the further restriction on $t$: $$\label{tmu} s \sim -u \gg M^2 \gg \mu^2 \geq -t$$ and similarly for Eq. (\[ukin\]) by $$\label{umu} s \sim -t \gg M^2 \gg \mu^2 \geq -u ~.$$ Basically in DLA, the invariant amplitudes $A_j$ depend on $s, u$ and $t$ through logarithms. Under the restriction imposed by Eqs. (\[tmu\], \[umu\]) then all $A_j$ depend only on logarithms of $s, M^2, \mu^2$ in the collinear kinematics. It is convenient to represent $A_j$ in the following form: $$\label{ampl} A_j(s, \mu^2, M^2) = A_j^{(QED)}(s, \mu^2) + {A'}_j(s, \mu^2, M^2) ~,$$ where $A_j^{(QED)}(s, \mu^2)$ accounts for QED DL contributions only, i.e. the contributions of Feynman graphs without virtual $W,Z$ bosons. To calculate $A_j^{(QED)}(s, \mu^2)$ we use the cut-off $\mu$, therefore the amplitudes $A_j^{(QED)}$ do not depend on $M$. In contrast, the amplitudes ${A'}_j(s, \mu^2, M^2)$ depend on both cut-offs. These amplitudes account for DL contributions of the Feynman graphs, with one or more $W,Z$ propagators. By technical reasons, it is convenient to introduce two auxiliary amplitudes. The first one, $\tilde{A}^{(QED)}_j(s, M^2)$, is the same QED amplitude but with a cut-off $M$. The second auxiliary amplitude, $\tilde{A}_j(s, M^2)$ accounts for all electroweak DL contributions and the cut-off $M$ is used to regulate both the virtual photons and the weak bosons infrared divergences. Beyond the Born approximation, the invariant amplitudes we have introduced depend on logarithms, the arguments of which can be chosen as in the following parameterization: $$\begin{aligned} \label{param} A_j^{(QED)} = A_j^{(QED)}(s, \mu^2) = A_j^{(QED)}(s/ \mu^2), ~\tilde{A}^{(QED)}_j = \tilde{A}^{(QED)}_j(s, M^2) = \tilde{A}^{(QED)}_j(s/ M^2), \\ \nonumber ~\tilde{A}_j = \tilde{A}_j(s, M^2) = \tilde{A}_j(s/ M^2), ~{A'}_j = {A'}_j (s, \mu^2, M^2) = {A'}_j (s/M^2,\eta) ~, \end{aligned}$$ with $$\label{eta} \eta \equiv \ln(M^2/\mu^2) ~.$$ Our aim is to calculate the amplitudes ${A'}_j$, whereas the amplitudes $A_j^{(QED)}, \tilde{A}^{(QED)}_j$ and $\tilde{A}_j(s/ M^2)$ are supposed to be known. The amplitudes $\tilde{A}_j$ were introduced and calculated in Ref. [@egt]. In order to define amplitudes ${A'}_j, \tilde{A}_j(s, M^2)$, the projection operators of Eq. (\[pr\]) have been used. The use of these operators is based on the fact that the $SU(2)\times U(1)$ symmetry for the electroweak scattering amplitudes takes place at energies much higher than the weak mass scale $M$. On the contrary, the QED amplitudes $A_j^{(QED)}$ and $\tilde{A}^{(QED)}_j$ are not $SU(2)$ invariant at any energy. Nevertheless, it is convenient to introduce “the QED invariant amplitudes” $A_j^{(QED)}, \tilde{A}^{(QED)}_j$ by explicit calculation of the forward and backward QED scattering amplitudes. Then inverting Eq. (\[amplquark\]), we construct the amplitudes $A_j^{(QED)}$ for $e^+e^-$- annihilation into quarks: $$\begin{aligned} \label{qedquark} A_1^{(QED)} = 2 A_F^{(QED)}(e^+e¯- \to d \bar{d}) - A_B^{(QED)}(e^+e¯- \to u \bar{u}) , ~~A_2^{(QED)} = A_B^{(QED)}(e^+e¯- \to u \bar{u}) , \\ \nonumber A_3^{(QED)} = 2A_F^{(QED)}(e^+e¯- \to u \bar{u}) - A_B^{(QED)}(e^+e¯- \to d \bar{d}) , ~~A_4^{(QED)} = A_B^{(QED)}(e^+e¯- \to d \bar{d}) ~ \end{aligned}$$ and inverting Eq. (\[ampllept\]) allows us to obtain $A_j^{(QED)}$ for $e^+e^-$- annihilation into leptons: $$\begin{aligned} \label{qedlept} A_1^{(QED)} = 2A_F^{(QED)}(e^+e¯- \to \mu^+ \mu^-), ~A_2^{(QED)} = 0, ~A_3^{(QED)} = - A_4^{(QED)} = A_B^{(QED)}(e^+e¯- \to \mu^+ \mu^-).\end{aligned}$$ Evolution equations for amplitudes $A_j$ in the collinear kinematics ==================================================================== We would like to discuss now the IREE for the amplitudes introduced earlier. The basic idea for constructing infrared evolution equations for the scattering amplitudes consists in introducing the infrared cut-offs in the transverse momentum space and evolving the scattering amplitudes with respect to them. This method does not involve analyzing the DL contributions of specific Feynman graphs but is based on quite general conceptions such as the analyticity of the scattering amplitudes and the dispersion relations which guarantees its applicability to a wide class of problems (see e.g. Ref. [@flmm] and Refs. therein). The essence of the method is the factorizing the DL contributions of virtual particles with the minimal transverse momenta. The IREE with two cut-offs for the electroweak amplitudes in the hard kinematics (\[hard\]) were obtained in Ref. [@flmm]. In the present section we construct the IREE for the $2\rightarrow 2$ - electroweak amplitudes in the Regge kinematics. According to Eqs. (\[mu\], \[M\]), we use two different cut-offs for the virtual photons and for the weak bosons. The amplitude $A_j$ is in the lhs of such an equation. The rhs contains several terms. In the first place, there is the Born amplitude $B_j$. In order to obtain the other terms in the rhs, we use the fact that the DL contributions of virtual particles with minimal transverse momenta $( \equiv k_{\perp})$ can be factorized. Furthermore, this $k_{\perp}$ acts as a new cut-off for the other virtual momenta. The virtual particle with $k_{\perp}$ (we call such a particle the softest one) can be either an electroweak bosons or a fermion. Let us suppose first that the softest particle is an electroweak boson. In this case, in the Feynman gauge, DL contributions come from the graphs where the softest propagator is attached to the external lines in every possible way whereas $k_{\perp}$ acts as a new cut-off for the blobs as shown in Fig. 2.  When the softest electroweak boson is a photon, the integration region over $k_{\perp}$ is $\mu^2 \ll k_{\perp}^2 \ll s$ and its contribution, $G^{\gamma}_j$, to the rhs of the IREE is: $$\begin{aligned} \label{gphot} G^{(\gamma)}_j = - \frac{1}{8\pi^2} b_j^{(\gamma)} &\Big(&\int_{\mu^2}^s \frac{dk_{\perp}^2 }{k_{\perp}^2} \ln(s/k_{\perp}^2)A^{(QED)}_j(s, k_{\perp}^2) + \int_{\mu^2}^{M^2} \frac{dk_{\perp}^2 }{k_{\perp}^2}\ln(s/k_{\perp}^2) {A'}_j(s, k_{\perp}^2, M^2) + \\ \nonumber &&\int_{M^2}^s \frac{dk_{\perp}^2 }{k_{\perp}^2} \ln(s/k_{\perp}^2) A'_j(s, k_{\perp}^2, k_{\perp}^2 )\, \Big) ~,\end{aligned}$$ where $$\begin{aligned} \label{bphot} b_1^{(\gamma)} = g^2 \sin^2 \theta_W \frac{(Y_2 - Y_1)^2}{4}, ~~~b_2^{(\gamma)} = g^2 \sin^2 \theta_W \Big[ \frac{1}{6} + \frac{(Y_2 - Y_1)^2}{4} \Big] ~,\\ \nonumber b_3^{(\gamma)} = g^2 \sin^2 \theta_W \frac{(Y_2 + Y_1)^2}{4}, ~~~b_4^{(\gamma)} = g^2\sin^2 \theta_W \Big[ \frac{1}{6} + \frac{(Y_2 + Y_1)^2}{4} \Big] ~. \\ \nonumber\end{aligned}$$ We have used the standard notations in Eq. (\[bphot\]): $g,~ g'$ are the Standard Model couplings, $Y_1 ~(Y_2)$ is the hypercharge of the initial (final) fermions and $\theta_W$ is the Weinberg angle. The logarithmic factors in the integrands of Eq. (\[gphot\]) correspond to the integration in the longitudinal momentum space. The amplitude $A'$ in the last integral of Eq. (\[gphot\]) does not depend on $\mu$ because $k_{\perp}^2 > M^2$. Therefore it can be expressed in terms of $\tilde{A}_j(s,k_{\perp}^2)$ and $\tilde{A}_j^{(QED)}(s,k_{\perp}^2)$ : $${A'}_j(s, k_{\perp}^2, k_{\perp}^2) = \tilde{A}_j(s, k_{\perp}^2)-\tilde{A}_j^{(QED)}(s,k_{\perp}^2)~.$$ When the softest boson is either a $Z$ or a $W$, its DL contribution can be factorized in the region $M^2 \ll k_{\perp}^2 \ll s$. This yields: $$\label{gwz} G_j^{(WZ)} = - \frac{1}{8\pi^2} b_j^{(WZ)} \int_{M^2}^s \frac{dk_{\perp}^2 }{k_{\perp}^2} \ln(s/k_{\perp}^2)\tilde{A}_j(s/k_{\perp}^2 ) ~,$$ with $$\label{bwz} b_j^{(WZ)} = b_j - b_j^{\gamma} ~$$ and the factors $b_j$ can be taken from Ref. [@egt]: $$\begin{aligned} \label{btot} b_1 = \frac{{g'}^2 (Y_1 - Y_2)^2}{4}, ~~b_2 = \frac{8g^2+{g'}^2 (Y_1 - Y_2)^2}{4}, \\ \nonumber ~~b_3=\frac{{g'}^2 (Y_1 + Y_2)^2}{4}, ~~b_4 = \frac{8g^2 + {g'}^2 (Y_1 + Y_2)^2}{4}~~.\end{aligned}$$ In Eq. (\[gwz\]) we have used the fact that the $W$ and the $Z$ bosons cannot be the softest particles for the amplitudes $A^{(QED)}_j$ since the integrations over the softest transverse momenta in $A^{(QED)}_j$ can go down to $\mu$, by definition. The sum of Eqs. (\[gphot\]) and (\[gwz\]) , $G_j$ can be written in the more convenient way: $$\label{gtot} G_j(s, \mu^2, M^2) = G^{(\gamma)}_j(s, \mu^2, M^2)+ G_j^{(WZ)}(s,M^2) = G_j^{(QED)}(s, \mu^2) - \tilde{G}^{(QED)}_j(s, M^2) + \tilde{G}_j(s, M^2) + {G'}_j(s, \mu^2, M^2) ~,$$ where $$\begin{aligned} \label{g} G^{(QED)}_j = - \frac{1}{8\pi^2} b_j^{(\gamma)} \int_{\mu^2}^s \frac{dk_{\perp}^2 }{k_{\perp}^2} \ln(s/k_{\perp}^2)A^{(QED)}_j(s/k_{\perp}^2) ~, \tilde{G}^{(QED)}_j = -\frac{1}{8\pi^2} b_j^{(\gamma)} \int_{M^2}^s \frac{dk_{\perp}^2 }{k_{\perp}^2} \ln(s/k_{\perp}^2)\tilde{A}^{(QED)}_j (s/k_{\perp}^2) ~, \\ \nonumber \tilde{G}_j = - \frac{1}{8\pi^2} b_j \int_{M^2}^s \frac{dk_{\perp}^2 }{k_{\perp}^2} \ln(s/k_{\perp}^2)\tilde{A}_j (s/k_{\perp}^2) ~, {G'}_j = -\frac{1}{8\pi^2} b_j^{(\gamma)} \int_{\mu^2}^{M^2} \frac{dk_{\perp}^2 }{k_{\perp}^2} \ln(s/k_{\perp}^2){A'}_j (s/k_{\perp}^2, M^2/k_{\perp}^2 ) ~. \end{aligned}$$ Eqs. (\[gtot\], \[g\]) account for DL contributions when the softest particle is an electroweak boson. However, the softest particle can also be a virtual fermion. In this case, DL contributions from the integration over the momentum $k$ of the softest fermion arise from the diagram shown in Fig. 3 where the amplitudes $A_j$ are factorized into two on-shell amplitudes in the $t$-channel. We denote this contribution by $Q_j(s, \mu^2, M^2)$.  The analytic expression for $Q_j$ is rather cumbersome. However it looks simpler when the Sudakov parameterization is introduced for the softest quark momentum $k$ (with $p_1$ and $p_2$ being the initial lepton momenta). $$\label{sudak} k = \alpha p_2 + \beta p_1 + k_{\perp}~.$$ After simplifying the spin structure, we obtain $$\label{q} Q_j(s, \mu^2, M^2) = c_j \int_{\mu^2}^s d k_{\perp}^2 \int \frac{d \alpha}{\alpha}\frac{ d \beta}{\beta} \frac{2k_{\perp}^2 }{(s\alpha\beta - k_{\perp}^2)^2} A_j(s\alpha, k_{\perp}^2, M^2) A_j(s\beta, k_{\perp}^2, M^2) ~,$$ where $$\label{c} c_1 = c_2 = -c_3 = -c_4 = \frac{1}{8 \pi^2}~.$$ Similarly to Eq. (\[gtot\]), $Q_j$ of Eq. (\[q\]) can be divided into the following simple contributions: $$\label{qtot} Q_j = Q_j^{(QED)} - \tilde{Q}_j^{(QED)} + \tilde{Q}_j + {Q'}_j ~,$$ where $$\label{qqed} Q_j^{(QED)} (s/\mu^2)= c_j \int_{\mu^2}^s d k_{\perp}^2 \int \frac{d \alpha}{\alpha}\frac{ d \beta}{\beta} \frac{k_{\perp}^2 }{(s\alpha\beta - k_{\perp}^2)^2} A_j^{(QED)}(s\alpha/ k_{\perp}^2) A_j^{(QED)}(s\beta/ k_{\perp}^2) ~,$$ $$\label{qqedtilde} \tilde{Q}_j^{(QED)}(s/ M^2) = c_j \int_{M^2}^s d k_{\perp}^2 \int \frac{d \alpha}{\alpha}\frac{ d \beta}{\beta} \frac{k_{\perp}^2 }{(s\alpha\beta - k_{\perp}^2)^2} \tilde{A}_j^{(QED)}(s\alpha/ k_{\perp}^2) \tilde{A}_j^{(QED)}(s\beta/ k_{\perp}^2) ~,$$ $$\label{qtilde} \tilde{Q}_j(s/ M^2) = c_j \int_{M^2}^s d k_{\perp}^2 \int \frac{d \alpha}{\alpha}\frac{ d \beta}{\beta} \frac{k_{\perp}^2 }{(s\alpha\beta - k_{\perp}^2)^2} \tilde{A}_j(s\alpha/ k_{\perp}^2) \tilde{A}_j(s\beta/ k_{\perp}^2) ~,$$ and $$\begin{aligned} \label{qprime} {Q'}_j(s/M^2, \eta) = c_j \int_{\mu^2}^{M^2} d k_{\perp}^2 \int \frac{d \alpha}{\alpha}\frac{ d \beta}{\beta} \frac{2k_{\perp}^2 }{(s\alpha\beta - k_{\perp}^2)^2} \Big(2 A_j^{(QED)}(s\alpha/ k_{\perp}^2 ) {A'}_j(s\beta/ k_{\perp}^2, M^2/k_{\perp}^2) \\ \nonumber + {A'}_j(s\alpha/ k_{\perp}^2, M^2/k_{\perp}^2) {A'}_j(s\beta/ k_{\perp}^2, M^2/k_{\perp}^2) \Big)~. \end{aligned}$$ Now we are able to write the IREE for amplitudes $A_j$. The general form it given by: $$\label{ireegeneral} A_j = B_j + G_j + Q_j ~.$$ Then using Eqs. (\[gtot\]) and (\[qtot\]) we can rewrite it as $$\label{iree} {A'}_j + A_j^{(QED)}= B_j^{(QED)} - \tilde{B}_j^{(QED)} + \tilde{B}_j + G_j^{(QED)} - \tilde{G}_j^{(QED)} + \tilde{G}_j + {G'}_j + Q_j^{(QED)} - \tilde{Q}_j^{(QED)} + \tilde{Q}_j + {Q'}_j ~.$$ Let us notice that $A_j^{(QED)}(s/\mu^2)$ obeys the equation $$\label{ireeqed} A_j^{(QED)} = B_j^{(QED)} + G_j^{(QED)} + Q_j^{(QED)}$$ and therefore $A_j^{(QED)}$ cancels out in Eq. (\[iree\]). Also, the auxiliary amplitudes $\tilde{A}_j$ and $\tilde{A}_j^{(QED)}$, obey similar equations: $$\label{ireem} \tilde{A}_j^{(QED)} = \tilde{B}_j^{(QED)} + \tilde{G}_j^{(QED)} + \tilde{G}_j^{(QED)}, ~~ \tilde{A}_j = \tilde{B}_j + \tilde{G}_j + \tilde{Q}_j ~.$$ The solutions to Eqs. (\[ireeqed\], \[ireem\]) are known. With the notations that we have used they can be taken directly from Ref. [@egt]. Hence, we are left with the only unknown amplitude ${A'}_j$ in Eq. (\[iree\]). Using Eqs. (\[ireeqed\], \[ireem\]), we arrive at the IREE for ${A'}_j$, namely: $$\label{ireeaprime} {A'}_j(s/M^2, \eta) = \tilde{A}_j(s/M^2) - \tilde{A}_j^{(QED)}(s/M^2) + {G'}_j(s/M^2, \eta) + {Q'}_j(s/M^2, \eta) ~.$$ In order to solve Eq. (\[ireeaprime\]), it is more convenient to use the Sommerfeld-Watson transform. As long as one considers the positive signature amplitudes, this transform formally coincides with the Mellin transform. It is convenient to use different forms of this transform for the invariant amplitudes we consider: $$\begin{aligned} \label{mellinaqed} A_j^{(QED)}(s/ \mu^2) &=& \int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\, \Big( \frac {s}{\mu^2}\Big)^{\omega} f_j^{(0)} (\omega) ~, \\ \nonumber \\ \label{mellinaqedtilde} \tilde{A}_j^{(QED)}(s/ M^2) &=& \int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\, \Big( \frac {s}{M^2}\Big)^{\omega} f_j^{(0)}(\omega) ~, \\ \nonumber \\ \label{mellinatilde} \tilde{A}_j(s/ M^2) &=& \int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\, \Big( \frac {s}{M^2}\Big)^{\omega} f_j(\omega) ~, \\ \nonumber \\ \label{mellinaprime} {A'}_j(s/M^2, \eta) &=& \int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\, \Big( \frac {s}{M^2}\Big)^{\omega} F_j (\omega, \eta) ~. \end{aligned}$$ Combining Eqs. (\[mellinaqed\]) to (\[mellinaprime\]) with Eq. (\[ireeaprime\]) we arrive at the following equation for the Mellin amplitude $F_j (\omega, \eta)$: $$\begin{aligned} \label{masterint} \int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\, \Big( \frac {s}{M^2}\Big)^{\omega} F_j (\omega, \eta) &=& \int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\, \Big( \frac {s}{M^2}\Big)^{\omega} [f_j(\omega) - f_j^{(0)}(\omega)] \\ \nonumber & & -\int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\, \Big( \frac {s}{M^2}\Big)^{\omega} \frac{1}{8\pi^2} b_j^{(\gamma)} \int_{\mu^2}^{M^2} \frac{dk_{\perp}^2 }{k_{\perp}^2} \ln(s/k_{\perp}^2)F_j (\omega, \eta' ) \\ \nonumber & &+\int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\, \Big( \frac {s}{M^2}\Big)^{\omega} c_j\int_{\mu^2}^{M^2} \frac{dk_{\perp}^2 }{k_{\perp}^2} \Big( 2f_j^{(0)}(\omega)F_j (\omega, \eta' ) + F_j^2(\omega, \eta') \Big) \Big] ~,\end{aligned}$$ where $\eta' = \ln(M^2/k_{\perp}^2)$. Differentiating Eq. (\[masterint\]) with respect to $\mu^2$ leads to the homogeneous partial differential equation for the on-shell amplitude $F_j (\omega, \eta)$: $$\label{master} \frac{\partial F_j}{\partial \eta} = - \frac{1}{8\pi^2} b_j^{(\gamma)} \Big( - \frac{\partial F_j}{\partial \omega} + \eta F_j\Big) + c_j \Big( 2 f_j^{(0)}(\omega) F_j + F_j^2 \Big)~,$$ where we have used the fact that $\ln(s/\mu^2)$, in Eq. (\[masterint\]), can be rewritten as $\ln(s/M^2) + \eta$ and that $\ln(s/M^2)$ corresponds to $- \partial/ \partial \omega$ . Solutions to the evolution equations for collinear kinematics ============================================================= Let us consider first the particular case when $b_1^{(\gamma)} = 0$. It contributes to the forward leptonic, $e^+e^- \to \mu^+\mu^-$ annihilation and corresponds, in our notations, to the option $$\label{leptons} ~~Y_1 = Y_2 = -1.$$ Let us notice that $A_j$ with $j = 1$ contributes also to the forward $e^+e^- \to d \bar{d}$ annihilation, though here $Y_1 = -1, Y_2 = 1/3$ and therefore $b_1^{(\gamma)} \neq 0$. In order to avoid confusion between these cases, we change our notations, denoting $\Phi_1 \equiv F_1,~ \phi_1 \equiv f_1$ and $\phi_1^{(0)} \equiv f_1^{(0)}$ when $Y_1 = Y_2 = -1$. We will also use notations $\Phi_{2,3,4}$ instead of $F_{2,3,4}$ when we discuss the annihilation into leptons. Then we denote $c \equiv c_1 = 1/(8 \pi^2)$. Therefore, the lepton amplitude $\Phi_1 (\omega, \eta)$ for the particular case (\[leptons\]) obeys the Riccati equation $$\label{eqf1} \frac{\partial \Phi_1}{\partial \eta} = c \Big( 2 \phi_1^{(0)}(\omega) \Phi_1 + \Phi_1^2 \Big)~,$$ with the general solution $$\label{f1general} \Phi_1 = \frac{e^{2c \phi_1^{(0)}\eta}}{C \phi_1^{(0)} - e^{2c \phi_1^{(0)}\eta}/2 \phi_1^{(0)} } ~.$$ In order to specify $C$, we use the matching (see Eq. (\[masterint\])) $$\label{match} \Phi_1 = \phi_1(\omega) - \phi_1^{(0)}(\omega)~,$$ when $\eta = 0$, arriving immediately at $$\label{Phi1} \Phi_1 = \frac{2\phi_1^{(0)}(\phi_1 -\phi _1^{(0)})e^{2c\phi_1^{(0)}\eta}} {\phi_1^{(0)} + \phi_1 - (\phi_1 -\phi_1^{(0)})e^{2c\phi_1^{(0)}\eta}}$$ and therefore to the following expression for the invariant amplitude $L_1 \equiv A_1$ when $Y_1 = Y_2 = -1$: $$\label{aleptf} L_1 = \int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\, \Big( \frac {s}{\mu^2}\Big)^{\omega} \phi_1^{(0)}(\omega) + \int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\, \Big( \frac {s}{M^2}\Big)^{\omega} \frac{2\phi_1^{(0)}(\phi_1 -\phi _1^{(0)})e^{2c\phi_1^{(0)}\eta}} {\phi_1^{(0)} + \phi_1 - (\phi_1 -\phi_1^{(0)})e^{2c\phi_1^{(0)}\eta}} ~.$$ Obviously, when $\mu \to M$, Eqs. (\[aleptf\]) converges to the same amplitude obtained with using only one cut-off. Indeed, substituting $\mu = M$ and $\eta = 0$ leads to $$\label{aleptf1} L_1 = \int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\, \Big( \frac {s}{M^2}\Big)^{\omega} \phi_1(\omega)~.$$ According to Eqs. (\[qedlept\], \[mellinaqed\]), the QED amplitude $\phi_1^{(0)}$ is easily expressed in terms of Mellin amplitude $\phi_F^{(0)}$ for the forward $e^+e^- \to \mu^+\mu^-$ annihilation: $$\label{phi0f} \phi_1^{(0)} = 2 \phi_F^{(0)}~.$$ The expression for $\phi_F^{(0)}$ can be taken from Refs. [@ggfl],[@flmm] and [@egt]: $$\begin{aligned} \label{phi0} \phi_F^{(0)} = 4\pi^2 \big( \omega - \sqrt{\omega^2 - \chi^2_0} \big), \end{aligned}$$ with $$\label{chi0} \chi^2_0 = 2\alpha/\pi.$$ On the other hand, the amplitudes $\phi_j$ were calculated in Ref. [@egt]. In particular, $$\label{phi1} \phi_1 = 4 \pi^2\big( \omega - \sqrt{\omega^2 - \chi^2} \big)~,$$ where $\chi^2$ is expressed through the electroweak couplings $g$ and $g'$: $$\label{chi} \chi^2 = [3 g^2 + {g'}^2]/(8 \pi^2) ~.$$ Next, let us solve Eq. (\[master\]) for the general case of non-zero factor $b_j^{(\gamma)}$. Then, this equation describes the backward $e^+ e^-$ annihilation into a lepton pair (e.g. $\mu^+ \mu^-$) and also the forward and backward annihilation into quarks. Eq. (\[master\]) looks simpler when $\omega$, $\eta$ are replaced by new variables $$\label{xy} x = \omega/\lambda_j,~~~y = \lambda_j \eta~,$$ with $\lambda_j = \sqrt{ b_j^{(\gamma)}/(8\pi^2)}$. Changing to the new variables, we arrive again at the Riccati equation: $$\label{riccatij} \frac{\partial F_j}{\partial \tau} = \big( \sigma - \tau \big) F_j -2q_j f^{(0)}_j F_j - q_j F^2_j ~,$$ where $\sigma = (x + y)/2,~~ \tau = (x - y)/2$ and $q_j = c_j /\lambda_j$. The general solution to Eqs. (\[riccatij\]) is $$\label{generalj} F_j = \frac{P_j(\sigma, \tau)}{ C(\sigma) + q_j Q_j(\sigma, \tau)}~,$$ where $C(\sigma)$ should be specified, $$\label{P} P_j(\sigma, \tau) = \exp \Big( \sigma\tau - \tau^2/2 - 2q_j \int_{\sigma}^{\sigma + \tau} d \zeta f_j^{(0)}(\zeta) \Big)$$ and $$\label{Q} Q_j (\sigma, \tau) = \int_{\sigma}^{\sigma + \tau} d \zeta P_j(\sigma, \zeta) ~.$$ The QED amplitudes $f^{(0)}_j$ can be obtained from the known expressions for the backward, $f^{(0)}_B$ and forward $f^{(0)}_j$ QED scattering amplitudes: $$\begin{aligned} \label{fzerob} f^{(0)}_B(x) = (4\pi\alpha e_q/ p_B^{(0)}) d \ln (e^{x^2/4} D_{p_B^{(0)}} (x)) /d x ~, \end{aligned}$$ where $D_p$ are the Parabolic cylinder functions with $ p_B^{(0)} = -2e_q/(1 + e_q)^2$ and $e_q = 1$ for the annihilation into muons, $e_q = 1/3 ~(2/3)$ for the annihilation into $d ~(u)$- quarks. Similarly, the QED forward scattering amplitudes for the annihilation into quarks are $$\begin{aligned} \label{fzerof} f^{(0)}_F(x) = (4\pi\alpha e_q / p_F^{(0)}) d \ln (e^{x^2/4} D_{p_F^{(0)}}(x))/d x ~, \end{aligned}$$ with $ p_B^{(0)} = 2e_q/(1 - e_q)^2$ . Let us stress that the forward amplitudes for the annihilation into leptons are given by Eq. (\[aleptf\]). The amplitude $f^{(0)}_{F,B}$ was obtained first in Ref. [@ggfl] for the backward scattering in QED. Obviously, the only difference between the formulae for $f_j(x)$ and $f^{(0)}_j(x)$ is in the different factors $a_j$, $p_j $ and $\lambda_j$. We can specify $C(\sigma)$, using the matching $$\label{matchj} F_j(\omega) = f_j(\omega) - f^{(0)}_j(\omega) ~,$$ when $\eta = 0$. The invariant amplitudes $f_j$ were calculated in Ref. [@egt]: $$\label{fj} f_j (x) = \frac{a_j}{ p_j} \frac{d \ln (e^{x^2/4} D_{p_j}(x)) }{d x} = a_j \frac{ D_{p_j - 1}(x))}{D_{p_j}(x))} ~.$$ Using Eq. (\[matchj\]) we are led to $$\label{Fj} F_j = \frac{\big( f_j(x + y) - f^{(0)}_j (x + y)\big) P(\sigma, \tau)} {P(\sigma, \sigma) - (f_j(x + y) - f^{(0)}_j (x + y)) \big( Q(\sigma, \sigma) - Q(\sigma, \tau) \big)} ~$$ and finally to $$\begin{aligned} \label{solutionaj} A_j(s/M^2, \eta) &=& \int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\, \Big( \frac {s}{\mu^2}\Big)^{\omega} f_j^{(0)} (\omega) + \\ \nonumber & &\int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\, \Big( \frac {s}{M^2}\Big)^{\omega} \frac{\big( f_j(x + y) - f^{(0)}_j (x + y)\big) P_j(\sigma, \tau)} {P_j(\sigma, \sigma) - (f_j(x + y) - f^{(0)}_j (x + y)) \big( Q_j(\sigma, \sigma) - Q_j(\sigma, \tau) \big)} ~. \end{aligned}$$ It is easy to check that when $\mu = M$, $A_j(s/M^2, \eta)$ coincides with the amplitude $\tilde{A}_j(s, M^2)$ obtained in Ref. [@egt]. Eqs. (\[aleptf\], \[solutionaj\]) describe all invariant amplitudes for $e^+e^-$ -annihilation into a quark or a lepton pair in the collinear kinematics (\[tmu\], \[umu\]). Scattering amplitudes at large values of $t$ and $u$ {#FB} ===================================================== In this section we calculate the scattering amplitudes $A$ when the restriction of Eqs. (\[tmu\], \[umu\]) for the kinematical configurations (\[tkin\], \[ukin\]) are replaced by $$\label{tm} s \gg M^2 \geq -t \gg \mu^2$$ and $$\label{um} s \gg M^2 \geq -u \gg \mu^2 ~.$$ In this kinematical regions it is more convenient to study the scattering amplitudes $A$ directly, rather than using the invariant amplitudes $A_j$. In order to unify the discussion for both kinematics (\[tm\], \[um\]), let us introduce $$\label{kappat} \kappa = -t ~,$$ when (\[tm\]) is considered and $$\label{kappau} \kappa = -u$$ for the other case (\[um\]). Using this notation, the same parameterization $A = A(s, \mu^2, M^2, \kappa)$ holds for both kinematics (\[tm\], \[um\]). Let us discuss now the evolution equations for $A$. As in the previous case, it is convenient to consider separately the purely QED part, $A^{(QED)}$ and the mixed part, $A'$: $$\label{separ} A(s, \kappa, \mu^2, M^2) = A^{(QED)}(s, \kappa, \mu^2) + A'(s, \kappa, \mu^2, M^2 ).$$ Generalizing Eq. (\[param\]), we can parameterize them as follows: $$A^{(QED)}(s, \kappa, \mu^2) = A^{(QED)}(s/\mu^2, \kappa/\mu^2),~~~~~~~ A'(s, \kappa, \mu^2, M^2 ) = A'(s/M^2, s/\mu^2, \kappa/ \mu^2, M^2/\mu^2 ).$$ In order to construct the IREE for $A^{(QED)}$ and $A'$, we should consider again all options for the softest virtual particles. The Born terms for the configurations (\[tm\]) and (\[um\]) do not depend on $\mu^2$ and vanish after differentiating on $\mu$. The same is true for the softest quark contributions. Indeed, the softest fermion pair yields DL contributions in the integration region $k_{\perp}^2 \gg \kappa$, which is unrelated to $\mu$. Hence, we are left with the only option for the softest particle to be an electroweak boson. The factorization region for this kinematics is $$\label{kappak} \mu^2 \ll k_{\perp}^2 \ll \kappa ~.$$ Obviously, only virtual photons can be factorized in this factorization region, which leads to a simple IREE: $$\begin{aligned} \label{ireekappa} \frac{\partial A^{(QED)}}{ \partial \rho} + \frac{\partial A^{(QED)}}{ \partial z} = - \lambda ({b^{(\gamma)} \rho+ h^{(\gamma)} z}) A^{(QED)},~~~~ \frac{\partial A'}{ \partial \rho} + \frac{\partial A'}{ \partial z} + \frac{\partial A'}{ \partial \eta'} = - \lambda ({b^{(\gamma)} \rho+ h^{(\gamma)} z}) A' \end{aligned}$$ where we have denoted $\rho = \ln(s/\mu^2)$, $z = \ln(\kappa/ \mu^2)$, $ \eta' = \ln(\eta) = \ln(M^2/\mu^2)$ and $\lambda=\alpha/2\pi$. The factors $b^{(\gamma)}$ and $h^{(\gamma)}$ are: $$\label{bht} h^{(\gamma)} = e_1{e'}_1 + e_2{e'}_2, ~~b^{(\gamma)} = e_1 e_2 +{e'}_1{e'}_2 - e_2{e'}_1 - e_1{e'}_2 ~$$ for the case (\[kappat\]), and $$\label{bhu} h^{(\gamma)} = - e_2{e'}_1 + e_1{e'}_2, ~~b^{(\gamma)} = e_1 e_2 +{e'}_1{e'}_2 +e_2{e'}_2 + e_1{e'}_1 ~$$ for the other case (\[kappau\]). The notations $e_i, {e'}_i$ in Eqs. (\[bht\], \[bhu\]) stand for the absolute values of the electric charges. They correspond to the notations of the external particle momenta introduced in Fig. 1. The terms proportional to $h^{(\gamma)}$ in Eq. (\[ireekappa\]) correspond to the Feynman graphs where the softest photons propagate in the $\kappa$-channels. Let us notice that for any kinematics we consider it holds $$\label{bhsum} b_j^{(\gamma)}+ h_j^{(\gamma)} = (1/2)[e_1^2 + e_2^2 + {e'}_1^2 +{e'}_2^2]$$ due to the electric charge conservation. In order to solve Eq. (\[ireekappa\]), we use the matching with the amplitude $\hat{A} (s, \mu^2, M^2)$ for the same process, however in the collinear kinematics: $$\label{matchkappa} A^{(QED)}(s, \mu^2, \kappa, M^2) = \hat{A}^{(QED)}(s,\mu^2),~~ A' (s, \kappa, \mu^2, M^2) = \hat{A}'(s, \mu^2, M^2)~,$$ when $ \kappa = \mu^2$. The solution to Eq. (\[ireekappa\]) is $$\label{akappageneral} A^{(QED)} = \psi^{(QED)}(\rho - z) e^{- \lambda b_j^{(\gamma)} \rho^2/2 - \lambda h_j^{(\gamma)} z^2/2},~~ A' = \psi'(\rho - z, \eta' -z) e^{-\lambda b_j^{(\gamma)} \rho^2/2 - \lambda h_j^{(\gamma)} z^2/2} .$$ Using the matching of Eq. (\[matchkappa\]) allows to specify $\psi$ and $\psi^{(QED)}$. After that we obtain: $$\label{akappaprime} A^{(QED)} = S' \hat{A}^{(QED)}(s/\kappa) ,~~~~ A' = S' \hat{A}'(s/M^2, M^2/\kappa) ~,$$ where $$\label{sprime} S' = e^{ -\lambda b_j^{(\gamma)}\rho z +\lambda ( b_j^{(\gamma)}- h_j^{(\gamma)}) z^2/2} ~.$$ We did not change $s/M^2$ to $s/\kappa$ in Eq. (\[akappaprime\]) because $M^2 \gg \kappa$. It is convenient to absorb the term $-\lambda b_j^{(\gamma)}\rho z$ into the amplitudes $\hat{A}^{(QED)}$ and $\hat{A}'$. Introducing, instead of $\omega$, the new Mellin variable $l = \omega + \lambda b_j^{(\gamma)} z$ (see Ref. [@egt] for details), we rewrite Eq. (\[akappaprime\]) as follows (for the sake of simplicity we keep the same notations for these new amplitudes $\hat{A}^{(QED)}$ and $\hat{A}'$): $$\label{akappa} A^{(QED)} = S \big( \hat{A}^{(QED)}(s/\kappa) + \hat{A}'(s, \kappa, \mu^2, M^2) \big)$$ with $S$ being the Sudakov form factor for the case under discussion. $S$ includes the softest, infrared divergent DL contributions. When the photon infrared cut-off $\mu$ is assumed to be greater than the masses of the involved fermions, this form factor is: $$\label{sudffbigmu} S = \exp\Big(- \frac{\lambda}{2} \big(b^{(\gamma)}+ h^{(\gamma)}\big) \ln^2(\kappa/\mu^2) \Big) ~.$$ However, in the case of $e^+e^-$ annihilation into quarks (muons), if the cut-off $\mu$ is chosen to be very small, less than the electron mass, $m_e$ the exponent in Eq. (\[sudffbigmu\]) should be changed to : $$\label{sudff} S = \exp\Big(- \frac{\lambda}{2} \big(b^{(\gamma)}+ h^{(\gamma)}\big) (\ln^2(\kappa/\mu^2) - \ln^2(m^2_e/\mu^2) - \ln^2(m^2/\mu^2) \Big) ~,$$ where $m$ is the mass of the produced quark or lepton (cf. Ref. [@Greco:1980mh]). If $m > \mu >m_e$, the last term in the exponent of Eq. (\[sudff\]) is absent. The kinematics with larger values of $\kappa$, e.g. $s \gg \kappa \gg M^2$, can be studied similarly, although it is more convenient to use the invariant amplitudes $\hat{A}_j$. The result is $$\label{akappam} \hat{A}_j = S_j\tilde{A}_j(s/\kappa) ~,$$ where $$\label{sudffm} S_j = \exp\Big[ -\frac{\lambda}{2} \Big( \big(b_j^{(\gamma)}+ h_j^{(\gamma)}\big) (\ln^2(\kappa/\mu^2) - \ln^2(m^2_e/\mu^2) - \ln^2(m^2/\mu^2) +\big(b_j - b_j^{(\gamma)} + h_j - h_j^{(\gamma)}\big) \ln^2(\kappa/M^2) \Big)\Big] ~$$ and $\tilde{A}(s/M^2)$ is the scattering amplitude of the same process in the limit of collinear kinematics and using a single cut-off $M$. These amplitudes were defined in Sect. 2. The factors $h_j$ given below were calculated in Ref [@egt]: $$\begin{aligned} \label{h} h_1 = g^2(3 + \tan^2 \theta_W Y_1 Y_2)/2, && h_2 = g^2 (-1 + \tan^2 \theta_W Y_1 Y_2)/2 ~, \\ \nonumber h_3 = g^2(3 - \tan^2 \theta_W Y_1 Y_2)/2, && h_4 = g^2(-1 - \tan^2 \theta_W Y_1 Y_2)/2 ~.\end{aligned}$$ The form factors $S$, $S_j$ include the soft DL contributions, with the cm energies of virtual particles ranging from $\mu^2$ to $\kappa$. Due to gauge invariance, the sums $b_j^{(\gamma)}+ h_j^{(\gamma)}$ and $b_j + h_j$ do not depend on $j$ and $S_j$ is actually the same for every invariant amplitude contributing to $A^{i i'}_{k' k}$ in the forward (backward) kinematics (see Ref. [@egt]). Obviously, in the case of the hard kinematics where (see Eq. (\[hard\])) $s \sim -u \sim -t$, i.e. $s \sim \kappa$, ladder graphs do not yield DL contributions. The easiest way to see this, is to notice that the factor $(s/\kappa)^{\omega}$ in the the Mellin integrals (\[mellinatilde\]) for amplitudes $\tilde{A}_j$ does not depend on $s$ in the hard kinematics, therefore all Mellin integrals do not depend on $s$. So, the only source of DL terms in this kinematics is given by the Sudakov form factor $S_j$ given by Eq. (\[sudffm\]). Therefore, we easily arrive at the known result $$\label{ahardm} A^{i i'}_{k' k} = B^{i i'}_{k' k} S_j~.$$ $B^{i i'}_{k' k}$ in Eq. (\[ahardm\]) stands for the Born terms. The electroweak Sudakov form factor (\[sudffm\]) with two infrared cut-offs was obtained in Ref. [@flmm]. Forward $e^+e^-$ annihilation into leptons ========================================== Eqs. (\[amplquark\], \[ampllept\], \[aleptf\]) and (\[solutionaj\]) give the explicit expressions for the scattering amplitudes of $e^+ e^-$-annihilation into quarks and leptons in the collinear kinematics. These expressions resume the DL contributions to all orders in the electroweak couplings and operate with two infrared cut-offs. In order to estimate the impact of the two-cuts approach, we compare these results to the formulae for the same scattering amplitudes obtained in Ref. [@egt] where one universal cut-off $M$ was used. We focus on the particular case of the scattering amplitudes for the forward $e^+e^-$ annihilation into leptons and restrict ourselves, for the sake of simplicity, to the collinear kinematics of Eq. (\[tmu\]). Other amplitudes, and other kinematics can be considered in a very similar way. Eqs. (\[ampllept\], \[aleptf\]) and (\[solutionaj\]) show that the scattering amplitude $L_F^{(\mu)}$ of the forward $e^+e^-$ into $\mu^-\mu^+$ is $$\begin{aligned} \label{amplmu} L_F^{(\mu)}&=& \int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\, \Big( \frac {s}{\mu^2}\Big)^{\omega} \phi_F^{(0)}(\omega) + \frac{1}{2} \int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\, \Big( \frac {s}{M^2}\Big)^{\omega} \frac{4\phi_F^{(0)}(\phi_1 -2\phi _F^{(0)})e^{4c\phi_F^{(0)}\eta}} {2\phi_F^{(0)} + \phi_1 - (\phi_1 -2\phi_F^{(0)})e^{4c\phi_F^{(0)}\eta}} \\ \nonumber &&+ \frac{1}{2} \int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\, \Big( \frac {s}{M^2}\Big)^{\omega} \frac{\phi_2(x + y) P_2(\sigma, \tau)} {P_2(\sigma, \sigma) - \phi_2(x + y)\big[Q_2(\sigma, \sigma) - Q_2(\sigma, \tau)\big]} ~.\end{aligned}$$ The first integral in this equation accounts for purely QED double-logarithmic contributions and depends on the QED cut-off $\mu$ whereas the next integrals sum up mixed QED and weak double-logarithmic terms and depend on both $\mu$ and $M$. The first and the second integrals in Eq. (\[amplmu\]) grow with $s$ whilst the last integral rapidly falls when $s$ increases. The point is that this term actually is the amplitude for the backward annihilation into muon neutrinos. It is easy to check that the QED amplitudes $\phi_F^{(0)}$ vanish when $\mu = M$ and the total integrand contains only $[\phi_1(\omega) + \phi_2 (\omega)]/2$. In contrast to Eq. (\[amplmu\]), purely QED contributions are absent in formulae for $e^+e^- $ annihilation into neutrinos. For example, the scattering amplitude $L_F^{(\nu)}$ of the forward $e^+e^- \to \nu_{\mu}\bar{\nu_{\mu}}$ -annihilation in the collinear kinematics is $$\begin{aligned} \label{amplnu} L_F^{(\nu)} = \frac{1}{2} \int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\, \Big( \frac {s}{M^2}\Big)^{\omega} &\Big[& \frac{\phi_3 (x + y)P_3(\sigma, \tau)} {P_3(\sigma, \sigma) - \phi_3(x + y) [Q_3(\sigma, \sigma) - Q_3(\sigma, \tau)]} + \\ \nonumber &&\frac{\phi_4 (x + y)P_4(\sigma, \tau)} {P_4(\sigma, \sigma) - \phi_4(x + y) [Q_4(\sigma, \sigma) - Q_4(\sigma, \tau)]} ~\Big] ~.\end{aligned}$$ Similarly to Eq. (\[amplmu\]), the integrand in Eq. (\[amplnu\]) is equal to $[\phi_3(\omega) + \phi_4 (\omega)]/2$ when $\mu = M$. Although formally Eqs. (\[amplmu\], \[amplnu\]) correspond to the exclusive $e^+e^-$ annihilation into two leptons, actually these expressions also describe the inclusive processes when the emission of photons with cm energies $< \mu$ is accounted for. Let us study the impact of our two-cut-offs approach on the scattering amplitude $L_F^{(\mu)}$ of Eq. (\[amplmu\]). As the last integral in Eq. (\[amplmu\]) rapidly falls with $s$, it is neglected in our estimates and we consider contributions of the first and the second integrals only. First we compare the one-loop and two-loop contributions. Such contributions can be easily obtained expanding the rhs of Eq. (\[amplmu\]) into a perturbative series. From Eqs. (\[phi0\]) and (\[phi1\]) one obtains that $$\begin{aligned} \label{phi01series} \phi_F^{(0)} &\approx& 2\pi^2 \Big( \frac{\chi_0^2}{\omega} + \frac{1}{4}\frac{\chi_0^4}{\omega^3} + \frac{1}{8}\frac{\chi_0^6}{\omega^5} + ... \Big) ~, \\ \nonumber \phi_1 &\approx& 2\pi^2 \Big( \frac{\chi^2}{\omega} + \frac{1}{4}\frac{\chi^4}{\omega^3} + \frac{1}{8}\frac{\chi^6}{\omega^5} + ... \Big) ~,\end{aligned}$$ with $\chi_0$, $\chi$ defined in Eqs. (\[chi0\], \[chi\]). Substituting these series into the first and the second integrals of Eq. (\[amplmu\]) and performing the integrations over $\omega$, we arrive at $$\label{afirstloop} L^{(1)} = \gamma_1^{(1)}\ln^2(s/\mu^2) + \gamma_2^{(1)} \ln(s/\mu^2)\ln(s/M^2) + \gamma_3^{(1)}\ln^2(s/M^2)$$ for the first-loop contribution to $L_F^{(\mu)}$ and $$\begin{aligned} \label{asecondloop} L^{(2)} &=& \gamma_1^{(2)} \ln^4(s/\mu^2) + \gamma_2^{(2)} \ln^3(s/\mu^2)\ln(s/M^2) + \\ \nonumber &&\gamma_3^{(2)} \ln^2(s/\mu^2)\ln^2(s/M^2) + \gamma_4^{(2)} \ln(s/\mu^2)\ln^3(s/M^2) + \gamma_5^{(2)} \ln^4(s/ M^2) \end{aligned}$$ for the second-loop contribution. The coefficients $\gamma_i^{(k)}$ are given below: $$\begin{aligned} \label{gammaik} \gamma_1^{(1)} = {\frac{{{\pi }^2}\,{{{{\chi}_0}}^4}}{4}},\, \gamma_2^{(1)} = {\frac{{{\pi }^2}\, \left( {{\chi}^4} - 4\,{{{{\chi}_0}}^4} \right) }{4}},\, \gamma_3^{(1)} = - {\frac{{{\pi }^2}\, \left( {{\chi}^4} - 6\,{{{{\chi}_0}}^4} \right) }{8}} ,\, \\ \nonumber \gamma_1^{(2)} = {\frac{{{\pi }^2}\,{{{{\chi}_0}}^6}}{96}}, \, \gamma_2^{(2)} = 0, \, \gamma_3^{(2)} = {\frac{{{\pi }^2}\,{{\chi}^2}\, \left( {{\chi}^4} - 4\,{{{{\chi}_0}}^4} \right) }{32}},\, \\ \nonumber \gamma_4^{(2)} = -{\frac{{{\pi }^2}\,\left( {{\chi}^6} - 6\,{{\chi}^2}\,{{{{\chi}_0}}^4} + 2\,{{{{\chi}_0}}^6} \right) }{24}}, \, \gamma_5^{(2)} = {\frac{{{\pi }^2}\, \left( 3\,{{\chi}^6} - 24\,{{\chi}^2}\,{{{{\chi}_0}}^4} + 14\,{{{{\chi}_0}}^6} \right) }{192}}.\end{aligned}$$ Let us compare the above results with those obtained with one universal cut-off $M$ only. We introduce the notation $\tilde{L}(s/M^2)$ for amplitude $L_F^{(\mu)}$ when one cut-off $M$ is used. The ratio $R^{1} = L^{1}(s,\mu,M)/\tilde{L}^{(1)}(s,M) $ of the first loop contributions to the amplitudes $L_F^{(\mu)}$ and $\tilde{L}$ is $$\label{R1} R^{(1)} = \frac{L^{(1)}} {\tilde{\gamma}^{1} \ln^2(s/M^2)} ~$$ where $\tilde{\gamma}^{1} = \pi^2 \chi^4/8$. Similarly the ratio $R^{(2)}$ of the second-loop contributions is $$\label{R2} R^{(2)} = \frac{L^{(2)}} {\tilde{\gamma}^{1} \ln^4(s/M^2)} ~,$$ where $\tilde{\gamma}^{2} = \pi^2 \chi^6/64$. Eqs. (\[R1\], \[R2\]) show explicitly that the difference between the one cut-off amplitude $\tilde{L}$ and the two cut-off amplitude $L_F^{(\mu)}$ grows with the order of the perturbative expansion, though rapidly decreasing with $s$. {width="9cm"} {width="9cm"} We can expect therefore that a sizable difference between $L_F^{(\mu)}$ and $\tilde{L}$ when all orders of the perturbative series are resumed. Asymptotics of the forward scattering amplitude for $e^+e^-$ annihilation into $\mu^+ \mu^-$. ============================================================================================= In order to estimate the effect of higher order DL contributions on the difference between the one-cut-off and two-cut-off amplitudes, it is convenient to compare their high-energy asymptotics. For the sake of simplicity, we present below such asymptotical estimates for the amplitude $L_F^{\mu}$ of the forward $e^+e^-$ annihilation into $\mu^+ \mu^-$ in the collinear kinematics (\[tmu\]). Calculations for the other amplitudes (\[solutionaj\]) can be done in a similar way. As well-known, the leading contribution to the asymptotic behavior is $L_F^{\mu} \sim s^{\omega_0}$, with ${\omega_0}$ being the rightmost singularity of the amplitude $L_F^{\mu}$. This amplitude contains the amplitudes $\phi_{1,2}^{(0)}$ and $\phi_{1,2}$ and therefore also their singularities. Eqs. (\[phi1\], \[phi0\]) show that the singularities of both $\phi_1$ and $\phi_1^{(0)}$ are the square root branching points. The rightmost singularity of $\phi_1^{(0)}$ is $\chi_0$ and the rightmost singularity of $\phi_1$ is $\chi$. They are defined in Eqs. (\[chi0\], \[chi\]). Obviously, $$\label{phi01as} \phi_1^{(0)}(\chi_0) = 4 \pi^2 \chi_0, ~\phi_1^{(0)}(\chi) = 4 \pi^2 \big( \chi - \sqrt{\chi^2 - \chi_0^2}\Big) \equiv 4 \pi^2 \big( \chi - \chi'\big) , ~\phi_1(\chi) = 4 \pi^2 \chi ~.$$ Combining Eqs. (\[amplmu\]) and (\[phi01as\]) and neglecting the last integral in Eq. (\[amplmu\]), we obtain the asymptotic formula for the forward leptonic invariant amplitude $A$: $$\label{leptas} L_F^{\mu} \sim 4 \pi^2 \Big(\frac{s}{\mu^2}\Big)^{\chi_0} \chi_0 + 4 \pi^2 \Big(\frac{s}{M^2}\Big)^{\chi} \frac{2(\chi - \chi')(2\chi' - \chi) e^{2 \eta(\chi - \chi')}} {3\chi - 2\chi' -(2\chi' - \chi) e^{2 \eta(\chi - \chi')}} .$$ The first term in Eq. (\[leptas\]) represents the asymptotic contribution of the QED Feynman graphs, the second term the mixing of QED and weak DL contributions. On the other hand, when the one-cut-off approach is used, the new amplitude ${\tilde{L}}_F^{\mu}$ asymptotically behaves as: $$\label{lepttilde} \tilde{L}_F^{\mu} \sim 4 \pi^2 \frac{\chi}{2}\Big(\frac{s}{M^2}\Big)^{\chi} ~.$$ Then defining $Z(s, \eta)$, as: $$\label{Z} L_F^{\mu} = \tilde{L}_F^{\mu} \big( 1 + Z(s, \eta)\big) ~,$$ it is easy to see that $$\label{Zas} Z (s, \eta) \sim \Big(\frac{s}{M^2}\Big)^{-\chi + \chi_0} \frac{2 \chi_0}{\chi} e^{\eta \chi_0} -1 + \frac{4(\chi - \chi')(2\chi' - \chi) e^{2 \eta(\chi - \chi')}} {\chi[3\chi - 2\chi' -(2\chi' - \chi) e^{2 \eta(\chi - \chi')}]} ~.$$ As $\chi_0 < \chi$, $Z(s)$ falls when $s$ grows. So, the one-cut-off and the two-cut-off approach lead to the same asymptotics, although at very high energies, say $\sqrt{s} \geq 10^6$ TeV. At lower energies, accounting for $Z$, the amplitude $L_F^{(\mu)}$ is increased by a factor of order 2. On the other hand, $Z$ strongly depends on the ratio $M/\mu$, which, of course, is related to the actual phenomenological conditions. To illustrate this dependence, we take $M = 100$ GeV and choose different values for $\mu$, ranging from $0.1$ to $1$ GeV. Then in Fig. 6 we plot $Z(s, \mu)$ for $\mu = 1$ GeV and $\mu = 0.5$ GeV. This shows that the variation is approximately 1.5 at energies in the interval from $0.5$ to $5$ TeV. {width="9cm"} It is also interesting to estimate the difference between the purely QED asymptotics of $L_F^{\mu}$ (the first term in the rhs of Eq. (\[leptas\])) and the full electroweak asymptotics. To this aim, we introduce $\Delta_{EW}$: $$\label{deltaew} L_F^{(\mu)} = \big(L_F^{(\mu)}\big)^{(QED)} (1 + \Delta_{EW})~.$$ From Eq. (\[leptas\]) we immediately get the following asymptotic behavior for $\Delta_{EW}$: $$\label{deltaewas} \Delta_{EW} \sim \Big(\frac{s}{M^2}\Big)^{\chi - \chi_0} ~~\frac{2(\chi - \chi')(2\chi' - \chi) e^{2 \eta(\chi - \chi')}} {3\chi - 2\chi' -(2\chi' - \chi) e^{2 \eta(\chi - \chi')}}$$ As $\chi > \chi_0$, $\Delta_{EW}$ grows with $s$, as shown in Fig. 7, Therefore the weak interactions contribution is approximately of the same size of the QED contribution, and their ratio rapidly increases as $\mu$ decreases. {width="9cm"} Summary and Outlook {#CONCLUSIONS} =================== Next future linear $e^+e^-$ colliders will be operating in a energy domain which is much higher than the electroweak bosons masses, so that the full knowledge of the scattering amplitudes for $e^+e^-$ annihilation into fermion pairs will be needed. In the present paper we have considered the high-energy non-radiative scattering amplitudes for $e^+e^-$ annihilation into leptons and quarks in the Regge kinematics (\[tkin\]) and (\[ukin\]). We have calculated these amplitudes in the DLA, using a cut-off $M$, with $M \geq M_Z \approx M_W$, for the transverse momenta of virtual weak bosons and an infrared cut-off $\mu$ for regulating DL contributions of virtual soft photons. We have obtained explicit expressions (\[Phi1\], \[solutionaj\]) for these amplitudes in the collinear kinematics (\[tmu\], \[umu\]) and Eqs. (\[akappa\], \[akappam\]) for the configuration where all Mandelstam variables are large. The basic structure of the expressions in the limit of collinear kinematics is quite clear. They consist of two terms: the first term presents the purely QED contribution, i.e. the one with virtual photon exchanges only, whereas the next term describe the combined effect of all electroweak boson exchanges. Obviously, in the limit when the cut-off $\mu \to M$, our expressions for the scattering amplitude converge to the much simpler expressions obtained in Ref [@egt] with one universal cut-off for all electroweak bosons. In order to calculate the electroweak scattering amplitudes, we derived and solved infrared equations for the evolution of the amplitudes with respect to the cut-offs $M$ and $\mu$. In order to illustrate the difference between the two methods, we have considered in more detail the scattering amplitude $L_F^{(\mu)}$ of the forward $e^+e^-$ annihilation into $\mu^+\mu^-$ and studied the ratios of the results obtained in the two approaches, first in one- and two-loop approximation and then to all orders to DLA. The ratios of the first- and second-loop DL results are plotted in Figs. 4 and 5. The total effect of higher-loop contributions is estimated comparing the asymptotic behaviors of the amplitudes. This is shown in Fig. 6. The effect of all electroweak DL corrections compared the QED ones is plotted in Fig. 7. It follows that accounting for all electroweak radiative corrections $L_F^{(\mu)}$ increases by up to factor of 2.5 at $\sqrt{s}\leq 1$ TeV, depending on the value of $M/\mu$. In formulae for the $2 \to 2$ - electroweak cross sections, one can put $M = M_W \approx M_Z$ whereas the value of $\mu$ is quite arbitrary. However it vanishes, when these expressions are combined with cross sections of the radiative $2 \to 2 + X$ processes. In the present paper we have considered the most complicated case of both the initial electron and the final quark or lepton being heft-handed (and their antiparticles right-handed). Studying other combinations of the helicities of the initial and final particles can be done quite similarly. We intend to use the results obtained in the present paper for further studying the forward-backward asymmetry at TeV energies, by including also the real radiative contributions. Basically, the QCD radiative corrections can give a big impact on the amplitudes of $e^+e^-$ - annihilation into hadrons. However, the perturbative QCD corrections cancel out of the expressions for the forward-backward asymmetry (see Ref. [@egt]) whereas the non-perturbative corrections describing hadronization of the produced $q \bar{q}$ - pairs can be accounted for in the same way as it was done in Ref. [@egt]. Acknowledgement =============== This work is supported by grants POCTI/FNU/49523/2002, SFRH/BD/6455/2001 and RSGSS-1124.2003.2 [99]{} B.I. Ermolaev, M. Greco and S.I. Troyan. Phys.Rev. D 67(2003)014017. V.S. Fadin, L.N. Lipatov, A.D. Martin, and M. Melles, Phys.Rev. D 61(2000)094002. P. Ciafaloni and D. Comelli. Phys Lett B 476(2000)49. M. Melles Phys. Lett. B 495(2000)81 . M. Hori, H. Kawamura and J. Kodaira. Phys. Lett. B 491(2000)275. W. Beenakker and A. Werthenbach. Nucl. Phys. B 630(2002)3. A. Denner, M. Melles and S. Pozzorini. Nucl.Phys. B 662(2003)299. J.H. Kuhn, A.A. Penin and V.A. Smirnov. Nucl.Phys.Proc.Suppl. 89(2000)94; Eur. Phys. J. C 17(2000)97. S. Moch. Nucl.Phys.Proc.Suppl. 116:23-27,2003. V.N. Gribov, V.G. Gorshkov, G.V. Frolov, L.N. Lipatov. Sov.J.Nucl.Phys. 6(1968)95; ibid 6(1968)262. M. Greco, G. Pancheri-Srivastava and Y. Srivastava, Nucl. Phys. B 171(1980)118 \[Erratum-ibid. B 197(1982)543\].

--- abstract: 'In the Galactic microquasars with double peak kHz quasi-periodic oscillations (QPOs), the ratio of the two frequencies is 3:2. This supports the suggestion that double peak kHz QPOs are due to a non-linear resonance between two modes of accretion disk oscillations. For the microquasars with known mass, we briefly compare the black hole spin estimates based on the orbital resonance model with the recently reported spin predictions obtained by fitting the spectral continua. Results of these two approaches are not in good agreement. We stress that if the spectral fit estimates are accurate and can be taken as referential (which is still questionable), the disagreement between the predicted and referential values would represent a rather generic problem for any relativistic QPO model, as no spin influence would appear in the observed $1/M$ scaling of the QPO frequencies. The epicyclic frequencies relevant in these models are often considered to be equal to those of a test particle motion. However modifications of the frequencies due to the disc pressure or other non-geodesic effects may play an important role, and the inaccuracy introduced in the spin estimates by the test particle approximation could be crucial.' date: '10th October and in revised form ??' --- Estimating the black hole spin from the resonance models ======================================================== The resonance model [@KluzniakAbramowicz2000] explains the twin peak QPOs as being caused by a 3:2 non-linear resonance between two global modes of oscillations in accretion flow in strong gravity. The modes in resonance are often assumed to be the epicyclic modes. The *orbital resonance model* [see @KluAbr:2003:aph] demonstrates that [*fluid accretion flows*]{} admit two linear quasi-incompressible modes of oscillations, radial and vertical, with corresponding eigenfrequencies equal to the radial and vertical epicyclic frequencies for free particles [@Ali-Gal:1981:GENRG2; @Now-Leh:1998:TheoryBlackHoleAccretionDisks:]. According to the resonance hypothesis, the two modes in resonance have eigenfrequencies $\nu_{\rm r}$ (radial epicyclic frequency) and $\nu_{\rm v}$ (vertical epicyclic frequency $\nu_{\theta}$ or Keplerian frequency $\nu_{\rm K}$). Several resonances of this kind are possible and have been discussed (see, e.g., @AbrKlu:2004). Formulae for the Keplerian $\nu_{\mathrm{K}}$ and the epicyclic frequencies $\nu_{\rm r}$ and $\nu_{\theta}$ in the field of a Kerr black hole with mass $M$ and spin $a$ are well known, and have the general form $$\label{eq:oneoverM:theory} \nu=\left ({{GM_0}\over {r_G^{~3}}}\right )^{1/2}f_\mathrm{i}(x,\,a)~~\doteq 32.3\left(\frac{M_0}{M_\odot}\right)f_\mathrm{i}(x,\,a)\,\mathrm{kHz},\quad\mathrm{i}\in{\mathrm{K},~\mathrm{r},\theta}$$ where $f_\mathrm{i}(x,a)$ are functions of a dimensionless black hole spin $a$ and a dimensioless radial coordinate . For a $n\!:\!m$ orbital resonance, the dimensionless resonance radius $x_{\mathrm{n}:\mathrm{m}}$ is determined as a function of spin $a$ by an equation $ \mathrm{n}\nu_{\rm r}\!= \mathrm{m}\nu_{\rm v}~(\nu_\mathrm{v}\!=\nu_\theta\,~\mathrm{or}~\,\nu_\mathrm{K}) $ [^1]. Thus, from the observed frequencies and from the estimated mass, one can determine the relevant spin (@AbramowiczKluzniak2001 [@TAKS]). We summarize the spin estimates for the three microquasars in Table \[table:1\]. by - \[table:1\] [ l l l l l ]{}  &\ Model for &     1550–564 &     1655–40 &     1655–40$^*$ &     1915+105\ \ 3:2 \[$\nu_{\theta},~\nu_r$\]   & +0.89 — +0.99 &+0.96 — +0.99 &+0.88 — +0.93 &+0.69 — +0.99\ 2:1 \[$\nu_{\theta},~\nu_r$\]   &+0.12 — +0.42 &+0.31 — +0.42 &+0.10 — +0.25 & [$\hspace{\sirkaA}$]{}-0.41 — +0.44\ 3:1 \[$\nu_{\theta},~\nu_r$\]   & +0.32 — +0.59 & +0.50 — +0.59 &+0.31 — +0.44 & [$\hspace{\sirkaA}$]{}-0.15 — +0.61\ $^{*}$ The two columns for GRO 1655–40 indicate numbers following from the two different mass analysis - [@Beer2002] vs. [@Greene2001]. Note that while the spin estimates from the 3:2 parametric resonance is for both the cases similar ($a\approx 0.9$), for the other models, the given mass range implies a large range of spins. Comparison with the fits of spectral continua ============================================= Except for one case, all the resonances considered in [@TAKS] are consistent with reasonable values of the black hole spin covering the range $a\in(0,~1)$. In particular, the 3:2 epicyclic parametric (internal) resonance model, supposed to be the most natural one in Einstein gravity [@Hor:2005:ASN:], implies the spin $a\sim 0.9$. The most recent results of the spectral fits correspond for GRO 1655–40  to the spin $a\in(0.65,~0.75)$, and for GRS 1915+105 to $a\!>0.98$ [@McC-etal:2006:APJ:]. Obviously, the value for GRS 1915+105 is in agreement with the prediction of the 3:2 parametric epicyclic resonance model, but the same prediction for GRO 1655–40 does not match the spectral fitting. No particular resonance model considered so far can cover the spectral limits to the spin for both microquasars. It could be interesting that the recently proposed 3:2 periastron precession resonance [@Bur:2005:RAG:] implies the spin of GRO 1655–40 to be $a\sim 0.7$. Nevertheless, eventuall periastron precession resonance requires the spin of GRS 1915+105 $a<0.8$ which is in strong disagreement with the spectral fitting limit, $a\!>0.98$. Troubles with the spin: $1/M$ scaling ===================================== In principle one cannot exlude the possibility of different mechanisms exciting the high frequency QPOs in different sources, but there are many indicies that the mechanism is the same or similar [e.g., @Kli:2005:ASN:; @Tor-etal:06]. @McClintockRemillard2003 found that the upper QPO frequency in microquasars scales well as $\nuU\! =\!2.793 ( {M_0/M_{\odot}})^{-1}\, \mathrm{kHz}$ which is in good agreement with the 1$/M$ scaling of the first term in equation (\[eq:oneoverM:theory\]). On the other hand the exact 1$/M$ scaling holds only for the fixed value of the spin $a$ as functions $f_\mathrm{i}$ in equation (\[eq:oneoverM:theory\]) are sensitive to the spin. The spectral limits to the spin for the two microquasars are *very different*: $a\!\sim0.7$ vs. $a\!>0.98$, and, in addition, functions $f_\mathrm{i}(a,x)$ are more sensitive to the value of the spin when it is close to $a\!=1$ [e.g., @tor-stu:05:AA]. Hence, if the spin values obtained from the spectral fits are correct, the observed high frequency QPOs do not show sensitivity to the spin $a$ under the assumption of a unified QPO model. This is a serious problem for any relativistic QPO model handling with the orbital and epicyclic frequencies (\[eq:oneoverM:theory\]). Requirement of a more realistic description =========================================== It was found recently that the pressure effects may have a strong influence on the oscillation frequencies. [@Sra:2005:ASN:] and [@Bla-etal:06] studied properties of the radial ad vertical epicyclic modes of slightly non-slender tori within Newtonian theory using the Paczyński-Wiita potential, and found the epicyclic frequencies to decrease with increasing thickness of the torus. The same behaviour was found for the resonant radius where the frequencies are in a 3:2 ratio, which on the contrary implies [*increase*]{} of the resonant frequencies. Considering the appropriate corrections to frequencies in the Kerr metric, one can reestimate the values of the spin using the resonance model. If the results in the Kerr metric were following the same trend as those in the Paczyński-Wiita case, the spin for some configurations can be [*lower*]{} than previously estimated. [^2] This research is supported by the Czech grant MSM 4781305903. , 2001, A&A 374L, 19A, astro-ph/010507 , 2004, AIP Conference Proceedings, Vol. 714, Edited by Kaaret, Philip, Frederick, K. Lamb, and Swank, Jean H., Melville, NY: AIP, 2004, p.21-28 , R. A., 2004, ApJ 609, L63 , 1981, Gen. Relativity Gravitation, 13, 899 , P. 2002, MNRAS 331, 351 Blaes, O. M., Šr[á]{}mkov[á]{}, E., Abramowicz, M. A., Klu[ź]{}niak, W. & Torkelsson, U., 2006, submitted to ApJ Bursa, M., in *Proceedings of RAGtime 6/7: Workshops on black holes and neutron stars, Opava, 16–18/18–20 September 2004/2005*, edited by S. Hledík and Z. Stuchlík, Silesian University in Opava, Opava, 2005, ISBN 80-7248-242-4 , 2001, ApJ 554, 1290 , 2005, *Astronomische Nachrichten* **326**, 845–848 Kluźniak, W., & Abramowicz, M. A., 2000, Phys. Rev. Lett. (submitted), astro-ph/0105057 , 2002, astro-ph/0203314 , 2003, astro-ph/0306213 , 2006, accepted for publication in ApJ, *astro-ph/0606076* , 1998, in [T]{}heory of [B]{}lack [H]{}ole [A]{}ccretion [D]{}isks, ed. M. A. Abramowicz, G. Bj[ö]{}rnsson, & J. E. Pringle (Cambridge: Cambridge University Press), 233–253 , 2005, *Astronomische Nachrichten* **326**, 835–837 , 2005, Astronomische Nachrichten **326**, 856 Török, G., Abramowicz, M. A., Klu[ź]{}niak, W. & Stuchl[í]{}k, Z., 2005, A&A, 436, 1 , 2005, A&A 37, 775-788 Török, G., Abramowicz, M. A., Kluźniak, W. & Stuchl[í]{}k, Z., 2006, in print, *astro-ph/0603847* , 2005, *Astronomische Nachrichten* **326**, 798–803 [^1]: Because of the properties of Kerr black hole spacetimes, *any* relativistic model of black hole QPOs should be rather sensitive to the spin $a$, however this sensitivity can be negligible on large scales of mass (@Abr-etal:2004:apj). [^2]: In case of the 3:2 parametric resonance, the maximal realistic increase of the resonant frequency due to the pressure effects is about 15 percent [@Bla-etal:06], which for GRO 1655-40 and the mass estimate by Beer & Podsiadlowski would lower the spin down to $a\sim$0.8.

--- author: - 'J. Klüter, U. Bastian, M. Demleitner, J. Wambsganss' date: 'Received 28 July, 2018; accepted 30 September, 2018' title: 'Prediction of astrometric microlensing events from [*Gaia*]{} DR2 proper motions' --- [Astrometric gravitational microlensing is an excellent tool to determine the mass of stellar objects. Using precise astrometric measurements of the lensed position of a background source in combination with accurate predictions of the positions of the lens and the unlensed source it is possible to determine the mass of the lens with an accuracy of a few percent.]{} [Making use of the recently published [*Gaia*]{} Data Release 2 (DR2) catalogue, we want to predict astrometric microlensing events caused by foreground stars with high proper motion passing a background source in the coming decades.]{} [We selected roughly 148000 high-proper-motion stars from [*Gaia*]{} DR2 with $\mu_{tot} > 150\,\mathrm{mas/yr}$ as potential lenses. We then searched for background sources close to their paths. Using the astrometric parameters of [*Gaia*]{} DR2, we calculated the future positions of source and lens. With a nested-intervals algorithm we determined the date and separation of the closest approach. Using [*Gaia*]{} DR2 photometry we determined an approximate mass of the lens, which we used to calculate the expected microlensing effects. ]{} [We predict 3914 microlensing events caused by 2875 different lenses between 2010 and 2065, with expected shifts larger than $0.1\,\mathrm{mas}$ between the lensed and unlensed positions of the source. Of those, 513 events are expected to happen between 2014.5 - 2026.5 and might be measured by [*Gaia*]{}. For 127 events we also expect a magnification between $1\,\mathrm{mmag}$ and $3\,\mathrm{mag}$.]{} Introduction ============ Gravitational lensing has become a powerful tool to study galactic and extragalactic objects [@2006AnP...518...43W]. It is used for example to investigate the mass distributions of galaxies, to determine the Hubble constant, to discover distant quasars, and to find extrasolar planets. Gravitational lensing describes the deflection and magnification of background sources by an intervening massive object [@1915SPAW...47..831E; @1936Sci....84..506E]. For stellar lenses (microlensing), two images of the source are created, a bright image close to the unlensed source position and a fainter image close to the lens. Both images merge into a so-called Einstein ring when the source is perfectly aligned with the lens. The characteristic size of this ring is given by the Einstein radius $$\theta_{E}=\sqrt{\frac{4GM_{L}}{c^{2}}\frac{D_{S}-D_{L}}{D_{L}D_{S}}}, \label{equation:theta_E}$$ where $M_{L}$ is the mass of the lens and $D_{S}$, $D_{L}$ are the distances between the observer and the source or the lens [@1924AN....221..329C; @1936Sci....84..506E; @1986ApJ...301..503P]. This is the most important quantity since it sets the scale for all lensing effects. For close-by stellar lenses (within $1\,\mathrm{kpc}$) and distant sources, the Einstein radius is typically of the order of a few milliarcseconds. This is much smaller than the angular resolution of most of the currently available instruments. Due to the relative motion of source, lens, and observer, magnification and image geometry change over time. Up to now, mostly photometric magnification has been monitored and investigated by surveys such as the Optical Gravitational Lensing Experiment [OGLE, @2003AcA....53..291U] or the Microlensing Observations in Astrophysics [MOA, @2001MNRAS.327..868B] and has also led to the discovery of many exoplanets [e.g. @2015AcA....65....1U], whereas the astrometric shift of the source was detected for the first time only recently [@2017Sci...356.1046S; @2018arXiv180701318Z]. Astrometric microlensing provides the possibility to measure the mass of a single star with a precision of about one percent [@1995AcA....45..345P]. Furthermore, astrometric microlensing events can be predicted from stars with a known proper motion. This is the aim of the present study. For the prediction of astrometric events, faint nearby stars with high proper motions are of particular interest. High proper motions are preferred because the covered sky area within a given time is larger, hence microlensing events are more likely. Nearby stars are preferred because their Einstein radius is larger and therefore the expected shift is also larger, and faint lenses are favourable since the measurement of the source position is less contaminated by the lens brightness. The first systematic search for astrometric microlensing events was done by [@2000ApJ...539..241S]. They found 146 candidates between 2005 and 2015. predicted 1118 candidates between 2012-2019. However, most of those predictions were based on erroneous proper motions in some of the catalogues used and only 49 events show reliable proper motions. High-accuracy proper motions are essential to make precise predictions. Today the Gaia mission provides the best data for such studies. Using the TGAS data of the first data release , [@2018MNRAS.478L..29M] predicted one event caused by a white dwarf in 2019. With the second data release from Gaia [[*Gaia*]{} DR2, @2018arXiv180409365G], we also have precise parallaxes, which are necessary to calculate the mass of a lens afterwards, as well as the proper motion of the background source. These big improvements in data quality and quantity made much more precise predictions possible. Using [*Gaia*]{} DR2, we reported two ongoing microlensing events in 2018 [@2018arXiv180508023K]. Further, [@2018arXiv180510630B] determine 76 microlensing events between 2014.5 and 2026.5, [@2018arXiv180511638M] predict 30 possible photometric microlensing events between 2015.5 and 2035.5, and [@2018arXiv180610003B] report the prediction of 2509 astrometric microlensing events until the year 2100. In the present paper we present our method of how to use the [*Gaia*]{} DR2 proper motions and parallaxes to predict astrometric microlensing events in the coming decades. In Sect. \[chapter:microlensing\] we explain the photometric and astrometric signatures of microlensing and describe how to determine the mass of the lens from the observation of the microlensing event. In Sect. \[chapter:method\], our method to find microlensing events is explained in detail. In Sect. \[chapter:results\] we present the events predicted by our search. Finally, we summarize our results and present conclusions in Sect. \[chapter:conclusion\]. Basics of microlensing {#chapter:microlensing} ====================== Photometric microlensing ------------------------ The magnification of a source due to the focusing of the light by an intervening lens is called photometric microlensing. The magnifications ($A_{-}$,$A_{+}$) of the two images $(+)$,$(-)$ only depend on the dimensionless impact parameter $\boldsymbol{u} = \boldsymbol{\Delta\theta}/\theta_{E} $, where $\boldsymbol{\Delta\theta}$ is the unlensed angular separation between lens and source. When both images are merged, which is usually the case when photometric effects are measurable, the total magnification can be determined via [@1986ApJ...301..503P] $$A = A_{+} + A_{-} = \frac{ u^{2}+2}{u\sqrt{u^{2}+4}} ,$$ where $u = \lvert \boldsymbol{u} \rvert $. For large impact parameters $(u\gg 1)$, it can be approximated by [@2000ApJ...534..213D] $$A \simeq 1+ \frac{2}{u^{4}}, \label{equation:magnification}$$ which shows a strong decline towards large separations. For bright, unresolved lenses, the flux of the lens $f_{L}$ has to also be taken into account. Considering this, the measured magnification is given by $$A_{lum} = \frac{f_{LS}+A}{f_{LS}+1} ,$$ where $f_{LS}=f_{L}/f_{S}$ is the flux ratio between lens and (unmagnified) source star. In units of magnitude it is given by $$\Delta m = 2.5 \cdot\log_{10}\left({\frac{f_{LS}+A}{f_{LS}+1}}\right). \label{equation:approx_mag}$$ Due to the strong decline with $u$, a measurable photometric magnification can only be observed when the impact parameter is small (i.e. on the order of the Einstein radius or smaller). Therefore, the timescale of a photometric microlensing event, given by the Einstein time, $$t_{E}=\frac{2\theta_{E}}{\mu_{rel}}$$ , is quite short. Here, $\mu_{rel}$ is the absolute value of the relative proper motion between source and lens. Typical values for $t_{E}$ are on the order of a few days or weeks. Astrometric microlensing ------------------------ ![ Top: Astrometric shift for an event with an Einstein radius of $\theta_{E} = 12.75$mas (black circle) and an impact parameter of u = 0.75. While the lens (red) passes a background star (black dot, fixed in origin) two images (blue) of the source are created due to gravitational lensing. This leads to a shift of the centre of light, shown in purple for a dark lens. In green, the centre of the combined light is shown for a flux ratio of $f_{LS} = 10$. The unlensed centre of the combined light is shown as a red dashed line. The black line connects the current positions of the snapshot. While the lens is moving in the direction of the red arrow, all other images are moving according to their individual arrows. The red, blue, and purple markers correspond to certain time steps . Bottom: Astrometric shift for different impact parameters. The black dot shows the fixed unlensed source position. The solid lines indicate the shift of the centre of light for a dark lens and the dashed lines indicate the shift of the brighter image. The maximum shift of the centre of light is reached at an angular distance of $u =\sqrt{2}$ (purple) [@1998ApJ...494L..23P], whereas the shift of the brightest image increases continuously with smaller distances. []{data-label="figure:shift"}](astroshift.png "fig:"){width="9cm"} ![ Top: Astrometric shift for an event with an Einstein radius of $\theta_{E} = 12.75$mas (black circle) and an impact parameter of u = 0.75. While the lens (red) passes a background star (black dot, fixed in origin) two images (blue) of the source are created due to gravitational lensing. This leads to a shift of the centre of light, shown in purple for a dark lens. In green, the centre of the combined light is shown for a flux ratio of $f_{LS} = 10$. The unlensed centre of the combined light is shown as a red dashed line. The black line connects the current positions of the snapshot. While the lens is moving in the direction of the red arrow, all other images are moving according to their individual arrows. The red, blue, and purple markers correspond to certain time steps . Bottom: Astrometric shift for different impact parameters. The black dot shows the fixed unlensed source position. The solid lines indicate the shift of the centre of light for a dark lens and the dashed lines indicate the shift of the brighter image. The maximum shift of the centre of light is reached at an angular distance of $u =\sqrt{2}$ (purple) [@1998ApJ...494L..23P], whereas the shift of the brightest image increases continuously with smaller distances. []{data-label="figure:shift"}](relshift "fig:"){width="9cm"} In astrometric microlensing, the change of the position of the background star is the signal of interest. This is shown in the top panel of Fig. \[figure:shift\]. The red line indicates a lens passing a background source (black dot, fixed in the origin of the coordinate system). The two images created by the microlensing are shown in blue. The bright image $(+)$ is always close to the source and the faint image $(-)$ is always close to the lens. Their positions relative to the lens can be described by $$\boldsymbol{\theta_{\pm}} = \frac{ u \pm \sqrt{(u^{2}+4)}}{2} \cdot \frac{\boldsymbol{u}}{u} \cdot{\theta_{E}} .$$ When the separation of the lensed images is too small to be resolved, only the position of the centre of light (purple line) can be measured. This can be expressed by $$\boldsymbol{\theta_c}= \frac{A_{+}\boldsymbol{\theta_{+}} +A_{-}\boldsymbol{\theta_{-}}}{A_{+}+A_{-}} =\frac{u^{2}+3}{u^{2}+2}\boldsymbol{u}\cdot{\theta_{E}}$$ and the corresponding shift is given by $$\delta\boldsymbol{\theta_{c}} = \frac{\boldsymbol{u}}{u^{2}+2} \cdot{\theta_{E}}. \label{equation:shift}$$ This is also a good approximation for the shift of the brightest image whenever $u > 5$, since in this case the second image is negligibly faint. The astrometric effect reaches a maximum value of $\delta\theta_{max} = 0.35 \theta_{E}$ at a separation of $u = \sqrt{2}$ (bottom panel of Fig. \[figure:shift\]). For smaller separations, the effect will decrease [@1998ApJ...494L..23P] . In the unresolved case, also luminous-lens effects usually have to be considered. The centre of light of the combined system (green line in Fig. \[figure:shift\], top panel ) can be expressed by $$\boldsymbol{\theta_{c,\,lum}} = \frac{A_{+}\boldsymbol{\theta_{+}} +A_{-} \boldsymbol{\theta_{-}}}{A_{+}+A_{-}+f_{LS}}$$ and the shift between lensed and unlensed position can be determined via $$\delta\boldsymbol{\theta_{c,\,lum}} = \frac{\boldsymbol{u}\cdot\theta_{E}}{1+f_{ls}}\,\frac{1+f_{LS}(u^{2}+3-u\sqrt{u^{2}+4})}{u^{2}+2+f_{LS}u\sqrt{u^{2}+4}} .$$ For large impact parameters ($u\gg\sqrt{2}$), when the photometric effect becomes negligible, this simplifies to [@2000ApJ...534..213D] $$\delta\theta_{c,\,lum} \simeq \frac{\delta\theta_{c}}{1+f_{ls}} .$$ By using space telescopes like Gaia, or telescopes with adaptive optics, luminous-lens effects can be neglected for most of the astrometric microlensing events, since the separation between lens and source is larger than the angular resolution . Such an instrument will measure the position of image (+). The shift compared to the unlensed position of the source can then be expressed by $$\delta\boldsymbol{\theta_{+}} = \frac{ \sqrt{(u^{2}+4)} - u}{2} \cdot \frac{\boldsymbol{u}}{u} \cdot{\theta_{E}}.$$ For large impact parameters the shift is proportional to $$\delta\theta_{+} \simeq \frac{\theta_{E}}{u} \label{equation:approx_shift}.$$ Therefore, with increasing separation the astrometric shift drops much more slowly than the photometric magnification, that is, with $1/u$ rather then with the fourth power (see Eq. (\[equation:magnification\])). This results in a measurable effect at large separations and consequently in a much longer timescale during which an astrometric microlensing event can be observed [@1996AcA....46..291P; @1996ApJ...470L.113M]. It can be described by [@2001PASJ...53..233H] $$t_{aml} = t_{E} \sqrt{\left(\frac{\theta_{E}}{\theta_{min}}\right)^{2} - u_{min}^{2}} ,$$ where $\theta_{min}$ is the precision threshold of the used instrument. We consider a value of $\theta_{min}= 0.1\, \mathrm{mas}$. With such high-precision instruments, some events can be observed over a period of many months or even a few years. Hence astrometric microlensing can also be directly measured by high-precision, long-term surveys like Gaia, if the lensed stars are observed at a sufficient number of epochs suitably distributed in time. Prediction of microlensing events {#chapter:method} ================================= For the prediction of astrometric microlensing events, we use a method similar to . The method consists of four steps: 1) Determine a list of high-proper-motion stars as potential lenses. 2) Find background sources close to their paths on the sky. 3) Forecast the exact position of source and lens stars from their current positions, proper motions, and parallaxes as well as determine the angular separation and epoch of the closest approach. 4) Calculate the expected microlensing effects, that is, the shifts of the background star positions. List of high-proper-motion stars -------------------------------- Due to its unprecedented accuracy, the [*Gaia*]{} DR2 provides the ideal catalogue for this task. [*Gaia*]{} DR2 contains roughly 170000 sources with proper motions $\mu_{tot} = \sqrt{\mu_{\alpha^{\star}}^{2}+\mu_\delta^{2}}$ larger than $150\,mas/yr$. As the Gaia Consortium has mentioned [@2018arXiv180409366L], DR2 contains a small proportion of erroneous astrometric solutions, most noticeably a set of unrealistically high proper motions or parallaxes. To clean up our target list, we therefore first neglect all sources with insignificant parallaxes $(\varpi < 8 \sigma_{\varpi})$. This and all other quality cuts used by us are shown in Table \[tab:cuts\]. Figure \[fig:px\_vs\_pm\] shows the absolute values of the proper motions and the parallaxes of the remaining high-proper-motion stars. Four different populations are clearly visible. The two lower ones are interpreted as the real populations of halo stars with a typical tangential velocity of $v_{tan} \sim 350\,\mathrm{km/s}$ (green line), and disk stars ($v_{tan} \sim 75 km/s$) (blue line), whereas the two upper populations (red lines) are incorrect data, since such stars do not exist — at least not in such numbers and at distances of 10pc or smaller. Why the faulty [*Gaia*]{} DR2 data show such sharp relations between parallax and proper motions is not yet known (private communication from the Gaia astrometry group). To exclude those faulty Gaia data, we neglect all stars with $\varpi/\mu_{tot} > 0.3\,\mathrm{yr}$. These suspicious data are also well separated in Fig. \[fig:n\_obs\], where the significance of the Gaia G flux (${G\_flux}/\sigma_{G\_flux}$) is plotted against the number of photometric observations by Gaia ($n_{obs}$). Hence we exclude all sources with $n_{obs}^{2}\cdot G\_flux/\sigma_{G\_flux}< 10^{6}$ (i.e below-left of the red line). Our final list contains $\sim 148\,000$ high-proper-motion stars, which are potential lenses. As expected, these nearby objects are quite evenly distributed over the sky (Fig. \[fig:aitoff\], top panel), whereas the rejected objects mainly cluster towards the Galactic disc/bulge or the Magellanic clouds (bottom panel). Background stars ---------------- For each of the roughly 148000 remaining high-proper-motion stars, we searched for background sources close to their paths. For this, we defined a box by using the position of the source at the epochs J2010.0 and J2065.5 with a half-width $w = 7''$ perpendicular to the direction of the proper motion. This box is illustrated in Fig. \[figure:window\]. The large box width is mainly adopted to account for potential motions of background sources. A widening shape would be more physically accurate, however, for simplicity we used the rectangular shape. The combination of a high-proper-motion foreground lens and a background source within the defined box is called a “candidate”. In the following, the source parameters are labelled with the prefix “\_”. We considered all [*Gaia*]{} DR2 sources, without a significantly negative parallax ($\text{Sou}\_ \varpi > -3 \cdot \text{Sou}\_ \sigma _{\varpi} -0.029\,\mathrm{mas}$) and with a standard error in the J2015.5 position below 10 mas $\left(\sqrt{\text{Sou\_}\sigma_{ra}^{2}+\text{Sou\_}\sigma_{dec}^{2}} < 10\,\mathrm{mas}\right)$ as potential background sources. For sources with non-significant negative parallaxes ($-0.029\,\mathrm{mas} > \text{Sou}\_\varpi + 3 \cdot \text{Sou}\_\sigma_{\varpi}$) or without parallax in [*Gaia*]{} DR2, we assumed a value of $\text{Sou}\_\varpi = -0.029 \,\mathrm{mas}$. This is the zero-point of Gaia’s parallaxes, as determined from a sample of known quasars [@2018arXiv180409376L]. By using this value, we were able to correct for this systematic error. For background sources that have only a two-parameter astrometric solution, we assumed a standard error in the proper motion of $\text{Sou}\_\sigma_{\mu_{ra,dec}} = 10\,\mathrm{mas/yr}$, and a parallax error of $\text{Sou}\_\sigma_{\varpi} =2\,\mathrm{mas}$. Roughly 90% of the five-parameter background sources have proper-motions and parallaxes below this value. To avoid binary stars and co-moving stars in our candidate list, we exclude pairs with common proper motion, that is, $$\lvert \vec{\mu}_{tot} - \text{Sou}\_\vec{\mu}_{tot}\rvert\, < 0.7\cdot \lvert \vec{\mu}_{tot} \rvert .$$ These criteria can only be used if the proper motion of the source is given in [*Gaia*]{} DR2. This is not the case for roughly $25\%$ of our events. Hence, these events have to be treated carefully, especially when the estimated date of closest approach is close to J2015.5. Nevertheless, most of them are expected to be real events. Further, we exclude candidates where the parallax of the source is larger than the parallax of the lens ($\text{Sou}\_\varpi > \varpi$), to avoid negative Einstein radii. We do not make a stronger cut for the parallax at this point since comparable parallaxes will lead to small Einstein radii anyway and hence to small astrometric shifts. ![Illustration of the window used in the search for background stars. The thick solid blue line indicates the proper motion of the lens (red star), and the origin is set to the J2015.5 position of the lens. The blue dashed line indicates the real motion, which includes the parallax (only five years are shown). When a background star (yellow star) is within the black box, it is considered as candidate. For the plot, the box is defined by the position of the lens in J2010.0 and J2015.5 and a half width of 7 arcsec. To account for the proper motion of the background source, the widening shape (dotted black line) would be more physically accurate.[]{data-label="figure:window"}](window.png){width="9cm"} Position forecast and determination of the closest approach ----------------------------------------------------------- For about 68000 candidates, we searched for the closest approach by calculating the positions of source and lens from [*Gaia*]{} DR2 positions, proper motions, and parallaxes. Since we are interested in the global minima and the periodic motion of the Earth may cause many local minima, we first neglected the Earth’s motion to calculate an approximate distance and time of the closest approach, using a nested-intervals algorithm. If the expected shift according to Eq. (\[equation:approx\_shift\]) for the approximate distance is larger than $0.03\,\mathrm{mas}$, the exact value is calculated by including the parallax. In order to account for the multiple minima, we searched for all local minima within $\pm1\,\mathrm{year}$ around the approximate time with intervals of roughly four weeks. It is possible that we considered two really close minima as one. However, the distances and dates of both minima should then be very similar. For all minima found, we determined the minimum separations and the epoch of the closest approaches, again using the nested-intervals algorithm. By comparing these values we selected the global minima. Since Gaia and the future James Web Space Telescope (JWST) are located at the Lagrange point L2, we repeated our study with a 1% larger parallax to take account of the larger heliocentric orbit at L2. As expected, the effects only differ when the smallest separation is small compared to the parallax. Approximate mass and Einstein radius ------------------------------------ In order to get a realistic value for the expected astrometric shifts of our candidates, we derived a rough approximation for the mass of each lens in the following way. First, we divided our candidates into three categories — white dwarfs (WD), main sequence stars (MS), and red giants (RG) — by using the following cuts in colour-magnitude space (see Fig. \[figure:cmd\]): $$\begin{aligned} WD:&\qquad G_{BP, abs} \ge 4\cdot (G-G_{RP})^{2}+4.5\cdot(G - G_{RP}) +6\\ RG:&\qquad G_{BP, abs} \le -3 \cdot (G-G_{RP})^{2}+8 \cdot (G-G_{RP}) - 1.3 \end{aligned} \label{colorcuts} .$$ ![Colour-magnitude diagram of all potential lenses with full [*Gaia*]{} DR2 photometry $G$, $G_{RP}$ , and $G_{Bp}$ The yellow dots indicate the lenses of the predicted events. All stars above the green line are considered as red giants and all sources below the red line as white dwarfs.[]{data-label="figure:cmd"}](CMD.png){width="9cm"} Here, $G_{BP,abs}$ represents the absolute magnitude determined via the distance modulus. Lenses without $G_{BP}$ and $G_{RP}$ magnitudes are assumed to be main-sequence stars. For white dwarfs and red giants, we used typical masses of $M_{WD} = (0.65\pm 0.15) \,M_{\odot}$ and $M_{RG} = (1.0\pm 0.5 \,M_{\odot})$, respectively, where the indicated uncertainties are used for the error calculus further below. For the main-sequence stars, we determined a relation between G magnitudes and stellar masses. We started with a list of temperatures, stellar radii, absolute V magnitudes, and V-Ic colours for different stellar types on the main sequence [@2013ApJS..208....9P]. We then translated these relations into the Gaia filter system using the colour relation from , $$G-V = -0.0257 - 0.0924 (V-Ic) -0.1623 (V-Ic)^{2} + 0.0090 (V-Ic)^{3} .$$ For the different stellar types, we calculated the stellar masses using the luminosity equation $$\frac{L}{L_{\odot}} = \left(\frac{R}{R_{\odot}}\right)^{2} \left(\frac{T}{T_{\odot}}\right)^{4},$$ and the mass-luminosity relations [@2005essp.book.....S] $$\begin{aligned} \text{for }L < 0.0304: &\,\frac{L}{L_{\odot}} = 0.23 \left(\frac{M}{M_{\odot}}\right)^{2.3},\\ \text{for }L > 0.0304: &\,\frac{L}{L_{\odot}} = \left(\frac{M}{M_{\odot}}\right)^{4} \end{aligned} .$$ Finally, we fitted two exponential functions to the data and got the equations $$\begin{aligned} \text{for } G_{abs} < 8.85:\\ \log\left(\frac{M}{M_{\odot}}\right) &= 0.00786\,G_{abs}^{2} -0.290\,G_{abs} + 1.18, \\ \text{for } 8.85 < G_{abs} < 15:\\ \log\left(\frac{M}{M_{\odot}}\right)& = -0.301\,G_{abs} + 1.89. \end{aligned} \label{Eq:Gmag}$$ In Fig. \[figure:fit\] the fitted relation and its residuals are displayed. The relative residuals in the interesting regime ($\sim 2< G_{abs} <\sim15$) are below $2\%$, which is amply sufficient for our purpose. However, in the error calculus below, we consider a mean error of $10\%$ to account also for the uncertainties in G magnitude, parallax, in the equations used and in the dependence on metallicity. We do not use a relation based on Gaia colours, for two reasons: first, some of our lenses do not have colour information in DR2, and second, our sample contains many metal-poor halo stars. Hence they appear much bluer, whereas the change in absolute magnitude is small. For $G_{abs} > 15.0$ ( i.e. $M < \sim 0.07 M_{\odot}$) we reach the area of brown dwarfs. Those stars cannot be described by the mass-luminosity relation. Hence, for them we chose a fixed mass of $(0.07 \pm 0.03)\,M_{\odot}$. We note that all of the calculated masses are only rough estimates in order to get an expectation of the Einstein radii, astrometric shifts, and magnifications of the forecast microlensing events. An exact and direct determination of their masses would not be a pre-requisite, but the goal of observing these events. ![Fitted $G_{abs}$- mass relation. The red points show the derived masses for different stellar types. The blue line shows the fitted relation. The two slopes are caused by the different luminosity mass relations. In the bottom part, the relative residuals after the fit are shown.[]{data-label="figure:fit"}](mass_relation.png){width="9cm"} Using the estimated masses $M_{L}$ and the [*Gaia*]{} DR2 parallaxes $\varpi$ and $\text{Sou\_}\varpi,$ we calculated the Einstein radii via the rewritten Eq. (\[equation:theta\_E\]), $$\theta_{E} = \sqrt{\frac{4GM_{L}}{c^{2}}\frac{\varpi-\text{Sou\_}\varpi}{1pc \cdot 1''}} = 2.854\,mas \sqrt{\frac{M_{L}}{M_{\odot}}\cdot\frac{\varpi-\text{Sou\_}\varpi}{1\,mas}} .$$ Finally, we computed the expected shifts ($\delta\theta_{c}$, $\delta\theta_{+}$ and $\delta\theta_{c,lum}$) and magnifications based on the equations in Sect. \[chapter:microlensing\]. We only selected those candidates where $\delta\theta_{+} > 0.1\,\mathrm{mas}$. application criteria ------------- ----------------------------------------------------------------------------------------------------------- Lenses $\mu_{tot}>150\,\mathrm{mas/yr}$ Lenses $\varpi/\sigma_{\varpi}>8$ Lenses $\varpi/\mu_{tot} < 0.3\,\mathrm{yr}$ Lenses $n_{obs}^{2} \cdot G\_flux / \sigma_{G\_flux} > 10^{6}$ Sources $(\text{Sou}\_\varpi+0.029\,\mathrm{mas})/\text{Sou}\_\sigma_{\varpi}> - 3$ Sources $\sqrt{\text{Sou}\_\sigma_{ra}^2+\text{Sou}\_\sigma_{dec}^2}< 10\,\mathrm{mas}$ Sources $\lvert \vec{\mu}_{tot} - \text{Sou}\_\vec{\mu}_{tot}\rvert\, < 0.7\cdot \lvert \vec{\mu}_{tot} \rvert$ Sources $\text{Sou}\_\varpi< \varpi $ Events $\delta\theta_{+} > 0.1\,\mathrm{mas}$ : Our quality cuts applied to the raw target list of high-proper-motion stars, background sources, and events. These quantetites based on the position ($ra,\,dec$, the total proper motion ($\mu_{tot}$), the parallax ($\varpi$), the number of photometric observations in G ($n_{obs}$) by Gaia, the G flux ($G\_flux$), the corresponding errors ($\sigma_{...}$) as well as the expected shift of the brightest image ($\delta\theta_{+}$). Parameters from the background source are indicated with a $\text{Sou}\_$ prefix.[]{data-label="tab:cuts"} Results: Astrometric microlensing events {#chapter:results} ======================================== We report the prediction of 3914 microlensing events by 2875 different lenses between J2010.0 and J2065.5. The past events are still of interest since Gaia possibly measured the shift of those events already. Due to the (small) motion of the background sources, some of them have a closest approach outside our original search interval in time (five earlier than J2010.0, 49 later than J2065.5) The properties of our sample and a few interesting events are discussed in the following. In Table \[tab:result\], 30 particularly interesting events are shown. The full catalogue of microlensing events can be accessed through the GAVO Data Center[^1], and through Virtual Observatory (look for “Astrometric Microlensing Events Predicted from Gaia DR2”). In the following, “shift” refers to the astrometric displacement of the brightest image only ($\delta\theta_{+}$) and “shift of the centre of light” refers to the combined centre of light ($\delta\theta_{c,lum}$) considering the luminous-lens effect. The full sample --------------- Figure \[fig:Aitoff\_events\] shows the distribution of all our 3914 events on the sky. Most of those are located towards the Galactic plane or the Large Magellanic Cloud, due to the high density of available background stars. For 1139 of the events the expected shift is smaller than three times its standard error. Insignificant shifts are mainly caused by the uncertainties in the positions of the sources due to their unknown proper motion or smallest separations below $100\,\mathrm{mas}$. Inspecting the G magnitude difference, in 210 events the source is brighter than the lens. Among the rest, 726 events have a source less than three magnitudes fainter than the lens, and for a total of 1050 events the sources are between three magnitudes and six magnitudes fainter than the lens. The remaining 1928 sources are more than six magnitudes fainter than the lens. These magnitude differences will change for different filters. The bright lenses tend to have large Einstein radii. Hence a measurable shift is also expected at larger separations, where the source might be detectable next to a bright star, even when the source is more than six magnitudes fainter. [@2018arXiv180701318Z] have shown that such observations are possible. The following numbers refer to the sample of 210 + 726 + 1050 events with a magnitude difference below $6\,\mathrm{mag}$. Figure \[fig:shift\_all\] shows the date of the closest approach and the expected astrometric shift. For 431, 201, and 54 **events**, respectively, we expect a shift of the brightest image larger than $0.5\,\mathrm{mas}$, $1\,\mathrm{mas,}$ and $3\,\mathrm{mas}$, respectively. Of them, 88, 18 and two have a minimum separation larger than $100\,\mathrm{mas, respectively}$. For 679 of the events the smallest separation is below $100\,\mathrm{mas}$. Considering luminous-lens effects, 198, 44, and 18 of those events have an expected shift of the centre of light larger than $0.1\,\mathrm{mas}$, $0.5\,\mathrm{mas}$ and $1\,\mathrm{mas}$, respectively. We note that the luminosity effects depend on the used filters, and modern telescopes with adaptive optics or interferometry can even resolve separations smaller than $100\,\mathrm{mas}$. [@2018arXiv180610003B] predicted 2509 events until the year 2100. Due to different selection criteria and time ranges, we only detect 656 of their events independently. For all common events, the predicted dates and impact parameters are similar, within the standard errors. Photometric microlensing effects of our astrometric microlensing events. ------------------------------------------------------------------------ [@2018arXiv180511638M] recently reported 30 possible photometric microlensing events in the next 20 years (J2015.5 to J2035.5). Twenty-four of their candidates are also listed in our sample, the other six have an absolute proper motion below $150\,\mathrm{mas/yr}$. We found 246 events in the same time range with a magnification greater than $A_{lum} -1 > 10^{-7}$ , which is the lowest magnification of their candidates. The typical photometric precision of photometric microlensing surveys is on the order of a few milli-magnitudes [@2015AcA....65....1U]. Hence, we assume a limit of $1\,\mathrm{mmag}$ to talk about photometric microlensing events. This criterion is only fulfilled for five of their events. For the same five events, a shift of the combined centre of light above $0.1\,\mathrm{mas}$ is expected. In our sample, 127 events fulfil this criterion, and for 20 events the magnification is above $0.1\,\mathrm{mag}$. For 104 and 18 of those, respectively, the motion of the background source is not known. For all of our photometric events, Fig. \[fig:mag\] shows the magnification and the predicted date. Since the predicted separation has to be really small, of the order $\Theta_{E}$, in order to produce a photometric effect, almost all predicted magnifications are not significant, especially when [*Gaia*]{} DR2 provides only a two-parameter solution for the background source. Furthermore, for many of these photometric events also the difference between the L2 magnification and the magnification seen from Earth is measurable. Candidates during the Gaia mission ---------------------------------- Since Gaia obtains many precise measurements over its mission time (from J2014.5 up to possibly J2024.5), events during this time are of special interest. During a slightly extended period of time (2026.5; to accommodate events starting during the late Gaia mission), we found 544 events with an astrometric shift above $0.1\,\mathrm{mas}$. For only 245 events, proper motions and parallaxes of the sources are known. The numbers for those events will be given in parentheses in the following. Of the events, 147 (62) have a minimum separation below $100\,\mathrm{mas}$ and will be (or were) blended for Gaia during the closest approach in the along-scan direction. In the across-scan direction they will be blended for a more extended time interval. For 44 (19) events the shift of the blended centre of light is larger than $0.1\,\mathrm{mas}$. For 29 (19) events we expect also a measurable magnification above $1\,\mathrm{mmag}$. The epoch and the astrometric shifts for our candidates during the Gaia mission are shown in Fig. \[fig:shift\_gaia\]. Since the expected timescales are on the order of a few years. it might be possible that Gaia observes the beginning or end of an event with a closest approach before 2014.5 or after 2024.5. [@2018arXiv180510630B] has recently reported 76 events during the Gaia mission life time (between J2014.5 and J2026.5). Independently, we discovered 60 of his events. The dates and distances of the common events are similar except for ten events where [@2018arXiv180510630B] listed the dates close to J2026.5 or J2014.5 and we expect the date a few years later or earlier. For events where the proper motion of the background source is known, also the given uncertainties are similar. In the case of unknown proper motions, our error estimates are much larger, since we assume an error of $\text{Sou\_}\sigma_{\mu_{ra,dec}} = 10\,\mathrm{mas/yr}$. The events which we did not reproduce either have a total proper motion of the lens below 150 mas/yr (eight cases), a positional uncertainty of the source above 10 mas (three cases), comparable proper motions (three cases), or are outside our defined box (two cases), we deliberately excluded from our sample. White dwarfs ------------ Our catalogue contains 486 events caused by 352 different white dwarfs. For 427 of those, the background source is less than six magnitudes fainter in the G band. Since white dwarfs are blue objects, using infrared filters will be more advantageous for possible follow-up observations. Of the events, 84 will happen between 2014.5 and 2026.5. For 98 of the events, the expected maximum shift is above $0.5\,\mathrm{mas}$ and for 53 above $1\,\mathrm{mas}$ (17 and 5 for the period 2014.5-2026.5). For 22 events also the blended centre of light will be shifted by at least $0.5\,\mathrm{mas}$. We also independently recovered the events of WD 1142-645 predicted by [@2018MNRAS.478L..29M], and the one of Stein 51B, which was already observed by [@2017Sci...356.1046S]. Proxima Centauri - the nearest ------------------------------- [@2014ApJ...782...89S] predicted two microlensing events of Proxima Centauri in October 2014 and February 2016 with a closest separation of $1600\,\mathrm{mas}$ and $500\,\mathrm{mas}$, respectively. By observing those events with VLT/SPHERE[^2] and HST/WFC3[^3], [@2018arXiv180701318Z] were able to determine the mass of Proxima Centauri, but with an uncertainty of about 40%. We did not recover either of those two events, since the background stars are not listed in [*Gaia*]{} DR2. However, we found 84 further microlensing events of Proxima Centauri until J2065.5. Nine of those have an expected shift larger than $1\,\mathrm{mas}$. With a G magnitude of $8.9\,\mathrm{mag,}$ Proxima Centauri is much brighter than the sources $(\Delta m \sim 6 - 12\,\mathrm{mag})$. Therefore a significant shift of the centre of light of the blended system cannot be observed. Due to the large Einstein radius of Proxima Centauri ($\Theta_{E} =\sim 27.1 \mathrm{mas}$), a shift of $1\,\mathrm{mas}$ can still be observed at a separation of $\sim 700\,\mathrm{mas}$ and for all sources with a separation smaller than $7000\,\mathrm{mas}$ a shift larger than $0.1\,\mathrm{mas}$ is expected. At this separation it is possible to observe background stars next to Proxima Centauri. Barnard’s star - the fastest ---------------------------- Barnard’s star is the fastest star on the sky. Hence the sky area passed by this star is the largest in our sample. Between J2010.0 and J2065.5 we found 37 astrometric microlensing events for Barnard’s star. Seven of those happen between 2014.5 and 2026.5, and so Gaia might measure the deflections. Barnard’s star has a G magnitude of 8.2 mag, and we determined an Einstein radius of $\sim 28.6\,\mathrm{mas}$. Due to its brightness, most of the sources are more than six magnitudes fainter. However, in 2035 it will pass by a $G = 11.8\,\mathrm{mag}$ star with a closest separation of $(335 \pm 13) \,\mathrm{mas}$. If it is possible to resolve source and lens, a shift of $(2.43 \pm 0.20)\,\mathrm{mas}$ is expected. The shift of the blended centre of light will be smaller than $0.1 \,\mathrm{mas}$. More information is given in Table \[tab:result\], event 10. Two photometric events in 2019 ------------------------------ In June 2019, a $G = 15.2\,\mathrm{mag}$ star ([*Gaia*]{} DR2 source id: 5862333044226605056) will pass a $G = 18.1\,\mathrm{mag}$ star with a closest separation of $(6.48\pm 3.4)\,\mathrm{mas}$. For this event we determined an Einstein radius of $(4.66\pm 0.24)\,\mathrm{mas}$. The blended centre of light will be shifted by $(0.18\pm 0.06)\,\mathrm{mas}$, and we expect a magnification of $(0.011\pm 0.014)\,\mathrm{mag}$. In November 2019 we expect a second photometric event when the $G = 17.2 \,\mathrm{mag}$ star 2MASS J13055171-7218081 ([*Gaia*]{} DR2 source id: 5840411363658156032) passes a $G = 18.2 \,\mathrm{mag}$ star with a closest separation of $(5.83 \pm 1.32)\,\mathrm{mas}$. We determined an Einstein radius of $(3.56\pm0.18)\,\mathrm{mas}$. The event will be magnified by $(0.032 \pm 0.019)\,\mathrm{mag}$, and the expected shift of the combined centre of light will be $(0.50 \pm 0.08)\,\mathrm{mas}$. Both events are listed in Table \[tab:result\] (lines 3 and 4). The event of 2MASS J13055171-7218081 was also predicted by [@2018arXiv180510630B] independently. Two astrometric events in 2018 ------------------------------ In [@2018arXiv180508023K], we already reported two ongoing astrometric microlensing events in Summer 2018 by Luyten 141-23 and Ross 322 Conclusion {#chapter:conclusion} ========== We determined a list of 148 000 high-proper-motion stars using [*Gaia*]{} DR2. We then searched for background sources close to their paths and found $\sim 68000$ candidates for astrometric microlensing events. For those, we computed the closest projected distances and the expected astrometric and photometric effects. The main difficulty in this process is to sort out probably erroneous DR2 data, while losing as few as possible valid events. We chose the rejection criteria such as to be confident that our list shows a small false positive rate, while not deleting too many promising predictions. Because of the large sample, we were not able to perform a Monte Carlo simulation to determine the uncertainties of our predictions, due to limited resources. Instead, we used an error propagation, which leads to robust values as long as the relative errors are small. At small separations, the derived uncertainties for shift and magnification tend to be overestimated. In total, we give predictions for 3914 microlensing events caused by 2875 different lens stars with an expected shift of the brighter image larger than $0.1\,\mathrm{mas}$. These include about 700 events, which were also predicted by [@2018arXiv180510630B], [@2018arXiv180511638M], and [@2018arXiv180610003B]. The independent detection of those shows the reliability of the respective methods. The standard errors of the predicted date of the closest approach is a few weeks for most of the events (and for the best events only a few hours). This is much smaller than the duration of the events. The standard error for the minimum separation is typically on the order of a few dozen milliarcseconds. Large uncertainties are mostly caused by unknown proper motions of the source. As expected, the standard errors increase with the time before and after J2015.5. However, for the year 2065, it is still possible to predict events with a separation significantly below $100 \mathrm{mas}$ ($d+3\sigma_{d} < 100\,\mathrm{mas}$). Typically the lens is much brighter than the source for 1928 events even with a G magnitude difference above 6 mag. These are hard to observe, but using more suitable wavelength bands the brightness differences can be reduced. Observations of the events and the determination of the lens masses will lead to a better understanding of mass relations for main sequence stars [@1991ApJ...371L..63P]. Perhaps even more interestingly, our sample contains also events caused by 352 different white dwarfs. The observation and subsequent mass determination of those will lead to a better understanding of white dwarfs, and the final phase of the evolution of stars. With [*Gaia*]{} DR 3 (expected in late 2020) we expect an improvement of the standard errors. In other words, the precision of the predicted events will increase. In addition, the number of background sources with five-parameter solutions will increase, which again leads to better predictions. Furthermore, [*Gaia*]{} DR3 will include detections and treatment of binary stars, while for [*Gaia*]{} DR2 all stars were treated as single. This too will help to make even more precise predictions for the paths of the lenses. With [*Gaia*]{} DR4 (expected in late 2022) the individual astrometric Gaia measurements will be published. Using such data it will be possible to perform a detailed reconstruction and modeling of past events and to even more precisely determine the masses of their lenses. This work has made use of results from the ESA space mission [*Gaia*]{}, the data from which were processed by the [*Gaia*]{} Data Processing and Analysis Consortium (DPAC). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the [*Gaia*]{} Multilateral Agreement. The [*Gaia*]{} mission website is: http://www.cosmos.esa.int/Gaia. Some of the authors are members of the [*Gaia*]{} Data Processing and Analysis Consortium (DPAC). This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This research made use of Astropy, a community-developed core Python package for Astronomy [@refId0]. This research made use of TOPCAT [@2005ASPC..347...29T], which was used to prepare ten figures in this paper. We gratefully acknowledge the technical support we received from the staff of the e-inf-astro project (BMBF Förderkennzeichen 05A17VH2). ------ ----------------------- -------------------------- ------------- ----------------- -------------- ----------------------- ------------ -------------- ----------- -------------------- -------------------------- ----------------------------------- -------------------- ----------------------------- ----------------- ---------------------- \# $source\,ID$ $\text{Sou\_}source\,ID$ ST $M$ $\theta_{E}$ $\sigma_{\theta_{E}}$ $T-2000$ $\sigma_{T}$ $d_{min}$ $\sigma_{d_{min}}$ $\delta\theta_{c,\,lum}$ $\sigma_{\delta\theta_{c,\,lum}}$ $\delta\theta_{+}$ $\sigma_{\delta\theta_{+}}$ $\Delta m$ $\sigma_{\Delta m }$ $M_{\odot}$ mas mas Jyear Jyear mas mas mas mas mas mas mag mag $1 $ $478978296199510912 $ $478978296204261248 $ $WD $ $0.65 $ $15.5 $ $1.8 $ $14.6923 $ $0.0024 $ $50.7 $ $1.1 $ $1.78 $ $0.27 $ $4.34 $ $0.67 $ $0.0055 $ $0.0023$ $2 $ $4733794485572154752$ $4733794485572154624$ $BD $ $0.07 $ $8.9 $ $1.9 $ $16.0744 $ $0.0099 $ $21.1 $ $7.7 $ $1.68 $ $0.61 $ $3.2 $ $1.3 $ $0.021 $ $0.027$ $3 $ $5862333044226605056$ $5862333048529855360$ $MS $ $0.400 $ $4.66 $ $0.24 $ $19.417 $ $0.012 $ $6.4 $ $3.5 $ $0.184 $ $0.069 $ $2.44 $ $0.76 $ $0.011 $ $0.015$ $4 $ $5840411363658156032$ $5840411359350016128$ $MS $ $0.174 $ $3.56 $ $0.18 $ $19.8392 $ $0.0029 $ $5.8 $ $1.3 $ $0.505 $ $0.079 $ $1.69 $ $0.27 $ $0.032 $ $0.020$ $5 $ $4687445500635789184$ $4687445599404851456$ $WD $ $0.65 $ $13.6 $ $1.7 $ $21.5001 $ $0.0062 $ $70.3 $ $1.9 $ $0.98 $ $0.16 $ $2.52 $ $0.41 $ $0.00101 $ $0.00044$ $6 $ $4248799013208327424$ $4248799013215266176$ $MS $ $0.219 $ $4.10 $ $0.23 $ $29.876 $ $0.039 $ $0.9 $ $10. $ $0.9 $ $3.8 $ $3.7 $ $4.4 $ $0.9 $ $7.8$ $7 $ $5918299904067162240$ $5918299908365843840$ $MS $ $0.113 $ $6.93 $ $0.35 $ $30.2480 $ $0.0032 $ $6.3 $ $2.8 $ $2.17 $ $0.17 $ $4.46 $ $0.84 $ $0.28 $ $0.22$ $8 $ $6130500670360253312$ $6130500567281038080$ $MS $ $0.318 $ $2.16 $ $0.33 $ $35.162 $ $0.072 $ $0.2 $ $7.4 $ $0.4 $ $4.3 $ $2.0 $ $3.5 $ $1. $ $24.$ $9 $ $5863711561290571008$ $5863711561290570112$ $MS $ $0.199 $ $3.39 $ $0.17 $ $35.333 $ $0.014 $ $5.0 $ $3.4 $ $1.179 $ $0.087 $ $1.70 $ $0.69 $ $0.13 $ $0.22$ $10$ $4472832130942575872$ $4472836292758713216$ $MS $ $0.184 $ $28.7 $ $1.5 $ $35.76441$ $0.00073 $ $335. $ $13. $ $0.0802 $ $0.0065 $ $2.43 $ $0.20 $ 3.69E-6 9.3E-7 $11$ $6074397471079635328$ $6074397471080379776$ $WD $ $0.65 $ $8.3 $ $1.1 $ $36.72 $ $0.26 $ $9.5 $ $40. $ $1.7 $ $2.6 $ $5. $ $10. $ $0.10 $ $0.87$ $12$ $5556349476589422336$ $5556349472293892224$ $WD $ $0.65 $ $7.94 $ $0.96 $ $37.173 $ $0.077 $ $15. $ $47. $ $2.1 $ $3.2 $ $3.7 $ $8.2 $ $0.06 $ $0.57 $ $13$ $5886942661427781760$ $5886942661374132096$ $MS $ $0.180 $ $3.69 $ $0.19 $ $39.266 $ $0.049 $ $0.2 $ $13. $ $0.5 $ $8.0 $ $3.5 $ $5.9 $ $1. $ $35.$ $14$ $5715906236031073280$ $5715906236031079040$ $WD $ $0.65 $ $7.85 $ $0.91 $ $40.049 $ $0.034 $ $20.0 $ $6.9 $ $2.06 $ $0.51 $ $2.71 $ $0.84 $ $0.026 $ $0.030$ $15$ $5332606522595645952$ $5332606277747043456$ $WD $ $0.65 $ $33.7 $ $3.9 $ $40.7896 $ $0.0021 $ $139.6$ $4.9 $ $0.0470 $ $0.0077 $ $7.7 $ $1.3 $ 3.4E-5 1.6E-5 $16$ $4118914220102650624$ $4118914185707335040$ $MS $ $0.319 $ $7.11 $ $0.36 $ $41.383 $ $0.014 $ $0.1 $ $13. $ $0.07 $ $5.1 $ $7.1 $ $6.5 $ $0.8 $ $92.$ $17$ $428051391503714432 $ $428051391509474816 $ $WD $ $0.65 $ $8.18 $ $0.97 $ $43.61 $ $0.11 $ $47. $ $29. $ $1.27 $ $0.71 $ $1.39 $ $0.83 $ $0.0017 $ $0.0040$ $18$ $5605383430285597696$ $5605383537671689856$ $WD $ $0.65 $ $15.0 $ $1.7 $ $43.9735 $ $0.0091 $ $207.6$ $4.5 $ $1.04 $ $0.17 $ $1.08 $ $0.18 $ 5.6E-5 2.7E-5 $19$ $6282457918962299776$ $6282457815883084928$ $WD $ $0.65 $ $19.9 $ $2.3 $ $45.466 $ $0.012 $ $22. $ $10. $ $1.53 $ $0.43 $ $11.7 $ $3.0 $ $0.032 $ $0.033$ $20$ $3365063724883180288$ $3365062964671171712$ $MS $ $0.119 $ $11.08 $ $0.56 $ $45.607 $ $0.011 $ $119. $ $14. $ $0.0095 $ $0.0013 $ $1.02 $ $0.14 $ 1.46E-6 7.1E-7 $21$ $1822711900548572544$ $1822711900548570624$ $MS $ $0.212 $ $2.37 $ $0.15 $ $49.80 $ $0.27 $ $0.3 $ $44. $ $0.3 $ $17. $ $2. $ $20. $ $0.8 $ $100. $ $22$ $4203875751318123904$ $4203875648238893696$ $BD $ $0.07 $ $5.1 $ $1.1 $ $53.538 $ $0.018 $ $11. $ $22. $ $1.2 $ $1.5 $ $2.1 $ $3.1 $ $0.04 $ $0.22$ $23$ $4117081643422165120$ $4117081467277401344$ $WD $ $0.65 $ $11.7 $ $1.4 $ $56.982 $ $0.027 $ $66. $ $25. $ $1.45 $ $0.54 $ $1.99 $ $0.77 $ $0.0014 $ $0.0021$ $24$ $6126095232211644160$ $6126095300931121024$ $WD $ $0.65 $ $5.68 $ $0.73 $ $57.33 $ $0.27 $ $4.6 $ $32. $ $1.52 $ $0.22 $ $3.8 $ $9.8 $ $0.2 $ $2.9 $ $25$ $6426625402561169024$ $6426625402559392896$ $MS $ $0.272 $ $4.94 $ $0.25 $ $58.252 $ $0.017 $ $4.5 $ $4.2 $ $1.12 $ $0.20 $ $3.2 $ $1.3 $ $0.16 $ $0.28$ $26$ $5243594081269535872$ $5243594253068231168$ $MS $ $0.175 $ $14.77 $ $0.74 $ $58.8661 $ $0.0076 $ $208.4$ $5.8 $ $0.00767 $ $0.00059 $ $1.042 $ $0.079 $ 3.93E-7 8.9E-8 $27$ $4484348145137238016$ $4484348145137243904$ $BD $ $0.07 $ $3.39 $ $0.72 $ $60.595 $ $0.097 $ $4. $ $28. $ $1.06 $ $0.96 $ $1.8 $ $6.3 $ $0.1 $ $2.1$ $28$ $6082407619449932032$ $6082407619449930880$ $MS $ $0.560 $ $6.04 $ $0.31 $ $62.295 $ $0.050 $ $0.07 $ $8.8 $ $0.07 $ $3.8 $ $6.0 $ $4.4 $ $0.8 $ $69.$ $29$ $2025071788687899776$ $2025071792959778688$ $WD $ $0.65 $ $8.06 $ $0.94 $ $62.64 $ $0.20 $ $0.3 $ $30. $ $2. $ $27. $ $8. $ $14. $ $2. $ $101.$ $30$ $4293315765165489536$ $4293315662070462080$ $BD $ $0.07 $ $9.8 $ $2.1 $ $65.727 $ $0.051 $ $79. $ $76. $ $0.0059 $ $0.0059 $ $1.19 $ $1.2 $ 2.3E-6 8.8E-6 ------ ----------------------- -------------------------- ------------- ----------------- -------------- ----------------------- ------------ -------------- ----------- -------------------- -------------------------- ----------------------------------- -------------------- ----------------------------- ----------------- ---------------------- [^1]: German Astrophysical Virtual Observatory,\ <http://dc.zah.uni-heidelberg.de/amlensing/q2/q/form>. [^2]: Very Large Telescope equipped with the SPHERE instrument [^3]: Hubble Space Telescope equipped with the Wide Field Camera 3

--- abstract: 'We report precise measurements of ground-state, $\lambda$-doublet microwave transitions in the hydroxyl radical molecule (OH). Utilizing slow, cold molecules produced by a Stark decelerator we have improved over the precision of the previous best measurement by twenty-five-fold for the F’ = 2 $\rightarrow$ F = 2 transition, yielding (1 667 358 996 $\pm$ 4) Hz, and by ten-fold for the F’ = 1 $\rightarrow$ F = 1 transition, yielding (1 665 401 803 $\pm$ 12) Hz. Comparing these laboratory frequencies to those from OH megamasers in interstellar space will allow a sensitivity of 1 ppm for $\Delta\alpha/\alpha$ over $\sim$$10^{10}$ years.' author: - 'Eric R. Hudson' - 'H. J. Lewandowski' - 'Brian C. Sawyer' - Jun Ye bibliography: - 'OH\_Microwave\_Bib.bib' title: Cold Molecule Spectroscopy for Constraining the Evolution of the Fine Structure Constant --- Current theories that attempt to unify gravity with the other fundamental forces predict spatial and temporal variations in the fundamental constants, including the fine structure constant, $\alpha$ [@Olive:2004]. Measurements of the variation of $\alpha$ by observation of multiple absorption lines from distant quasars are currently not in agreement [@Webb:2001; @Quast:2004]. Due to the use of spatially diverse absorbers, these measurements are sensitive to relative Doppler shifts. Therefore an independent confirmation of the variation of $\alpha$ is important. Recently, there has been much interest in using OH megamasers in interstellar space to constrain the evolution of fundamental constants [@Darling:2003; @Chengalur:2003; @Kanekar:2004] with several important advantages. Specifically, it has been shown that the sum and difference of the $\Delta$F = 0 (F is total angular momentum) transition frequencies in the ground $\lambda$-doublet of OH depend on $\alpha$ as $\alpha^{0.4}$ and $\alpha^{4}$, respectively [@Darling:2003]. Thus, by comparing the values measured from OH megamasers to laboratory values it is possible to constrain $\alpha$ over cosmological time. Most importantly, the multiple lines (that have different dependence on $\alpha$) arising from a single localized source differentiate the relative Doppler shift from the $\Delta\alpha/\alpha$ measurement. Furthermore, because of the unique properties of the $\lambda$-doublet, the $\Delta$F = 0 transitions are extremely insensitive to magnetic fields. However, as pointed out by Darling [@Darling:2003], for the current limits on $\Delta\alpha/\alpha$ the change in the relevant measurable quantities is on the order of 100 Hz, which prior to this work was the accuracy of the best laboratory based measurement. Thus, Darling called on the community to produce a more precise measurement of the OH $\lambda$-doublet microwave transitions to allow for tighter constraints on $\Delta\alpha/\alpha$. Despite the prominence of the OH radical in molecular physics, the previous best measurement of the OH ground $\lambda$-doublet, performed by ter Meulen and Dymanus [@TerMeulen:1971], has stood for over 30 years. This lack of improvement was due to the relatively slow progress in the center-of-mass motion control of molecules, which limited the maximum field interrogation time, and thus the spectroscopic resolution. The ability of a Stark decelerator [@Bethlem:1999; @Bochinski:2003] to provide slow, cold pulses of molecules makes it an ideal source for molecular spectroscopy [@VanVeldhoven:2004]. In this work, a Stark decelerator is used along with standard microwave spectroscopy techniques to perform the best measurement to date of the $\Delta$F = 0, $\lambda$-doublet microwave transitions in OH, which along with appropriate astrophysical measurements can be used to constrain $\Delta\alpha/\alpha$ with a sensitivity of 1 ppm over the last $\sim$$10^{10}$ years. In its ro-vibronic ground state OH is a Hund’s case (a) molecule with a $^2\Pi$ configuration and total molecule-fixed angular momentum of $\Omega = \frac{3}{2}$. For the most abundant isotopomer (O$^{16}$H) the oxygen has no nuclear spin and the hydrogen carries a nuclear spin of $\frac{1}{2}$ leading to two total spin states with F = 1 and 2. Because the unpaired electron in OH has one unit of orbital angular momentum, these ro-vibronic ground states are ‘$\lambda$-doubled’, leading to the closely spaced opposite parity $\lambda$-doublet states shown in Fig. \[Figure1\](a) labeled as $|F, m_F, \rm{parity}\rangle$. Though molecules are decelerated only in the $|2,\pm2,+\rangle$ and $|2,\pm1,+\rangle$ states, the field-free region from the hexapole to the microwave cavity leads to an equal redistribution of the population among the five magnetic sub-levels of the F = 2 upper doublet state. In this work, both $\Delta$F = 0 electric dipole transitions between the upper and lower doublet states are studied. Though these transitions exhibit large Stark shifts (as is necessary for Stark deceleration), they show remarkably small Zeeman shifts. In fact, because the magnetic dipole operator respects parity, one expects the $\Delta$F = 0 transitions of a pure case (a) molecule to show no magnetic field dependence. Nonetheless, because the hyperfine splitting differs in the upper and lower doublet, and part of the $|m_F|$ = 1 magnetic dipole moment comes from mixing with the other hyperfine component, there is a small quadratic shift of the $|2,\pm1,+\rangle$ $\rightarrow$ $|2,\pm1,-\rangle$ transition frequency (150 Hz/Gauss$^2$). Furthermore, because OH is not completely case (a) (*i.e.* the electron’s orbital angular momentum and spin are slightly decoupled from the axis) the g-factor is slightly larger in the lower doublet [@Radford]. Thus, transitions between the $m_F >$ 0 ($m_F <$ 0) components are blue-shifted (red-shifted) relative to the zero field value. Hence for experiments probing both positive and negative $m_F$ components such as this work, the g-factor difference leads to a broadening and eventually a bifurcation in the lineshape for increasing magnetic field. This effect, explored at relatively large fields (0.6 - 0.9 T), was found to shift the $\Delta$F = 0 transition frequencies of the $|2,\pm2,+\rangle, |2,-1,+\rangle$, and $|1,1,+\rangle$ states at a rate of sign$(m_F)\times2.7$ kHz/Gauss, and the $|1,-1,+\rangle$ and $|2,1,+\rangle$ states at a rate of sign$(m_F)\times0.9$ kHz/Gauss [@Radford]. For even population distribution this effect leads to a shift of -450 Hz/Gauss and 900 Hz/Gauss for the 2$\rightarrow$2 and 1$\rightarrow$1 transitions, respectively. Shown in Fig. \[Figure1\](b) is a schematic of our experimental setup. OH molecules seeded in Xenon are created in a pulsed discharge [@Lewandowski:2004] producing a molecular pulse with a mean speed of 410 m/s and 10% longitudinal velocity spread. Following a skimmer is an electrostatic hexapole used to transversely couple the molecules into our Stark decelerator, which is described in detail elsewhere [@Bochinski:2003; @Bochinski:2004; @Hudson:2004]. The Stark decelerator is used in a variety of operating conditions, yielding pulses of molecules at a density of 10$^{6}$ cm$^{-3}$ with mean speeds chosen between 410 m/s and 50 m/s with longitudinal temperatures between 1 K and 5 mK, respectively. Once the molecular pulse exits the decelerator it is focused by a second hexapole to the detection region. Located between the second hexapole and the detection region is a 10 cm long cylindrical microwave cavity with its axis aligned to the molecular beam. The microwave cavity is operated near the TM$_{010}$ mode, such that the electric field is extremely uniform over the region sampled by the molecules [@Jackson]. The microwave cavity is surrounded by highly magnetically permeable material to provide shielding from stray magnetic fields. From measurements of the magnetically sensitive F = 2$\rightarrow$F = 1 transition frequency the residual field in the cavity was determined to be $<$ 2 milliGauss. This small field results in an absolute Zeeman shift of $<$ 0.9 Hz and $<$ 1.8 Hz for the 2$\rightarrow$2 and 1$\rightarrow$1 transitions, respectively. Note that this is a cautious upper-bound on the Zeeman shift, since it is based on work performed at $\sim$10$^7$ larger magnetic field where the angular momentum decoupling is enhanced. Furthermore, it was verified within experimental resolution that there were no transition frequency shifts due to stray electric fields. Two independently switchable microwave synthesizers, both referenced to a Cesium standard, are used to provide the microwave radiation for driving the $\lambda$-doublet transitions. For probing the $|2,m_F,+\rangle$ $\rightarrow$ $|2,m_F,-\rangle$ $\lambda$-doublet transition (2$\rightarrow$2) only one synthesizer is needed since the molecules enter the cavity in the F = 2 state. Accordingly, two synthesizers are needed for probing the 1$\rightarrow$1 transition. The first synthesizer transfers molecules from the F = 2 upper doublet state to the F = 1 lower doublet state. The second synthesizer then probes the 1$\rightarrow$1 transition. After the molecules exit the cavity the population of the upper doublet is probed by laser-induced fluorescence (LIF). For the LIF measurement the molecules are excited along the $^2\Sigma_{1/2} (v = 1) \leftarrow$ $^2\Pi_{3/2} (v = 0)$ line at 282 nm by light produced from a doubled pulsed-dye laser. This excited state decays primarily along $^2\Sigma_{1/2} (v = 1) \rightarrow$ $^2\Pi_{3/2} (v = 1)$ at 313 nm. The red-shifted fluorescence is collected and imaged onto a photomultiplier tube coupled to a multi-channel scaler. To characterize the performance of the microwave cavity the population in the upper doublet was recorded as a function of applied microwave pulse length as shown in Fig. \[RabiFlop\](a) (so-called Rabi-flopping). For the data shown, the molecules were decelerated to 200 m/s and a microwave pulse resonant with the 2$\rightarrow$2 transition was applied such that its midpoint time coincided with the molecules being at the cavity center. Thus, as the pulse length was increased the molecules encountered a pulse that grew symmetrically about the cavity center. Because the LIF detection scheme is sensitive to all molecules in the upper doublet and the applied microwave field simultaneously drives transitions between the different magnetic sub-levels, the Rabi-flopping signal is more complicated than the traditional $\sin^2(\frac{\omega'_Rt_p}{2})$. Here $\omega'_R$ is the effective Rabi frequency given in terms of the detuning, $\delta$, and Rabi frequency, $\omega_R$, as $\omega'_R = \sqrt{\delta^2 + \omega_R^2}$, and $t_p$ is the microwave pulse length. As shown in Fig. \[Figure1\](a) for the 2$\rightarrow$2 transition (solid arrows), both the $|m_F|$ = 2 and $|m_F|$ = 1 magnetic sub-levels are driven by the microwave field (the $|m_F|$ = 0 level has a zero transition moment). Because the electric dipole transition moment of the $|m_F|$ = 2 is twice that of $|m_F|$ = 1 [@Avdeenkov:2002], the Rabi-flopping signal exhibits beating. The calculated individual magnetic sub-level contributions are shown at the bottom of Fig. \[RabiFlop\](a) with their sum represented by the dashed line plotted over the data. The data points were determined by comparing the populations in the upper doublet at the detection region with and without the microwave field applied. Clearly, the behavior is exactly as expected until $t_p \geq$ 300 $\mu$s when $\omega_R$ of both the $|m_F|$ = 2 and $|m_F|$ = 1 transitions appears to decrease. This reduction in $\omega_R$ is the result of the electric field diminishing near the cavity end-caps [@Efieldnote]. Alternatively to fixing the molecular velocity, $v$, and varying $t_p$, the Stark decelerator allows $v$ to be varied while applying a fixed (spatial) length microwave pulse. This allows a check of systematics associated with beam velocity. Data taken in this manner is shown in Fig. \[RabiFlop\]b, where a microwave pulse resonant with the 2$\rightarrow$2 transition was applied for the entire time the molecules were in the cavity. The microwave power for this measurement was chosen such that molecules with a 400 m/s velocity underwent one complete population oscillation (*i.e.* a 2$\pi$ pulse for the $|m_F| = 1$ and a 4$\pi$ pulse for the $|m_F|$ = 2 transitions). This was done so that population revivals occurred at velocities that were integer sub-multiples of 400 m/s. While the behavior agrees well with the expected (dotted line), there are two noticeable deviations. First, for 270 m/s $\leq v \leq$ 400 m/s the fringe visibility is less than expected. This is because, as detailed in our earlier work [@Bochinski:2004; @Hudson:2004], molecules with these relatively high velocities have not been decelerated out of the background molecular pulse. Thus, molecules with a large distribution of speeds (as compared to the decelerated molecules) are detected, leading to reduced contrast. Second, substantial decoherence is observed for $v \leq$ 130 m/s. The source of this decoherence has been experimentally determined as the result of microwave radiation leaking from the cavity and being reflected off the decelerator back into the cavity. It is interesting to note that since the metal detection tube acts as a waveguide with a cut-off frequency much higher than that applied, no radiation leaks from the rear of the cavity. Thus, future experiments should include a small waveguide section on both sides of the cavity to prevent leakage. Furthermore, because this decoherence is due to reflected radiation it is microwave power dependent, and presents no problem for the transitions frequency measurements, which use $<$ 10% of the microwave power used in Fig. \[RabiFlop\], such that no noticeable decoherence occurs. Because two magnetic sub-levels with differing $\omega_R$’s undergo the 2$\rightarrow$2 transition, the traditional Rabi-spectroscopy method of applying a $\pi$ pulse over the length of the cavity is not optimal. Thus, for measurements of the 2$\rightarrow$2 transition frequency the microwave power was chosen such that for $\delta$ = 0 the maximum contrast was produced with as small a microwave power as possible (*i.e.* transfer the molecules to the first dip in Fig. \[RabiFlop\](a) over the length of the cavity). Representative data for the 2$\rightarrow$2 transition is shown in Fig. \[LineGraph\](a) where the driving frequency, f, is varied under a fixed microwave power. Data points in this graph were generated by comparing the population in the upper doublet at the detection region with and without the probing microwave field. The fit to the data (solid line) is generated from the typical Rabi lineshape formula except contributions from the different magnetic sub-levels are included. For probing the 1$\rightarrow$1 transition it is necessary to prepare the molecules in a F = 1 level since they originate in the $|2,m_F,+\rangle$ level from the Stark decelerator. This is accomplished by using the first 70 $\mu$s the molecules spend in the cavity to drive them on the satellite 2$\rightarrow$1 line, yielding molecules in the $|1,\pm1,-\rangle$ and $|1,0,-\rangle$ states (downward dotted arrows in Fig. \[Figure1\]). The remaining time the molecules spend in the cavity (depending on $v$) is then used to probe the 1$\rightarrow$1 transition frequency by applying a $\pi$ pulse, which transfers only the $|m_F|$ = 1 molecules to the upper doublet (upward dotted arrows in Fig. \[Figure1\]). Representative data for the 1$\rightarrow$1 transition is shown in Fig. \[LineGraph\]b. In contrast to the 2$\rightarrow$2, line the LIF signal is maximum on resonance because molecules are being transferred into the detected upper doublet. Also, because only 40% of the population participate in the transitions (*i.e.* the $|m_F| = 1$) the contrast between on and off resonance is reduced relative to the 2$\rightarrow$2 transition, leading to a slightly larger error for determining the center frequency. For both panels of Fig. \[LineGraph\] the molecules were decelerated to 200 m/s yielding linewidths of 2 kHz. A typical fit determines the center frequency within 20 to 50 Hz, depending on the transition. The results of several measurements of both the 2$\rightarrow$2 and 1$\rightarrow$1 center frequencies are displayed in Fig. \[FreqCum\]. Each point and its error bar represents the result of a fit to a measured lineshape like those shown in Fig. \[LineGraph\]. Using the standard error of each fit as a weight the mean and standard error of the transition frequencies are found to be (1 667 358 996 $\pm$ 4) Hz and (1 665 401 803 $\pm$ 12) Hz for the 2$\rightarrow$2 and 1$\rightarrow$1, respectively. For comparison, the lightly hatched boxes represent the bounds set on the transition frequency by the previous best measurement [@TerMeulen:1971], while the limits produced by this measurement are displayed as darker cross-hatched boxes. The transition frequencies reported here are limited only by statistical uncertainties. The Ramsey technique of separated pulses inside the same microwave cavity was also used to measure the transition frequencies as seen in Fig. \[Ramsey\] with a resolution comparable to the reported Rabi measurements. However, because the molecular pulses are extremely monochromatic, minimal gain was observed in the recovered signal-to-noise ratio. This technique will of course be critical for any future molecular fountain clock. The molecular clock could enjoy reduced systematic shifts, such as the magnetic field insensitive transitions demonstrated in this work. It will be most interesting to compare an atomic clock against a molecular one which depends differently on the fine structure constant. In summary, microwave spectroscopy was performed on slow, cold molecular pulses produced by a Stark decelerator resulting in the most precise measurement of the OH $\Delta$F = 0, $\lambda$-doublet transitions. These results along with appropriate astrophysical measurement of OH megamasers can be used to produce constraints on $\Delta\alpha/\alpha$ with a sensitivity of 1 ppm over the last $\sim$$10^{10}$ yr. At the same time the use of cold molecules for the most precise molecular spectroscopy has been demonstrated. Specifically, by producing slow, cold molecular packets the Stark decelerator allows increased interrogation time, while virtually eliminating any velocity broadening. The authors are indebted to Steven Jefferts and John L. Hall for crucial contributions. This work is supported by NSF, NIST, DOE, and the Keck Foundation.

--- author: - 'T. Lebzelter' - 'P.R. Wood' date: 'Received ; accepted ' title: 'Long period variables in 47Tuc: direct evidence for lost mass' --- Introduction ============ It is now well established that pulsating red giant stars lie on a series of up to six parallel period-luminosity (PL) sequences (e.g. Wood et al. [@wm99], Wood [@wood00]; Ita et al. [@ita04]; Soszyński et al. [@sos04]; Fraser et al. [@fra05]). The stars known to populate these sequences are generally field variables so that their metallicities, masses and ages are not known individually, although the luminosities are well known because the stars lie in stellar systems at a known distance, such as the SMC and LMC. The dispersion in the PL relations is much larger than the amplitude of the pulsation, especially for the small amplitude variables, and this dispersion is almost certainly a consequence of the dispersion in mass and metallicity at a given pulsation period. Comparison of the PL relations with theory (Wood et al. [@wm99]) shows that four of the PL sequences (sequences A, B and C of Wood et al. [@wm99], or sequences A, B, C and C$^{\prime}$ of Ita et al. [@ita04]) can be broadly explained by radial pulsation in the lower order modes. However, a real test of the theoretical models requires comparison with stars of known metallicity, age and (initial) mass since the fits of theory to data in Wood et al. ([@wm99]) and Ita et al. ([@ita04]) are not particularly good. Globular clusters are well suited for carrying out such a comparison, and for studying the relation between pulsation, mass loss and stellar evolution along the giant branch, since fundamental parameters like initial mass, luminosity and metallicity are well known. However, only a small number of pulsating red giants, hereinafter referred to as long-period variables or LPVs, is known in any given globular cluster (Clement et al.[@clement01]), with 47 Tuc having the most known LPVs (14). This is an insufficient number of variables for clearly defining PL sequences, especially as many of the periods of the LPVs are poorly determined. The origin of this problem is that most search programs for variable stars in globular clusters were designed to optimize detection and period determination for variables with periods of about one day or less. LPVs with their periods of a few ten to a few hundred days have therefore not been well surveyed and the existing samples are far from complete. To rectify this situation, we have started a search program for long period variables in Galactic Globular clusters (Lebzelter et al. [@lebz04]). The relatively large number of already known or proposed long period variables in 47Tuc made it a good starting point. Here, we present the results for 47Tuc (NGC 104), the first cluster analyzed. We adopt the following properties for 47 Tuc: $(m-M)_{V}$=13.5$\pm$0.08 (Gratton et al. [@gratton03]); metallicity \[Fe/H\]$=$$-$0.66 (Carretta & Gratton [@cg97]); interstellar reddening $E(B-V)$$=$0.024; an age of 11.2$\pm$1.1Gyr (Gratton et al.[@gratton03]); and a turnoff mass between 0.86 and 0.9$M_{\sun}$. In a recent paper discussing radial velocity variations in known LPVs in 47 Tuc (Lebzelter et al.[@LWHJF05]) we gave more discussion of the properties of 47 Tuc and its variables and we refer the reader to that paper for details. Observations ============ Mount Stromlo data ------------------ Our monitoring program of 47Tuc started in August 2002 at the 50inch telescope at Mount Stromlo. This telescope was equipped with a two channel camera used earlier for the MACHO experiment (Alcock et al.[@macho92]). The camera obtained two images in two broad band ranges at the same time (Marshall [@macho94]). These passbands did not correspond to standard filters but the blue one had a mean wavelength similar to Johnson $V$. The camera covers a field of about 0.5 square degree on the sky with a pixel scale of 0.62 arcseconds. The whole cluster could thus be observed with one observation, and most of the cluster was on one CCD of the MACHO camera (there are four CCDs per filter band). Observations were obtained once to twice a week. A few frames were lost due to technical problems or bad seeing conditions. Our monitoring came to an early end after about five months when Mount Stromlo Observatory was destroyed by a bush fire. Altogether we collected 15 useable frames over this time span. All observations were done in queue observing mode. For the determination of light curves and the detection of variables, only the blue frames were used. The light amplitude of long period variables is typically larger in the blue than in the red (e.g. Fox [@fox82]) making the detection of variables and the determination of periods easier. Furthermore, many bright stars on the red frame were over-exposed and thus not useable. The blue detector had an area of bad pixels as well as a few scattered dead pixels. Consequently, due to small positional shifts between the different observing nights some stars could not be measured on all 15 frames. CTIO data --------- From August 2003 to January 2004, we continued the monitoring with ANDICAM at CTIO’s 1.3m telescope operated by the SMARTS consortium. The size of the CCD field is 6x6 arcminutes and the pixel scale is 0.37 arcsec/pixel. A description of the camera is given in DePoy et al.([@andi03]). Due to the smaller FOV we had to make a mosaic of images to cover most of the cluster. Even then, we were limited to the central part, and some of the outer areas covered by the Mount Stromlo data could not be observed. Observations were done in $V$ and $I_{\rm c}$. Observations were scheduled roughly once per week giving a total of 16 useable frames. As for the Mount Stromlo data, observations were done in queue observing mode. Other data ---------- Six observations of the cluster in $V$ and $I_{\rm c}$ were taken in service mode with the WFI at the ESO 2.2m. These observations were obtained between June and September 2002. Four additional observations taken in July/August 2003 were kindly provided by Laszlo Kiss and collaborators. These observations in $V$ and $I_{\rm c}$ were obtained with the Siding Spring Observatory 40-inch telescope, helping to reduce the gap between the Mount Stromlo and the CTIO data set. For a different project (Lebzelter et al.[@LWHJF05]), we did a short photometric monitoring of some parts of 47 Tuc with ANDICAM at CTIO’s 1.3m telescope in March to May 2002. These data were obtained in the $V$-filter with the old ANDICAM CCD which had severe limitations (only half of the chip was useable). These data were used only in rare cases. Data reduction ============== For the Mount Stromlo and the CTIO data, flatfield and bias correction was done as part of the standard data pipeline. The 40inch data were reduced applying standard data reduction with MIDAS, while the WFI data were reduced with the corresponding IRAF package. For the detection of variables and the measurement of the light curves on the Mount Stromlo and CTIO frames we used the image subtraction code ISIS 2.1 by Alard ([@alard00]). First, the two data sets were analyzed separately. The reference flux of the identified variables required to produce light curves from the image subtraction measurements was derived using the PSF fitting software written by Ch.Alard for the DENIS project. A description of this code can be found in Schuller et al.([@Schuller03]). For the photometric calibration of the CTIO data we used standard stars from Landolt’s field RU149 (Landolt [@Landolt92]). Photometric accuracy of the resulting light curves could be analyzed by comparing measurements of variables in overlapping parts of the CTIO mosaic. Typical deviations were of the order of 0.007mag. We then used about 40 non-variable cluster stars ranging in $V-I$ between 0.9 and 2.1 to link the Mount Stromlo measurements to those from CTIO. Typical errors in the Mount Stromlo magnitudes were of the order of 0.013mag. For various reasons (field of view, pixel scale, damaged parts of the CCD, depth of the observation) not all variables could be measured on both the CTIO and the Mount Stromlo data. In these cases only one of the data sets could be used for the analysis. Our aim was to detect and measure long period variables on the upper part of the giant branch only. Thus we selected for analysis only those variables that varied on time scales of more than approximately 30 days with a total light amplitude of at least 0.1mag in $V$ (or the blue MACHO pass band). For stars with shorter periods, our sampling was not sufficient to estimate a useful period. We searched for periods in the selected light curves using the Fourier analysis code Period98 (Sperl [@sperl98]). A maximum of two periods was derived for each star. First, the two data sets from Mount Stromlo and CTIO were analyzed separately. The observations from ESO and SSO, measured using PSF fitting, were inserted in the combined light curves. Then a period search was done on the whole light curve. Generally, the agreement between the periods derived from these three data sets agreed very well. Naturally, stars with periods exceeding the length of one of the data sets or with high irregularity in their light curve lead to deviating results. The uncertainties of the periods calculated from a least square fit to the complete data sets are in most cases less than a few percent. Of course the semiregular nature of most of the stars in our sample means that the derived period is representative only for the current variability behaviour. For the long period cases, we used only the period derived from the combined light curve. Some stars were classified as irregular if no consistent period could be found. These stars will not be discussed further in this paper. The variables {#var_section} ============= Periods have previously been published in the literature for the LPVs V1–V8, V11, V13, V18 and V21 (we use the nomenclature of Clement et al.[@clement01]). Light curves for these stars derived from the current observations are given in Lebzelter et al.([@LWHJF05]), along with some discussion of the origin and reliability of the published periods. These periods are given in column 7 of Table\[variables\]. Beside these stars, a few more red giants have been identified as variables before, namely V15, V16, V17, V19, V20, V22, V23, V25, V27, V28 and A19. For some of these (V15 to V17, V25 and V28) periods were given by Fox ([@fox82]). A19 was reported variable by Lloyd-Evans ([@LE74]), but no period was given. Later variability surveys, like those of Kaluzny et al.([@kaluzny98]) and Weldrake et al. ([@weldrake04]), have focused on binaries and RRLyr stars and thus typically were not suitable for determining periods of the stars on the upper giant branch due to overexposure. The long period variables they report are most likely members of the Small Magellanic Cloud. Table\[variables\] lists the variables detected or characterized in the course of this study. Our sample includes 22 variables detected for the first time, and we give the first period determinations for six further stars previously known to be variable. All these periods are listed in column 6 of Table\[variables\]. We do not include in Table\[variables\] any previously known variables that we were not able to monitor (because they fell on CCD defects or outside our field of view). As nomenclature for new variables, we use LWxx with the numbering going from east to west. Near infrared $J$ and $K$ magnitudes were extracted from the 2MASS database and they are listed in columns 4 and 5. The 2MASS coordinates are given in columns 2 and 3. [lcccccccccc]{} Name & $\alpha$ (2000) & $\delta$ (2000) & $V$ & $V-I_{\rm c}$ & $J$ & $K$ & P \[d\] & P$_{literature}$ & Remark\ V1 & 00 24 12.4 & $-$72 06 39 & 13.15 & 3.86 & 7.45 & 6.21 & 221 & 212 & 1\ V2 & 00 24 18.4 & $-$72 07 59 & - & - & 7.52 & 6.29 & 203 & 203 & 1,2\ V3 & 00 25 15.9 & $-$72 03 54 & 12.63 & 3.54 & 7.49 & 6.27 & 192 & 192 & 1\ V4 & 00 24 00.3 & $-$72 07 26 & 12.34 & 2.62 & 7.87 & 6.69 & 165 & 82, 165 & 1,3\ V5 & 00 25 03.7 & $-$72 09 31 & 11.80 & 1.99 & 8.65 & 7.47 & 50 & 60 & 1,4\ V6 & 00 24 25.5 & $-$72 06 30 & 11.74 & 1.94 & 8.54 & 7.43 & 48 & 48 & 1\ V7 & 00 25 20.6 & $-$72 06 40 & 11.83 & 2.30 & 8.18 & 6.97 & 52 & 52 & 1,4\ V8 & 00 24 08.3 & $-$72 03 54 & 12.01 & 2.39 & 7.94 & 6.70 & 155 & 155 & 1\ V11 & 00 25 09.0 & $-$72 02 17 & 12.03 & 2.70 & 7.91 & 6.71 & 160: & 52, 100 & 4,5\ V13 & 00 22 58.3 & $-$72 06 56 & 12.36 & 2.23 & 8.79 & 7.70 & 40 + long & 40 & 1,4,8\ V16 & 00 25 23.2 & $-$72 11 05 & 11.65 & 2.0 & 8.35 & 7.23 & 41, 88: & 45 &\ V18 & 00 25 09.2 & $-$72 02 39 & 11.67 & 1.95 & 8.59 & 7.47 & 83: & & 1,4\ V19 & 00 24 14.8 & $-$72 04 44 & 11.46 & 1.70 & 8.79 & 7.61 & 83 & &\ V20 & 00 24 14.5 & $-$72 05 09 & 12.30 & 2.53 & 8.08 & 6.94 & 232: & & 11\ V21 & 00 23 50.1 & $-$72 05 50 & 12.41 & 2.85 & 8.07 & 6.78 & 76: + long & & 1\ V22 & 00 24 08.9 & $-$72 03 00 & 11.80 & 1.99 & 8.38 & 7.18 & 62 & & 8\ V23 & 00 24 29.5 & $-$72 09 08 & 11.77 & 2.08 & 8.68 & 7.53 & 52 + long & &\ V25 & 00 23 58.9 & $-$72 02 35 & 11.96 & 2.50 & 8.20 & 7.03 & 44 & 42 & 9\ V27 & 00 24 15.2 & $-$72 04 36 & 12.11 & 2.52 & 7.97 & 6.77 & 69 + long & &\ A19 & 00 24 21.8 & $-$72 04 13 & 12.18 & 2.57 & 8.08 & 6.79 & 60 & &\ LW1 & 00 23 22.3 & $-$72 05 40 & 11.83 & 2.08 & 8.39 & 7.19 & 39 & & 9\ LW2 & 00 23 29.2 & $-$72 06 20 & 11.90 & 2.23 & 8.26 & 7.05 & 60 + long & & 9\ LW3 & 00 23 47.4 & $-$72 06 53 & 11.92 & 2.35 & 8.13 & 6.89 & 107 & &\ LW4 & 00 23 51.3 & $-$72 03 49 & 11.80 & 1.98 & 8.56 & 7.37 & 32 + long: & & 9\ LW5 & 00 23 53.2 & $-$72 04 16 & 11.74 & 1.96 & 8.53 & 7.32 & 40 & &\ LW6 & 00 23 54.7 & $-$72 03 39 & 11.85 & 2.00 & 8.51 & 7.30 & 41 & & 9\ LW7 & 00 23 56.9 & $-$72 05 33 & 11.66 & 1.86 & 8.26 & 7.18 & 42 & & 10\ LW8 & 00 23 57.7 & $-$72 05 30 & 11.82 & 1.88 & 8.10 & 7.12 & 27: + long & & 9\ LW9 & 00 23 58.2 & $-$72 05 49 & 12.34 & 2.77 & 7.97 & 6.74 & 74 & & 9\ LW10 & 00 24 02.6 & $-$72 05 07 & 12.22 & - & 7.57 & 6.40 & 110:, 221: & & 2,10,11\ LW11 & 00 24 03.2 & $-$72 04 51 & 11.95 & 1.79 & 8.47 & 7.40 & 36: & & 11\ LW12 & 00 24 04.0 & $-$72 05 10 & 11.96 & 2.35 & 8.12 & 6.89 & 61, 116 & & 11\ LW13 & 00 24 07.9: & $-$72 04 32:& 12.56 & 2.93 & 7.44: & 6.25: & 65: & & 6,11\ LW14 & 00 24 09.4: & $-$72 04 49:& 11.71 & 1.99 & 8.42 & 7.39 & 50 & & 7,10\ LW15 & 00 24 11.2 & $-$72 05 09 & 11.76 & 2.05 & 8.32 & 7.13 & 46 + long & & 9\ LW16 & 00 24 13.6 & $-$72 04 52 & 11.88 & 1.89 & 8.39 & 7.32 & 29 + long & & 9\ LW17 & 00 24 16.3 & $-$72 01 31 & 11.81 & 1.99 & 8.43 & 7.20 & 41 & & 9\ LW18 & 00 24 20.5 & $-$72 04 50 & 11.91 & 2.33 & 7.95 & 6.77 & 65 + long & &\ LW19 & 00 24 23.2 & $-$72 04 23 & 11.85 & 2.17 & 8.38 & 7.16 & 40 + long & & 9,10\ LW20 & 00 24 52.1 & $-$71 56 11 & - & - & 8.31 & 7.11 & 49 & & 2,8\ LW21 & 00 25 23.2 & $-$72 11 05 & - & - & 8.44 & 7.28 & 38 & & 2,8\ LW22 & 00 25 30.1 & $-$72 04 32 & 12.06 & 2.48 & 8.16 & 6.93 & 63 + long & &\ Notes: Column 2 and 3 give coordinates from 2MASS. Column 4 is the mean V brightness. J and K colours are from 2MASS converted to the AAO system, if not stated otherwise. If there is an indication for a long secondary period exceeding the time span of the monitoring, the star is marked with ’+long’. Uncertain values are marked with a colon.\ Remarks: (1) see Lebzelter et al. ([@LWHJF05]); (2) no $V$ or $V-I_{\rm c}$, not in the CMD (Fig.\[lctime\]); (3) currently only the longer period is visible; (4) photometry in the CMD from Fox ([@fox82]) – see text; (5) current periodicity unclear, see Lebzelter et al. ([@LWHJF05]) for details; (6) close neighbour on 2MASS images, NIR fluxes and position possibly wrong; (7) not in 2MASS point source catalogue, coordinates calculated from our images, $K$ magnitude from Origlia et al. ([@origlia02]); (8) only on Mount Stromlo frames; (9) no Mount Stromlo data; (10) stars with near infrared excess detected by Origlia et al. ([@origlia02]); (11) see text in Section\[var\_section\] for comments on the periods. \[variables\] Light variation versus time are presented in Figure\[lctime\], together with the fits from Fourier analysis. For selected variables with a reliable period determination and no long secondary period we show the light curve plotted against phase in Figure\[lcphase\]. In Fig.\[lctime\] obvious deviations of the data from the fit illustrate the semiregular nature of many stars of the sample. It can be seen that quite a large fraction of variables shows a long secondary period lasting several hundred to several thousand days. As these periods exceed the length of the time span monitored, we could not determine their length accurately and thus we do not give them in Table\[variables\]. They are included in the fits for illustrative purposes only. The formal ratio between the long and the short period in the plot ranges between 4 (LW4) and 100 (LW15). In most cases the second period is 10 or more times longer than the shorter period. The Fourier fits to the stars V20, LW10, LW11, LW12 and LW13 deserve special comment. V20 and LW10 both appear to have periods of about 220-230 days but, because of the time gap in our observations, these periods are quite uncertain. Figs.\[lctime\] and \[lcphase\] suggest that these two stars have light curves with bumps during rising light, a feature found in the light curves of several miras in the solar neighbourhood. For LW10 the semi-period shows up clearly in the Fourier analysis although this is not the case for V20. The fit of LW11 is rather bad and the period determined is very uncertain. In LW12, the period and amplitude of the light curve clearly changed between the Mount Stromlo and CTIO observations and we use separate Fourier fits to the two sets of observations. The Mount Stromlo data set gave a period of 61d while the CTIO data give 116d. The fit to LW13 is poor, and the plotted fit curves include a phase change of 20 days between the Mount Stromlo and CTIO observations. Finally, we note that the 2MASS observations could not separate LW13 from its close companion. Multiperiodicity is a well known phenomenon among red variables in the solar neighbourhood (e.g. Percy et al.[@percy03], Kiss et al.[@kiss99]). Long secondary periods are well known among long period variables, but a physical explanation for them has not yet been found (Wood, Olivier & Kawaler [@wok04]). In Figure\[cmd\] we show the position of the variables in the colour-magnitude diagram using magnitudes from the CTIO images or, for a few stars, an alternative source given below. Only the giant branches and the horizontal branch are shown. Beside the variables listed in Table\[variables\], we also marked stars detected as variables but slightly below our amplitude cutoff criterion and stars with large irregularities in the light curve so that a period could not be determined. All $V$ and $V-I_{\rm c}$ values of the variables are mean values derived from our light curves. For V5, V7, V11, V13 and V18 no CTIO data have been obtained. Instead, we give the values from Fox ([@fox82]). V3 was also not observed at CTIO, but $V$ and $I_K$ light curves were obtained by Eggen([@eggen75]) and we use the mean magnitudes from his observations, converting $V-I_K$ values to $V-I_{\rm c}$ using the transform in Bessell([@bessell79]). V2 and LW10 are missing because the stars were saturated or unmeasurable on some of our $I$ frames. Figure\[cmd\] illustrates that all variables in our sample, with the possible exception of V19, are almost certainly on the upper giant branch of 47Tuc (rather than in the SMC or Galactic halo). V19 is located above and to the left of the 47Tuc AGB. There is no indication for a long secondary period in this star (see Fig.\[lctime\]) which may lead to a non-representative measurement[^1]. At the blue end of the giant branch where V19 is found, its amplitude of more than 0.8mag is rather untypical. Possible explanations may be that it has a very close blue neighbour that was not separated from V19 in our images, or it may be a star in the Galactic halo. The most interesting aspect of Figure\[cmd\] is that it clearly shows that all stars on the giant branch of 47Tuc redder than $V-I_{\rm c}$$\approx$1.8 are variable. Finally, we comment on the connection between LPV pulsation and mass loss in 47Tuc. Origlia et al.([@origlia02]) listed five stars in 47Tuc with a high infrared excess, indicating that they are surrounded by some dust. All five are located within our investigated field. Identification is not simple as Origlia et al. provide only near infrared finding charts of low spatial resolution. Three stars from their list can be identified unequivocally with optical counterparts (stars number 1, 3 and 5 from Origlia et al.). All three are variables, namely LW10, LW14 and LW19, respectively. The source number 4 from Origlia et al. is probably related to the variable star LW7. The fifth star listed in Origlia et al. (their number 2) could not be identified with any LPV. It is the star with the weakest infrared excess in their list. It is surprising that the star (LW19) with the largest mass loss rate in Origlia’s list (number 5) is neither one of the brightest variables nor has it a long period. A similar comment can be made about V18, the variable with the largest infrared excess in a survey of 47Tuc by Ramdani & Jorissen ([@rj01], see also Lebzelter et al.[@LWHJF05]). There is obviously no strict correlation between pulsation period and mass loss rate. The $K$-log$P$ diagram ====================== Given the large number of LPVs now known in 47 Tuc, we are now in a position to examine the PL relations in the cluster. We do this in the form of the $K$-$\log P$ relation, which is shown in Figure\[klp\_obs\]. Generally, we do not have $K$ light curves for the LPVs in 47 Tuc so we can not readily calculate mean $K$ magnitudes. However, a small number of the variables have been observed many times (see Fox [@fox82], Menzies & Whitelock [@mw85] and Frogel, Persson & Cohen [@fpc81]) and good estimates of the mean $K$ magnitude and $K$ amplitude can be obtained. For each star plotted on Figure\[klp\_obs\], the $K$ magnitude is the mean of the maximum and minimum observed $K$ magnitude. For stars without multiple observations, the 2MASS $K$ magnitude has been used. All $J$ and $K$ magnitudes have been converted to the AAO system using the conversions in Allen & Cragg ([@ac83]) and Carpenter ([@car01]): these are the values shown in Table \[variables\]. A comparison of the $K$ amplitude with the $V$ light amplitude obtained in the present study shows that the $K$ amplitude is approximately 20% of the $V$ amplitude. The error bars in Figure\[klp\_obs\] have a full length of 20% of the full visual amplitude of the pulsation mode associated with each point. Hence, they should represent the $K$ amplitude of the variables. Also shown in Figure\[klp\_obs\] are the sequences of LPVs in the LMC as given by Ita et al. ([@ita04]). The magnitudes have been adjusted to bring them from an LMC distance modulus of 18.55 to a 47Tuc distance modulus of 13.50. The large-amplitude Mira variables fall close to sequence C, while the bulk of the smaller amplitude variables fall near, but not necessarily on, sequences B$^{+}$, B$^{-}$ and C$^{\prime}$. The lack of variables on sequences A$^{+}$ and A$^{-}$ may be real or it may be that these stars have amplitudes that are below our detection limit of $\sim$0.1 mag. It is notable in Figure\[klp\_obs\] that the 47Tuc LPVs do not appear to fall exactly on the LMC sequences. For example, the majority of the smaller amplitude variables brighter than $K$=7 fall between sequences B$^{+}$ and C$^{\prime}$ while fainter than $K$=7, most variables fall between sequences A$^{-}$ and B$^{-}$ with a small number appearing to fall on sequence C$^{\prime}$. This is probably a result of the different masses of the stars in the 47Tuc and LMC samples. We address this possibility in the next section. If we neglect V19 which appears anomalously blue (see Sect. \[var\_section\]), stars seem to evolve up to $K$$\approx$6.7 in the 47Tuc equivalents of sequences B and C$^{\prime}$, and to then switch to sequence C for the final stage of evolution. Using the results in Wood et al.([@wm99]), this switch seems to correspond to a transition in pulsation from a low order radial overtone mode to the fundamental radial mode (compare also Lebzelter et al.[@LWHJF05]). The tip of the RGB in the $K$-$\log P$ diagram occurs at $K=$7.1 (see Fig.\[klp\_th\]) at log$L$/L$_{\odot}$=3.355. From Figs.\[klp\_obs\] and \[hrd\] we can say that about half of the variables are more luminous than the RGB tip and therefore must be AGB stars. The stars below the RGB tip could be either on the AGB or the RGB. Pulsation models ================ Description of the models ------------------------- With a substantial set of LPVs now known in 47Tuc, we are in a position to make theoretical models for the period-luminosity laws. For our models, we adopt the following parameters for 47Tuc (see the Introduction): distance modulus $(m-M)_{V}$=13.5, reddening $E(B-V)$$=$0.024, helium mass fraction Y=0.27, metal abundance Z=0.004 and main-sequence turnoff mass 0.9M$_{\odot}$. Since we are computing pulsation periods, it is important to get the radii (and hence $T_{\rm eff}$) of the models correct. In order to realize this condition, the mixing-length in the convection theory was set so that the models coincided with the giant branch of 47Tuc. Figure \[hrd\] shows the models in the HR-diagram. The giant branch of the models clearly passes through the region of the giant branch occupied by the variables. [rccccrrrr]{} $L$/L$_{\odot}$ & $M$/M$_{\odot}$ & $M_{\rm c}$/M$_{\odot}$ & $\ell$/H$_p$ & logT$_{\rm eff}$ & P$_0$ & P$_1$ & P$_2$ & P$_3$\ \ 794 & 0.9000 & 0.4002 & 1.80 & 3.5987 & 27.6 & 18.1 & 12.5 & 9.6\ 1000 & 0.9000 & 0.4136 & 1.80 & 3.5906 & 36.1 & 23.2 & 15.8 & 12.3\ 1259 & 0.9000 & 0.4274 & 1.80 & 3.5818 & 47.7 & 29.8 & 20.1 & 16.2\ 1585 & 0.9000 & 0.4417 & 1.80 & 3.5724 & 64.0 & 38.2 & 25.6 & 21.8\ 1995 & 0.9000 & 0.4565 & 1.80 & 3.5621 & 87.3 & 49.2 & 33.1 & 29.4\ 2239 & 0.9000 & 0.4640 & 1.80 & 3.5567 & 102.6 & 56.0 & 38.2 & 34.0\ \ 794 & 0.9000 & 0.4737 & 1.80 & 3.6034 & 26.5 & 17.3 & 11.9 & 9.2\ 1000 & 0.9000 & 0.4783 & 1.80 & 3.5948 & 34.8 & 22.2 & 15.2 & 11.8\ 1259 & 0.9000 & 0.4841 & 1.80 & 3.5857 & 46.2 & 28.6 & 19.3 & 15.7\ 1585 & 0.9000 & 0.4913 & 1.80 & 3.5760 & 62.1 & 36.9 & 24.7 & 21.2\ 1995 & 0.9000 & 0.5005 & 1.80 & 3.5656 & 84.8 & 47.6 & 32.1 & 28.5\ 2512 & 0.9000 & 0.5120 & 1.80 & 3.5546 & 117.7 & 61.6 & 43.0 & 37.8\ 3162 & 0.9000 & 0.5265 & 1.80 & 3.5427 & 166.8 & 79.9 & 59.8 & 48.9\ 3981 & 0.9000 & 0.5448 & 1.80 & 3.5304 & 242.8 & 104.2 & 84.9 & 63.4\ \ 794 & 0.8441 & 0.4002 & 1.90 & 3.6034 & 27.5 & 18.1 & 12.5 & 9.6\ 1000 & 0.8340 & 0.4136 & 1.90 & 3.5949 & 36.5 & 23.4 & 16.0 & 12.7\ 1259 & 0.8166 & 0.4274 & 1.90 & 3.5853 & 49.5 & 30.5 & 20.6 & 17.3\ 1585 & 0.7915 & 0.4417 & 1.90 & 3.5746 & 69.2 & 40.0 & 27.2 & 24.2\ 1995 & 0.7679 & 0.4565 & 1.90 & 3.5629 & 98.7 & 52.7 & 37.4 & 32.7\ 2239 & 0.7628 & 0.4640 & 1.90 & 3.5572 & 117.7 & 60.2 & 44.3 & 37.3\ \ 794 & 0.7371 & 0.4737 & 1.90 & 3.6045 & 29.9 & 19.1 & 13.1 & 10.5\ 1000 & 0.7344 & 0.4783 & 1.90 & 3.5956 & 39.9 & 24.6 & 16.7 & 14.1\ 1259 & 0.7301 & 0.4841 & 1.90 & 3.5862 & 53.9 & 31.7 & 21.5 & 19.0\ 1585 & 0.7244 & 0.4913 & 1.90 & 3.5762 & 74.0 & 40.9 & 28.5 & 25.3\ 1995 & 0.7118 & 0.5005 & 1.90 & 3.5654 & 104.6 & 53.2 & 39.4 & 32.9\ 2512 & 0.6903 & 0.5120 & 1.90 & 3.5539 & 154.2 & 70.0 & 56.8 & 42.8\ 3162 & 0.6567 & 0.5265 & 1.90 & 3.5441 & 237.1 & 93.6 & 81.9 & 57.0\ 3981 & 0.6057 & 0.5448 & 1.90 & 3.5478 & 321.2 & 123.2 & 99.9 & 74.1\ 4217 & 0.5904 & 0.5500 & 1.90 & 3.5571 & 320.6 & 124.6 & 96.1 & 73.9\ 4467 & 0.5743 & 0.5556 & 1.90 & 3.5789 & 291.6 & 113.7 & 81.3 & 66.7\ 4571 & 0.5675 & 0.5579 & 1.90 & 3.6012 & 266.7 & 98.6 & 67.0 & 57.3\ Notes: $\ell$/H$_p$ is the ratio of mixing-length to pressure scale height. P$_0$, P$_1$, P$_2$ and P$_3$\ are the linear periods (in days) of the fundamental, 1st, 2nd and 3rd overtone modes. \[puls\_mods\] The properties of the models are given in Table \[puls\_mods\]. The linear non-adiabatic pulsation models were created with the pulsation code described in Fox & Wood ([@fw82]), updated to include interior opacities of Iglesias & Rogers ([@ir93]) and low temperature opacities of Alexander & Ferguson ([@af94]). This code uses a mixing length theory of convection that explicitly treats variation of the convective velocity with time. The core mass $M_{\rm c}$ was obtained from the $L$-$M_{\rm c}$ relation of Boothroyd & Sackmann ([@bs88]). The most luminous RGB model corresponds to the RGB tip luminosity of the 0.9M$_{\odot}$, Z=0.004 tracks of Fagotto et al. ([@fbbc94]). The models with mass loss were constructed according to the following prescriptions. Firstly, note that without mass loss the giant branch should extend to very high luminosities well beyond log$L$/L$_{\odot}$=10$^{4}$$L$/L$_{\odot}$. Since the most luminous stars on the 47Tuc giant branch have $L$/L$_{\odot}$$\approx$4000$L$/L$_{\odot}$, it is assumed that mass loss dissipates the hydrogen-rich envelope at about this luminosity. Evolutionary models with mass loss were constructed adopting a Reimers’ mass loss law (Reimers [@r75]), and the evolution rates on the the giant branch given by the evolutionary tracks of Fagotto et al. ([@fbbc94]). The Reimers mass loss rate was multiplied by a factor $\eta$ which was adjusted so that the stars left the AGB at $L$/L$_{\odot}$$\approx$4000 $L$/L$_{\odot}$ (a value $\eta$ = 0.33 was required). The resulting models are shown in the HR-diagram Figure \[hrd\] and the masses of the models are given in Table \[puls\_mods\]. The $K$-$\log P$ diagram for models both with and without mass loss is compared with the observed $K$-$\log P$ sequences in Figure \[klp\_th\]. We now consider the small and large amplitude sequences separately. Small amplitude variables ------------------------- It is clear that the models without mass loss fail to reproduce the observed periods for the smaller amplitude pulsators (the overtone pulsators, which have $\log P < 2$). In fact the model sequences avoid the observed sequences. Since the model calculations are for linear pulsation models, they should fit the small amplitude variable sequences. A simple way to get the observed and model sequences to agree would be to make the giant branch cooler at a given luminosity. Since the pulsation period for the overtones $P \propto R^{\frac{3}{2}}$ (Fox & Wood [@fw82]), and since $L=4\pi\sigma R^2 T_{\rm eff}^4$, at a given luminosity $P \propto T_{\rm eff}^-3$. In order to fit the theoretical overtone sequences to the observed sequences, $\log P$ needs to increase by $\sim$0.15, which means that $\log T_{\rm eff}$ needs to decrease by $\sim$0.05. As can be seen from Figure \[hrd\], this is far greater than any likely uncertainty in $\log T_{\rm eff}$. We therefore conclude that models that have not undergone considerable mass loss since the main-sequence can not explain the pulsation periods of the overtone LPVs in 47Tuc. In contrast to the models without mass loss, the overtone pulsation periods for the stars that have undergone mass loss agree well with the observations. The periods of the mass loss models are longer than those of the models without mass loss due to the lower stellar mass ($P \propto M^{-\frac{1}{2}}$), and the slope of the $K$-$\log P$ relations is smaller due to the decrease in mass with luminosity. The decreased slope is particularly prominent for the second overtone which moves from having a period close to that of the 3rd overtone at low luminosities to a period close to that of the 1st overtone at high luminosities. Large amplitude variables ------------------------- In contrast to the small amplitude variables, the large-amplitude variables i.e. the Miras are consistent with theoretical models without mass loss, while the models with mass loss do not fit the observed $K$-$log P$ sequence. (We ignore V19 in this comparison since Figure \[cmd\] suggests this star is a Galactic halo star rather than a 47Tuc star.) We believe this contradiction can be explained by the nonlinear effects in the pulsation of red giants with large amplitudes. Some preliminary nonlinear pulsation calculations made for these models show that the full-amplitude pulsation periods are considerably shorter than the linear periods due to a change in the envelope structure associated with large amplitude pulsation (see also Ya’Ari & Tuchman [@yt96]). An example of the change in period of one of these models, perturbed with a low amplitude and allowed to reach limiting amplitude, is shown in Figure \[p\_change\]. Thus the periods of [*nonlinear*]{} fundamental mode models may be able to explain the periods of the Miras in 47Tuc, although this needs further exploration. The nonlinear models will be described elsewhere. Evolution of pulsation mode with luminosity ------------------------------------------- Figure \[klp\_th\] provides an indication of the pulsation evolution of stars evolving up the giant branch. Ignoring V19 (as noted above), it seems that stars start pulsating at $K \sim 7.7$ in the 1st to 3rd overtone, then evolve up to higher luminosities to $K \sim 6.9$ where the stars transit to fundamental mode pulsation. Further evolution is in the fundamental mode, until mass loss terminates AGB evolution at $K \sim 6.2$. This is broadly consistent with what is expected from the linear non-adiabatic growth rates of the models. These are shown plotted against luminosity in Figure \[gr\]. It should be remembered that the theoretical growth rates for these highly convective stars are very uncertain and should be regarded as indicative only. The general behaviour shown in Figure \[gr\] is that, at low luminosities the 2nd overtone has the highest growth rate, at intermediate luminosities the 1st overtone has the highest growth rate, and at high luminosities the fundamental mode grows most rapidly. The 2nd overtone also has a high growth rate at high luminosities but it is likely to be overwhelmed by the fundamental mode in the nonlinear case. This suggests an evolution from 2nd to 1st overtone and then to fundamental mode. This is similar to the observed situation, although the overtones co-exist rather than following a distinct progression with luminosity. Summary and Conclusions ======================= We have identified 22 new variable red giants in 47Tuc and determined periods for another 8 previously known variables. All red giants redder than $V$-$I_{\rm c}$=1.8 are variable at the limits of our detection threshold, which corresponds to $\delta V$$\approx$0.1 mag. This colour limit corresponds to a luminosity log$L$/L$_{\odot}$=3.15 and it is considerably below the tip of the RGB at log$L$/L$_{\odot}$=3.35. In the $K$-log$P$ diagram, the 47Tuc variables do not closely follow the ridge lines of PL relations seen for LPVs in the MCs, indicating that the PL relations are mass dependent. Linear non-adiabatic modelling was used to try to reproduce the observed PL relations, especially for the low amplitude pulsators where linear calculations should be appropriate. It was shown that models without mass loss can not reproduce the observed PL laws for the low amplitude pulsators. Models that lose sufficient mass to terminate AGB evolution near $L \sim 4000 L$/L$_{\odot}$ do reproduce the observed PL relations for low amplitude variables. This is the first time that measurements of the masses of stars on the AGB have shown that mass loss of the order of 0.3 M$_{\odot}$ occurs along the RGB and AGB. The linear pulsation periods do not agree well with the observed periods of the large amplitude Mira variables, which pulsate in the fundamental mode. The solution to this problem appears to be that the nonlinear pulsation periods in these low mass stars are considerably shorter than the linear pulsation periods due to a rearrangement of stellar structure caused by the pulsation. Although such effects have been seen in pulsation models before, the 47Tuc stars studied here provide the first observational evidence for this effect. The observations show that stars evolve up the RGB and first part of the AGB pulsating in low order overtone modes, then switch to fundamental mode at high luminosities. The linear non-adiabatic growth rates of models suggest that such behaviour should occur but the models at this stage are only indicative. It is hoped that future improved models including the effect of turbulent viscosity (e.g. Olivier & Wood [@ow05]) will allow a reliable determination of the growth rates and mode selection processes in red giant stars, as well as an estimation of the effect of nonlinear pulsation on the pulsation period. We are indebted to Laszlo Kiss for kindly providing additional images for our time series of 47 Tuc. We wish to thank Christophe Alard for support with the image subtraction code ISIS and for providing his PSF fitting software. We thank Brian Schmidt for organizing the monitoring of 47Tuc at Mount Stromlo Observatory, and the queue mode observers at ESO 2.2m and CTIO 1.3m. This project obtained data via the NOAO share of the SMARTS consortium. TL has been supported by the Austrian Academy of Science (APART programme). PRW has been partially supported by a grant from the Australian Research Council. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. 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--- abstract: | Background : In coupled-channel models the poles of the scattering S-matrix are located on different Riemann sheets. Physical observables are affected mainly by poles closest to the physical region but sometimes shadow poles have considerable effect, too. Purpose : The purpose of this paper is to show that in coupled-channel problem all poles of the S-matrix can be calculated with properly constructed complex-energy basis. Method : The Berggren basis is used for expanding the coupled-channel solutions. Results : The location of the poles of the S-matrix were calculated and compared with an exactly solvable coupled-channel problem: the one with the Cox potential. Conclusions : We show that with appropriately chosen Berggren basis poles of the S-matrix including the shadow ones can be determined. author: - 'R. M. Id Betan' - 'A.T. Kruppa' - 'T. Vertse' title: 'Shadow poles in a coupled-channel problem calculated with Berggren basis' --- Introduction ============ The physics of coupled-channel (CC) models spans many research areas, ranging from traditional nuclear physics [@atk1; @atk2; @tho09] to atomic physics [@mar04; @nyg06], to chiral perturbation theory combined with multichannel approach [@cie13; @dot13], and to hypernuclei physics [@miy99]. As in the scattering theory in general, also in scattering CC models the exploitation of the analytic properties of the S-matrix is a basic tool in describing the scattering processes [@ede66]. The S-matrix as a function of energy is analytic over a Riemann surface of many sheets [@bad82; @gra00; @gra03]. The analytic continuation of the physical S-matrix may have poles on the unphysical sheets. These poles correspond to the eigenvalues of the Scrödinger equation with various special boundary conditions. The eigenfunctions belonging to these generalized eigenvalues are referred as resonance/virtual states of the system. On the physical sheet the eigenfunctions are asymptotically decreasing i.e. they are bound states. The asymptotic form of the resonance/virtual states are not bounded. In the following the term resonant state will be used for all discrete (bound, resonance and virtual) states. The main question in CC scattering model is how the S-matrix poles affect the scattering observables. Obviously, they are affected by poles closest to the physical region. However, it has been recognized long ago that poles far from the physical regions (so called shadow poles) may have large effects on observables [@ede64; @fra64]. This phenomena occurs in atomic [@dij08; @rak06; @pot88], nuclear [@hal87; @csot93; @pea89] and particle physics [@mor87]. The effect of the shadow poles was studied also for confined two dimensional electron gas [@cat07]. As for the finding all types of poles of the S-matrix i.e. solving the CC equation with generalized asymptotic condition there are two approaches: either the task is modified so as to incorporate the special conditions then is solved with orthodox methods or the task is considered as it is and the solving methods are generalized. The most straightforward example for the first approach is the complex scaling method which has a long history [@cs]. In a recent nuclear three-body application [@atk] the back-rotation problem of this method is solved. The usage of the complex scaling method to CC problem sometimes requires slight extensions as e.g. when shadow poles are to be found [@csot93a]. A good and nowadays often used approach of the second kind is the series expansion of the wave function on an extended basis. The basis includes the bound states, selected resonance/virtual states as well as complex energy scattering states of the potential. The existence of such basis was proven by T. Berggren [@Be68]. Berggren basis were used in the solution of many-body shell model calculations [@2002IdBetan; @2002Michel], especially when weakly bound or unbound states of certain nuclei observed in radioactive beam facilities were to be described by Gamow shell model or complex energy shell model [@Mi09]. In certain applications [@Bet04; @ve89] it was shown that anti-bound states can also be included as basis states. The application of the Berggren bases in some model means that the wave function of the studied system is expanded on this basis. The Berggren basis was used also in CC problems: the radial coupled HFB equations were solved with it [@mic08] and also the CC Lane equation describing isobaric analog states [@Bet08]. The aim of this paper is to show that in CC problem [*all poles*]{} of the S-matrix can be calculated with properly constructed Berggren basis. To this end we reproduce the results of an exactly solvable CC problem: the one with the Cox potential. This CC problem can be solved exactly [@1964Cox; @pup08; @spa06; @sam07] so there is no ambiguity in the location of the S-matrix poles. The analytic solution of the Cox potential offers a unique opportunity for testing the results of the numerical solution of the CC problem. The paper is organized as follows. Section \[sec:cox\] describes the CC Cox problem and its analytic solution. In Section  \[sec:berg\] we discuss how to use the Berggren basis for the solution of a CC problem. Here we discuss the main point of the paper: how to choose the Berggren basis in order to get all poles of the S-matrix. Section \[sec:applications\] presents the results for the CC Cox problem both analytic and numerical ones. Finally, conclusions are contained in Section. \[sec:conclusions\]. The Cox coupled-channel problem {#sec:cox} =============================== The Cox potential ----------------- The two-channel radial Schrödinger-equation with energy $E$ in reduced units reads $$\label{eq.h} H \psi(r,E) = K^2 \psi(r,E),$$ where $K={\rm diag}(k_1,k_2)$, $k_i=\sqrt{E-\Delta_i}$ denotes the channel wave numbers, $\Delta_i$ are the threshold energies. We will use as in [@pup08] $\Delta_1=0$ and $\Delta_2=\Delta>0$. The notation ${\rm diag}(a_1,a_2)$ means a two by two diagonal matrix with elements $a_i$ in the main diagonal. The Hamiltonian is $$\label{hami} H = \left( \begin{array}{cc} -\frac{d^2}{dr^2} + v_{11}(r) & v_{12}(r) \\ v_{21}(r) & -\frac{d^2}{dr^2} + v_{22}(r) \end{array} \right)$$ and the solution forms a vector $$\label{solvector} \psi = \left( \begin{array}{cc} \psi_1 \\ \psi_2 \\ \end{array} \right)~.$$ The Schrödinger equation in Eq. has two matrix value Jost solution from which the Jost matrix can be constructed defining both the scattering and bound state solutions. We classify the solutions of Eq. (\[eq.h\]) as it is done in Ref. [@pup08]. We call a solution a *bound state* when the zero of the determinant of the Jost matrix $k_1$ and $k_2$ are both pure positive imaginary numbers. We call the solution a *virtual state* or *anti-bound state*, when the zero of the Jost-matrix determinant corresponds to a real energy below the thresholds and the zero is lying on the imaginary $k_i$ axes, but not all of them are located on the positive imaginary axis. Finally, we call the solution a *resonance* if the zero is not lying on any of the imaginary $k_i$ axes, hence the corresponding energy is complex or if real then it is above at least one of the thresholds. The derivation of the Cox potential and how to solve it exactly are given in [@pup08; @spa06; @sam07]. To make the paper self-contained we collect some formulas. The Cox [@1964Cox] interaction matrix $$\label{cox_v} V(r) = \left( \begin{array}{cc} v_{11}(r) & v_{12}(r) \\ v_{21(r)} & v_{22}(r) \\ \end{array} \right)~.$$ is given by $$V(r)=-{\cal K}+2 {\cal K}^{1/2}(I+X(r))^{-1} {\cal K}^{1/2},$$ where $I$ is the 2x2 unit matrix and $$\begin{aligned} &X(r)={\rm diag}(\exp(-\kappa_1r),\exp(-\kappa_2 r)) X_0\nonumber\\ &\times {\rm diag} (\exp(-\kappa_1 r),\exp(-\kappa_2 r)).\end{aligned}$$ The symmetric 2x2 matrix $X_0$ contains the parameters of the potential. The factorization wave numbers $\kappa_1$ and $\kappa_2$ are positive parameters and they satisfy the condition $\kappa_2^2-\kappa_1^2=\Delta.$ (we will use $\kappa_1=\kappa$ as independent parameter). The matrix $\cal K$ is related to the factorization energies by ${\cal K}={\rm diag}(\kappa_1,\kappa_2)$. Of course, the interaction matrix is symmetric. The determinant of the Jost matrix is given by [@pup08] $$\label{eq.f} f(k,p)=\frac{(k+i\, \alpha_1)(p+i\, \alpha_2)+\beta^2}{(k+i\, \kappa_1)(p+i\, \kappa_2)}.$$ Here we denote for convenience the channel wave numbers as $k_1=k$ and $k_2=p$. The threshold condition reads $k^2-p^2=\Delta$. The zeros of the function $f(k,p)$ determine the position of the bound, virtual and resonance states. Interestingly the Cox potential depends on the factorization wave numbers but the eigenenergies are independent from $\kappa_i$. The connection between the parameterization $\alpha_1$, $\alpha_2$, $\beta$ and $X_0$ is given by the equation $$U_0=\left(\begin{array}{cc} \alpha_1 & \beta \\ \beta & \alpha_2 \\ \end{array}\right)={\cal K}^{1/2}(I-X_0)(I+X_0)^{-1}{\cal K}^{1/2}$$ and the inverse relation is $$X_0={\cal K}^{-1/2}({\cal K}-U_0)({\cal K}+U_0)^{-1}{\cal K}^{1/2}.$$ Reduced inverse problem\[sec.inverse\] -------------------------------------- In a *direct problem* we calculate the eigenenergies and the corresponding channel wave numbers $(k_i,p_i)$ of the Cox CC equations for a given set of potential parameters $\alpha_1$, $\alpha_2$, $\beta$, and $\Delta$. In the *inverse problem* we fix a few eigenenergies or other characteristics of the problem and search for the parameters of the potential which give back the fixed characteristics. A very convenient approach was introduced in Ref. [@2007Pupasov]. In this approach one first fixes two solutions and the other two solutions are obtained in closed form. Let us fix $(k_1,p_1)$ and $(k_2,p_2)$ as the first two zeros of the Jost-matrix determinant $f(k,p)$ then from we get $$\begin{aligned} (k_1+i\, \alpha_1)(p_1+i\, \alpha_2)+\beta^2 &=&0 \\ (k_2+i\, \alpha_1)(p_2+i\, \alpha_2)+\beta^2 &=&0\end{aligned}$$ and we obtain the parameters $\alpha_1$ and $\alpha_2$ in terms of $(k_1,p_1)$ and $(k_2,p_2)$ and $\beta$ $$\begin{aligned} \label{alpha_sol1} \alpha_1 &=& \frac{1}{2} \left[ i(k_1+k_2) \pm \sqrt{-\Delta^2_k - 4 \beta^2 \frac{\Delta_k}{\Delta_p}} \right] \label{eq.a1} \\ \alpha_2 &=& \frac{1}{2} \left[ i(p_1+p_2) \mp \sqrt{-\Delta^2_p - 4 \beta^2 \frac{\Delta_p}{\Delta_k}} \right] \label{eq.a2},\end{aligned}$$ where $\Delta_k = k_2 - k_1$ and $\Delta_p = p_2 - p_1$. The signs define two sets of solutions which correspond to the same $(k_1,p_1)$ and $(k_2,p_2)$ but two different $(k_3,p_3)$ and $(k_4,p_4)$ roots. In selecting the solution either the upper or the lower signs are to be taken. The other two roots $(k_3,p_3)$ and $(k_4,p_4)$ are determined in Ref. [@2007Pupasov] as follows. $$\begin{aligned} k_3 &=& \frac{1}{2} \left[ \mp i \sqrt{-\Delta^2_k - 4 \beta^2 \frac{\Delta_k}{\Delta_p}} + \sqrt{D_k} \right] \label{eq.k3} \\ p_3 &=& \frac{1}{2} \left[ \mp i \sqrt{-\Delta^2_p - 4 \beta^2 \frac{\Delta_p}{\Delta_k}} + \sqrt{D_p} \right] \label{eq.p3} \\ k_4 &=& \frac{1}{2} \left[ \mp i \sqrt{-\Delta^2_k - 4 \beta^2 \frac{\Delta_k}{\Delta_p}} - \sqrt{D_k} \right] \label{eq.k4} \\ p_4 &=& \frac{1}{2} \left[ \mp i \sqrt{-\Delta^2_p - 4 \beta^2 \frac{\Delta_p} {\Delta_k}} - \sqrt{D_p}\right] \label{eq.p4},\end{aligned}$$ where $D_k = \Delta^2_k + 4 \beta^2 \frac{\Delta_p}{\Delta_k} + 4 k_1 k_2$ and $D_p = \Delta^2_p + 4 \beta^2 \frac{\Delta_k}{\Delta_p} + 4 p_1 p_2$. Note the difference in sign of the second term in $p_3$ and $p_4$ with respect to Eq. (50) in Ref. [@2007Pupasov]. Solution using Berggren basis {#sec:berg} ============================= We calculate the eigenvalues of the CC Cox problem by diagonalizing the Hamiltonian (\[hami\]) in the Berggren bases of the potentials $v_{11}(r)$ and $v_{22}(r)$. We consider two auxiliary problems $$\label{ham_bas} \left [-\frac{d^2}{dr^2}+v_{ii}(r)-E_n^{(i)}\right ] u^{(i)}_n(r)=0\ \ \ i=1,2.$$ The states of the Berggren basis are the solutions of Eq. and for each channel they are composed of the resonant basis states which are eigensolutions of Eq. (\[ham\_bas\]) with purely outgoing wave boundary condition, i.e. they correspond to the poles of the $S$-matrix in that channel. The resonant states (bound,virtual and resonance solutions) with energy $E_n^{(i)}$ are denoted by $u^{(i)}_n(r)$. Beside the resonant states the basis contains scattering states along a complex contour $L$. The scattering solutions are denoted by $u^{(i)}(r,E)$ or $u^{(i)}(r,k)$ if we use the energy $E$ or wave number $k$, respectively. The shape of the contour $L$ is restricted by certain rules [@Be68]. In the $k$ plane the contour has to go through the origin and has to be symmetric to the origin, i.e. if $k$ is on the contour $L$ then $-k$ should be on the contour $L$ too. Asymptotically i.e. for large $|k|$ values the contour has to go back to the real axis and remain there. Only those resonant states have to be included into the Berggren basis whose wave numbers are above the contour $L$. Similar relations should be hold for the $p$ contour of the second channel. (See Figures \[r2sheet\] and \[r3sheet\] the wave numbers to be included into the basis should be in the shaded area.) ![\[r2sheet\] (Color online) Illustration of the contours $L$ (thick lines) on the $k$ and $p$ planes for the second Riemann sheet. Resonance states can be determined with contours similar to the upper part. Both virtual and resonance states can be calculated with contours similar to the lower part. The double roles of the shaded areas are explained in the text.](Fig1.eps){width="1.\columnwidth"} The completeness relation of the Berggren basis reads $$\begin{aligned} \label{eq:delb} \delta(r-r^\prime)=\sum_{n=b,d,v}{u^{(i)}_n(r)}u^{(i)}_n(r^\prime) \nonumber \\ +\int_{L} dk~ {u^{(i)}(r,k)}u^{(i)}(r^\prime,k)~.\end{aligned}$$ In this relation (and later) the notation $n=b,d,v$ means that the sum over $n$ runs through all bound states, decaying resonances and virtual states in the shaded area of Fig. \[r2sheet\]. The integral in Eq. (\[eq:delb\]) is over the scattering states along $L$. The completeness relation in Eq. (\[eq:delb\]) for chargeless particles was introduced in Ref. [@Be68] and its validity has been shown for charged particles too [@Michel; @Michel08]. Since we can not handle the continuum part exactly the complex contour is discretized and truncated in order to have a finite number of contour states. The renormalization of the discretized contour states was introduced first in Ref. [@Li96] and it is performed as in Ref. [@Mi09]. We use as discretization points $E_k^{(i)}$ ($k=1,\ldots,N_i$) the abscissas of a Gaussian quadrature procedure. The corresponding weights of the quadrature points are denoted by $h^{(i)}_k$. After discretizing the integral in Eq. (\[eq:delb\]) an approximate completeness relation for the finite number of basis states reads $$\label{eq:finb} \delta(r-r^\prime)\approx \sum_{n=b,d,v,c}^{M_i} {w^{(i)}_n(r,E^{(i)}_n)}w^{(i)}_n(r^\prime,E^{(i)}_n),$$ where $c$ labels the discretized scattering states from the contour $L$ and $M_i$ is the sum of the resonant (bound, virtual and resonant) states contained inside the the shaded area plus $N_i$ number of discretized continuum states. If $E^{(i)}_n$ is a scattering energy from the contour $L$ then the scattering state of the discretized continuum is denoted by $w^{(i)}_n(r,E^{(i)}_n)=\sqrt{h^{(i)}_n} u^{(i)}_n(r,E^{(i)}_n)$. If however $E^{(i)}_n$ corresponds to a normalized resonant state of the potential $v_{ii}(r)$ then $w^{(i)}_n(r,E^{(i)}_n)= u^{(i)}_n(r)$. The set of Berggren vectors form a bi-orthonormal basis in the truncated space $$\label{eq:biort} <\tilde {w}^{(i)}_n|w^{(i)}_m>=\delta_{n,m}~.$$ Having fixed the Berggren basis the solution is approximated in the form $$\psi_i(r)=\sum_{k=1}^{M_i}C^{(i)}_kw^{(i)}_k({ r},E^{(i)}_k)\ \ \ i=1,2.$$ Using Eq. (\[eq.h\]) we get the following set of linear equations for $C^{(i)}_k$ $$\begin{aligned} \label{eq:secul1} &(E^{(1)}_k-E)C^{(1)}_k+\sum_{m=1}^{M_1}\langle\tilde w^{(1)}_k| v_{12}|w^{(2)}_m\rangle C^{(2)}_m=0\nonumber \\ &k=1,\ldots,M_1\end{aligned}$$ and $$\begin{aligned} \label{eq:secul2} &(E^{(2)}_k-(E-\Delta))C^{(2)}_k+\sum_{m=1}^{M_2}\langle\tilde w^{(2)}_k| v_{21}|w^{(1)}_m\rangle C^{(1)}_m=0\nonumber\\ &k=1,\ldots,M_2.\end{aligned}$$ These two equations can be combined into one matrix eigenvalue equation. By diagonalizing the matrix of the Hamiltonian we get complex eigenvalues $E_\nu$ $\nu=1,\ldots M_1+M_2$. Some complex/real eigenvalues $E_\nu$ can be identified as resonant states of the CC problem. The identification in this case is easy because we should find the $E_\nu$ eigenvalue being closest to the exact value. In general this task is more complicated, some methods can be found in Ref.[@Mi09] . ![\[r3sheet\] (Color online) Similar to Fig. \[r2sheet\] but for the third Riemann sheet.](Fig2.eps){width="1.\columnwidth"} In two-channel case we have a Riemann surface with four sheets. Let us define the four Riemann sheets in terms of the sign of the imaginary parts of the $k$ and the $p$ wave numbers. The Riemann sheets can be labeled by a two-term sign string $({\rm sgn}({\rm Im} k),{\rm sgn}({\rm Im} p))$. We follow the standard notations introduced in Refs. [@fra64; @bad82]. The first sheet is the physical one and it is signed by $(+,+)$. The second sheet is $(-,+)$ and these two levels are connected if $0<E<\Delta$. The third and fourth sheets are identified by $(-,-)$ and $(+,-)$, respectively. These two sheets are also connected if $0<E<\Delta$. If the energy $E$ is above the threshold $\Delta$ the topological structure changes: sheets one and three as well as two and four are connected. The location of a resonant state determines the asymptotic behavior of its wave function. Bound state from the first Riemann sheet has square integrable wave functions in both channels. However a resonant state from the second Riemann sheet have such a wave functions that the first component asymptotically diverges and the second channel has bound state type behavior. For resonant states from the third Riemann level both components of the wave function diverge asymptotically. When we diagonalize the CC Cox potential in Berggren bases sometimes we have to take different contours in the complex $k$ and $p$ planes in order to determine the solutions we are interested in. As we discussed earlier the shape of the complex contour $L$ determines which resonant states of the potential $v_{ii}(r)$ should be included into the Berggren basis of the given channel. However the shape of the contours also determines the Riemann sheets and we are able to find only the resonant CC states on that Riemann sheets. The CC resonant states of a given calculation are in the shaded areas both in the $k$ and $p$ planes. If both the $k$ and the $p$ contours remain on the real axis we can find only bound states on the physical $(+,+)$ sheet. In order to locate resonant states on the second Riemann sheet $(-,+)$ we have to use contours of the form displayed on Fig. \[r2sheet\]. Contours similar to the upper part can be used only for calculation of resonance states of the second Riemann sheet and for bound states of the first Riemann sheet. If a virtual state is located on the second Riemann level then the contours have to look like as displayed on the lower part of Fig. \[r2sheet\]. Of course, also resonance states on the second Riemann sheet can be calculated using contours similar to the lower part of Fig. \[r2sheet\]. This type of contours however discard some CC bound state from the first Riemann level. If resonant states located on the third Riemann sheet are to be determined, the shape of the contours depicted in Fig. \[r3sheet\] have to be used. Only resonance states can be determined by contours similar to the upper part of Fig. \[r3sheet\]. The lower part is appropriate for calculation aimed at obtaining virtual states and resonance states located on the third Riemann level. We mention that using contours corresponding to the upper part of Fig. \[r3sheet\] bound and resonance states of all Riemann sheets can be determined simultaneously. However numerically it is favorable to use simpler contours. For resonant state on the second Riemann sheet the accuracy of the numerical calculation is better for contours on Fig. \[r2sheet\] then for contours of Fig. \[r3sheet\]. Simpler contours for states located on the fourth Riemann sheet can be similarly constructed. Applications {#sec:applications} ============ At certain parameters of the Cox potential the two channels decouple. Since the eigenvalue problem is exactly solvable, the accuracy of our program for solving the single-channel problem can be conveniently checked. This program integrates the differential equation in Eq.(\[ham\_bas\]) numerically. It is important that we solve the single-channel problem accurately, since the discrete basis states are calculated using this program. The inaccuracy of the basis states would spoil the numerical results of the CC system. Therefore we deal with the solution of the single-channel case first. Then we will consider a CC system having a resonance state above the first threshold and below the second one. The threshold $\Delta$ and $\beta$ are fixed to the values $1$ and $0.1$, respectively for the applications considered here. Single-channel solutions ------------------------ If we take the parameter $\beta=0$ then the CC equation with Cox potential reduces to two uncoupled equations. In this case the zeros of the Jost determinant (\[eq.f\]) are $k=-i\, \alpha_1$ and $p=-i\, \alpha_2$. We compare the analytical results with that of our numerical calculations in Table \[table.uncop\]. Note that the exact value of ${\rm Im}(k)$ has opposite sign than the value of the parameter $\alpha_1$. $\alpha_1$ ${\rm Im}(k)$ exact ${\rm Im}(k)$ numerical ------------ --------------------- ------------------------- 0.8 -0.8 -0.801800 0.7 -0.7 -0.700028 0.6 -0.6 -0.599525 0.5 -0.5 -0.499555 0.4 -0.4 -0.399984 0.3 -0.3 -0.299996 0.2 -0.2 -0.199998 0.1 -0.1 -0.099998 -0.1 +0.1 0.100001 -0.2 +0.2 0.200000 -0.3 +0.3 0.299999 -0.4 +0.4 0.399999 : \[table.uncop\] Wave numbers of the anti-bound/bound states calculated exactly and numerically for the single-channel problem with potential $v_{11}(r)$ using $\beta=0$. To calculate the eigenvalues we used the highly reliable Fortran program ANTI [@Ix95] which is based on Ixaru’s method [@Ix84] for the numerical solution of the differential equation . This program reproduces the exact results reasonably well in most of the cases given in Table \[table.uncop\]. The agreements are best for the bound state cases and the anti-bound wave numbers are also reproduced well, although the deviation from the exact value has increased gradually as the $\alpha_1$ value has increased. In solving numerically the problem the diagonal potentials $v_{11}(r)$ and $v_{22}(r)$ are cut to zero at a reasonable large $R_{{\rm max}}$. Beyond $R_{{\rm max}}$ the potential is considered to be zero. The results should be at most slightly dependent on the chosen value of $R_{{\rm max}}$. With a cut-off radius $R_{{\rm max}}=13$ our numerical result is ${\rm Im}(k)=-0.76950$ for $\alpha_1=0.76938$, which deviates from the exact value in the fourth decimal digit. With the same $\alpha_1$ value if we changed the cut-off radius value to smaller or larger values we got slightly different ${\rm Im}(k)$ values. (For $R_{{\rm max}}=12$ we got ${\rm Im}(k)=-0.77031$. For $R_{{\rm max}}=14$ we got ${\rm Im}(k)=-0.768499$.) So we found that the wave number of the anti-bound state depends only weakly on the cut-off radius of the diagonal potential. This is in agreement with the finding in Ref.[@Da12] for a cut-off Woods-Saxon potential. The pole energy of the $S$-matrix is determined from the condition that the logarithmic derivatives of the internal and the external solutions of the equation(\[ham\_bas\]) are equal at a matching distance $R_{{\rm match}}$. See e.g. Ref.[@Da12; @Ix95; @Ve82]. The internal solution is regular in $r=0$, while the external solution is a purely outgoing wave at $R_{{\rm max}}$. In principle the pole energy should not depend on $R_{{\rm match}}$. In our calculation the value of $R_{{\rm match}}$ influenced only the fifth decimal digit of ${\rm Im}(k)$ if we used a value in the range $R_{{\rm match}}\in [1,5]$. These comparisons of the exact and approximate energies give some hint on the limits of accuracy we can expect between the exact and approximate results of the CC calculations. We certainly can not expect better agreement for the CC case than we got for the single-channel case. Coupled-channel: exact solutions -------------------------------- In order to obtain the exact solution to the CC problem we will appeal to the inverse procedure introduced in section \[sec.inverse\]. Let us consider a resonance solution of the Cox potential with complex energy $E_r-iE_i$ so that $0 < E_r < \Delta$ and $E_i>0$. We will determine $(k_1,p_1)$ and $(k_2,p_2)$ from the complex energy solutions $E_1=E_r - i E_i$ and $E_2=E_1^*=E_r + i E_i$, which correspond to the wave numbers $k_1=k_r + i k_i$ and $k_2=-k^*_1=-k_r + i k_i$ with $k_r>0$ and $k_i<0$. The relations between the real and imaginary parts of the energy and wave numbers are $$\begin{aligned} k_r &=& \frac{1}{\sqrt{2}} \left[ E_r + \sqrt{E_r^2 + E_i^2} \right]^{1/2} \\ k_i &=&-\frac{E_i}{\sqrt{2}} \left[ E_r + \sqrt{E_r^2 + E_i^2} \right]^{-1/2} \end{aligned}$$ (note that Eq. (52) of Ref. [@2007Pupasov] is wrong). Using the threshold condition $k^2-p^2=\Delta$ we can determine $p_r$ and $p_i$ with $p_1=p_r + i p_i$ and $p_2=-p^*_1=-p_r + i p_i$. The sign of $p_r$ is determined by noticing that from Eq. (31) of Ref. [@2007Pupasov] we can get $k_r p_r<0$, while the sign of $p_i$ is determined by the condition $p_r p_i<0$ for $E_i>0$ (which also implies $k_i p_i<0$). Considering these restrictions we have $$\begin{aligned} p_r &=& \frac{-1}{\sqrt{2}} \left[-(\Delta-E_r) + \sqrt{(\Delta-E_r)^2+E_i^2} \right]^{1/2} \\ p_i &=& \frac{E_i}{\sqrt{2}} \left[-(\Delta-E_r) + \sqrt{(\Delta-E_r)^2+E_i^2} \right]^{-1/2}.\end{aligned}$$ Taking for example $E_r=0.4$ and $E_i=0.01$ as in [@2007Pupasov] we get $k_r=0.632550$, $ k_i=-0.007905$ and $p_r=-0.006455$ $p_i=0.774624$. The exact solutions for the upper sign in Eqs. (\[eq.k3\]–\[eq.p4\]) are displayed in Table \[table.cor\]. Now the CC problem has two resonances and two anti-bound states. According to Table \[table.cor\] the resonances $E_1$ and $E_2$ are located in the second Riemann sheet. The anti-bound $E_3$ state is on the second sheet too while the second anti-bound solution $E_4$ is a shadow pole on the third Riemann sheet. $E_j$ $k_j$ $p_j$ ------------ ----------------------- ----------------------- 0.4-i0.1 0.632550 -i 0.007905 -0.006455 +i 0.774624 0.4+i0.1 -0.632550 -i 0.007905 0.006455 +i 0.774624 -0.5604738 -i 0.748648 i 1.24919 -0.5995714 -i 0.774302 -i 1.26473 : \[table.cor\] Exact energies ($E_j$ for $j=1,\dots ,4$) and corresponding wave numbers in the CC Cox potential with parameters given in the text. Coupled-channel: approximate solutions\[sec:path\] -------------------------------------------------- The approximate solutions are obtained by direct diagonalization of the Cox potential on Berggren basis as it is described in section \[sec:berg\]. For this we need to find the parameters $\alpha_1$ and $\alpha_2$ of the potential which give the energies of the exact solution of the previous section. Using Eqs. and and the upper signs we get $\alpha_1=0.769379934$, $\alpha_2=-0.766852669$. channel Re$E^{(i)}_n$ Im$k^{(i)}_n$ --------- --------------- --------------- 1 -0.573887 -0.757553 2 -0.5292006 0.769419 : \[table.spe\] Energies and wave numbers of discrete basis states in the first and in the second channel. The solutions of CC equations using Berggren bases are carried out as follows. Two Berggren bases are calculated using the diagonal potentials $v_{11}(r)$ and $v_{22}(r)$, respectively, with the $\kappa=1$ parameter value. The parameter $\kappa$ affects the shape of the radial potentials but does not affect the CC eigenenergies. For $\kappa=1$ we found that the unperturbed potential $v_{11}(r)$ has an anti-bound state and the unperturbed potential $v_{22}(r)$ has a bound state. The actual values of the energies and wave numbers of the resonant basis states are given in Table \[table.spe\]. The decaying CC resonance state at the energy $E_1=0.4\, -\, i\, 0.01$ is on the second Riemann sheet . The shape of the contour therefore should look like as the upper part of Fig. \[r2sheet\], since the contour should go below $k_1$ and $k_1$ should be in the shaded area. The basis in the first channel has no bound state therefore it is formed from the complex continuum states only. In the second channel the basis contains the unperturbed bound state and a continuum which can be taken along the real $p$-axis. Diagonalizing the Cox potential using these bases, we got a decaying resonance at the energy $E=0.400047 - i\, 0.0100011$. The corresponding wave function is displayed in Fig. \[wffig\]. The form of the wave function follows from the rules discussed in section \[sec:berg\]. The real and the imaginary parts of the first channel wave function show resonant behavior, i.e. they both diverge asymptotically. The real and the imaginary parts of the second channel wave function however both falls asymptotically as a bound state wave function does. Beside the resonance states at $E_1$ and $E_2=E_1^*$ there is an anti-bound state at the energy $E_3=-0.5604738$. It is also in the second Riemann sheet. In order to calculate this state the contour should be taken similar to the one in the lower part of the Fig. \[r2sheet\], since $k_3$ should be in the shaded area therefore we had to modify the contour used before. We still have two possibilities for selecting the basis in this channel. If the contour crosses the imaginary $k$-axis far from the origin, say at $k=(0,-1.2)$ then the anti-bound basis state will be in the shaded area and should be included into the Berggren basis. Therefore the Berggren basis in the first channel contains the unperturbed anti-bound state at $k_n=(0,-0.757553)$ and a set of discretized complex $k$ scattering states and in the second channel the unperturbed bound state and the real $p$ scattering states are in the basis. If we use this basis the diagonalization of the Cox potential gives a CC virtual state at energy $-0.561467 - i\, 0.494 \times 10^{-7}$ which is very close to the exact $E_3$ value. The other option for choosing the contour is that we cross the imaginary $k$-axis just between the exact $k_3=(0,-0.749648)$ and the unperturbed anti-bound state at $k_n=(0,-0.757553)$. If we cross the imaginary axis at $(0,-0.75)$ then the unperturbed anti-bound state will be outside the shaded area therefore it won’t be included in the basis. By using this basis in the first channel (the basis in the second channel remains unchanged) the diagonalization gives a CC virtual state at energy $-0.561008 - i\, 0.413 \times 10^{-3}$ which is also very close to the exact $E_3$ value. This later basis shows an example for a case in which a correlated anti-bound state is produced by diagonalization with Berggren bases in which only bound state and complex scattering states are included. The small imaginary parts of the energies in the results of the diagonalization in both cases are due to the numerical errors of the numerical procedures used. They are beyond the accuracies of the errors of the single channel calculation of the anti-bound basis state for $\alpha_1=0.8$. In the Cox potential there is an another anti-bound solution at the energy $E_4=-0.599544$. This state lies on the third Riemann sheet so it is a shadow pole. To be able to expand this state we have to use a contour which is similar to the one in the lower part of Fig. \[r3sheet\]. The Berggren basis in the first channel contains the unperturbed anti-bound state. Because of the symmetry requirement of the complex $L$ contour to the origin, in the second channel the unperturbed bound state should be excluded from the basis. Now we have no alternative in choosing the contour since $k_4=(0,-0.774302)$ lies lower than the negative of the imaginary part of the bound state at $-k_n^{(2)}=-0.769419$. The bound pole now is in the not shadowed part. But a finite number of discretized scattering states with complex $p$ wave number are naturally included in the basis. The diagonalization of the Cox potential in these bases gives an anti-bound CC state at the energy $-0.600357 - i\, 0.149 \times 10^{-4}$, which is very close to the exact $E_4$ value. The deviation from the exact value is again within the accuracy of reproducing the exact single particle basis states. ![\[wffig\] The wave function of the first resonance state located in the second Riemann sheet and calculated with the Berggren expansion method. The upper and lower parts show the first and second channel components, respectively. The imaginary part of the second component is practically zero.](Fig3.eps){width="0.8\columnwidth"} The Berggren basis allows the calculation of more than one resonant states simultaneously with a properly chosen basis. As an interesting example we consider the first resonance state from the second Riemann sheet and the shadow anti-bound state from the third Riemann sheet. Numerically we will show that although the states are on different sheets they can be calculated simultaneously by using the same bases. Because we want to calculate an anti-bound state in the third Riemann sheet we have to use contours similar to the one in the lower part of Fig. \[r3sheet\]. As we discussed before these contours however might exclude some resonance states located in the second Riemann sheet. Therefore we have to choose the crossing points of the contours and the positive imaginary axes carefully, since the wave numbers of the resonance CC state should be in the shaded area of Fig. \[r3sheet\]. The Berggren basis in the first channel is formed by the unperturbed anti-bound state and a set of scattering states along the $k$ contour, while in the second channel the basis is composed of the unperturbed bound state and a set of complex $p$ scattering states. The diagonalization of the Cox potential using these bases gives the following eigenenergies $0.400214 - i\, 0.0098868$ and $-0.599100 - i\, 0.501 \times 10^{-3}$ simultaneously. Although these numerical results received by these bases are still quite close to the exact values of $E_1$ and $E_4$, the quality of the approximation is a little bit poorer than the ones we presented earlier with bases adjusted to the energies individually. Nevertheless the accuracies are still inside the ones of the single channels basis states. Conclusions {#sec:conclusions} =========== We considered the exactly solvable CC problem of the Cox potential and gave numerical examples that by using the Berggren expansion method we are able to reproduce all poles of the S-matrix even the shadow ones resulted by the exact calculation. It was shown that the proper choice of the complex contours are very important since it determines which CC states can be calculated by diagonalization of the Hamiltonian in the bases. With suitably chosen contours we were able to calculate poles on different Riemann sheets simultaneously. 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--- abstract: 'The large number of top quarks produced at the LHC and possible future hadron colliders allows to study rare decays of this particle. In many well motivated models of new physics, for example in non-minimal composite-Higgs models, the existence of scalar singlets can induce new flavor-violating top decays surpassing the Higgs contribution by orders of magnitude. We study the discovery prospects of rare top decays within such models and develop new search strategies to test these interactions in top pair-produced events at the LHC. We demonstrate that scales as large as $10$–$50$ TeV can be probed. Improvements by factors of $\sim 1.5$ and $\sim 3$ can be obtained at $\sqrt{s} = 27$ TeV and $\sqrt{s} = 100$ TeV colliders respectively.' bibliography: - 'notes.bib' --- IPPP/18/41 [Top quark FCNCs in extended Higgs sectors]{} **Shankha Banerjee, Mikael Chala and Michael Spannowsky**\ [*[Institute of Particle Physics Phenomenology, Physics Department, Durham University, Durham DH1 3LE, UK]{}*]{} Introduction ============ Processes mediated via Flavour Changing Neutral Currents (FCNC) are very rare within the Standard Model (SM) of particle physics. Any experimental evidence of such processes will thus serve as a clear signal for new physics. These possible rare processes have garnered strong interest in both the theoretical and experimental communities and have triggered numerous studies in search for FCNCs, particularly in the top production [@Tait:1996dv; @Guasch:2006hf; @Plehn:2009it; @Blanke:2013zxo; @Degrande:2014tta; @Goldouzian:2014nha; @Backovic:2015rwa] and decay channels [@Agashe:2009di; @Mele:1998ag; @Greljo:2014dka; @Azatov:2014lha; @Khachatryan:2015att; @Botella:2015hoa; @Bardhan:2016txk; @Badziak:2017wxn; @Khachatryan:2016sib; @Gabrielli:2016cut; @CMS-PAS-TOP-17-017; @Aaboud:2018nyl; @Aaboud:2017mfd; @Aaboud:2018pob; @Papaefstathiou:2017xuv]. Looking for such rare processes in the top sector is very lucrative because top quarks are copiously produced at high-energy hadron colliders, and therefore a large number of events can be expected even if the FCNC top decays are very rare. Furthermore, because of its large mass, the top quark is inherently connected to the Electroweak Symmetry Breaking (EWSB) sector where new physics effects are more likely to be present. Numerous experimental searches have been carried out in the top FCNC sector. Some of these searches include $t\rightarrow \gamma c/u$ [@Khachatryan:2015att], $t\rightarrow gc/u$ [@Khachatryan:2016sib], $t\rightarrow Zc/u$ [@CMS-PAS-TOP-17-017; @Aaboud:2018nyl] and $t\rightarrow hc/u$ [@Aaboud:2017mfd; @Aaboud:2018pob] in single top production or top-pair production. The present experimental bounds on these decays are $\mathcal{B}(t\rightarrow \gamma c \; (u)) < 1.7 \; (0.13) \times 10^{-3}$ [@Khachatryan:2015att], $\mathcal{B}(t\rightarrow gc \; (u)) < 4.1 \; (0.2) \times 10^{-4}$ [@Khachatryan:2016sib], $\mathcal{B}(t\rightarrow Z c \; (u)) < 2.4 \; (1.7) \times 10^{-4}$ [@CMS-PAS-TOP-17-017; @Aaboud:2018nyl] and $\mathcal{B}(t\rightarrow h c \; (u)) < 2.2 \; (2.4) \times 10^{-3}$ [@Aaboud:2017mfd; @Aaboud:2018pob]. Similar conclusions can also be seen from the following references [@Craig:2012vj; @Harnik:2012pb; @Backovic:2015rwa]. *A priori*, however, there might be other particles lurking around the Electroweak (EW) scale, into which the top quark can possibly decay. Out of the different possibilities, scalar singlets, $S$, constitute a prime example at the level of dimension-four interactions. When the singlet mixes with the SM-like Higgs boson, its production is strongly constrained owing to the increasingly precise Higgs signal strength measurements (central values reaching unity along with decreasing uncertainties) in various channels [@Khachatryan:2016vau; @Banerjee:2015hoa] and also from measurements of the $W$-boson mass [@Alcaraz:2006mx; @Aaltonen:2012bp; @D0:2013jba; @Robens:2015gla; @Lopez-Val:2014jva]. For a relatively small mixing parameter of $\sin\theta$ varying between 0.2 and 0.35 [@Peskin:2012we; @Craig:2014lda], one can however, have a wide range of allowed singlet mass. Such scalar particles are predicted by some of the best motivated models of new physics, including supersymmetric extensions (*e.g.,* the NMSSM [@Ellwanger:2009dp]) and Composite Higgs Models (CHM) [@Dimopoulos:1981xc; @Kaplan:1983fs; @Kaplan:1983sm; @Panico:2015jxa]. Moreover, we will see that these scalars can induce FCNCs significantly larger than those mediated by the SM-like Higgs boson [@Zhang:2013xya]. The reason for the above is three-fold; *(i)* The top FCNCs mediated by a new scalar singlet are generally suppressed by one less power of the heavy physics scale, *(ii)* in principle, the scalar singlet can have a larger decay width into cleaner final states, such as $\ell^+\ell^-$, $b\overline{b}$ or $\gamma\gamma$ and *(iii)* in broad classes of models (*for example* in CHMs), Higgs mediated FCNCs are forbidden in first approximation [@Agashe:2009di]. Altogether, top FCNCs mediated by new scalars might be well within the reach of the LHC. Presently, there are no direct limits on $t \to q S$. However, one can have strong constraints from $D^0-\bar{D}^0$ oscillations [@Bona:2007vi; @Harnik:2012pb; @Agashe:2013hma] which always come about as a product of two $S$ Yukawas, $Y_{ct}$ and $Y_{ut}$ (and also $Y_{uc}$). In order to circumvent these constraints, one can always fall back upon scenarios where $Y_{ut}$ is negligibly small. We will argue in section \[sec:models\], that the $ut$ FCNCs can be vanishingly small compared to their $ct$ counterpart in explicit models. In this work, we scrutinise the reach of the LHC for top FCNCs in top-pair produced events. We consider the standard leptonic decay of one of the tops, while the other is assumed to decay into $S c$, with either $S\rightarrow b\overline{b}$ or $S\rightarrow\gamma\gamma$ (leptonic decays will be analysed in a later work). In principle, the current experimental searches for $t\rightarrow hc$ might be also sensitive to these signals. However, these searches are only optimised for a $125$ GeV scalar resonance and not for the whole range of masses, in which $S$ can potentially lie. Moreover, the latest experimental strategies (see *e.g.* Ref. [@CMS:2017cck]) rely on trained BDTs which make them hard to recast for arbitrary scalar masses. In light of these issues, we develop new dedicated analyses tailored for each mass point. The structure of the paper is as follows. In section \[sec:eft\] we outline a model-independent introduction to the signals of interest. We follow this up in section \[sec:models\], where we discuss concrete realisations, involving both strongly and weakly coupled models of new physics. This allows us to establish well-motivated benchmark points (BP). We go on to discuss the analysis for the $b\overline{b}$ channel in section \[sec:bb\] and in section \[sec:gg\], we present the corresponding analysis for the $S \to \gamma\gamma$ mode for the 14 TeV LHC machine. We finally conclude in section \[sec:conclusions\], where we also provide an outlook for the high energy colliders by commenting on naive estimations of the reach of hadron colliders at $\sqrt{s} = 27$ TeV and $\sqrt{s} = 100$ TeV. Effective Lagrangian {#sec:eft} ==================== Let us consider a scenario where the SM Higgs sector is extended by a gauge singlet, $S$, having a mass $m_S$ in the EW regime. For low energies, we write the relevant Yukawa Lagrangian as follows [^1] $$\label{eq:lag} \mathcal{L} = -\mathbf{\overline{q_L}}\bigg(\mathbf{Y} + \mathbf{Y'}\frac{|H|^2}{f^2} + \mathbf{\tilde{Y}}\frac{S}{f}\bigg) \tilde{H} \mathbf{u_R} + \text{h.c.}~,$$ where $H = [\phi^+, (h + \phi^0)/\sqrt{2}]^t$, is the SM-like Higgs doublet, $\mathbf{q_L}$ ($\mathbf{u_R}$) denotes the left-handed (right-handed) quarks, $\mathbf{Y}, \mathbf{Y'}, \; \textrm{and} \; \mathbf{\tilde{Y}}$ are arbitrary flavour matrices, $v\sim 246$ GeV, is the Higgs vacuum expectation value (*vev*) and $f\gtrsim \mathcal{O}(\textrm{TeV})$ is the new physics scale. In general, the flavour matrices are not aligned, and thus the FCNCs can arise in the EW phase. Among various new physics effects, these induce top flavour-violating decays, *viz.*, $t\rightarrow h c$ or $t\rightarrow S c$. In general, the latter dominates over the former, because *(i)* $t \to h c$ is further suppressed by an additional factor of $1/f$ and *(ii)* in several UV-complete models, $\mathbf{Y}$ and $\mathbf{Y'}$ are approximately aligned. Finally, after the EWSB, one obtains $$\begin{aligned} \label{lag:lag1} \mathcal{L} &= -\frac{v}{\sqrt{2}}\bigg[\mathbf{\overline{q_L}}\mathbf{Y}\left(1+\frac{h}{v}\right)\mathbf{u_R} + \frac{S}{f}\mathbf{\overline{q_L}}\mathbf{\tilde{Y}}\mathbf{u_R} + \mathcal{O}\left(\frac{1}{f^2}\right)\bigg]\supset \tilde{g} \frac{m_t}{f} \overline{t_L} S c_R + \text{h.c.},\end{aligned}$$ where $m_t\sim 173$ GeV is the top mass and $g$ is an $\mathcal{O}(1)$ coupling. Such interactions can be tested to a high accuracy through rare top decays. Upon using Eq. \[lag:lag1\], one obtains the partial width of $t \to Sc$ as follows $$\Gamma(t\rightarrow Sc) = \frac{\tilde{g}^2}{32\pi}\frac{v^2}{f^2}m_t\bigg(1-\frac{m_S^2}{m_t^2}\bigg)^2~,$$ and thus for a benchmark point with $\tilde{g}\sim 1$ and $f\sim 1$ TeV, one obtains $\mathcal{B}(t\rightarrow Sc) \sim \Gamma(t\rightarrow Sc)/\Gamma_t^{\text{SM}} \sim 0.03$ with $\Gamma_t^{\text{SM}}\sim 1.4$ GeV [@Patrignani:2016xqp]. A full exploration of this decay in singly or pair-produced top quarks at colliders, depends also on how $S$ decays to SM particles. Motivated by CHMs (as discussed below), we consider a scenario where $S$ couples to the SM fermions, $\psi$, via $\frac{c_{\psi} m_\psi}{f} S\bar{\psi}\psi$ and to the photons via $\frac{c_{\gamma}\alpha}{4\pi f} S F_{\mu\nu}\tilde{F}^{\mu\nu}$, where $c_{\psi}$ and $c_{\gamma}$ are arbitrary $\mathcal{O}(1)$ couplings and $\alpha$ is the fine-structure constant. Thus, in the regime, $m_S \gg m_\psi$, one obtains at leading order $$\Gamma(S\rightarrow\psi\psi) = \frac{N_c}{8\pi}\frac{c_{\psi}^2 m_\psi^2}{f^2}m_S~ \; \textrm{and} \; \quad \Gamma(S\rightarrow\gamma\gamma) = \frac{c_\gamma^2 \alpha^2}{64 \pi^3 f^2}m_S^3~.$$ Thus, one finds the relation $\mathcal{B}(S\rightarrow\gamma\gamma)/\mathcal{B}(S\rightarrow \overline{\psi}\psi) \sim \frac{\alpha^2}{\pi^2} (m_S/m_\psi)^2$. The suppression factor driven by the small electromagnetic coupling can thus be partially compensated upon scaling with the free parameter $m_S$. $\mathcal{B}(S\to \gamma\gamma)$ can be significantly larger than the $\mathcal{B}(h\to \gamma\gamma) \sim 2 \times 10^{-3}$, with the precise value being model dependent. Explicit models {#sec:models} =============== The Lagrangian in Eq. \[eq:lag\] appears naturally in several UV-complete models, for example in CHMs. In these classes of models, $H$ and $S$ are pseudo Nambu-Goldstone Bosons (pNGBs) arising in a new global symmetry breaking $\mathcal{G}/\mathcal{H}$ at a scale $\sim f$. A prime example is the CHM based on the coset $SO(6)/SO(5)$ [@Gripaios:2009pe], which is the smallest one that admits four-dimensional UV completion [@Ferretti:2013kya]. The generators of this coset can be chosen as $$\begin{aligned} \label{eq:base} T^{mn}_{ij} &= -\frac{\mathtt{i}}{\sqrt{2}} (\delta^m_i\delta^n_j-\delta^n_i\delta^m_j)~, &m<n\in[1,5]~,\\[-1mm] X^{m6}_{ij} &= -\frac{\mathtt{i}}{\sqrt{2}} (\delta^m_i\delta^6_j-\delta^6_i\delta^m_j)~, &m\in[1, 5]~.\end{aligned}$$ Among these, $X^{16}-X^{46}$ expand the coset space of the Higgs doublet, while the broken generator associated to $S$ is provided by $X^{56}$. The SM fermions do not couple directly to the (fully composite) Higgs. Instead, the latter couples to composite fermionic resonances, which in turn mix with the SM fermions, thus explicitly breaking the global symmetry. The Yukawa Lagrangian depends therefore on the quantum numbers of the aforementioned fermionic resonances. For concreteness, we will assume that these fields transform in the fundamental representation $\mathbf{6}$ of $SO(6)$. The latter can be decomposed as $\mathbf{6} = 1 + 1 + \mathbf{4}$ under the custodial symmetry group $SO(4)$. Let us assume that $u_R^i$ is embedded in both singlets, whereas the one for $q_L^i$ is fixed. These are listed as $$\begin{aligned} U_{R_1}^i = (0, 0, 0, 0, \mathtt{i} u_R^i, 0)~, \quad U_{R_2}^i &= (0, 0, 0, 0, 0, u_R^i)~\\ \mathrm{and}~ \quad Q_L^i &= \frac{1}{\sqrt{2}}(\mathtt{i} b_L^i, b_L^i, \mathtt{i} t_L^i, -t_L^i, 0, 0)~.\end{aligned}$$ Using the corresponding Goldstone matrix $$\text{U}=\left[\begin{array}{cccc} 1_{3\times 3} & & & \\ &1-h^2/(f^2+\Pi) & -hs/(f^2+\Pi) & h/f\\ &-hs/(f^2+\Pi) & 1 -S^2/(f^2+\Pi) & S/f\\ &-h/f & -S/f & \Pi/f^2 \end{array}\right]~, \Pi = f^2\left(1-\frac{h^2}{f^2}-\frac{S^2}{f^2}\right)^{1/2}~,$$ one obtains the Yukawa Lagrangian $$\begin{aligned} \nonumber L &= -f y^{(1)}_{ij}\overline{(\text{U}^T~ Q_L^i)_6}~(\text{U}^T~ U_{R_1}^j)_6 -f y^{(2)}_{ij}\overline{(\text{U}^T~ Q_L^i)_6}~(\text{U}^T~ U_{R_2}^j)_6 +\text{h.c.} \\ &=-\frac{1}{\sqrt{2}}\overline{q_L^i} h t_R^j\left[ y_{ij}^{(2)}\left(1-\frac{h^2}{f^2}-\frac{S^2}{f^2}\right)^{1/2} + \mathtt{i} y_{ij}^{(1)} \frac{S}{f}\right] + \text{h.c.}~, \end{aligned}$$ which, to leading order, reads $$L = -\frac{1}{\sqrt{2}} \overline{q_L^i} h t_R^j \left[y_{ij}^{(2)} -y_{ij}^{(2)} \frac{h^2}{2f^2} + \mathtt{i} y_{ij}^{(1)}\frac{S}{f}+\cdots\right]+\text{h.c.}$$ Hence, we obtain $\mathbf{Y}_{ij} = - \mathbf{Y'} = y^{(2)}_{ij}$ and $\mathbf{\tilde{Y}}_{ij} = \mathtt{i} y_{ij}^{(1)}$. Thus, to leading order, scalar mediated FCNCs are only driven by $S$, provided that $y_{ij}^{(1)}$ and $y_{ij}^{(2)}$ are not aligned. Even in that case, the FCNCs would still arise in the presence of higher-dimensional operators, and then undergo suppression by a factor of $1/g_*^2$ (with $g_*$ being a strong coupling) just as in the minimal CHM [@Agashe:2009di; @Panico:2015jxa]. Similar results hold for other representations, with the exception of those that respect a $S\rightarrow -S$ parity and those for which the shift symmetry of $S$ remains unbroken, examples being $q_L^i$ in the $\mathbf{6}$, $t_R^i$ in the $\mathbf{15}$ [@Chala:2018qdf]. We note that although $\mathbf{\tilde{Y}}_{ij}$ is in principle arbitrary, one can easily expect sizeable top-charm couplings and still have small top-up FCNCs. Indeed, despite being not directly measurable, in common viable *ansätze*, $\mathbf{Y}$ is hierarchical and nearly block-diagonal, with the maximal mixing occurring in the top-charm sector [@Branco:1999tw; @Roberts:2001zy]. It can therefore be diagonalised as $\mathbf{Y}\rightarrow \mathbf{L}^\dagger \mathbf{Y}\mathbf{R}$, with $\mathbf{L}$ and $\mathbf{R}$ being block-diagonal as well. Moreover, in CHMs the aforementioned hierarchy reflects the fact that heavier fermions couple stronger to the composite sector, so not only to $H$ but also to $S$. One can then easily expect a similar block-diagonal structure for $\mathbf{\tilde{Y}}$. As a consequence, $\mathbf{L}\mathbf{\tilde{Y}}\mathbf{R}$ is also block-diagonal with only the top-charm mixing. Hence, we concentrate on the $t\rightarrow Sc$ decay channel. However, because we will not use any explicit $c$-tagging in our analyses, our results can easily be translated for the $t\rightarrow Su$ mode. We define three Benchmark Points (BP), each including $m_S = 20, 50, 80, 100, 120$ and $150$ GeV, as follows $$\begin{aligned} \nonumber \text{BP 1}: \quad \tilde{g} = 1.0~, \quad f =~\, 2~\text{TeV}\quad\Longrightarrow\quad\mathcal{B}(t\rightarrow Sc)\sim 10^{-3}-10^{-2}~;\\\nonumber \text{BP 2}: \quad \tilde{g} = 1.0~, \quad f = 10~\text{TeV}\quad\Longrightarrow\quad\mathcal{B}(t\rightarrow Sc)\sim 10^{-4}-10^{-3}~;\\ \text{BP 3}: \quad \tilde{g} = 0.1~, \quad f =~\, 2~\text{TeV}\quad\Longrightarrow\quad\mathcal{B}(t\rightarrow Sc)\sim 10^{-5}-10^{-4}~; \end{aligned}$$ We note that, although being *a priori* relatively light, values of $m_S < m_h/2\sim 62.5$ GeV are not necessarily excluded by the LHC constraints on the Higgs width [@Khachatryan:2016ctc]. Actually, the latter translates into an upper bound $\Gamma(h\rightarrow S S) \lesssim 10$ MeV. Given that for a quartic coupling $\lambda_{HS} S^2 |H|^2$ one obtains $\Gamma(h\rightarrow S S)\sim \lambda_{HS}^2/(32\pi) \times v^2/m_h$, we can evade the aforementioned bound provided $\lambda_{HS} < 0.05$. Similarly, derivative interactions $\sim h S \partial h \partial S/f^2$, unavoidable in CHMs, contribute to the Higgs width with an effective $\lambda_{HS}\sim 4 m_h^2/f^2\lesssim 0.05$ for a scale, $f \gtrsim 1.2$ TeV. Field Relevant Lagrangian Diagram $\mathbf{\tilde{Y}}_{ij}/f^2$ ---------------------------------- -------------------------------------------------------------------------------------------- --------- ------------------------------- \[-0.4cm\] $Q = (1, 2)_{1/6}$ $L_Q = -m_Q\overline{Q}Q + (\alpha_i^Q\overline{Q} S q^i_L$ \[-0.6cm\] $+ \tilde{\alpha}_j^Q\overline{Q}\tilde{H}u_R^j+\text{h.c.})$ \[-0.4cm\] $U = (1, 1)_{2/3}$ $L_U = -m_U\overline{U}U + ( \alpha_i^U\overline{U}Hq_L^i$ \[-0.6cm\] $+\tilde{\alpha}_j^U\overline{U} S u_R^j + \text{h.c.})$ \[-0.4cm\] $\Phi = (1, 2)_{1/2}$ $L_\Phi = -\frac{1}{2}m_\Phi^2\Phi^2 + (\alpha_{ij}^\Phi\overline{q_L^i}\tilde{\Phi}u_R^j$ \[-0.6cm\] $+\kappa S\Phi^\dagger H + \text{h.c.})$ \[0.3cm\] : *Single field extensions of the SM supplemented with $S$ that induce the FCNC of interest at low energy at tree level. The numbers in parenthesis and the subscript denote the $SU(3)_c$ and $SU(2)_L$ representations and the hypercharge, respectively. From the top left and clockwise, the different diagram legs represent $q_L^i$, $t_R^j$, $H$ and $S$, respectively.*[]{data-label="tab:weak"} If we consider only weakly-coupled extensions of the SM$+S$, the Lagrangian in Eq. \[eq:lag\] can also be induced *at tree level* by the fields listed in Tab. \[tab:weak\]. In particular, this means that the NMSSM [@Ellwanger:2009dp], which extends the SM scalar sector with an additional $SU(2)_L$ doublet with $Y=1/2$ (required by SUSY itself) as well as with a singlet (in order to avoid the $\mu$-problem [@Kim:1983dt]), fits naturally into the targets of our analysis. LHC prospects for $t\rightarrow Sc, S\rightarrow b\overline{b}$ {#sec:bb} =============================================================== In this section, we focus on the scenario where the scalar singlet decays to a pair of $b$-quarks, yielding a final state comprised of at least four jets, three of them required to be $b$-tagged and exactly one isolated lepton. As described above, we quantify our results in terms of six benchmark masses, *viz.*, $m_S = 20, 50, 80, 100, 120 \; \textrm{and} \; 150$ GeV. Our ultimate goal in this section is to derive an upper bound on $\mathcal{B}(t \to S c, S \to b \bar{b})$ at 95 % Confidence Level (CL). We have fixed the $b$-tagging efficiency to 70%. The $c \; (\ell) \to b$ mistag rate has been taken as 10% (1%). The most dominant real background ensues from semi-leptonic $t\bar{t} b \bar{b}$ production. Besides, the fully leptonic channel from the aforementioned production mode also contributes. The major fake backgrounds that we consider are the semi-leptonic (and leptonic) $t\bar{t}$ merged up to one extra matrix element parton, the $Wb\bar{b}$ process merged up to two extra matrix element partons and $Zb\bar{b}$ also merged up to two extra partons with the $Z$-boson decaying leptonically. For the analysis framework, we use [MG5\_aMC@NLO v2.6.0]{} [@Alwall:2014hca] for generating the signal and background samples. We employ the MLM merging scheme [@Mangano:2006rw] embedded in this framework, with appropriate parameter choices. We use very loose parton level cuts, *viz.*, $p_T(j) > 15$ GeV, $p_T(b) > 15$ GeV and $p_T(\ell) > 10$ GeV, as well as $|\eta(j)| < 4$, $|\eta(b)| < 4$ and $|\eta(\ell)| < 3$. Moreover, we require the $\Delta R$ separations to be zero for each pair at the generation level. The cross sections of $t\overline{t}$, $Wb\overline{b}$, $Zb\overline{b}$ and $t\overline{t}b\overline{b}$ are multiplied by $K$-factors of $1.6$, $2.3$, $1.25$ and $1.13$, respectively. While the first one can be found in Ref. [@kfactors1], the other three have been estimated by computing in `MG5_aMC@NLO` at NLO in QCD. We use the `NNPDF 2.3` [@Ball:2012cx] at leading order. The analyses in this section and in the next are carried out for the 14 TeV LHC. ![\[fig:sbb1\]*Left) The reconstructed top mass from the closest $b$-pair. Center) The transverse mass $m_T$. Right) The reconstructed mass of the scalar $S$.*](mTOPdelR.pdf "fig:"){width="0.32\columnwidth"} ![\[fig:sbb1\]*Left) The reconstructed top mass from the closest $b$-pair. Center) The transverse mass $m_T$. Right) The reconstructed mass of the scalar $S$.*](mT.pdf "fig:"){width="0.32\columnwidth"} ![\[fig:sbb1\]*Left) The reconstructed top mass from the closest $b$-pair. Center) The transverse mass $m_T$. Right) The reconstructed mass of the scalar $S$.*](mETAdelR.pdf "fig:"){width="0.32\columnwidth"} Furthermore, we shower the samples with `Pythia 8` [@Sjostrand:2014zea] [^2]. Finally, at the analysis level, we construct the jets employing the anti-$k_T$ [@Cacciari:2008gp] algorithm with a jet parameter $R=0.4$ in the [FastJet]{} [@Cacciari:2011ma] framework. All the jets are required to have $p_T > 30$ GeV and to lie within a pseudorapidity range of $|\eta| < 2.5$. Leptons must have a $p_T > 10$ GeV and $|\eta| < 2.5$. For the isolation, we require that the total hadronic activity around the lepton within a cone of $\Delta R = 0.2$ is less than 10 % of its transverse momentum. All the aforementioned selected objects are also required to be separated by $\Delta R > 0.4$. Cuts 20 GeV 50 GeV 80 GeV 100 GeV 120 GeV 150 GeV ------------------------------------- -------- -------- -------- --------- --------- --------- Basic 0.014 0.050 0.051 0.056 0.063 0.063 $|\eta_{(b,\ell,j)}| < 2.5$ 0.83 0.88 0.86 0.87 0.86 0.82 $\Delta R(\text{all pairs}) > 0.4$ 0.96 0.94 0.93 0.93 0.94 0.94 $|m_{t}^{\Delta R} - m_t| <$ 50 GeV 0.29 0.63 0.57 0.55 0.49 0.41 $m_T < 200$ GeV 0.72 0.56 0.87 0.85 0.83 0.74 : \[tab:sbb1\] *Efficiency after each cut for the six signal benchmark points.* Cuts $t\bar{t}$ (SL) $t\bar{t}$ (LL) $Wb\bar{b}$ $Zb\bar{b}$ $t\bar{t}b\bar{b}$ (SL) $t\bar{t}b\bar{b}$ (LL) ------------------------------------- ----------------- ----------------- ------------- ------------- ------------------------- ------------------------- Basic 0.0038 0.0016 0.00032 0.00016 0.11 0.073 $|\eta_{(b,\ell,j)}| < 2.5$ 0.78 0.69 0.74 0.71 0.90 0.85 $\Delta R(\text{all pairs}) > 0.4$ 0.95 0.94 0.95 0.95 0.96 0.91 $|m_{t}^{\Delta R} - m_t| <$ 50 GeV 0.49 0.32 0.27 0.33 0.31 0.28 $m_T < 200$ GeV 0.80 0.58 0.56 0.70 0.63 0.53 : \[tab:sbb2\] *Efficiency after each cut for the six dominant backgrounds. SL (LL) denotes semi (di)-leptonic decays.* After selecting these events, we look for the closest pair (in terms of $\Delta R$ separation) of $b$-tagged jets and reconstruct the top-quark mass $m_t^{\Delta R}$ with the additional hardest jet which is not $b$-tagged. We require this variable to be within a window of 50 GeV from $m_t$. With the remaining $b$-tagged jet, we construct the transverse mass variable $m_T$ and require it to be less than $200$ GeV. We show the distributions of $m_t^{\Delta R}$, $m_T$ and the reconstructed scalar mass, $m_S^{\Delta R}$, after the basic cuts (which include the aforementioned $p_T$ cuts as well as a requirement for 3 $b$-tagged jets, at least an additional light jet and one isolated lepton) for two signal benchmark points and four dominant backgrounds, in Fig. \[fig:sbb1\]. The mass-independent cutflow tables for the six benchmark points and six backgrounds are listed in Tables \[tab:sbb1\] and \[tab:sbb2\] respectively. To optimise each signal region, we impose an additional cut, *viz.*, $0.8\, m_S < m_{S}^{\Delta R} < m_S + 10$ GeV. In Table \[tab:sbb3\], we list the final efficiencies for each signal region after this additional cut on top of the aforementioned ones. Finally, we show our results in Fig. \[fig:sbb2\]. The left plot shows the 95 % upper limit on $BR(t \to S c, S \to b \bar{b})$ and the right plot shows the minimum integrated luminosity to test the aforementioned branching ratio to $10^{-4}$ at 95 % CL. $m_S$ \[GeV\] Signal $t\bar{t}$ (SL) $t\bar{t}$ (LL) $Wb\bar{b}$ $Zb\bar{b}$ $t\bar{t}b\bar{b}$ (SL) $t\bar{t}b\bar{b}$ (LL) --------------- -------- ----------------- ----------------- ------------- ------------- ------------------------- ------------------------- 20 8.2 0.12 0.037 0.017 0.0094 4.0 1.5 50 110 1.8 0.35 0.093 0.056 37 17 80 140 3.4 0.60 0.080 0.070 51 24 100 120 3.7 0.59 0.066 0.062 49 24 120 96 3.1 0.47 0.052 0.042 41 19 150 51 1.4 0.23 0.025 0.019 22 11 : \[tab:sbb3\] *Efficiencies ($\times 10^4$) after the final cut, $0.8\, m_S < m_{S}^{\Delta R} < m_S + 10$ GeV, for each signal benchmark point and for the corresponding backgrounds. SL (LL) denotes semi (di)-leptonic decays.* ![\[fig:sbb2\] *Left) Branching ratios that can be tested in the $b\overline{b}$ channel. Superimposed are the theoretical expectations in the three BPs. Right) Luminosity required to test $\mathcal{B}(t\rightarrow S c, S\rightarrow b\overline{b}) = 10^{-4}$. Superimposed are $\mathcal{L} = 300$ fb$^{-1}$ and $3000$ fb$^{-1}$.*](bbChannel.pdf "fig:"){width="0.50\columnwidth"} ![\[fig:sbb2\] *Left) Branching ratios that can be tested in the $b\overline{b}$ channel. Superimposed are the theoretical expectations in the three BPs. Right) Luminosity required to test $\mathcal{B}(t\rightarrow S c, S\rightarrow b\overline{b}) = 10^{-4}$. Superimposed are $\mathcal{L} = 300$ fb$^{-1}$ and $3000$ fb$^{-1}$.*](LbbChannel.pdf "fig:"){width="0.48\columnwidth"} LHC prospects for $t\rightarrow Sc, S\rightarrow \gamma\gamma$ {#sec:gg} ============================================================== We closely follow the previous section in terms of the analysis framework. Here we focus on the scenario in which the scalar decays to a pair of photons, yielding the final state with at least two jets, with one being $b$-tagged, one isolated lepton and two isolated photons. Similar to the leptons, we require the photons to have $p_T > 10$ GeV and require them to lie within a pseudorapidity range of $|\eta| < 2.5$. We demand the photons to be isolated with the exact same criteria as for the leptons as discussed in the section above. The $\Delta R > 0.4$ cuts between pairs of all the selected objects are also used for this study. The dominant backgrounds for this channel are the semi-leptonic and di-leptonic $t\bar{t}h$ processes and the QCD-QED production of $t\bar{t}\gamma\gamma$. The cross section of the former is scaled by a $K$-factor of $1.68$, what takes into account the NLO corrections to both the production and the $h$ decay [@kfactors2]. For the second, we use a conservative $K$-factor of $2$. We also include the $W\gamma\gamma$ background matched up to two hard jets. However, despite having a cross section of order $\mathcal{O}(0.1)$ pb, it becomes irrelevant after imposing all cuts. Consequently, we do not show explicit numbers for this process hereafter. ![\[fig:distrgamma\]*Left) The reconstructed top mass from the hardest two photons and the hardest jet. Right) The transverse mass $m_T$.* ](mTOPgaga.pdf "fig:"){width="0.49\columnwidth"} ![\[fig:distrgamma\]*Left) The reconstructed top mass from the hardest two photons and the hardest jet. Right) The transverse mass $m_T$.* ](mTgaga.pdf "fig:"){width="0.49\columnwidth"} The selection level cuts up until the transverse mass are identical to the previous section. However, because of a much sharper diphoton mass resolution, we demand a very narrow window of 3 GeV around the scalar mass. The shape of the reconstructed top mass distributions as well as $m_T$ in this case are shown in Fig. \[fig:distrgamma\] after the basic cuts (which includes objects selected with the $p_T$ requirement as mentioned above along with the selection criteria of exactly one $b$-tagged jet, at least one additional jet, one isolated lepton and two or more isolated photons). The cutflows are listed in Tables \[tab:sgg1\], \[tab:sgg2\] and \[tab:sgg3\]. The 95% CL upper limit on the branching ratio $\mathcal{B}(t \to S c, S \to \gamma \gamma)$ is shown in Fig. \[fig:sgg2\] along with the minimum integrated luminosity required to probe a branching ratio of $10^{-6}$. In this analysis we have added a new $m_S$ mass point of $125$ GeV, where the $t\overline{t}h$ background is much larger. The dominance of this latter process is apparent in the figure. Cuts 20 GeV 50 GeV 80 GeV 100 GeV 120 GeV 125 GeV 150 GeV ------------------------------------ -------- -------- -------- --------- --------- --------- --------- Basic 0.18 0.18 0.18 0.17 0.16 0.15 0.12 $|\eta_{(b,\ell,j,\gamma)}| < 2.5$ 0.91 0.91 0.91 0.90 0.89 0.88 0.83 $\Delta R(\text{all pairs}) > 0.4$ 0.62 0.91 0.91 0.90 0.88 0.88 0.87 $|m_t^{reco}-m_t| < 50$ GeV 0.81 0.78 0.77 0.74 0.65 0.61 0.33 $m_T < 200$ GeV 0.93 0.92 0.92 0.93 0.93 0.94 0.94 : \[tab:sgg1\] *Efficiency after each cut for the seven signal benchmark points.* Cuts $t\bar{t}\gamma\gamma$ (SL) $t\bar{t}\gamma\gamma$ (LL) $t\bar{t}h$ (SL) $t\bar{t}h$ (LL) ------------------------------------ ----------------------------- ----------------------------- ------------------ ------------------ Basic 0.18 0.12 0.26 0.16 $|\eta_{(b,\ell,j,\gamma)}| < 2.5$ 0.94 0.92 0.94 0.89 $\Delta R(\text{all pairs}) > 0.4$ 0.86 0.72 0.88 0.79 $|m_t^{reco}-m_t| < 50$ GeV 0.37 0.36 0.38 0.37 $m_T < 200$ GeV 0.65 0.50 0.71 0.60 : \[tab:sgg2\] *Efficiency after each cut for the four dominant backgrounds. SL (LL) denotes semi (di)-leptonic decays.* $m_S$ \[GeV\] Signal $t\bar{t}\gamma\gamma$ (SL) $t\bar{t}\gamma\gamma$ (LL) $t\bar{t}h$ (SL) $t\bar{t}h$ (LL) --------------- -------- ----------------------------- ----------------------------- ------------------ ------------------ 20 760 13 5.5 0.15 0.20 50 1100 27 9.9 0.40 0.25 80 1000 19 6.8 0.45 0.35 100 940 13 5.0 0.20 0.25 120 740 6.4 3.5 0.25 0.35 125 660 5.0 2.6 570 240 150 280 2.3 1.1 0.00 0.00 : \[tab:sgg3\] *Efficiencies ($\times 10^4$) after the final cut, $|m_{\gamma\gamma} - m_S| < 3$ GeV, for each signal benchmark point and for the corresponding backgrounds. SL (LL) denotes semi (di)-leptonic decays.* ![\[fig:sgg2\] *Left) Branching ratios that can be tested in the $\gamma\gamma$ channel. Superimposed are the theoretical expectations in the three BPs. Right) Luminosity required to test $\mathcal{B}(t\rightarrow St, S\rightarrow \gamma\gamma) = 10^{-6}$. Superimposed are $\mathcal{L} = 300$ fb$^{-1}$ and $3000$ fb$^{-1}$.*](ggChannel.pdf "fig:"){width="0.50\columnwidth"} ![\[fig:sgg2\] *Left) Branching ratios that can be tested in the $\gamma\gamma$ channel. Superimposed are the theoretical expectations in the three BPs. Right) Luminosity required to test $\mathcal{B}(t\rightarrow St, S\rightarrow \gamma\gamma) = 10^{-6}$. Superimposed are $\mathcal{L} = 300$ fb$^{-1}$ and $3000$ fb$^{-1}$.*](LggChannel.pdf "fig:"){width="0.49\columnwidth"} Conclusions {#sec:conclusions} =========== Flavour-violating top decays into scalar singlets, $S$, mediated by new physics at a scale $f\gtrsim \mathcal{O}(\textrm{TeV})$, dominate strongly over the ones involving the SM-like Higgs boson. From an effective-field theory point of view, the main reason for the dominance of the new scalar is due to the fact that the latter proceeds via effective operators of dimension six and is hence suppressed by $1/f^2$, whereas the former is already present at dimension five (and hence suppressed only by $1/f$). Moreover, the singlet can be much lighter than the Higgs, the corresponding top decay being therefore kinematically enhanced. Since such scalar particles are predicted in several new physics models, we designed novel analyses dedicated for the upcoming runs of the LHC, to search for $t\rightarrow Sc, S\rightarrow b\overline{b}/\gamma\gamma$ in events pertaining to top pair production. We restricted our study of $S$ masses varying between $20$ GeV $<m_S<150$ GeV. In the $S \to b \bar{b}$ channel, the highest reach is obtained for $m_S\sim 80$ GeV, for which we can probe $\mathcal{B}(t\rightarrow Sc, S\rightarrow b\overline{b}) > 10^{-4}$ at $95$ % CL with an integrated luminosity of $\mathcal{L} = 3$ ab$^{-1}$. The reach is about a factor of $5$ smaller for low masses. This is due to the fact that at low masses the two $b$-quarks ensuing from the scalar $S$, do not always form two resolved $b$-jets, and hence upon requiring three $b$-tagged jets, we incur a reduction in the efficiency of the signal. However, one might consider a fat jet in the framework of a boosted analysis to overcome this difficulty. On the other hand, for large masses, the invariant mass of the two $b$-tagged jets closest in $\Delta R$ separation, do not always peak at $m_S$. In the $\gamma\gamma$ channel, the sensitivity of the signal is considerably less dependent on $m_S$, given an excellent resolution of the di-photon mass spectrum. We find that a branching ratio, $\mathcal{B}(t\rightarrow Sc, S\rightarrow \gamma\gamma) > 10^{-7}$ can be tested at the 95 % CL with the same integrated luminosity. We note that, if $\mathcal{B}(S\rightarrow\gamma\gamma)$ is as large as $\sim 1\%$, then we can indirectly probe new physics scales as large as $\sim 50$ TeV. Furthermore, we note that the bound obtained in this channel for $m_S\sim m_h$ agrees well with the results obtained for $t\rightarrow hc$ listed in previous works [@Papaefstathiou:2017xuv] (which utilise significantly different search strategies and statistical approaches). Reference [@Papaefstathiou:2017xuv] also showed that the increase in sensitivity at a 100 TeV collider can be roughly estimated by scaling the signal and background cross-sections and the luminosity. In this particular channel, the dominant background is $t\overline{t}\gamma\gamma$ for masses of the singlet, $m_S$, well separated from $m_h$. It turns out that the increase in cross section in this background at $\sqrt{s} = 27$ TeV ($100$ TeV) with respect to that at $\sqrt{s} = 14$ TeV is similar to that in the signal, and of order $\sim 4$ ($\sim 40$). Thus, assuming an integrated luminosity of $10$ ab$^{-1}$, we expect an increase in significance of order $4/\sqrt{4}\times\sqrt{10/3}\sim 3.7$ ($40/\sqrt{40}\times\sqrt{10/3}\sim 11.5$). This implies that one can expect up to an order of magnitude improvement in the bound on $\mathcal{B}(t\rightarrow Sc, S\rightarrow \gamma \gamma)$. Similar results will hold for the $b\overline{b}$ channel. Acknowledgments {#acknowledgments .unnumbered} ================ We acknowledge Shilpi Jain, Maria Ramos, Jose Santiago and Jakub Scholtz for useful discussions. MC is supported by the Royal Society under the Newton International Fellowship programme. SB is supported by a Durham Junior Research Fellowship COFUNDed between Durham University and the European Union under grant agreement number 609412. [^1]: We note that in the absence of effective operators, the interaction of $S$ with the fermions, is negligible. Indeed, such interactions only arise at one loop (and only if $S$ is not a pseudo-scalar; otherwise there is an accidental symmetry $S\rightarrow -S$, provided $CP$ is conserved). Moreover, these are proportional to the *vev* of $S$, which triggers the mixing with the Higgs boson and are therefore severely constrained by current Higgs measurements [@Banerjee:2015hoa; @Robens:2015gla; @Lopez-Val:2014jva]. In addition, the FCNC currents will be further suppressed by the GIM mechanism. Thus, we would expect $\mathcal{B}(t\rightarrow Sc)$ to be several orders of magnitude smaller than $\mathcal{B}(t\rightarrow hc)$, which in the SM is predicted to be smaller than $10^{-13}$  [@Mele:1998ag]. [^2]: We did not find any significant differences when comparing with another setup using `MG5_aMC@NLO v2.1.1` with the showering done with `Pythia 6`.

--- abstract: 'The distribution of the inner orbital periods of solar-type main-sequence (MS) triple star systems is known to be peaked at a few days, and this has been attributed to tidal evolution combined with eccentricity excitation due to Lidov-Kozai oscillations. Solar-type MS quadruple star systems also show peaks in their inner orbital period distributions at a few days. Here, we investigate the natural question whether tidal evolution combined with secular evolution can explain the observed inner orbital period distributions in quadruple stars. We carry out population synthesis simulations of solar-type MS quadruple star systems in both the 2+2 (two binaries orbiting each other’s barycentre) and 3+1 (triple orbited by a fourth star) configurations. We take into account secular gravitational and tidal evolution, and the effects of passing stars. We assume that no short-period systems are formed initially, and truncate the initial orbital period distributions below 10 d accordingly. We find that, due to secular and tidal evolution, the inner orbital period distributions develop tails at short periods. Although qualitatively consistent with the observations, we find that our simulated orbital period distributions only quantitatively agree with the observations for the 3+1 systems. The observed 2+2 systems, on the other hand, show an enhancement of systems around 10 d, which is not reproduced in the simulations. This suggests that the inner orbital periods of 2+2 systems are not predominantly shaped by tidal and secular evolution, but by other processes, most likely occurring during the stellar formation and early evolution.' author: - | Adrian S. Hamers$^{1}$[^1]\ $^{1}$Institute for Advanced Study, School of Natural Sciences, Einstein Drive, Princeton, NJ 08540, USA bibliography: - 'literature.bib' date: 'Accepted 2018 October 20. Received 2018 October 13; in original form 2018 August 13' title: Shrinking orbits in hierarchical quadruple star systems --- \[firstpage\] (stars:) binaries (including multiple): close – stars: kinematics and dynamics – gravitation Introduction {#sect:introduction} ============ It has been known for over a decade that the orbital period distribution of isolated solar-type main-sequence (MS) binaries is different from their counterparts with tertiary companions. Specifically, the presence of a tertiary star implies a peak in the period distribution at $\sim 3\,{\mathrm{d}}$, which is not present in the distribution of isolated binaries . This has been attributed to shrinkage of the binary orbit due to tides enhanced by high eccentricities, which are driven by secular Lidov-Kozai (LK) oscillations (; see for a review). This process, known as LK cycles with tidal friction, has been studied in detail by numerous authors , and may be responsible for producing a large fraction of short-period binaries that would otherwise not be expected to form due to the larger sizes of the stars during the pre-MS. Triple systems constitute about 10% of all stellar systems with solar-type MS stars in the solar neighbourhood (e.g., @2014AJ....147...86T [@2014AJ....147...87T]). A smaller, but non-negligible fraction of stellar systems, about 1%, is composed of quadruple star systems. The latter are known to occur in the 2+2 (two binaries orbiting each other’s barycentre) and 3+1 (triple orbited by a fourth star) configurations. The long-term dynamics of hierarchical quadruple systems are complicated, and have been considered by a number of authors [@2013MNRAS.435..943P; @2015MNRAS.449.4221H; @2016MNRAS.461.3964V; @2017MNRAS.470.1657H; @2018MNRAS.476.4234F; @2018MNRAS.474.3547G; @2018MNRAS.478..620H]. In particular, it has been shown that the efficiency to attain high eccentricities in these quadruple systems is higher compared to equivalent triples, i.e., if two stars would be replaced by a single star. Similarly to triple stars, the inner orbital period distributions of solar-type MS quadruple stars show an enhancement at a few to several tens of days [@2008MNRAS.389..925T]. Therefore, a natural question is whether tidal friction combined with secular evolution can explain this enhancement, in analogy to triple star systems. Naively, one might expect this process to be very efficient given the higher efficiency to attain high eccentricities in quadruple compared to triple stars. However, one should also take into account that the secular evolution of these systems, in the parameter space in which the enhancement is large, is typically chaotic [@2015MNRAS.449.4221H; @2017MNRAS.470.1657H; @2018MNRAS.474.3547G], and the time-scale for reaching high eccentricities can be long, i.e., longer than the MS lifetimes. The aim of this paper is to study the formation of close binaries in solar-type MS quadruple systems through tidal and secular evolution. We assume that no short-period systems are formed initially, and accordingly truncate the initial orbital period distributions below 10 d. We carry out population synthesis simulations of quadruple stars in both the 2+2 and 3+1 configurations, taking into account secular gravitational and tidal evolution, and the effects of passing stars in the field. The plan of the paper is as follows. In §\[sect:meth\], we describe the numerical algorithm used for our simulations. We illustrate two notable consequences of tidal migration in quadruple systems in §\[sect:examples\]. The initial conditions and assumptions for the population synthesis simulations are discussed in §\[sect:IC\]. We present our results in §\[sect:pop\_syn\] and discuss them in §\[sect:discussion\], and we conclude in §\[sect:conclusions\]. Numerical algorithm {#sect:meth} =================== Symbol Description Initial value(s) and/or distribution in population synthesis ------------------------- ------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- [*Quadruple*]{} [*system*]{} $m_1$ Mass of the primary star in orbit 1. $1\,{\mathrm{M}_\odot}$ $m_2$ Mass of the secondary star in orbit 1. $m_1 q_1$, where $q_1$ has a flat distribution and with$0.1<m_2/{\mathrm{M}_\odot}<1$. $m_3$ Mass of star 3. $q_2 (m_1+m_2)$, where $q_2$ has a flat distribution, and with $0.1<m_3/{\mathrm{M}_\odot}<1$. $m_4$ Mass of star 4. $q_2 m_3$, and with $0.1<m_4/{\mathrm{M}_\odot}<1$. $R_i$ Radius of star $i$. $(m_i/{\mathrm{M}_\odot})^{0.8}\,{\mathrm{R}_\odot}$ [@1994sse..book.....K] $P_{\mathrm{s},i}$ Spin period of star $i$. $10\,{\mathrm{d}}$ [@2007ApJ...669.1298F] $\theta_{\mathrm{s},i}$ Obliquity (spin-orbit angle) of star $i$. $0^\circ$ $t_{\mathrm{V},i}$ Viscous time-scale of star $i$. $5\,{\mathrm{yr}}$ [@2007ApJ...669.1298F] $k_{\mathrm{AM},i}$ Apsidal motion constant of star $i$. 0.014 [@2007ApJ...669.1298F] $r_{\mathrm{g},i}$ Gyration radius of star $i$. 0.08 [@2007ApJ...669.1298F] $P_{\mathrm{orb},i}$ Orbital period of orbit $i$. \(1) Gaussian distribution in $\mathrm{log}_{10}(P_{\mathrm{orb},i}/{\mathrm{yr}})$ with mean 5.03 and standard deviation 2.28 [@2010ApJS..190....1R] or (2) an Öpic law (@opic1924; flat distribution in $\mathrm{log}_{10} a_i$). In both cases, the orbital periods range between $10$ and $10^9$ d, and the systems are subject to dynamical stability constraints [@2001MNRAS.321..398M], and $a_i(1-e_i^2) > a_\mathrm{crit}$ for orbits 1 and 2 (2+2), or orbit 1 (3+1). Here, $a_\mathrm{crit}$ is the semimajor axis corresponding to an orbital period of 10 d. $a_i$ Semimajor axis of orbit $i$. Computed from $P_{\mathrm{orb},i}$ and the $m_i$ using Kepler’s law. $e_i$ Eccentricity of orbit $i$. \(1) Rayleigh distribution between 0.01 and 0.95 with an rms width of 0.33 [@2010ApJS..190....1R]. (2) Flat distribution between 0.01 and 0.95. $i_i$ Inclination of orbit $i$. $0$-$180^\circ$ (flat distribution in $\cos i_i $) $i_{ij}$ Inclination of orbit $i$ relative to orbit $j$. $0$-$180^\circ$ (flat distribution in $\cos i_{ij} $) $\omega_i$ Argument of periapsis of orbit $i$. $-180$-$180^\circ$ (flat distribution in $\omega_i$) $\Omega_i$ Longitude of the ascending node of orbit $i$. $-180$-$180^\circ$ (flat distribution in $\Omega_i$) [*Flybys*]{} $M_\mathrm{per}$ Mass of the perturbers. $0.1$-$80\,{\mathrm{M}_\odot}$ with a Kroupa initial mass function [@1993MNRAS.262..545K], corrected for gravitational focusing and a stellar age of 10 Gyr. $Z_i$ Metallicity of perturbers. $0.02$ $n_\star$ Stellar number density. $0.1 \, \mathrm{pc^{-3}}$ [@2000MNRAS.313..209H] ${R_\mathrm{enc}}$ Encounter sphere radius. $0, 10^4\,{\,\textsc{au}}$ $\sigma_\star$ One-dimensional stellar velocity dispersion. $30\,\mathrm{km\,s^{-1}}$ We model the long-term evolution of stellar hierarchical quadruple systems in both the ‘2+2’ (two binaries orbiting each other’s barycentre) and ‘3+1’ (a triple orbited by a fourth star) configurations. An illustration of the configurations is given in [Fig.]{}\[fig:configurations\]. We give an overview of our notation in Table \[table:IC\], in which we also summarize in the third column the assumptions made in the population synthesis study (§s\[sect:IC\] and \[sect:pop\_syn\]). Throughout, we will refer to the ‘inner’ orbits as orbit 1 or 2 for the 2+2 configuration, and orbit 1 for the 3+1 configuration. The ‘outermost’ or ‘outer’ orbit is orbit 3 in both configurations, and the ‘intermediate’ orbit is orbit 2 in the 3+1 configuration. Our numerical algorithm is similar to that of @2018MNRAS.478..620H, with the exception that we do not use a dedicated code to model the stellar evolution. Since our focus is on solar-type MS stars, the stellar parameters (mass and radius) do not change significantly during 10 Gyr, so it is reasonable to assume these parameters to be constant. Below we give a brief description of the numerical algorithm. For more details, we refer to @2018MNRAS.478..620H and references therein. Secular dynamical evolution {#sect:meth:sec} --------------------------- To model the secular dynamics, we use <span style="font-variant:small-caps;">[SecularMultiple]{}</span> [@2016MNRAS.459.2827H], which is a generalization of a code developed earlier for 3+1 quadruple systems [@2015MNRAS.449.4221H]. The <span style="font-variant:small-caps;">[SecularMultiple]{}</span> code is based on an expansion of the Hamiltonian of the system in terms of ratios of separations of orbits on different levels (e.g., in [Fig.]{}\[fig:configurations\], the small expansion parameters are $r_1/r_3$ and $r_2/r_3$ for the 2+2 configuration, and $r_1/r_2$ and $r_2/r_3$ for the 3+1 configuration, respectively, which $r_i$ is the separation of orbit $i$). The Hamiltonian is subsequently orbit averaged, and the orbit-averaged equations of motion are solved numerically. In the integrations, we include terms up to and including octupole order (third order in the separation ratios) for interactions involving three binaries, and up to and including dotriacontupole order (fifth order in the separation ratios) for pairwise interactions. Post-Newtonian (PN) corrections are included in each orbit to the 1 and 2.5PN orders (i.e., including relativistic precession, and energy and angular-momentum loss due to gravitational wave radiation). Any ‘cross’ terms, i.e., PN terms involving more than one orbit simultaneously [@2013ApJ...773..187N], are neglected. We note that although they are included for completeness, the 2.5PN terms are not important for solar-type MS stars. The 1PN terms can be important by quenching secular evolution due to apsidal precession in the inner orbits. Tidal evolution {#sect:meth:tides} --------------- Tidal evolution is modelled with the equilibrium tide model . Specifically, we use equations (81) and (82) of @1998ApJ...499..853E, with the non-dissipative terms $X$, $Y$ and $Z$ given explicitly by equation (10)-(12) of @2001ApJ...562.1012E, and the dissipative terms $V$ given explicitly by equations (A7)-(A11) of @2009MNRAS.395.2268B. In these equations, we include the effects of dissipative tides, orbital precession due to tidal bulges, and orbital precession due to stellar rotation (assuming uniform rotation, but allowing the spins to be non-parallel to the orbit). The spin vectors of all stars are tracked and the spins are not confined to be parallel with the orbit, although we initialize the spins to be parallel with the orbit (i.e., the initial obliquity $\theta_{\mathrm{s},i}=0^\circ$), with a spin period of $P_{\mathrm{s},i} = 10\,{\mathrm{d}}$. We assume a constant viscous time-scale $t_{\mathrm{V},i}$ for the stars, which we set to $t_{\mathrm{V},i}=5\,{\mathrm{yr}}$. The apsidal motion constants are set to $k_{\mathrm{AM},i}=0.014$, and the gyration radii are set to $r_{\mathrm{g},i}=0.08$ (i.e., the moment of inertia is $I_i = r_{\mathrm{g},i} m_i R_i^2$). The values for $P_{\mathrm{s},i}$, $t_{\mathrm{V},i}$, $k_{\mathrm{AM},i}$, and $r_{\mathrm{g},i}$ are adopted from @2007ApJ...669.1298F. We remark that many uncertainties exist regarding the efficiency of tidal dissipation (see, e.g., for a review). In particular, we do not take into account dynamical tides, which could be important at high eccentricities, when the orbits are close to parabolic (@1977ApJ...213..183P; the transition eccentricity is around 0.8, @1995ApJ...450..732M). A more sophisticated tidal model, e.g., in which the equilibrium tide is used for low eccentricities and the formalism of @1977ApJ...213..183P for high eccentricities such as in @2018ApJ...854...44M, is beyond the scope of this paper. Flybys {#sect:meth:flybys} ------ We include the gravitational effects of stars passing by the quadruple system using the impulsive approximation, i.e., the stars in the quadruple system can be considered to be fixed in space whereas the perturber imparts velocity kicks on each of the components. These kicks lead to changes of the orbits, in principle affecting all orbital elements, but most significantly the semimajor axes and eccentricities of wide orbits ($a_i\gtrsim 10^4\,{\,\textsc{au}}$). Our method for computing the effects of impulsive flybys on the orbits of the system is the same as in @2017AJ....154..272H and @2018MNRAS.478..620H; for details, we refer to the latter papers. We adopt the same parameters as in @2018MNRAS.478..620H, i.e., we assume a locally homogeneous stellar background with a number density $n_\star = 0.1 \, \mathrm{pc^{-3}}$ [@2000MNRAS.313..209H], a one-dimensional velocity dispersion $\sigma_\star = 30\,\mathrm{km\,s^{-1}}$, and a Kroupa mass function [@1993MNRAS.262..545K] between 0.1 and 80 ${\mathrm{M}_\odot}$, corrected for gravitational focusing and stellar evolution. The correction for stellar evolution is carried out by replacing the initial mass with the final mass after 10 Gyr of stellar evolution assuming solar metallicity and using the <span style="font-variant:small-caps;">SeBa</span> stellar evolution code in <span style="font-variant:small-caps;">AMUSE</span> . The radius of the sphere used to compute the encounter rate and encounter properties, ${R_\mathrm{enc}}$, is set to either ${R_\mathrm{enc}}=0$ (no encounters), or ${R_\mathrm{enc}}= 10^4\,{\,\textsc{au}}$, such that most encounters with the widest orbit (orbit 3) are impulsive, whereas not too large as to be computationally too inhibitive. We reject sampled encounters that are secular in nature (i.e., the speed of the perturber is much slower than the orbital speed in the quadruple system), since the effects of secular encounters are typically negligible compared to those of impulsive encounters (Hamers, unpublished). We neglect the effects of encounters in the intermediate regime between secular and impulsive encounters, although some encounters in this regime could be important [@2010ApJ...725..353D]. Stopping conditions {#sect:meth:sc} ------------------- We impose a number of stopping conditions in our simulations. When high eccentricities are reached in the inner orbits, Roche lobe overflow (RLOF) can be triggered, leading to mass transfer, and possibly to the coalescence of two stars. If the mass transfer and secular time-scales are comparable, then the resulting evolution can be complicated, since the two effects are competing (RLOF tends to circularize the orbits, and secular evolution tends to increase the eccentricity). This complicated process is beyond the scope of the paper. Instead, we simply check for the onset of RLOF in eccentric orbits, and stop the simulation. Furthermore, we stop the simulation when the system becomes dynamically unstable. This is most likely to occur in 3+1 systems, when orbit 2 is driven to high eccentricity due to the torque of orbit 3, driving dynamical instability of the orbit pair 1-2. If a dynamical instability is triggered, then stars may collide or could become ejected from the system, destroying the hierarchy of the system. The implemented stopping conditions are as follows. 1. One of the stars fills its Roche lobe, either after tidal evolution in a circular orbit, or in an eccentric orbit. We use the fits of @2007ApJ...660.1624S, in particular, equations (47) through (52) evaluated at periapsis, to determine the instantaneous Roche lobe radius. The latter equations give the Roche lobe radius as a function of orbital phase, spin frequency, eccentricity and mass ratio, as a correction to the Roche lobe radius fits of @1983ApJ...268..368E evaluated at periapsis (i.e., $a$ in @1983ApJ...268..368E is replaced by $a[1-e]$). We note that, for MS stars, RLOF always occurs prior to a physical collision. 2. One of the orbit pairs becomes dynamically unstable. To evaluate stability, we use the criterion of @2001MNRAS.321..398M, which is assumed to be correct for quadruple systems if two of the bodies are appropriately replaced by a single body. In the case of 2+2 systems, we apply the stability criterion to the 1-3 and 2-3 orbit pairs; in the case of 3+1 systems, we apply the stability criterion to the 1-2 and 2-3 orbit pairs (see also §\[sect:IC:orbits\]). We remark that a breakdown of the secular equations of motion could already occur before dynamical stability. This caveat is discussed in §\[sect:discussion:sub\]. 3. The age of the system exceeds 10 Gyr. 4. The CPU wall time exceeds 24 hr. Some systems, in particular for the 3+1 configuration, take excessively long to integrate due to very short secular time-scales compared to 10 Gyr. Although we terminate the integration of these systems for practical reasons, we show below in §\[sect:discussion:exceed\] that the majority of these systems are not expected to lead to tidal migration, and the stopping condition therefore does not strongly affect our results. We note that conditions (i) and (ii), which depend on the instantaneous orbital semimajor axes and eccentricities, are implemented as root finding conditions within the set of ordinary differential equations (which are solved using the <span style="font-variant:small-caps;">CVODE</span> routine, @1996ComPh..10..138C, which supports root finding). Therefore, there is no risk of the stopping conditions being missed in the simulations due to a finite number of output snapshots. Examples of tidal migration {#sect:examples} =========================== Here, we discuss two notable examples of tidal migration as found in the population synthesis simulations (§\[sect:pop\_syn\]). The initial conditions of the two examples are listed in Table\[table:IC\_example\]. For other parameters such as the viscous time-scale, we refer to Table\[table:IC\]. ---------------------- ---------------------- ---------------------- ---------------------- ------------------- ------------------- ------------------- ------- ------- ------- ------- ------- ------- ------------ ------------ ------------ ------------ ------------ ------------ $m_1$ $m_2$ $m_3$ $m_4$ $a_1$ $a_2$ $a_3$ $e_1$ $e_2$ $e_3$ $i_1$ $i_2$ $i_3$ $\omega_1$ $\omega_2$ $\omega_3$ $\Omega_1$ $\Omega_2$ $\Omega_3$ ${\mathrm{M}_\odot}$ ${\mathrm{M}_\odot}$ ${\mathrm{M}_\odot}$ ${\mathrm{M}_\odot}$ ${\,\textsc{au}}$ ${\,\textsc{au}}$ ${\,\textsc{au}}$ deg deg deg deg deg deg deg deg deg 1.0 0.7 0.4 0.1 3.6 0.1 21.3 0.08 0.23 0.32 13.8 55.5 102.0 59.2 -77.9 -94.6 169.6 6.5 -128.6 1.0 0.2 0.9 0.7 0.7 82.8 5949.4 0.50 0.42 0.24 129.6 67.7 115.5 82.0 149.5 -106.8 40.5 85.9 153.7 ---------------------- ---------------------- ---------------------- ---------------------- ------------------- ------------------- ------------------- ------- ------- ------- ------- ------- ------- ------------ ------------ ------------ ------------ ------------ ------------ Double migration in a 2+2 system {#sect:examples:1} -------------------------------- First, we show in [Fig.]{}\[fig:example1\] an example for the 2+2 configuration in which ‘double migration’ occurs, i.e., both orbits 1 and 2 shrink to $P_{\mathrm{orb,i}}<10\,{\mathrm{d}}$. In the figure, we plot, as a function of time in the top-left panel, the semimajor axes (dashed lines), periapsis distances $a_i(1-e_i)$ (solid lines), and stellar radii (solid, dashed, dotted, and dot-dashed lines for stars 1 through 4). The top-right panel shows the eccentricities, the bottom-left panel shows the ratio of LK time-scales, i.e., [@2017MNRAS.470.1657H] $$\begin{aligned} \label{eq:R_2p2} \mathcal{R}_{2+2} &\equiv \frac{t_\mathrm{LK,13}}{t_\mathrm{LK,23}} \simeq \left ( \frac{a_2}{a_1} \right)^{3/2} \left ( \frac{m_1+m_2}{m_3+m_4} \right )^{3/2},\end{aligned}$$ and the bottom-right panel shows the mutual inclinations of orbit pairs 1-3 and 2-3. We note that, when $\mathcal{R}_{2+2}$ is close to unity (roughly speaking, within an order of magnitude), secularly chaotic behaviour and particularly high eccentricities are to be expected (@2017MNRAS.470.1657H; similar behaviour applies to the 3+1 configuration in terms of $\mathcal{R}_{3+1}$, see the second example below in §\[sect:examples:2\]). Initially, orbit 1 is highly inclined with respect to orbit 3, with $i_{13,\mathrm{i}} \simeq 95.2^\circ$. This triggers high-amplitude eccentricity oscillations in orbit 1, which cause its semimajor axis to gradually decrease. Consequently, the ratio $\mathcal{R}_{2+2}$ gradually increases. Initially, orbit 2 is inclined with respect to orbit 3 by $i_{23,\mathrm{i}} \simeq 133.4^\circ$, and the resulting maximum eccentricities in orbit 2 are not sufficiently high to trigger tidal migration. However, at $\simeq 1400\,\mathrm{Myr}$, due to the tidal migration of orbit 1, $\mathcal{R}_{2+2}$ reaches $\sim 0.1$, and the two orbits (1 and 2) become dynamically coupled due to secular resonances. Consequently, the eccentricity is excited in orbit 2, causing tidal migration in that orbit, and which initially decreases $\mathcal{R}_{2+2}$. As orbit 1 continues to shrink, $\mathcal{R}_{2+2}$ increases again. Finally, the orbits circularize due to tides with $a_{1,\mathrm{f}} \simeq 4\times 10^{-2}\,{\,\textsc{au}}$, and $a_{2,\mathrm{f}} \simeq 1\times 10^{-2}\,{\,\textsc{au}}$. This example shows that tidal shrinkage of one orbit can trigger the second orbit to shrink as well. Due to the gradual shrinkage of the first orbit, secular resonances are almost guaranteed to occur. Note, however, that the occurrence of this phenomenon does require $\mathcal{R}_{2+2}$ to pass within roughly an order of magnitude during tidal migration. Switching of dynamical regimes due to migration in a 3+1 system {#sect:examples:2} --------------------------------------------------------------- In the second example, we show how tidal migration in the inner orbit of a 3+1 system can bring the system into a different dynamical regime. We show the evolution in [Fig.]{}\[fig:example2\], which is similar to [Fig.]{}\[fig:example1\], except that the bottom-left panel now shows the ratio of LK time-scales [@2017MNRAS.470.1657H] $$\begin{aligned} \label{eq:R_3p1} \mathcal{R}_{3+1} &\equiv \frac{t_\mathrm{LK,12}}{t_\mathrm{LK,23}} \simeq \left ( \frac{a_2^3}{a_1 a_3^2} \right)^{3/2} \left ( \frac{m_1+m_2}{m_1+m_2 + m_3} \right )^{1/2} \frac{m_4}{m_3} \left ( \frac{1-e_2^2}{1-e_3^2} \right)^{3/2}.\end{aligned}$$ Orbits 1 and 2 are initially mildly mutually inclined, i.e., $i_{12,\mathrm{i}} \simeq 75.0^\circ$. Nevertheless, since orbit 1 is relatively tight ($a_{1,\mathrm{i}}\simeq0.7\,{\,\textsc{au}}$), the maximum eccentricity reached during the secular oscillations is high enough to gradually shrink it, thereby gradually increasing $\mathcal{R}_{3+1}$, which was initially small ($\mathcal{R}_{3+1}\simeq 2\times10^{-3}$). After $\simeq 4500\,\mathrm{Myr}$, orbit 1 shrinks significantly, increasing $\mathcal{R}_{3+1}$ to $\mathcal{R}_{3+1}\sim 10^{-1}$. At the same time, the eccentricity oscillations in orbit 2, which initially had small amplitude, increase significantly, with $e_2$ reaching $e_2\simeq 0.97$. The enhanced eccentricity of orbit 2 has some effect on orbit 1; around $6000\,\mathrm{Myr}$, orbit 1 is slightly excited in eccentricity, but due to tidal evolution it immediately circularizes. This process, which causes several ‘bumps’ in $a_1$ around $6000\,\mathrm{Myr}$, continues several times until $a_1$ becomes small enough to become completely decoupled from orbit 2. The enhanced eccentricity of orbit 2 after tidal migration of orbit 1 can be understood by noting that the eccentricity oscillations in orbit 2 due to the secular torque of orbit 3 are initially (partially) quenched due to apsidal precession imposed by orbit 1. This phenomenon is the same as described by @2015MNRAS.449.4221H. Orbits 2 and 3 are initially inclined by $i_{23,\mathrm{i}} \simeq 81.2^\circ$; therefore, eccentricity oscillations with amplitude $\sim \sqrt{1-5/3 \cos^2 (i_{23,\mathrm{i}})}\simeq 0.98$ are to be expected. However, due to apsidal precession induced by orbit 1, the actual maximum eccentricity reached is $\simeq 0.46$. This is reproduced with the dark blue dotted line in [Fig.]{}\[fig:example2\], which shows a semianalytic calculation of the maximum eccentricity of orbit 2 using the method based on the secular Hamiltonian described in section 3.4.2 of @2015MNRAS.449.4221H ([-@2015MNRAS.449.4221H]; the approach is similar to the analytic method to compute the maximum eccentricity in the presence of additional sources of apsidal motion in triple systems, see, e.g., @2002ApJ...578..775B [@2003ApJ...598..419W; @2011ApJ...741...82T; @2015MNRAS.447..747L; @2018MNRAS.481.4907G]). After orbit 1 shrinks, the apsidal precession rate imposed by orbit 1 on orbit 2 decreases, thereby increasing the maximum eccentricity in orbit 2 to $\simeq 0.97$, close to the expected value in the equivalent three-body case. Before and after tidal migration, the semianalytic calculation (dark blue dotted line) is consistent with the numerical solutions to the equations of motion. In this example, the enhanced eccentricity of orbit 2 in response to the shrinking of the innermost orbit does not dramatically affect the resulting evolution (although, as noted above, the final semimajor axis of the innermost orbit is somewhat decreased). There are, however, other cases (not shown explicitly here) in which the eccentricity of orbit 2 can become large enough after tidal migration of the innermost orbit to trigger dynamical instability of the system. This shows that dynamical instability in 3+1 systems can be triggered not only by mass loss in evolving systems (e.g., @2018MNRAS.478..620H), but also due to tidal evolution during the MS. Population synthesis set-up {#sect:IC} =========================== In this section, we describe the methodology used to generate the systems for the population synthesis simulations. A summary is given in the third column of Table\[table:IC\]. We sample $N_\mathrm{MC}=10^3$ systems for the 2+2 and 3+1 systems, both with and without the effects of flybys (§\[sect:meth:flybys\]), and with two different assumptions about the orbital distributions. Masses {#sect:IC:masses} ------ Our focus is on systems with solar-type MS stars. We set the mass of the primary (most massive) star to $m_1 = 1\,{\mathrm{M}_\odot}$. The MS time-scale for this mass (and assuming solar metallicity) is $\simeq 10\,\mathrm{Gyr}$, which is also the integration time in our simulations. The secondary mass, $m_2$ ($m_2<m_1$), is sampled from a flat mass ratio distribution, $q_1 = m_2/m_1$ . From $m_1$ and $m_2$, we sample $m_3$ according to $m_3=(m_1+m_2)q_2$, where $q_2$ has a flat distribution. Lastly, we sample $m_4$ according to $m_4=q_2 m_3$. We reject any sampled combination of masses $m_2$, $m_3$, and $m_4$ if the masses do not satisfy the restrictions $0.1<m_i/{\mathrm{M}_\odot}<1$ for $i=2,3,4$. We note that this approach implies that the masses of stars 3 and 4 are correlated with those of stars 1 and 2. Orbits {#sect:IC:orbits} ------ We adopt two different assumptions about the orbital distributions, in order to establish the sensitivity of our results on the underlying assumed distributions. In case (1), we draw three orbital periods from a Gaussian distribution in $\mathrm{log}_{10}(P_{\mathrm{orb},i}/{\mathrm{yr}})$, with a mean of 5.03 and a standard deviation of 2.28, and $1<\mathrm{log}_{10}(P_{\mathrm{orb},i}/{\mathrm{yr}})<9$ [@2010ApJS..190....1R]. The lower limit is approximately the period for which the stars are expected to merge during the pre-MS. The upper limit of $10^{9}\,{\mathrm{yr}}$ is 10 times lower than the usual limit of $10^{10}\,{\mathrm{yr}}$ in order to better match observed quadruple systems (see §\[sect:IC:comp\] below). The corresponding semimajor axes are computed according to the configuration using Kepler’s law. In addition, three eccentricities are drawn from a Rayleigh distribution between 0.01 and 0.95 with an rms width of 0.33 [@2010ApJS..190....1R]. In case (2), we adopt Öpic’s law [@opic1924], i.e., flat distributions in $\log_{10}(a_i)$, subject to the same orbital period ranges, i.e,. $1<\mathrm{log}_{10}(P_{\mathrm{orb},i}/{\mathrm{yr}})<9$. The eccentricity distributions are assumed to be flat in this case, again with $0.01<e_i<0.95$. In both cases, we impose stability criteria to ensure that the systems are dynamically stable using the criterion of @2001MNRAS.321..398M ([-@2001MNRAS.321..398M]; implicitly assuming that this criterion also applies to quadruple systems). Note that @2018MNRAS.474...20H recently reinvestigated the criterion of @2001MNRAS.321..398M, and found that it predicts stability against ejections reasonably well for a wide range of parameters. We also impose conditions to ensure that the inner orbits would not evolve due to tides in the absence of secular evolution (i.e., as isolated binaries). Specifically, let the criterion of @2001MNRAS.321..398M, as applied to a triple system with the inner and outer orbits indicated with ‘in’ and ‘out’, respectively, be denoted with $a_\mathrm{out}/a_\mathrm{in} > f_\mathrm{MA01}(m_\mathrm{in},m_\mathrm{out},e_\mathrm{out})$, where the latter function is given by $$\begin{aligned} f_\mathrm{MA01}(m_\mathrm{in},m_\mathrm{out},e_\mathrm{out}) \equiv \frac{2.8}{1-e_\mathrm{out}} \left [ \left (1+\frac{m_\mathrm{out}}{m_\mathrm{in}} \right ) \frac{1+e_\mathrm{out}}{\sqrt{1-e_\mathrm{out}}} \right ]^{2/5}.\end{aligned}$$ For 2+2 systems, we impose $$\begin{aligned} a_2/a_1 &> f_\mathrm{MA01}(m_1+m_2,m_3+m_4,e_3); \\ a_3/a_2 &> f_\mathrm{MA01}(m_3+m_4,m_1+m_2,e_3); \\ a_1\left(1-e_1^2\right) &> a_\mathrm{crit}; \\ a_2\left(1-e_2^2\right) &> a_\mathrm{crit},\end{aligned}$$ whereas for 3+1 systems, we require $$\begin{aligned} a_2/a_1 &> f_\mathrm{MA01}(m_1+m_2,m_3,e_2); \\ a_3/a_2 &> f_\mathrm{MA01}(m_1+m_2+m_3,m_4,e_3); \\ a_1\left(1-e_1^2\right) &> a_\mathrm{crit}.\end{aligned}$$ Here, $a_\mathrm{crit}$ is the semimajor axis corresponding to the smallest allowed orbital period of 10 d. The restrictions on the semilatus recti, $a_i\left(1-e_i^2\right)$, ensure that the inner orbits do not shrink to an orbital period less than 10 d due to tides. This implies that any shrinkage of the inner orbits to less than 10 d observed in the simulations can be fully ascribed to secular evolution (i.e., tides triggered by enhanced eccentricity due to secular evolution). The initial orbital orientations for all configurations are assumed to be random, i.e., for each orbit $i$, flat distributions are assumed for $\cos(i_i)$, $\omega_i$ and $\Omega_i$, where $i_i$, $\omega_i$ and $\Omega_i$ are the inclination, argument of periapsis, and longitude of the ascending node, respectively, of orbit $i$ (the orbital elements are defined with respect to an arbitrary fixed frame). Comparison to the Multiple Star Catalogue {#sect:IC:comp} ----------------------------------------- The statistics of orbital properties of quadruple star systems are much less constrained than those of isolated binary (and triple) stars. Nevertheless, here we briefly compare the distributions of sampled systems to observations. We take data from the Multiple Star Catalogue (MSC; ), selecting systems with primary masses $0.5<m_1/{\mathrm{M}_\odot}<1.5$ and all orbits known, giving 87 (66) systems for the 2+2 (3+1) configuration. We compare our sampling methodology to the MSC in terms of the orbital period distributions (Figs.\[fig:IC\_comp\_per1\] and \[fig:IC\_comp\_per2\]), and the mass distributions ([Fig.]{}\[fig:IC\_comp\_m\]). In these figures, the top (bottom) panel corresponds to the 2+2 (3+1) configuration. The first orbital sampling method (lognormal orbital period and Rayleigh eccentricity distributions; see [Fig.]{}\[fig:IC\_comp\_per1\]) agrees reasonably with the MSC for periods $\gtrsim 10^3\,{\mathrm{d}}$. There is a clear excess of systems with inner periods between $\sim 1$ and $\sim 10^2\,{\mathrm{d}}$, as noted before by @2008MNRAS.389..925T. Evidently, our aim in §\[sect:pop\_syn\] is to establish whether tidal migration coupled with secular evolution can reproduce such an excess of short-period systems. The second sampling method (flat distributions in $\log_{10} a_i$ and the eccentricities; see [Fig.]{}\[fig:IC\_comp\_per2\]) compares less favourably to the MSC at periods $\gtrsim 10^3\,{\mathrm{d}}$, especially for the outermost orbits. The median $m_1$ in the extracted sample of the MSC agrees with the assumed $m_1=1\,{\mathrm{M}_\odot}$ (see [Fig.]{}\[fig:IC\_comp\_m\]). The other observed masses, $m_2$, $m_3$ and $m_4$, tend to be somewhat larger than the sampled masses. However, we do not anticipate a very strong dependence of our results on the masses. Population synthesis results {#sect:pop_syn} ============================ Here, we present the results from the population synthesis simulations. The initial conditions were discussed in §\[sect:IC\]. We first focus on the occurrences of tidal migration and other outcomes (§\[sect:pop\_syn:frac\]). We then consider the inner orbital period distributions mediated by tidal and secular evolution (§\[sect:pop\_syn:in\]). Other orbital properties are considered in §\[sect:pop\_syn:orb\], and in §\[sect:pop\_syn:time\] we discuss the migration times. Outcome fractions {#sect:pop_syn:frac} ----------------- ----------------------------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- NF F NF F NF F NF F NF F NF F NF F NF F $\mathrm{No\,Mig.}$ 0.790 0.772 0.842 0.805 0.763 0.727 0.822 0.764 0.507 0.576 0.577 0.590 0.420 0.535 0.521 0.528 $\mathrm{Mig. \,O1}$ 0.015 0.019 0.034 0.041 0.019 0.025 0.034 0.041 0.093 0.084 0.131 0.122 0.104 0.086 0.144 0.131 $\mathrm{Mig. \,O2}$ 0.026 0.032 0.025 0.024 0.028 0.038 0.030 0.027 — — — — — — — — $\mathrm{Mig. \,O1+O2}$ 0.002 0.001 0.004 0.004 0.002 0.003 0.004 0.006 — — — — — — — — $\mathrm{RLOF\,O1}$ 0.055 0.063 0.033 0.043 0.059 0.073 0.038 0.056 0.141 0.142 0.117 0.119 0.152 0.151 0.127 0.131 $\mathrm{RLOF\,O2}$ 0.106 0.098 0.053 0.053 0.115 0.109 0.057 0.061 — — — — — — — — $\mathrm{Dyn.\, Inst.\,O1}$ 0.001 0.006 0.003 0.018 0.001 0.012 0.004 0.022 0.127 0.138 0.100 0.105 0.138 0.142 0.102 0.118 $\mathrm{Dyn.\, Inst.\,O2}$ 0.002 0.007 0.002 0.011 0.002 0.010 0.002 0.021 0.027 0.027 0.018 0.039 0.029 0.040 0.019 0.056 $\mathrm{Unbound\,Flyby}$ — 0.001 — 0.001 — 0.001 — 0.001 — 0.000 — 0.000 — 0.000 — 0.000 $\mathrm{Time\,exceeded}$ 0.003 0.001 0.004 0.000 0.011 0.002 0.009 0.001 0.105 0.033 0.057 0.025 0.157 0.046 0.087 0.036 ----------------------------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- In Table\[table:fractions\], we show the fractions of various outcomes in the simulations. We distinguish between the following outcomes, which are partially based on the stopping conditions (§\[sect:meth:sc\]). - Migration of an inner orbit, i.e., $P_{\mathrm{orb},i}<10\,{\mathrm{d}}$, where $i$ can be 1 or 2 for the 2+2 configuration, or 1 for the 3+1 configuration. - RLOF in orbit 1 or 2 (2+2 configuration), or orbit 1 only (3+1 configuration). - Dynamical instability in orbit 1 or 2 with respect to their parent according to the criterion of @2001MNRAS.321..398M. Note that, in the 2+2 configuration, the parent to orbits 1 and 2 is orbit 3, whereas for the 3+1 configuration the parent of orbit 1 is orbit 2, and the parent of orbit 2 is orbit 3. - One or more of the orbits in the quadruple system (typically, the outer orbit) become unbound due to an impulsive encounter. This is a rare occurrence, and evidently only applies if ${R_\mathrm{enc}}>0$. - The integration time exceeds the set CPU wall time of 24 hr (§\[sect:meth:sc\]). These systems potentially present an uncertainty in the results. However, we show below in §\[sect:discussion:exceed\] that this likely does not affect the final orbital period distributions in the simulations. - In all other cases, we mark the outcome as ‘no migration’. For each outcome, we show the corresponding fraction, based on the $N_\mathrm{MC}=10^3$ simulations, for the 2+2 and 3+1 configurations, and various sets of simulations: with the first (‘Orb. Distr. (1)’) and second (‘Orb. Distr. (2)’) assumptions about the orbital distributions (see §\[sect:IC:orbits\]), and with (‘F’) and without (‘NF’) the inclusion of flybys. Also, we show fractions after 10 Gyr of evolution (if applicable; otherwise, the end time is the stopping condition time), or after a random time between 0 and 10 Gyr ($t_x$), appropriate for continuous star formation. The most likely outcome in all cases is ‘no migration’. Note that the eccentricities may still have been excited (see §\[sect:pop\_syn:orb:e\] below). In a few per cent for the 2+2 systems, and in up to $\sim 14\%$ for the 3+1 systems, tidal migration occurs in orbit 1 or 2. In the 2+2 configuration, double migration, of which we showed an example in §\[sect:examples:1\], occurs in a few tenths of per cents, i.e., is relatively rare. RLOF is triggered in up to $\sim 11\%$ for the 2+2 systems, and $\sim 15\%$ for the 3+1 systems. In the 2+2 systems, dynamical instability rarely occurs, which can be understood by noting that dynamical instability can only be triggered in our simulations by flybys, or, more rarely, if the outer orbit eccentricity, $e_3$, is enhanced, which can only occur due to high-order terms (higher than quadrupole-order), and therefore only in compact systems. For the 3+1 systems, dynamical instability is relatively common for the orbit 1-2 pair (up to $\sim 14\%$). This can be attributed to enhanced eccentricity of orbit 2 if it is inclined relative to orbit 3. As noted above, the unbinding of the system due to flybys rarely occurs. The run time is exceeded mostly for 3+1 systems, and this is further addressed in §\[sect:discussion:exceed\]. There are typically no major differences between the results for the two assumptions about the orbital distributions (Orb. Distr. (1)’ and ‘Orb. Distr. (2)’). Typically, the second assumed orbital distributions (flat distributions in $\log_{10}a_i$ and $e_i$) tend to lead to fewer stronger interactions (the ‘no migration’ fractions are larger, and all other fractions are lower). This can be attributed to the generally wider orbits compared to the first assumed orbital distributions (see §\[sect:IC:orbits\]). As can be natively expected, flybys tend to increase the occurrence of strong interactions (decreasing the no-migration fractions, and increasing the others). Inner orbital period distributions {#sect:pop_syn:in} ---------------------------------- In [Fig.]{}\[fig:pop\_syn\_p\_ins\], we show histograms of the inner orbital period distributions in the simulations. The data shown correspond to the ‘no migration’ and ‘migration’ outcomes of Table\[table:fractions\]. For the 2+2 configuration, we make a distinction between orbits 1 (thicker black lines), and 2 (thinner red lines). Initial (final) distributions are shown with dashed (solid) lines. The four sets of panels (each set containing two panels for the two configurations) correspond to whether or not flybys were included, and which orbital distribution was assumed, indicated above the top panel. We first note that the inner orbital periods for the 3+1 configuration tend to be shorter compared to the 2+2 configuration, although the underlying distributions for all orbits were assumed to be the same. This can be ascribed to the requirement of dynamical stability and the hierarchy of the system: for 2+2 systems, only two tiers (or levels) of orbits need to fit within $\sim 10$ decades of orbital period, whereas for the 3+1 configuration, three tiers need to fit within the same orbital period range. This was illustrated in Figs.\[fig:IC\_comp\_per1\] and \[fig:IC\_comp\_per2\], which also show that this property is consistent with observed quadruple systems. For both configurations, the initial orbital period distributions were cut off at 10 d; evidently, due to tidal and secular evolution, the distributions develop tails at periods below 10 d, similar to previous Monte Carlo studies of tidal migration in triple systems [@2007ApJ...669.1298F; @2009ApJ...697.1048P; @2014ApJ...793..137N]. These tails are more prominent for the 3+1 systems compared to the 2+2 systems, which is also reflected by the higher migration fractions for the 3+1 configuration in Table\[table:fractions\]. This can be attributed to two effects: (1) typically tighter initial inner orbits for the 3+1 systems, due to the reasons given above; (2) typically stronger secular evolution (i.e., higher eccentricities are reached and therefore tidal migration is more efficient) for the innermost orbit in the 3+1 system, due to the possibility of enhanced outer orbit eccentricity (orbit 2 for the 3+1 configuration). The enhanced secular evolution in the 3+1 systems is also corroborated by the significantly higher RLOF fractions for these systems (see Table\[table:fractions\]). We note that there are no major qualitative differences in the final orbital period distributions when different assumptions are made about the orbital distributions, and whether or not flybys are included. In [Fig.]{}\[fig:pop\_syn\_p\_ins\_obs\], we show the same data as shown in [Fig.]{}\[fig:pop\_syn\_p\_ins\], but now comparing, in terms of the cumulative inner orbital period distributions, the data from the simulations (red lines; with the thin and thick lines corresponding to the initial and final distributions, respectively) to the observations (the MSC, , with black dashed lines; see also §\[sect:IC:comp\]). Here, we combine the data of orbits 1 and 2 for the 2+2 configuration. We compare the simulated final orbital periods to the observations using two-sided Kolmogorov-Smirnov (K-S) tests [@Kolmogorov_33; @smirnov_48]; the $D$ and $p$ statistics are indicated in each panel. From [Fig.]{}\[fig:pop\_syn\_p\_ins\_obs\], we observe that, although the simulated distributions are enhanced at short periods ($<10\,{\mathrm{d}}$) due to secular and tidal evolution, the observed distributions are more peaked at short periods, in particular for the 2+2 systems. For the latter, the observed distribution contains a sharp rise in systems around 10 d, which is not reproduced in the simulations. Such a sharp rise is less prominent in the observed distribution for 3+1 systems, and the match with the simulations is notably better. Quantitatively, the $p$ values for the 2+2 configuration are $\lesssim 0.05$, whereas for the 3+1 configuration, the $p$ values are generally larger, up to $\simeq 0.44$, although in the case of orbital distribution assumption (1) and flybys included, the $p$ value is low, $p\simeq0.02$. The best statistical agreement with the observations is reached for the 3+1 configuration with no flybys and the second assumption about the orbital distributions (flat in $\log_{10} a_i$ and $e_i$). We further discuss the implications of this result in §\[sect:discussion:per\]. Other orbital properties {#sect:pop_syn:orb} ------------------------ Here, we discuss orbital properties other than pertaining to the innermost orbital periods. We focus on the simulations with the first assumption about the orbital distributions and with no flybys included. Generally, there are no major differences in these distributions between these assumptions. ### Semimajor axes {#sect:pop_syn:orb:sma} In [Fig.]{}\[fig:pop\_syn\_sma\], we show the cumulative semimajor axis distributions of the migrating systems (specifically, orbit 1; top two panels) in the simulations with the first assumption about the orbital distributions, and no flybys included. Solid, dashed and dotted lines correspond to orbits 1, 2 and 3, respectively. The initial and final distributions are shown with the thin blue and thicker red lines, respectively. The initial distributions of [*all*]{} systems (i.e., not just the migrating systems) are shown with thin black lines. The bottom two panels correspond to the systems in which the maximum CPU wall time was exceeded (see §\[sect:discussion:exceed\] for discussion). Evidently, for the migrating systems, the final distribution of $a_1$ is peaked around a few $\times 10^{-2}$, corresponding to an orbital period of a few days. The initial $a_1$ of the migrating systems are somewhat smaller compared to all systems, which can be attributed to the fact that tighter orbits are more susceptible to tidal migration. For 2+2 systems, this translates into typically smaller $a_3$, to compensate for the smaller initial $a_1$. Note that, for the 2+2 systems, the initial and final distributions of $a_2$ are slightly distinct, with the final $a_2$ being slightly more compact. This can be attributed to tidal evolution in orbit 2. ### Eccentricities {#sect:pop_syn:orb:e} In [Fig.]{}\[fig:pop\_syn\_e\], we show cumulative eccentricity distributions of the ‘no migration’ (top two panels) and ‘orbit 1 migration’ outcomes. The top two panels (‘no migration’) show that secular evolution can still significantly enhance the eccentricities of orbits 1 and 2 for both configurations. In the bottom two panels, the final distribution of $e_1$ is peaked around zero, with some exceptions (especially for the 3+1 configuration, for which the absolute number of migrating systems is larger). In those exceptions, tidal migration occurs, but the orbit is not yet completely circularized by 10 Gyr. ### Ratios of LK time-scales {#sect:pop_syn:orb:R} As shown in previous studies [@2015MNRAS.449.4221H; @2017MNRAS.470.1657H; @2018MNRAS.474.3547G], secularly chaotic behaviour and particularly high eccentricities are to be expected if the ratio of LK time-scales corresponding to a particular configuration is close to unity. In [Fig.]{}\[fig:pop\_syn\_R\], we show the cumulative distributions of $\mathcal{R}_{2+2}$ (equation \[eq:R\_2p2\]) and $\mathcal{R}_{3+1}$ (equation \[eq:R\_3p1\]) in the top and bottom panels, respectively, for several outcomes in the simulations (indicated in the legend). In particular, for the 2+2 systems, the migrating systems tend to have LK time-scale ratios that are more concentrated towards unity than the overall population. This applies similarly to the systems in which RLOF occurs, and is consistent with the expectation that eccentricity excitation peaks if the LK time-scales are comparable. ### Inclinations In [Fig.]{}\[fig:pop\_syn\_incl\], we show distributions of the inclinations relative to the parents of orbits 1 and 2 for the non-migrating (top two panels) and migrating (orbit 1; bottom two panels) systems. Note that for the 2+2 configuration, these inclinations are $i_{13}$ and $i_{23}$ for orbits 1 and 2, respectively; for the 3+1 configuration, they are $i_{12}$ and $i_{23}$. We recall that the initial orientations were assumed to be random, i.e., the initial distributions of the mutual inclinations were assumed to be flat in their cosine. The non-migrating systems show some paucity of final inner inclinations near $90^\circ$, and an enhancement near $\sim 50^\circ$ and $130^\circ$. This can be understood by noting that high inclinations trigger secular interactions, and in which case peaks in the distributions are expected around $50^\circ$ and $130^\circ$. In the case of migration of orbit 1, $i_{13,\mathrm{i}}$ ($i_{12,\mathrm{i}}$) tends to be concentrated around $90^\circ$ for the 2+2 (3+1) configurations, as expected based on three-body secular evolution. However, in particular for the 3+1 systems, $i_{12,\mathrm{i}}$ does not have to be close to $90^\circ$ — values as small as a few tens of degrees are sufficient. This is consistent with @2017MNRAS.470.1657H, who showed that high eccentricities can be reached even for small initial mutual inclinations, provided that the ratio of LK time-scales is close to unity. Migration times {#sect:pop_syn:time} --------------- Lastly, we show in [Fig.]{}\[fig:stop\_times\] the cumulative distributions of the migration times for the migrating systems (orbit 1 or 2). We also show cumulative distributions of the stopping times for various other outcomes. The median migration time in our simulations is $\sim 1\,\mathrm{Gyr}$. Also shown are the times at which RLOF was triggered in orbit 1 or 2. RLOF is typically triggered early in the simulations, with a median time of $\sim 10\,\mathrm{Myr}$ and $\sim 0.1\,\mathrm{Myr}$ for the 2+2 and 3+1 configurations, respectively. Discussion {#sect:discussion} ========== Implications of the inner orbital period distributions {#sect:discussion:per} ------------------------------------------------------ We found in §\[sect:pop\_syn:in\] that our simulated inner orbital period distributions match the observations reasonably for 3+1 systems. However, the observations show the presence of a large population of 2+2 systems with inner periods around 10 d, which is not reproduced in the simulations. In our simulations, we made the simplifying assumption that no systems are formed at the MS with inner periods shorter than 10 d. This assumption is based on the argument that short-period systems would merge during their pre-MS evolution. However, the pre-MS evolution of short-period binaries in multiple systems is poorly understood. Evidently, if the initial inner orbital period distribution were closer to the observed distribution, then our simulations could be made to better match the observations. We do remark that for the 3+1 systems, the observed systems are reasonably described by secular and tidal evolution alone. This difference between the 2+2 and 3+1 systems could hint at different formation scenarios between 2+2 and 3+1 systems, i.e., sequential formation versus cascade (hierarchical) fragmentation (e.g., @2018AJ....155..160T). Time-exceeded systems {#sect:discussion:exceed} --------------------- As mentioned in Sections\[sect:meth:sc\] and \[sect:pop\_syn:frac\], in a number of systems, in particular for the 3+1 configuration, the integration proceeded very slowly. For practical reasons, the CPU wall time was limited to 24 hr. The fraction of terminated systems is less than 1 per cent for the 2+2 systems, but up to $\sim 16\%$ for the 3+1 systems. In the bottom two panels of [Fig.]{}\[fig:pop\_syn\_sma\], we show the distributions of the semimajor axes for the ‘time-exceeded’ systems. For reference, we note that the top two panels of [Fig.]{}\[fig:pop\_syn\_sma\] show the distributions for the migrating systems (orbit 1). The time-exceeded systems tend to have much tighter orbits compared to other systems, in particular the migrating systems. This can be understood by noting that the secular time-scales are shorter for more compact systems, implying that these systems are computationally demanding. Specifically, for the 2+2 configuration, $a_3\sim 20{\,\textsc{au}}$ for the time-exceeded systems, which is significantly smaller than the typical $a_3$ for migrating systems ($a_3 \sim 3\times10^2\,{\,\textsc{au}}$). For the 3+1 configuration, typically $a_2\sim 10\,{\,\textsc{au}}$ for the time-exceeded systems, whereas $a_2\sim 10^2\,{\,\textsc{au}}$ for the migrating systems. Other large differences between the migrating and time-exceeded systems are illustrated by the distributions of the LK time-scale ratios ([Fig.]{}\[fig:pop\_syn\_R\]), and the migration/stopping times ([Fig.]{}\[fig:stop\_times\]). We conclude that the existence of the time-exceeded systems likely does not significantly affect our results of the orbital period distributions. RLOF {#sect:discussion:RLOF} ---- We stopped our simulations at the onset of RLOF (see §\[sect:meth:sc\]). RLOF is expected to lead to mass transfer or common-envelope evolution, typically resulting in significant shrinkage of the orbit, or possibly even coalescence of two stars. The evolution can be complicated if the time-scales of mass transfer and secular evolution are comparable. Such evolution is beyond the scope of this work, but certainly merits further investigation. Nevertheless, we can remark that, if the result of RLOF is a tight orbit, then this would imply an enhancement of the inner orbital period distribution at short orbital periods ($<10\,{\mathrm{d}}$). This could help to reduce tensions between the observed and simulated orbital period distributions (see §\[sect:pop\_syn:in\]). On the other hand, coalescence of the stars would transform the system into a triple, and the observed inner orbital period distribution of quadruple stars would not apply in that case (however, it would affect the inner orbital period distribution of triple stars). Another related caveat is that the stellar radii may have been larger during the pre-MS phase (however, pre-MS phase evolution is still not fully understood, and some recent studies have shown that accretion during the pre-MS could affect the evolution compared to the standard picture of contraction along the Hayashi line, @1961PASJ...13..450H, resulting in only modestly larger radii compared to the zero-age MS by a few tens of per cent, see, e.g., ). Larger radii during the pre-MS phase would trigger more RLOF in our simulations, in which zero-age MS radii were assumed. If RLOF were triggered during the pre-MS, then the stars would likely merge, and it would be more appropriate to identify the system as a triple system. Nevertheless, to investigate the potential effect of larger pre-MS radii, we carried out additional population synthesis simulations for a short duration of 1 Myr and with the primary star radius enlarged to $R_1=5 \,{\mathrm{R}_\odot}$ (taken to be constant during the 1 Myr integration). This enlarged radius is based on Fig. 2 of , and should give an upper limit to the effect of the larger primary star radius during the pre-MS (we do not consider a larger secondary star radius, since its mass and hence radius are smaller). The RLOF (star 1) fractions in these short pre-MS simulations are $\sim0.03$ ($\sim0.14$) for the 2+2 (3+1) systems. In simulations with a constant $R_1 = 1 \,{\mathrm{R}_\odot}$, the RLOF (star 1) fractions during the first 1 Myr are $\sim0.02$ ($\sim0.09$) for the 2+2 (3+1) systems. This implies that the larger primary radius increases the RLOF (star 1) fractions during the first Myr by a factor of $\sim1.5$, i.e., the pre-MS evolution does not affect the occurrence of RLOF by more than $\sim 50\%$. Breakdown of the averaging approximation {#sect:discussion:sub} ---------------------------------------- The algorithm used in our integrations (see §\[sect:meth:sec\]) is based on an averaging of the Hamiltonian (and thus the equations of motion) over all three orbits. In reality, there exist short-term osculating eccentricity variations depending on the ratio of the outer orbital period to the secular time-scale (e.g., @2014MNRAS.439.1079A [@2016MNRAS.458.3060L; @2018MNRAS.476.4234F; @2018MNRAS.481.4907G]). The averaging approximation can break down in various cases, for example when the time-scale for the angular momentum or eccentricity vector to change is comparable to some of the orbital periods [@2014ApJ...781...45A]. In addition, the inner orbit precession frequency can be close to, but still longer than the outer orbital period in some cases (such as in the Earth-Moon system), in which case evection terms may become important. In case of comparable inner orbit precession frequency and outer orbital period, the evection resonance can come into play (e.g., ). Another example is the occurrence of mean-motion resonance in 2+2 systems [@2018MNRAS.475.5215B]. These effects are beyond the scope of this work. Galactic tides {#sect:discussion:gal_tide} -------------- We did not consider the effects of galactic tides in our simulations. The typical outermost orbit in our simulation has a semimajor axis of $\sim 10^3\,{\,\textsc{au}}$, up to $\sim 10^4\,{\,\textsc{au}}$ (see, e.g., the top two panels of [Fig.]{}\[fig:pop\_syn\_sma\]), which is significantly below $10^5\,{\,\textsc{au}}$, the separation at which galactic tides are expected to become important. We conclude that galactic tides are not important for the MS evolution of solar-type quadruple stars. However, we note that mass loss can drive orbital expansion in evolving quadruple star systems (e.g., @2018MNRAS.478..620H). Therefore, galactic tides are potentially important in such systems. Conclusions {#sect:conclusions} =========== We studied the formation of short-period orbits through tidal and secular evolution in hierarchical quadruple systems containing solar-type MS stars. We considered the 2+2 (two binaries orbiting each other’s barycentre) and 3+1 (triple orbited by a fourth star) configurations (see [Fig.]{}\[fig:configurations\]). In addition to secular gravitational and tidal evolution, we took into account the effects of encounters with passing stars. Our main conclusions are given below. 1\. In our simulations, the initial inner orbital periods were longer than 10 d. Due to secular and tidal evolution, the inner orbital periods shrank to $<10\,{\mathrm{d}}$ in a few per cent of systems for the 2+2 configuration, and up to 14% of systems for the 3+1 configuration. The higher migration efficiency for the 3+1 configuration can be attributed to typically tighter initial inner orbits, and typically stronger secular evolution for these systems. RLOF is triggered due to high eccentricity in up to $\sim15\%$ of systems, and occurs most frequently in the 3+1 systems. Dynamical instability of the system occurs most commonly for the 3+1 systems. For the latter, in most cases dynamical stability is triggered by increased eccentricity of the intermediate orbit (orbit 2) due to the secular torque of the outermost orbit (orbit 3). 2\. Through examples, we have shown that, in the 2+2 configuration, tidal shrinkage of one orbit can trigger the second orbit to shrink as well, leading to two short-period orbits (‘double migration’). For the 3+1 configuration, we have shown that the eccentricity of the intermediate orbit (orbit 2) can become enhanced in response to the shrinking of the innermost orbit, due to a reduction in the apsidal precession in orbit 2 imposed by orbit 1 [@2015MNRAS.449.4221H]. This can affect the subsequent evolution of the inner orbit (in particular, further shrinking the inner orbit), but also triggering dynamical instability of the system. This shows that dynamical instability in 3+1 systems can be triggered not only by mass loss in evolving systems (e.g., @2018MNRAS.478..620H), but also due to tidal evolution during MS evolution. 3\. Our simulated inner orbital period distributions compare reasonably to the observations for the 3+1 systems. However, for 2+2 systems, the observed inner orbital period distribution shows a significant enhancement at $\sim 10\,{\mathrm{d}}$, which is not reproduced by the simulations. This suggests that the inner orbital periods of 2+2 systems are not predominantly set by tidal and secular evolution, but by other processes, most likely occurring during the stellar formation and early evolution. 4\. The migrating systems in our simulations show a preference for similar LK time-scales in the appropriate orbit pairs, reflecting the fact that high eccentricities are induced due to coupled secular evolution in these cases. 5\. Our simulated inner orbital period distributions are not strongly dependent on whether or not flybys are taken into account. We also considered a set of simulations with different assumptions about the orbital distributions, which yielded no qualitatively significantly different results. Quantitatively, flybys can enhance (decrease) the migration fractions by a few tenths of per cent for the 2+2 (3+1) configuration. Acknowledgements {#acknowledgements .unnumbered} ================ I thank the anonymous referee for helpful comments. I gratefully acknowledge support from the Institute for Advanced Study, and The Peter Svennilson Membership. \[lastpage\] [^1]: E-mail: hamers@ias.edu

--- abstract: 'Let $\Omega$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and $\mathcal{M}_{\Omega}$ be the higher-dimensional Marcinkiewicz integral associated with $\Omega$. In this paper, the author considers the complete continuity on weighted $L^p(\mathbb{R}^n)$ spaces with $A_p(\mathbb{R}^n)$ weights, weighted Morrey spaces with $A_p(\mathbb{R}^n)$ weights, for the commutator generated by ${\rm CMO}(\mathbb{R}^n)$ functions and $\mathcal{M}_{\Omega}$ when $\Omega$ satisfies certain size conditions.' address: | Guoen Hu, Department of Applied Mathematics, Zhengzhou Information Science and Technology Institute\ Zhengzhou 450001, P. R. China author: - Guoen Hu title: Weighted complete continuity for the commutator of Marcinkiewicz integral --- [^1]. Introduction ============ As an analogy of the classicial Littlewood-Paley $g$-function, Marcinkiewicz [@mar] introduced the operator $$\mathcal{M}(f)(x)=\Big(\int^{\pi}_0\frac{|F(x+t)-F(x-t)-2F(x)|^2}{t^3}\,{\rm d}t\Big)^{\frac{1}{2}},$$ where $F(x)=\int^x_0f(t){\rm d}t.$ This operator is now called Marcinkiewicz integral. Zygmund [@zy] proved that $\mathcal{M}$ is bounded on $L^p([0,\,2\pi])$ for $p\in (1,\,\infty)$. Stein [@st] generalized the Marcinkiewicz operator to the case of higher dimension. Let $\Omega$ be homogeneous of degree zero, integrable and have mean value zero on the unit sphere $S^{n-1}$. Define the Marcinkiewicz integral operator $\mathcal{M}_\Omega$ by $$\begin{aligned} \mathcal{M}_\Omega(f)(x)= \Big(\int_0^\infty|F_{\Omega, t}f(x)|^2\frac{{\rm d}t}{t^3}\Big)^{\frac{1}{2}},\end{aligned}$$where $$F_{\Omega, t}f(x)= \int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-1}}f(y){\rm d}y$$ for $f\in \mathcal{S}(\mathbb{R}^n)$. Stein [@st] proved that if $\Omega\in {\rm Lip}_{\alpha}(S^{n-1})$ with $\alpha\in (0,\,1]$, then $\mathcal{M}_\Omega$ is bounded on $L^p(\mathbb{R}^n)$ for $p\in (1,\,2]$. Benedek, Calderón and Panzon showed that the $L^p(\mathbb{R}^n)$ boundedness $(p\in (1,\,\infty)$) of $\mathcal{M}_\Omega$ holds true under the condition that $\Omega\in C^1(S^{n-1})$. Using the one-dimensional result and Riesz transforms similarly as in the case of singular integrals (see [@cz]) and interpolation, Walsh [@wal] proved that for each $p\in (1,\,\infty)$, $\Omega\in L(\ln L)^{1/r}(\ln \ln L)^{2(1-2/r')}(S^{n-1})$ is a sufficient condition such that $\mathcal{M}_\Omega$ is bounded on $L^{p}(\mathbb{R}^n)$, where $r=\min\{p,\,p'\}$ and $p'=p/(p-1)$. Ding, Fan and Pan [@dfp] proved that if $\Omega\in H^1(S^{n-1})$ (the Hardy space on $S^{n-1}$), then $\mathcal{M}_\Omega$ is bounded on $L^p(\mathbb{R}^n)$ for all $p\in (1,\,\infty)$; Al-Salmam, Al-Qassem, Cheng and Pan [@aacp] proved that $\Omega\in L(\ln L)^{1/2}(S^{n-1})$ is a sufficient condition such that $\mathcal{M}_\Omega$ is bounded on $L^p(\mathbb{R}^n)$ for all $p\in(1,\,\infty)$. Ding, Fan and Pan [@dfp2] considered the boundedness on weighted $L^p({\mathbb R}^n)$ with $A_p(\mathbb{R}^n)$ when $\Omega\in L^q(S^{n-1})$ for some $q\in (1,\,\infty]$, where and in the following, for $p\in [1,\,\infty)$, $A_p(\mathbb{R}^n)$ denotes the weight function class of Muckenhoupt, see [@gra2] for the definitions and properties of $A_p(\mathbb{R}^n)$. For other works about the operator defined by (1.1), see [@a; @aacp; @cfp; @dfp; @dly1; @dxy] and the related references therein. The commutator of $\mathcal{M}_{\Omega}$ is also of interest and has been considered by many authors (see [@tw; @hy; @dly; @chenlu; @h2]). Let $b\in {\rm BMO}(\mathbb{R}^n)$, the commutator generated by $\mathcal{M}_\Omega$ and $b$ is defined by $$\begin{aligned} \qquad\mathcal{M}_{\Omega, b}f(x)=\bigg(\int_0^\infty\Big|\int_{|x-y|\leq t}\big(b(x)-b(y)\big) \frac{\Omega(x-y)}{|x-y|^{n-1}}f(y)dy\Big|^2\frac{dt}{t^3}\bigg)^{\frac{1}{2}}.\end{aligned}$$ Torchinsky and Wang [@tw] showed that if $\Omega\in {\rm Lip}_\alpha(S^{n-1})$ ($\alpha\in (0,\,1]$), then $\mathcal{M}_{\Omega, b}$ is bounded on $L^p(\mathbb{R}^n)$ with bound $C\|b\|_{{\rm BMO}(\mathbb{R}^n)}$ for all $p\in (1,\,\infty)$. Hu and Yan [@hy] proved the $\Omega\in L(\ln L)^{3/2}(S^{n-1})$ is a sufficient condition such that $\mathcal{M}_{\Omega,\,b}$ is bounded on $L^2$. Ding, Lu and Yabuta [@dly] considered the weighted estimates for $\mathcal{M}_{\Omega,\,b}$, and proved that if $\Omega\in L^{q}(S^{n-1})$ for some $q\in (1,\,\infty]$, then for $p\in (q',\,\infty)$ and $w\in A_{p/q'}(\mathbb{R}^n)$, or $p\in (1,\,q)$ and $w^{-1/(p-1)}\in A_{p'/q'}(\mathbb{R}^n)$, $\mathcal{M}_{\Omega,\,b}$ is bounded on $L^p(\mathbb{R}^n,\,w)$. Chen and Lu [@chenlu] improved the result in [@hy] and showed that if $\Omega\in L(\ln L)^{3/2}(S^{n-1})$, then $\mathcal{M}_{\Omega,\,b}$ is bounded on $L^p(\mathbb{R}^n)$ with bound $C\|b\|_{{\rm BMO}(\mathbb{R}^n)}$ for all $p\in (1,\,\infty)$. Let ${\rm CMO}(\mathbb{R}^n)$ be the closure of $C^{\infty}_0(\mathbb{R}^n)$ in the ${\rm BMO}(\mathbb{R}^n)$ topology, which coincide with ${\rm VMO}(\mathbb{R}^n)$, the space of functions of vanishing mean oscillation introduced by Coifman and Weiss [@cw], see also [@b]. Uchiyama [@u] proved that if $T$ is a Calderón-Zygmund operator, and $b\in {\rm BMO}(\mathbb{R}^n)$, then the commutator of $T$ defined by $$[b,\,T]f(x)=b(x)Tf(x)-T(bf)(x),$$ is a compact operator on $L^p(\mathbb{R}^n)$ $(p\in (1,\,\infty)$) if and only if $b\in {\rm CMO}(\mathbb{R}^n)$. Chen and Ding [@chend] considered the compactness of $\mathcal{M}_{\Omega,\,b}$ on $L^p(\mathbb{R}^n)$, and proved that if $\Omega$ satisfies certain regularity condition of Dini type, then for $p\in (1,\,\infty)$, $\mathcal{M}_{\Omega,\,b}$ is compact on $L^p(\mathbb{R}^n)$ if and only if $b\in {\rm CMO}(\mathbb{R}^n)$. Using the ideas from [@chenhu], Mao, Sawano and Wu [@msw] consider the compactness of $\mathcal{M}_{\Omega,\,b}$ when $\Omega$ satisfies the size condition that $$\begin{aligned} \sup_{\zeta\in S^{n-1}}\int_{S^{n-1}}|\Omega(\eta)|\Big(\ln\frac{1}{|\eta\cdot\zeta|}\Big)^{\theta}{\rm d}\eta<\infty,\end{aligned}$$ and proved that if $\Omega$ satisfies (1.3) for some $\theta\in (3/2,\,\infty)$, then for $b\in {\rm CMO}(\mathbb{R}^n)$ and $p\in \big(4\theta/(4\theta-3),\,4\theta/3\big)$, $\mathcal{M}_{\Omega,\,b}$ is compact on $L^p(\mathbb{R}^n)$. Recently, Chen and Hu [@chenhu2] proved that if $\Omega\in L(\log L)^{\frac{1}{2}}(S^{n-1})$, then for $b\in {\rm CMO}(\mathbb{R}^n)$ and $p\in (1,\,\infty)$, $\mathcal{M}_{\Omega,\,b}$ is completely continuous on $L^p(\mathbb{R}^n)$. Our first purpose of this paper is to consider the complete continuity on weighted $L^p(\mathbb{R}^n)$ spaces $\mathcal{M}_{\Omega,\,b}$ when $\Omega\in L^q(S^{n-1})$ for some $q\in (1,\,\infty]$. To formulate our main result, we first recall some definitions. Let $\mathcal{X}$ be a normed linear spaces and $\mathcal{X}^{*}$ be its dual space, $\{x_k\}\subset \mathcal{X}$ and $x\in \mathcal{X}$, If for all $f\in \mathcal{X}^*$, $$\lim_{k\rightarrow \infty}|f(x_k)-f(x)|=0,$$ then $\{x_k\}$ is said to converge to $x$ weakly, or $x_k\rightharpoonup x$. Let $\mathcal{X}$, $\mathcal{Y}$ be two Banach spaces and $S$ be a bounded operator from $\mathcal{X}$ to $\mathcal{Y}$. - If for each bounded set $\mathcal{G}\subset \mathcal{X}$, $S\mathcal{G}=\{Sx:\,x\in \mathcal{G}\}$ is a strongly pre-compact set in $\mathcal{Y}$, then $S$ is called a compact operator from $\mathcal{X}$ to $\mathcal{Y}$; - if for $\{x_k\}\subset\mathcal{X}$ and $x\in\mathcal{X}$, $$x_k\rightharpoonup x\,\,\hbox{in}\, \,\mathcal{X}\Rightarrow \|Sx_k-Sx\|_{\mathcal{Y}}\rightarrow 0,$$ then $S$ is said to be a completely continuous operator. It is well known that, if $\mathcal{X}$ is a reflexive space, and $S$ is completely continuous from $\mathcal{X}$ to $\mathcal{Y}$, then $S$ is also compact from $\mathcal{X}$ to $\mathcal{Y}$. On the other hand, if $S$ is a linear compact operator from $\mathcal{X}$ to $\mathcal{Y}$, then $S$ is also a completely continuous operator. However, if $S$ is not linear, then $S$ is compact do not imply that $S$ is completely continuous. For example, the operator $$Sx=\|x\|_{l^2}$$ is compact from $l^2$ to $\mathbb{R}$, but not completely continuous. Our first result in this paper can be stated as follows. \[t1.2\] Let $\Omega$ be homogeneous of degree zero, have mean value zero on $S^{n-1}$ and $\Omega\in L^q(S^{n-1})$ for some $q\in (1,\,\infty]$. Suppose that $p$ and $w$ satisfy one of the following conditions - $p\in (q',\,\infty)$ and $w\in A_{p/q'}(\mathbb{R}^n)$; - $p\in (1,\,q)$ and $w^{-1/(p-1)}\in A_{p'/q'}(\mathbb{R}^n)$; - $p\in (1,\,\infty)$ and $w^{q'}\in A_{p}(\mathbb{R}^n)$. Then for $b\in {\rm CMO}(\mathbb{R}^n)$, $\mathcal{M}_{\Omega,\,b}$ is completely continuous on $L^p(\mathbb{R}^n,\,w)$. Our argument used in the proof of Theorem \[t1.2\] also leads to the complete continuity of $\mathcal{M}_{\Omega,\,b}$ on weighted Morrey spaces. Let $p\in (0,\,\infty)$, $w$ be a weight and $\lambda\in (0,\,1)$. The weighted Morrey space $L^{p,\,\lambda}(\mathbb{R}^n,\,w)$ is defined as $$L^{p,\,\lambda}(\mathbb{R}^n,\,w)=\{f\in L_{\rm loc}^p(\mathbb{R}^n):\,\|f\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}<\infty\},$$ with $$\|f\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}=\sup_{y\in \mathbb{R}^n,\,r>0}\Big(\frac{1}{\{w(B(y,\,r))\}^{\lambda}}\int_{B(y,\,r)}|f(x)|^pw(x)\,{\rm d}x\Big)^{1/p},$$¡¡ here $B(y,\,r)$ denotes the ball in $\mathbb{R}^n$ centered at $y$ and having radius $r$, and $w(B(y,\,r))=\int_{B(y,\,r)}w(z){\rm d}z$. For simplicity, we use $L^{p,\,\lambda}(\mathbb{R}^n)$ to denote $L^{p,\,\lambda}(\mathbb{R}^n,1)$. The Morrey space $L^{p,\,\lambda}(\mathbb{R}^n)$ was introduced by Morrey \[17\]. It is well-known that this space is closely related to some problems in PED (see [@rv; @shen1]), and has interest in harmonic analysis (see [@ax] and the references therein). Komori and Shiral [@ks] introduced the weighted Morrey spaces and considered the properties on weighted Morrey spaces for some classical operators. Chen, Ding and Wang [@cdw] considered the compactness of $\mathcal{M}_{\Omega,\,b}$ on Morrey spaces. They proved that if $\lambda\in (0,\,1)$, $\Omega\in L^q(S^{n-1})$ for $q\in (1/(1-\lambda),\,\infty]$ and satisfies a regularity condition of $L^q$-Dini type, then $\mathcal{M}_{\Omega,\,b}$ is compact on $L^{p,\,\lambda}(\mathbb{R}^n)$. Our second purpose of this paper is to prove the complete continuity of $\mathcal{M}_{\Omega,\,b}$ on weighted Morrey spaces with $A_p(\mathbb{R}^n)$ weights. \[t1.3\] Let $\Omega$ be homogeneous of degree zero, have mean value zero on $S^{n-1}$ and $\Omega\in L^q(S^{n-1})$ for some $q\in (1,\,\infty]$. Suppose that $p\in (q',\,\infty)$, $\lambda\in (0,\,1)$ and $w\in A_{p/q'}(\mathbb{R}^n)$; or $p\in (1,\,q')$, $w^{r}\in A_1(\mathbb{R}^n)$ for some $r\in (q',\,\infty)$ and $\lambda\in (0,\,1-r'/q)$. Then for $b\in {\rm CMO}(\mathbb{R}^n)$, $\mathcal{M}_{\Omega,\,b}$ is completely continuous on $L^{p,\,\lambda}(\mathbb{R}^n,\,w)$. The proofs of Theorems \[t1.2\] involve some ideas used in [@chenhu] and a sufficient condition of strongly pre-compact set in $L^p(L^2([1,\,2]), l^2;\,\mathbb{R}^n,\,w)$ with $w\in A_p(\mathbb{R}^n)$. To prove Theorem \[t1.3\], we will establish a lemma which clarify the relationship of the bounds on $L^p(\mathbb{R}^n,\,w)$ and the bounds on $L^{p,\,\lambda}(\mathbb{R}^n,\,w)$ for a class of sublinear operators, see Lemma 4.1 below. We make some conventions. In what follows, $C$ always denotes a positive constant that is independent of the main parameters involved but whose value may differ from line to line. We use the symbol $A\lesssim B$ to denote that there exists a positive constant $C$ such that $A\le CB$. For a set $E\subset\mathbb{R}^n$, $\chi_E$ denotes its characteristic function. Let $M$ be the Hardy-Littlewood maximal operator. For $r\in (0,\,\infty)$, we use $M_r$ to denote the operator $M_rf(x)=\big(M(|f|^r)(x)\big)^{1/r}.$ For a locally integrable function $f$, the sharp maximal function $M^{\sharp}f$ is defined by $$M^{\sharp}f(x)=\sup_{Q\ni x}\inf_{c\in\mathbb{C}}\frac{1}{|Q|}\int_{Q}|f(y)-c|{\rm d}y.$$ Approximation ============= Let $\Omega$ be homogeneous of degree zero, integrable on $S^{n-1}$. For $t\in [1,\,2]$ and $j\in\mathbb{Z}$, set $$\begin{aligned} K^j_t(x)=\frac{1}{2^j}\frac{\Omega(x)}{|x|^{n-1}}\chi_{\{2^{j-1}t<|x|\leq 2^jt\}}(x).\end{aligned}$$ As it was proved in [@drf], if $\Omega\in L^q(S^{n-1})$ for some $q\in (1,\,\infty]$, then there exists a constant $\alpha\in (0,\,1)$ such that for $t\in[1,\,2]$ and $\xi\in\mathbb{R}^n\backslash\{0\}$, $$\begin{aligned} |\widehat{K^j_t}(\xi)|\lesssim \|\Omega\|_{L^{q}(S^{n-1})}\min\{1,\,|2^j\xi|^{-\alpha}\}.\end{aligned}$$ Here and in the following for $h\in\mathcal{S}'(\mathbb{R}^n)$, $\widehat{h}$ denotes the Fourier transform of $h$. Moreover, if $\int_{S^{n-1}}\Omega(x'){\rm d}x'=0$, then $$\begin{aligned} |\widehat{K^j_t}(\xi)|\lesssim \|\Omega\|_{L^{1}(S^{n-1})}\min\{1,\,|2^j\xi|\}.\end{aligned}$$ Let $$\widetilde{\mathcal{M}}_{\Omega}f(x)=\Big(\int^2_1\sum_{j\in\mathbb{Z}}\big|F_{j}f(x,\,t)\big|^2{\rm d}t\Big)^{\frac{1}{2}},$$with $$F_jf(x,\,t)=\int_{\mathbb{R}^n}K^j_t(x-y)f(y){\rm d}y.$$ For $b\in {\rm BMO}(\mathbb{R}^n)$, let $\widetilde{\mathcal{M}}_{\Omega,\,b}$ be the commutator of $\widetilde{\mathcal{M}}_{\Omega}$ defined by $$\widetilde{\mathcal{M}}_{\Omega,\,b}f(x)=\Big(\int^2_1\sum_{j\in\mathbb{Z}}\big|F_{j,\,b}f(x,\,t)\big|^2{\rm d}t\Big)^{1/2},$$with $$F_{j,\,b}f(x,\,t)=\int_{\mathbb{R}^n}\big(b(x)-b(y)\big)K^j_t(x-y)f(y){\rm d}y.$$ A trivial computation leads to that $$\begin{aligned} \mathcal{M}_{\Omega}f(x)\approx \widetilde{\mathcal{M}}_{\Omega}f(x),\,\, \widetilde{\mathcal{M}}_{\Omega,\,b}f(x)\approx\widetilde{\mathcal{M}}_{\Omega,\,b}f(x).\end{aligned}$$ Let $\phi\in C^{\infty}_0(\mathbb{R}^n)$ be a nonnegative function such that $\int_{\mathbb{R}^n}\phi(x){\rm d}x=1$, ${\rm supp}\,\phi\subset\{x:\,|x|\leq 1/4\}$. For $l\in \mathbb{Z}$, let $\phi_l(y)=2^{-nl}\phi(2^{-l}y)$. It is easy to verify that for any $\varsigma\in (0,\,1)$, $$\begin{aligned} |\widehat{\phi_l}(\xi)-1|\lesssim \min\{1,\,|2^l\xi|^{\varsigma}\}.\end{aligned}$$Let $$F_{j}^lf(x,\,t)=\int_{\mathbb{R}^n}K^j_t*\phi_{j-l}(x-y)f(y)\,{\rm d}y.$$ Define the operator $\widetilde{\mathcal{M}}_{\Omega}^l$ by $$\begin{aligned} \widetilde{\mathcal{M}}_{\Omega}^lf(x)=\Big(\int^2_1\sum_{j\in\mathbb{Z}}\big|F_{j}^lf(x,\,t)\big|^2 {\rm d}t\Big)^{\frac{1}{2}}.\end{aligned}$$ This section is devoted to the approximation of $\widetilde{\mathcal{M}}_{\Omega}$ by $\widetilde{\mathcal{M}}_{\Omega}^l$. We will prove following theorem. \[t2.2\] Let $\Omega$ be homogeneous of degree zero and have mean value zero. Suppose that $\Omega\in L^q(S^{n-1})$ for some $q\in (1,\,\infty]$, $p$ and $w$ are the same as in Theorem \[t1.2\], then for $l\in\mathbb{N}$, $$\|\widetilde{\mathcal{M}}_{\Omega}f-\widetilde{\mathcal{M}}_{\Omega}^{l}f\|_{L^p(\mathbb{R}^n,\,w)}\lesssim 2^{-\varrho_pl}\|f\|_{L^{p}(\mathbb{R}^n,\,w)},$$ with $\varrho_p\in (0,\,1)$ a constant depending only on $p$,$n$ and $w$. To prove Theorem \[t2.2\], we will use some lemmas. \[l2.2\] Let $\Omega$ be homogeneous of degree zero and belong to $L^q(S^{n-1})$ for some $q\in (1,\,\infty]$, $K_t^j$ be defined as in (2.1). Then for $t\in [1,\,2]$, $l\in\mathbb{N}$, $R>0$ and $y\in \mathbb{R}^n$ with $|y|<R/4$, $$\begin{aligned} \sum_{j\in\mathbb{Z}}\sum_{k=1}^{\infty}(2^kR)^{\frac{n}{q'}}\Big(\int_{2^kR<|x|\leq 2^{k+1}R}\big|K^j_{t}*\phi_{j-l}(x+y)-K^j_{t}*\phi_{j-l}(x)\big|^{q}{\rm d}x\Big)^{\frac{1}{q}}\lesssim l.\end{aligned}$$ For the proof of Lemma \[l2.2\], see [@wat]. \[l2.4\]Let $\Omega$ be homogeneous of degree zero and $\Omega\in L^q(S^{n-1})$ for some $q\in (1,\,\infty]$, $p\in (1,\,q)$ and $w^{-1/(p-1)}\in A_{p'/q'}(\mathbb{R}^n)$. Then $$\begin{aligned} \quad\Big\|\Big(\sum_{j\in\mathbb{Z}}|K^j_t*\phi_{j-l}*f_j|^2\Big)^{\frac{1}{2}}\Big\|_{L^p(\mathbb{R}^n,\,w)}\lesssim \Big\|\Big(\sum_{j\in\mathbb{Z}}|f_j|^2\Big)^{\frac{1}{2}}\Big\|_{L^p(\mathbb{R}^n,\,w)}.\end{aligned}$$ Let $M_{\Omega}$ be the maximal operator defined by $$\begin{aligned} M_{\Omega}h(x)=\sup_{r>0}\frac{1}{|B(x,\,r)|}\int_{B(x,\,r)}|\Omega(x-y)h(y)|{\rm d}y.\end{aligned}$$ We know from the proof of Lemma 1 in [@duo] that for $p\in (1,\,2]$, $$\begin{aligned} \Big\|\Big(\sum_{j\in\mathbb{Z}}|M_{\Omega}f_j|^2\Big)^{\frac{1}{2}}\Big\|_{L^p(\mathbb{R}^n,\,w)}\lesssim \Big\|\Big(\sum_{j\in\mathbb{Z}}|f_j|^2\Big)^{\frac{1}{2}}\Big\|_{L^p(\mathbb{R}^n,\,w)},\end{aligned}$$ provided that $p\in (q',\,\infty)$ and $w\in A_{p/q'}(\mathbb{R}^n)$, or $p\in (1,\,q)$ and $w^{-1/(p-1)}\in A_{p'/q'}(\mathbb{R}^n)$. On the other hand, it is easy to verify that $$|K^j_t*\phi_{j-l}*f_j(x)|\lesssim M_{\Omega}Mf_j(x).$$ The inequality (2.9), together with the weighted vector-valued inequality of $M$ (see Theorem 3.1 in [@aj]), proves that (2.7) hold when $p\in (1,\,2]$, $p\in (q',\,\infty)$ and $w\in A_{p/q'}(\mathbb{R}^n)$, or $p\in (1,\,q)$ and $w^{-1/(p-1)}\in A_{p'/q'}(\mathbb{R}^n)$. This, via a standard duality argument, shows that (2.7) holds when $p\in(2,\,\infty)$, $p\in (1,\,q)$ and $w^{-1/(p-1)}\in A_{p'/q'}(\mathbb{R}^n)$. [*Proof of Theorem \[t2.2\]*]{}. We employ the ideas used in [@wat]. By Fourier transform estimates (2.2) and (2.5), and the Plancherel theorem, we know that $$\begin{aligned} \|\widetilde{\mathcal{M}}_{\Omega}f-\widetilde{\mathcal{M}}_{\Omega}^{l}f\|_{L^2(\mathbb{R}^n)}^2 &=&\int^2_1\Big\|\Big(\sum_{j\in\mathbb{Z}}\big|F_lf(\cdot,\,t)-F_{j}^lf(\cdot,\,t)\big|^2\Big)^{\frac{1}{2}}\Big\|_{L^2(\mathbb{R}^n)}^2{\rm d}t\\ &=&\int^2_1\sum_{j\in\mathbb{Z}}\int_{\mathbb{R}^n}|\widehat{K_t^j}(\xi)|^2|1-\widehat{\phi_{j-l}}(\xi)|^2|\widehat{f}(\xi)|^2{\rm d}\xi{\rm d}t\\ &\lesssim&2^{-\alpha l}\|f\|_{L^2(\mathbb{R}^n)}^2.\end{aligned}$$ Now let $p$ and $w$ be the same as in Theorem \[t1.2\]. Recall that $\mathcal{M}_{\Omega}$ is bounded on $L^p(\mathbb{R}^n,\,w)$ and so is $\widetilde{\mathcal{M}}_{\Omega}$. Thus, by interpolation with changes of measures of Stein and Weiss [@stw], it suffices to prove that $$\begin{aligned} \|\widetilde{\mathcal{M}}_{\Omega}^{l}f\|_{L^p(\mathbb{R}^n,\,w)}\lesssim l\|f\|_{L^{p}(\mathbb{R}^n,\,w)}.\end{aligned}$$ We now prove (2.10) for the case $p\in (1,\,q)$ and $w^{-1/(p-1)}\in A_{p'/q'}(\mathbb{R}^n)$. Let $\psi\in C^{\infty}_0(\mathbb{R}^n)$ be a radial function such that ${\rm supp}\,\psi\subset \{1/4\leq |\xi|\leq 4\}$ and $$\sum_{i\in\mathbb{Z}}\psi(2^{-i}\xi)=1,\,\,|\xi|\not =0.$$ Define the multiplier operator $S_i$ by $$\widehat{S_if}(\xi)=\psi(2^{-i}\xi)\widehat{f}(\xi).$$ Set $${\rm E}_1f(x)=\sum_{m=-\infty}^0\Big(\int^2_1\sum_{j}\Big|K^j_t*\phi_{j-l}*(S_{m-j}f)(x)\Big|^2{\rm d}t\Big)^{\frac{1}{2}},$$ $${\rm E_2}f(x)=\sum_{m=1}^\infty\Big(\int^2_1\sum_{j}\Big|K^j_t*\phi_{j-l}*(S_{m-j}f)(x)\Big|^2{\rm d}t\Big)^{\frac{1}{2}}.$$ It then follows that for $f\in \mathcal{S}(\mathbb{R}^n)$, $$\Big\|\Big(\int^2_1\sum_{j}\big|K^j_t*\phi_{j-l}*f(x)\big|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^p(\mathbb{R}^n)}\leq \sum_{i=1}^2\|{\rm E}_if\|_{L^p(\mathbb{R}^n)}.$$ We now estimate the term ${\rm E}_1$. By Fourier transform estimate (2.3), we know that $$\begin{aligned} &&\Big\|\Big(\int^2_1\sum_{j}\Big|K^j_t*\phi_{j-l}*(S_{m-j}f)(x)\Big|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^2(\mathbb{R}^n)}^2\\ &&\quad=\int^2_1\int_{\mathbb{R}^n}\sum_{j\in\mathbb{Z}}\Big|K^j_t*\phi_{j-l}*(S_{m-j}f)(x)\Big|^2{\rm d}x{\rm d}t\nonumber\\ &&\quad\lesssim\sum_{j\in\mathbb{Z}}\int_{\mathbb{R}^n}|2^j\xi||\psi(2^{-m+j}\xi)|^2|\widehat{f}(\xi)|^2{\rm d}\xi\nonumber\\ &&\quad\leq 2^{2m}\|f\|_{L^2(\mathbb{R}^n)}^2.\nonumber\end{aligned}$$ On the other hand, applying the Minkowski inequality, Lemma \[l2.4\] and the weighted Littlewood-Paley theory, we have that $$\begin{aligned} &&\Big\|\Big(\int^2_1\sum_{j}\Big|K^j_t*\phi_{j-l}*(S_{m-j}f)(x)\Big|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^p(\mathbb{R}^n,\,w)}^2\\ &&\quad\leq \int^2_1\Big(\int_{\mathbb{R}^n}\Big(\sum_{j\in\mathbb{Z}}\Big|K^j_t*\phi_{j-l}*(S_{m-j}f)(x)\Big|^2\Big)^{p/2}w(x){\rm d}x\Big)^{2/p}{\rm d}t\nonumber\\ &&\quad\leq \|f\|_{L^p(\mathbb{R}^n,\,w)}^2,\,\,p\in[2,\,\infty).\nonumber\end{aligned}$$ To estimate $$\Big\|\Big(\int^2_1\sum_{j}\Big|K^j_t*\phi_{j-l}*(S_{m-j}f)(x)\Big|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^p(\mathbb{R}^n,\,w)}$$ for $p\in (1,\,2)$, we consider the mapping $\mathcal{F}$ defined by $$\mathcal{F}:\,\,\{h_j(x)\}_{j\in\mathbb{Z}}\longrightarrow \{K^j_t*\phi_{j-l}*h_j(x)\}.$$ Note that for any $t\in (1,\,2)$, $$\big|K^j_t*\phi_{j-l}*h_j(x)\big|\lesssim MM_{\Omega}h_j(x).$$ We choose $p_0\in (1,\,p)$ such that $w^{-1/(p_0-1)}\in A_{p_0'/q'}(\mathbb{R}^n)$. Then by the weighted estimates for $M_{\Omega}$ (see [@duo]), we have that $$\begin{aligned} &&\int_{\mathbb{R}^n}\int^2_1\sum_{j\in \mathbb{Z}}\big|K^j_t*\phi_{j-l}*h_j(x)\big|^{p_0}{\rm d}tw(x){\rm d}x \lesssim\int_{\mathbb{R}^n}\sum_{j\in\mathbb{Z}}|h_j(x)|^{p_0}w(x){\rm d}x.\end{aligned}$$ Also, we have that $$\sup_{j\in \mathbb{Z}}\sup_{t\in[1,\,2]}\big|K^j_t*\phi_{j-l}*h_j(x)\big|\lesssim\sup_{j\in\mathbb{Z}}|h_j(x)|.$$ which implies that for $p_1\in (1,\,\infty)$, $$\begin{aligned} &&\Big\|\sup_{j\in \mathbb{Z}}\sup_{t\in[1,\,2]}\big|K^j_t*\phi_{j-l}*h_j\big|\Big\|_{L^{p_1}(\mathbb{R}^n,\,w)}\lesssim \Big\|\sup_{j\in\mathbb{Z}}|h_j|\Big\|_{L^{p_1}(\mathbb{R}^n,\,w)}.\end{aligned}$$ By interpolation, we deduce from the inequalities (2.13) and (2.14) (with $p_0\in (1,\,2)$, $p_1\in (2,\,\infty)$ and $1/p=1/2+(2-p_0)/(2p_1)$) that $$\Big\|\Big(\int^2_1\sum_{j\in \mathbb{Z}}\big|K^j_t*\phi_{j-l}*h_j\big|^{2}{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^p(\mathbb{R}^n,\,w)}\lesssim\Big\|\Big(\sum_{j\in\mathbb{Z}}|h_j|^{2}\Big)^{\frac{1}{2}} \Big\|_{L^p(\mathbb{R}^n,\,w)},$$ and so $$\begin{aligned} &&\Big\|\Big(\int^2_1\sum_{j}\Big|K^j_t*\phi_{j-l}*(S_{m-j}f)\Big|^2{\rm d}t\Big)^{\frac{1}{2}} \Big\|_{L^p(\mathbb{R}^n,w)}\\ &&\quad\lesssim \Big\|\Big(\sum_{j\in\mathbb{Z}}|S_{m-j}f|^2\Big)^{\frac{1}{2}}\Big\|_{L^p(\mathbb{R}^n,\,w)} \lesssim \|f\|_{L^p(\mathbb{R}^n,\,w)},\,p\in (1,\,2).\nonumber\end{aligned}$$ This, along with (2.12), states that for $p\in (1,q)$, $$\begin{aligned} \Big\|\Big(\int^2_1\sum_{j}\Big|K^j_t*\phi_{j-l}*(S_{m-j}f)\Big|^2{\rm d}t\Big)^{\frac{1}{2}} \Big\|_{L^p(\mathbb{R}^n,w)}\lesssim \|f\|_{L^p(\mathbb{R}^n,w)}.\end{aligned}$$ Again by interpolating, the inequalities (2.11) and (2.15) give us that for $p\in (1,\,q)$, $$\Big\|\Big(\int^2_1\sum_{j}\Big|K^j_t*\phi_{j-l}*(S_{m-j}f)(x)\Big|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^p(\mathbb{R}^n,\,w)}\\ \lesssim 2^{t_p m}\|f\|_{L^p(\mathbb{R}^n,\,w)}.$$ with $t_p\in (0,\,1)$ a constant depending only on $p$. Therefore, $$\|{\rm E}_1f\|_{L^p(\mathbb{R}^n,\,w)}\lesssim \|f\|_{L^p(\mathbb{R}^n,\,w)}.$$ We consider the term ${\rm E}_2$. Again by the Plancherel theorem and the Fourier transform estimates (2.2) and (2.5), we have that $$\begin{aligned} &&\Big\|\Big(\int^2_1\sum_{j\in\mathbb{Z}}\Big|K^j_t*\phi_{j-l}*(S_{m-j}f)(x)\Big|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|^2_{L^2(\mathbb{R}^n)}\\ &&\quad=\int^2_1\sum_{j\in\mathbb{Z}}\int_{\mathbb{R}^n}|\widehat{K_t^j}(\xi)|^2|\psi(2^{-m+j}\xi)|^2|\widehat{f}(\xi)|^2{\rm d}\xi{\rm d}t\nonumber\\ &&\quad\lesssim\sum_{j\in\mathbb{Z}}\int_{\mathbb{R}^n}|2^j\xi|^{-2\alpha}|2^{j-l}\xi|^{\alpha}\psi(2^{-m+j}\xi)|^2|\widehat{f}(\xi)|^2{\rm d}\xi\nonumber\\ &&\quad\lesssim2^{-m\alpha}\|f\|_{L^2(\mathbb{R}^n)}^2.\nonumber\end{aligned}$$ As in the inequality (2.15), we have that $$\begin{aligned} \qquad\Big\|\Big(\int^2_1\sum_{j\in\mathbb{Z}}\Big|K^j_t*\phi_{j-l}*(S_{m-j}f)(x)\Big|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^p(\mathbb{R}^n,w)} \lesssim \|f\|_{L^p(\mathbb{R}^n,w)}.\end{aligned}$$ Interpolating the inequalities (2.16) and (2.17) then shows that $$\begin{aligned} \Big\|\Big(\int^2_1\sum_{j\in\mathbb{Z}}\Big|K^j_t*\phi_{j-l}*(S_{m-j}f)(x)\Big|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^p(\mathbb{R}^n,\,w)}\lesssim 2^{-t_pm}\|f\|_{L^p(\mathbb{R}^n,w)}.\end{aligned}$$ This gives the desired estimate for ${\rm E}_2$. Combining the eastimates for ${\rm E}_1$ and ${\rm E}_2$ then yields (2.10) for the case $p\in (1,\,q)$ and $w^{-1/(p-1)}\in A_{p'/q'}(\mathbb{R}^n)$. We now prove (2.10) for the case of $p\in (q',\,\infty)$ and $w\in A_{p/q'}(\mathbb{R}^n)$. By a standard argument, it suffices to prove that $$\begin{aligned} M^{\sharp}(\widetilde{\mathcal{M}}_{\Omega}^lf)(x)\lesssim lM_{q'}f(x),\end{aligned}$$ To prove (2.18), let $x\in\mathbb{R}^n$ and $Q$ be a cube containing $x$. Decompose $f$ as $$f(y)=f(y)\chi_{4nQ}(y)+f(y)\chi_{\mathbb{R}^n\backslash 4nQ}(y)=:f_1(y)+f_2(y).$$ It is obvious that $\widetilde{\mathcal{M}}_{\Omega}^l$ is bounded on $L^{q'}(\mathbb{R}^n)$. Thus, $$\begin{aligned} \frac{1}{|Q|}\int_{Q}\widetilde{\mathcal{M}}_{\Omega}^lf_1(y){\rm d}y\lesssim\Big(\frac{1}{|Q|}\int_{Q}\big\{\widetilde{\mathcal{M}}_{\Omega}^lf_1(y)\big\}^{q'}{\rm d}y\Big)^{1/q'}\lesssim M_{q'}f(x).\end{aligned}$$ Let $x_0\in Q$ such that $\widetilde{\mathcal{M}}_{\Omega}^lf_2(x_0)<\infty$. For $y\in Q$, it follows from Lemma \[l2.2\] that $$\begin{aligned} &&\Big|\widetilde{\mathcal{M}}_{\Omega}^lf_2(y)-\widetilde{\mathcal{M}}_{\Omega}^lf_2(y_0)\Big|\\ &&\quad\lesssim \Big(\int^2_1\sum_{j\in\mathbb{Z}}\big|K^j_t*\phi_{j-l}*f_2(y)-K^j_t*\phi_{j-l}*f_2(x_0)\big|^2{\rm d}t\Big)^{\frac{1}{2}}\nonumber\\ &&\quad\lesssim\Big(\int^2_1\Big(\sum_{j\in\mathbb{Z}}\big|K^j_t*\phi_{j-l}*f_2(y)-K^j_t*\phi_{j-l}*f_2(x_0)\big|\Big)^2{\rm d}t\Big)^{\frac{1}{2}}\nonumber\\ &&\quad\lesssim lM_{q'}f(x).\nonumber\end{aligned}$$ Combining the estimates (2.19) and (2.20) leads to that $$\inf_{c\in\mathbb{C}}\frac{1}{|Q|}\int_{Q}\big|\widetilde{\mathcal{M}}_{\Omega}^lf(y)-c\big|{\rm d}y\lesssim lM_{q'}f(x)$$ and then establish (2.18). Finally, we see that (2.10) holds for the case of $p\in (1,\,\infty)$ and $w^{q'}\in A_{p}(\mathbb{R}^n)$, if we invoke the interpolation argument used in the proof of Theorem 2 in [@kw]. This completes the proof of Theorem \[t2.2\]. Proof of Theorem \[t1.2\] ========================= We begin with some preliminary lemmas. \[l3.1\] Let $\Omega$ be homogeneous of degree zero and belong to $L^1(S^{n-1})$, $K_t^j$ be defined as in (2.1). Then for $l\in\mathbb{N}$, $t\in [1,\,2]$, $s\in (1,\,\infty]$, $j_0\in\mathbb{Z}_-$ and $y\in \mathbb{R}^n$ with $|y|<2^{j_0-4}$, $$\begin{aligned} &&\sum_{j>j_0}\sum_{k\in\mathbb{Z}}2^{kn/s}\Big(\int_{2^k<|x|\leq 2^{k+1}}\big|K^j_{t}*\phi_{j-l}(x+y)-K^j_{t}*\phi_{j-l}(x)\big|^{s'}{\rm d}x\Big)^{\frac{1}{s'}}\\ &&\quad\lesssim 2^{l(n+1)}2^{-j_0}|y|.\nonumber\end{aligned}$$ For the proof of Lemma \[l3.1\], see [@chenhu2]. For $t\in [1,\,2]$ and $j\in \mathbb{Z}$, let $K_t^j$ be defined as in (2.1), $\phi$ and $\phi_l$ (with $l\in\mathbb{N}$) be the same as in Section 2. For $b\in{\rm BMO}(\mathbb{R}^n)$, let $\widetilde{\mathcal{M}}_{\Omega,\,b}^l$ be the commutator of $\widetilde{\mathcal{M}}_{\Omega}^{l}$ defined by $$\widetilde{\mathcal{M}}_{\Omega,\,b}^lf(x)=\Big(\int^2_1\sum_{j\in\mathbb{Z}}\big|F_{j,\,b}^lf(x,\,t)\big|^2{\rm d}t\Big)^{\frac{1}{2}},$$with $$F_{j,\,b}^lf(x,\,t)=\int_{\mathbb{R}^n}\big(b(x)-b(y)\big)K^j_t*\phi_{j-l}(x-y)f(y)\,{\rm d}y.$$ For $j_0\in\mathbb{Z}$, define the operator $\widetilde{\mathcal{M}}_{\Omega}^{l,\,j_0}$ by $$\widetilde{\mathcal{M}}_{\Omega}^{l,\,j_0}f(x)=\Big(\int^2_1\sum_{j\in\mathbb{Z}:j> j_0}\big|F_{j,\,b}^lf(x,\,t)\big|^2{\rm d}t\Big)^{\frac{1}{2}},$$ and the commutator $\widetilde{\mathcal{M}}_{\Omega,\,b}^{l,\,j_0}$ $$\widetilde{\mathcal{M}}_{\Omega,\,b}^{l,\,j_0}f(x)=\Big(\int^2_1\sum_{j\in\mathbb{Z}:j> j_0}\big|F_{j,\,b}^lf(x,\,t)\big|^2{\rm d}t\Big)^{\frac{1}{2}},$$ with $b\in {\rm BMO}(\mathbb{R}^n)$. \[l3.2\] Let $\Omega$ be homogeneous of degree zero and integrable on $S^{n-1}$. Then for $b\in C^{\infty}_0(\mathbb{R}^n)$, $l\in \mathbb{N}$, $j_0\in\mathbb{Z}_-$, $$\big|\widetilde{\mathcal{M}}_{\Omega,\,b}^{l,\,j_0}f(x)-\widetilde{\mathcal{M}}_{\Omega,\,b}^lf(x)\big|\lesssim 2^{j_0}MM_{\Omega}f(x).$$ Let $b\in C^{\infty}_0(\mathbb{R}^n)$ with $\|\nabla b\|_{L^{\infty}(\mathbb{R}^n)}=1$. For $t\in [1,\,2]$, by the fact that ${\rm supp}\,K_t^j*\phi_{j-l}\subset \{x:\,2^{j-2}\leq |x|\leq 2^{j+2}\}$, it is easy to verify that $$\begin{aligned} &&\sum_{j\leq j_0}\int_{\mathbb{R}^n}\big|K_{t}^j*\phi_{j-l}(x-y)\big||x-y||f(y)|{\rm d}y\\ &&\quad\lesssim \sum_{j\leq j_0}\sum_{k\in\mathbb{Z}}2^k\int_{2^k<|x-y|\leq 2^{k+1}}\big|K_{t}^j*\phi_{j-l}(x-y)\big||f(y)|{\rm d}y\\ &&\quad\lesssim \sum_{j\leq j_0}\sum_{|k-j|\leq 3}2^k\int_{2^k<|x-y|\leq 2^{k+1}}\big|K_{t}^j*\phi_{j-l}(x-y)\big||f(y)|{\rm d}y\\ &&\quad\lesssim2^{j_0} M_{\Omega}Mf(x).\end{aligned}$$ Thus, $$\begin{aligned} &&\Big|\widetilde{\mathcal{M}}_{\Omega,\,b}^{l,\,j_0}f(x)-\widetilde{\mathcal{M}}_{\Omega,\,b}^lf(x)\Big|^2\\ &&\quad\leq \sum_{j<j_0}\int^2_1\Big|\int_{\mathbb{R}^n}\big(b(x)-b(y)\big)K^j_t*\phi_{j-l}(x-y)f(y)\Big|^2{\rm d}t\\ &&\quad\lesssim\int^2_1\Big(\sum_{j\leq j_0}\int_{\mathbb{R}^n}|x-y|\big|K_{t}^j*\phi_{j-l}(x-y)f(y)|{\rm d}y\Big)^2{\rm d}t\\ &&\quad\lesssim \{2^{j_0} M_{\Omega}Mf(x)\}^2.\end{aligned}$$ The desired conclusion now follows immediately. Let $p,\,r\in[1,\,\infty)$, $q\in [1,\,\infty]$ and $w$ be a weight, $L^{p}(L^q([1,\,2]),\,l^r;\,\mathbb{R}^n,\,w)$ be the space of sequences of functions defined by $$L^{p}(L^q([1,\,2]),\,l^r;\,\mathbb{R}^n,\,w)=\big\{\vec{f}=\{f_k\}_{k\in \mathbb{Z}}:\, \|\vec{f}\|_ {L^{p}(L^q([1,\,2]),\,l^r;\,\mathbb{R}^n,\,w)}<\infty\big\},$$ with $$\|\vec{f}\|_{L^{p}(L^q([1,\,2]),\,l^r;\,\mathbb{R}^n,\,w)}=\Big\|\Big(\int^2_1\Big(\sum_{k\in \mathbb{Z}}|f_k(x,\,t)|^r\Big)^{\frac{q}{r}}{\rm d}t\Big)^{1/q}\Big\|_{L^{p}(\mathbb{R}^n,\,w)}.$$ With usual addition and scalar multiplication, $L^{p}(L^q([1,\,2]),\,l^{r};\,\mathbb{R}^n,\,w)$ is a Banach space. \[l3.4\] Let $p\in (1,\,\infty)$ and $w\in A_p(\mathbb{R}^n)$, $\mathcal{G}\subset L^p(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)$. Suppose that $\mathcal{G}$ satisfies the following five conditions: - $\mathcal{G}$ is bounded, that is, there exists a constant $C$ such that for all $\{f_k\}_{k\in\mathbb{Z}}\in \mathcal{G}$, $\|\vec{f}\|_{L^p(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)}\leq C$; - for each fixed $\epsilon>0$, there exists a constant $A>0$, such that for all $\{f_k\}_{k\in\mathbb{Z}}\in \mathcal{G}$, $$\Big\|\Big(\int^2_1\sum_{k\in\mathbb{Z}}|f_k(\cdot,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\chi_{\{|\cdot|>A\}}(\cdot)\Big\|_{ L^p(\mathbb{R}^n,\,w)}<\epsilon;$$ - for each fixed $\epsilon>0$ and $N\in\mathbb{N}$, there exists a constant $\varrho>0$, such that for all $\{f_k\}_{k\in\mathbb{Z}}\in \mathcal{G}$, $$\Big\|\sup_{|h|\leq \varrho}\Big(\int^2_1 \sum_{|k|\leq N}|f_k(x,\,t)-f_k(x+h,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^{p}(\mathbb{R}^n,\,w)}< \epsilon;$$ - for each fixed $\epsilon>0$ and $N\in\mathbb{N}$, there exists a constant $\sigma\in (0,\,1/2)$ such that for all $\{f_k\}_{k\in\mathbb{Z}}\in \mathcal{G}$, $$\Big\|\sup_{|s|\leq \sigma}\Big(\int^2_1 \sum_{|k|\leq N}|f_k(\cdot,\,t+s)-f_k(\cdot,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^{p}(\mathbb{R}^n,\,w)}< \epsilon,$$ - for each fixed $D>0$ and $\epsilon>0$, there exists $N\in\mathbb{N}$ such that for all $\{f_k\}_{k\in\mathbb{Z}}\in \mathcal {G}$, $$\Big\|\Big(\int^2_1\sum_{|k|>N}|f_k(\cdot,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\chi_{B(0,\,D)}\Big\|_{L^p(\mathbb{R}^n,\,w)}<\epsilon.$$ Then $\mathcal{G}$ is a strongly pre-compact set in $L^p(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)$. We employ the argument used in the proof of [@ccp Theorem 5], with some refined modifications. Our goal is to prove that, for each fixed $\epsilon>0$, there exists a $\delta=\delta_{\epsilon}>0$ and a mapping $\Phi_{\epsilon}$ on $L^p(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)$, such that $\Phi_{\epsilon}(\mathcal{G})=\{\Phi_{\epsilon}(\vec{f}):\,\vec{f}\in \mathcal G\}$ is a strong pre-compact set in the space $L^p(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)$, and for any $\vec{f}$, $\vec{g}\in \mathcal{G}$, $$\begin{aligned} &&\|\Phi_{\epsilon}(\vec{f})-\Phi_{\epsilon}(\vec{g})\|_{L^p(L^2([1,2]),l^{2};\,\mathbb{R}^n,w)}<\delta\\ &&\quad\Rightarrow \|\vec{f}-\vec{g}\|_{L^p(L^2([1,2]),l^{2};\,\mathbb{R}^n,w)}<8\epsilon.\nonumber\end{aligned}$$ If we can prove this, then by Lemma 6 in [@ccp], we see that $\mathcal{G}$ is a strongly pre-compact set in $L^p(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)$. Now let $\epsilon>0$. We choose $A>1$ large enough as in assumption (b), $N\in\mathbb{N}$ such that for all $\{f_k\}_{k\in\mathbb{Z}}\in \mathcal {G}$, $$\Big\|\Big(\int^2_1\sum_{|k|>N}|f_k(\cdot,\,t)|^2{\rm d}t\Big)^{1/2}\chi_{B(0,\,2A)} \Big\|_{L^p(\mathbb{R}^n,\,w)}<\epsilon.$$ Let $\varrho\in (0,\,1/2)$ small enough as in assumption (c) and $\sigma\in (0,\,1/2)$ small enough such that (d) holds true. Let $Q$ be the largest cube centered at the origin such that $2Q\subset B(0,\,\varrho)$, $Q_1,\,\dots,\,Q_J$ be $J$ copies of $Q$ such that they are non-overlapping, and $\overline{B(0,\,A)}\subset \overline{\cup_{j=1}^JQ_j}\subset B(0,\,2A)$. Let $I_1,\,\dots,\,I_L\subset [1,\,2]$ be non-overlapping intervals with same length $|I|$, such that $|s-t|\leq \sigma$ for all $s,\,t\in I_j$ $(j=1,\,\dots,\,L)$ and $\cup_{j=1}^NI_j=[1,\,2]$. Define the mapping $\Phi_{\epsilon}$ on $L^p(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)$ by $$\begin{aligned} \Phi_{\epsilon}(\vec{f})(x,\,t)&=& \Big\{\dots,0,\,\,\dots,\,0,\,\sum_{i=1}^J\sum_{j=1}^Lm_{Q_i\times I_j}(f_{-N})\chi_{Q_i\times I_j}(x,t),\\ &&\quad\sum_{i=1}^J\sum_{j=1}^Lm_{Q_i\times I_j}(f_{-N+1})\chi_{Q_i\times I_j}(x,t),\dots,\\ &&\quad\sum_{i=1}^J\sum_{j=1}^Lm_{Q_i\times I_j}(f_{N})\chi_{Q_i\times I_j}(x,t),0,\dots\Big\},\end{aligned}$$ where and in the following, $$m_{Q_i\times I_j}(f_k)=\frac{1}{|Q_i|}\frac{1}{|I_j|}\int_{Q_i\times I_j}f_k(x,\,t){\rm d}x{\rm d}t.$$ We claim that $\Phi_{\epsilon}$ is bounded on $L^p(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)$. In fact, if $p\in [2,\,\infty)$, we have by the Hölder inequality that $$\begin{aligned} |m_{Q_i\times I_j}(f_k)|&\leq &\Big(\frac{1}{|Q_i||I_j|}\int_{I_j\times Q_i}|f_k(y,\,t)|^pw(y) {\rm d}y{\rm d}t\Big)^{\frac{1}{p}}\\ &&\quad\times\Big(\frac{1}{|Q_i|}\int_{Q_i}w^{-\frac{1}{p-1}}(y){\rm d}y\Big)^{\frac{1}{p'}} ,\end{aligned}$$and $$\begin{aligned} &&\sum_{|k|\leq N}\Big(\frac{1}{|Q_i||I_j|}\int_{I_j}\int_{Q_i}|f_k(y,\,t)|^pw(y) {\rm d}y{\rm d}t\Big)^{2/p}\\ &&\quad \lesssim N^{1-2/p}\Big(\sum_{|k|\leq N}\frac{1}{|Q_i||I_j|}\int_{I_j\times Q_i}|f_k(y,\,t)|^pw(y) {\rm d}y{\rm d}t\Big)^{2/p}.\end{aligned}$$ Therefore, $$\begin{aligned} \|\Phi_{\epsilon}(\vec{f})\|_{L^{p}(L^2([1,\,2]),l^{2};\mathbb{R}^n,w)}^{p} &=& \sum_{i=1}^J\sum_{j=1}^L\int_{I_j}\int_{Q_i}\Big(\sum_{|k|\leq N}|m_{Q_i\times I_j}(f_k)|^2\Big)^{p/2}w(x){\rm d}x{\rm d}t\\ &\lesssim& N^{p/2-1}\sum_{i=1}^J\sum_{j=1}^L\int_{I_j}\int_{Q_i}\sum_{|k|\leq N}|f_k(y,t)|^pw(y){\rm d}y{\rm d}t \\ &\leq& N^{p/2}\sum_{i=1}^J\sum_{j=1}^L\int_{I_j}\int_{Q_i}\Big\{\sum_{|k|\leq N}|f_k(y,t)|^2\Big\}^{\frac{p}{2}}w(y){\rm d}y{\rm d}t \\ &\leq &N^{p/2}\|\vec{f}\|_{L^{p}(L^2([1,\,2]),\,l^2;\,\mathbb{R}^n,\,w)}^{p}.\nonumber\end{aligned}$$ On the other hand, for $p\in (1,\,2)$ and $w\in A_{p}(\mathbb{R}^n)$, we choose $\gamma\in (0,\,1)$ such that $w\in A_{p-\gamma}(\mathbb{R}^n)$. Note that $$\begin{aligned} &&\sup_{-N\leq k\leq N}\sup_{t\in [1,\,2]}\Big|\sum_{i=1}^J\sum_{j=1}^Lm_{Q_i\times I_j}(f_{k})\chi_{Q_i\times I_j}(x,t)\Big|\lesssim\sup_{k\in \mathbb{Z}}\sup_{t\in [1,\,2]}|f_k(x,\,t)|,\end{aligned}$$ which implies that for $p_1\in (1,\,\infty)$, $$\begin{aligned} \|\Phi_{\epsilon}(\vec{f})\|_{L^{p_1}(L^{\infty}([1,\,2]),\,l^{\infty};\,\mathbb{R}^n,\,w)}\lesssim \|\vec{f}\|_{L^{p_1}(L^{\infty}([1,\,2]),\,l^{\infty};\,\mathbb{R}^n,\,w)}.\end{aligned}$$ We also have that for $p_0=p-\gamma$,$$|m_{Q_i\times I_j}(f_k)|\leq \Big(\frac{1}{|Q_i||I_j|}\int_{I_j}\int_{Q_i}|f_k(y,t)|^{p_0}w(y){\rm d}y{\rm d}t\Big)^{\frac{1}{p_0}}\Big(\frac{1}{|Q_i|}\int_{Q_i}w^{-\frac{1}{p_0-1}}(y){\rm d}y\Big)^{\frac{1}{p_0'}} ,$$and so $$\begin{aligned} \|\Phi_{\epsilon}(\vec{f})\|_{L^{p_0}(L^{p_0}([1,\,2]),\,l^{p_0};\,\mathbb{R}^n,\,w)}\lesssim \|\vec{f}\|_{L^{p_0}(L^{p_0}([1,\,2]),\,l^{p_0};\,\mathbb{R}^n,\,w)}.\end{aligned}$$ By interpolation, we can deduce from (3.2) and (3.3) that in this case $$\begin{aligned} \|\Phi_{\epsilon}(\vec{f})\|_{L^{p}(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)}\lesssim\|\vec{f}\|_{L^{p}(L^2([0,\,1]),\,l^2,\,\mathbb{R}^n,\,w)}^{p}.\end{aligned}$$ Our claim then follows directly, and so $\Phi_{\epsilon}(\mathcal{G})=\{\Phi_{\epsilon}(\vec{f}): \vec{f}\in \mathcal{G}\}$ is strongly pre-compact in $L^p(L^2([1,\,2]),l^{2};\mathbb{R}^n,\,w)$. We now verify (3.1). Denote $\mathcal{D}=\cup_{i=1}^JQ_i$ and write $$\begin{aligned} &&\big\|\vec{f}\chi_{\mathcal{D}}-\Phi_{\epsilon}(\vec{f}) \big\|_{L^p(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)}\\&&\quad\leq \Big\|\Big(\int^2_1\sum_{|k|\leq N}\Big|f_k(\cdot,\,t)\chi_{\mathcal{D}}-\sum_{i=1}^{J}\sum_{j=1}^{L}m_{Q_i\times I_j} (f_k)\chi_{Q_i\times I_j}(x,t)\Big|^2{\rm d}t\Big)^{1/2}\Big\|_{L^p(\mathbb{R}^n,\,w)}\\ &&\qquad+\Big\|\Big(\int^2_1\sum_{|k|>N}\big|f_k(\cdot,\,t)\big|^2\Big)^{\frac{1}{2}}\chi_{B(0,\,2A)}\Big\|_{L^p(\mathbb{R}^n,\,w)}.\end{aligned}$$ Noting that for $x\in Q_i$ with $1\leq i\leq J$, $$\begin{aligned} &&\Big\{\int^2_1\sum_{|k|\leq N}\big|f_k(x,\,t)\chi_{\mathcal{D}}(x)-\sum_{u=1}^{J}\sum_{v=1}^Lm_{Q_u\times I_v}(f_k)\chi_{Q_u\times I_v}(x,t)\big|^2{\rm d}t\Big\}^{\frac{1}{2}}\\ &&\quad\lesssim |Q|^{-1/2}|I|^{-1/2} \Big\{\sum_{j=1}^L\int_{I_j}\int_{Q_i}\int_{I_j}\sum_{|k|\leq N}\big|f_k(x,\,t)-f_k(y,\,s)\big|^2\,{\rm d}y {\rm d}s{\rm d}t\Big\}^{\frac{1}{2}}\\ &&\quad\lesssim |Q|^{-1/2}\Big\{\int_{2Q}\int_{1}^2 \sum_{|k|\leq N}|f_k(x,\,s)-f_k(x+h,\,s)|^2{\rm d}s\,{\rm d}h\Big\}^{\frac{1}{2}}\\ &&\qquad+ |I|^{-1/2}\Big\{\sum_{j=1}^L\int_{I_j}\int_{I_j} \sum_{|k|\leq N}|f_k(x,\,t)-f_k(x,\,s)|^2{\rm d}t\,{\rm d}s\Big\}^{\frac{1}{2}}\\ &&\quad\lesssim\sup_{|h|\leq \varrho}\Big(\int^2_1 \sum_{|k|\leq N}|f_k(x,\,t)-f_k(x+h,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\\ &&\qquad+\sup_{|s|\leq \sigma}\Big(\int^2_1 \sum_{|k|\leq N}|f_k(x,\,t+s)-f_k(x,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}},\end{aligned}$$ we then get that $$\begin{aligned} \sum_{i=1}^J\int_{Q_i}\Big\{\int^2_1 \sum_{|k|\leq N}\big|f_k(x,\,t)-\sum_{l=1}^{J}m_{Q_l}(f_{k}) \chi_{Q_l}(x)\big|^2\,{\rm d}t\Big\}^{p/2}w(x)\,{\rm d}x \lesssim 2\epsilon.\end{aligned}$$ It then follows from the assumption (b) that for all $\vec{f}\in \mathcal{G}$, $$\begin{aligned} \|\vec{f}-\Phi_{\epsilon}(\vec{f})\|_{L^p(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)}&\leq& \big\|\vec{f}\chi_{\mathcal{D}}-\Phi_{\epsilon}(\vec{f}) \big\|_{L^p(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)}\\ &+&\Big\|\Big(\int^2_1\sum_{k\in\mathbb{Z}}|f_k(\cdot,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\chi_{\{|\cdot|>A\}}(\cdot)\Big\|_{L^p(\mathbb{R}^n,w)}\\ &<&3\epsilon. \end{aligned}$$ Noting that $$\begin{aligned} \|\vec{f}-\vec{g}\|_{L^p(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)}&\leq& \|\vec{f}-\Phi_{\epsilon}(\vec{f})\|_{L^p(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)}\\ &&+\|\Phi_{\epsilon}(\vec{f})-\Phi_{\epsilon}(\vec{g})\|_{L^p(L^2([1,\,2]),l^{2};\mathbb{R}^n,w)}\\ &&+\|\vec{g}-\Phi_{\epsilon}(\vec{g})\|_{L^p(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)},\end{aligned}$$ we then get (3.1) and finish the proof of Lemma \[l3.4\]. [*Proof of Theorem \[t1.2\]*]{}. Let $j_0\in\mathbb{Z}_-$, $b\in C^{\infty}_0(\mathbb{R}^n)$ with ${\rm supp}\, b\subset B(0,\,R)$, $p$ and $w$ be the same as in Theorem \[t1.2\]. Without loss of generality, we may assume that $\|b\|_{L^{\infty}(\mathbb{R}^n)}+\|\nabla b\|_{L^{\infty}(\mathbb{R}^n)}=1.$ We claim that - for each fixed $\epsilon>0$, there exists a constant $A>0$ such that $$\Big\|\Big(\int^2_1\sum_{j\in\mathbb{Z}}|F_{j,\,b}^{l}f(x,\,t)|^2{\rm d}t\Big)^{1/2}\chi_{\{|\cdot|>A\}}(\cdot)\Big\|_{L^p(\mathbb{R}^n,\,w)}<\epsilon\|f\|_{L^p(\mathbb{R}^n,\,w)};$$ - for $s\in (1,\,\infty)$,$$\begin{aligned} && \Big(\int^2_1 \sum_{j>j_0}|F_{j,\,b}^{l}f(x,\,t)-F_{j,\,b}^{l}f(x+h,\,t)|^2{\rm d}t\Big)^{1/2}\\ &&\quad\lesssim 2^{-j_0}|h|\Big(\widetilde{\mathcal{M}}_{\Omega}^{l,\,j_0}f(x)+2^{l(n+1)}M_sf(x)\Big);\nonumber\end{aligned}$$ - for each $\epsilon>0$ and $N\in\mathbb{N}$, there exists a constant $\sigma\in (0,\,1/2)$ such that $$\begin{aligned} &&\Big\|\sup_{|s|\leq \sigma}\Big(\int^2_1 \sum_{|j|\leq N}|F_{j,b}^{l}f(x,s+t)-F_{j,b}^{l}f(x,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^{p}(\mathbb{R}^n,w)}\\ &&\quad< \epsilon\|f\|_{L^p(\mathbb{R}^n,w)};\nonumber\end{aligned}$$ - for each fixed $D>0$ and $\epsilon>0$, there exists $N\in\mathbb{N}$ such that $$\begin{aligned} \Big\|\Big(\int^2_1\sum_{j>N}|F_{j,\,b}^{l}f(\cdot,\,t)|^2{\rm d}t\Big)^{1/2}\chi_{B(0,\,D)}\Big\|_{L^p(\mathbb{R}^n,\,w)}<\epsilon\|f\|_{L^p(\mathbb{R}^n,\,w)}.\end{aligned}$$ We now prove claim (i). Let $t\in [1,\,2]$. For each fixed $x\in\mathbb{R}^n$ with $|x|>4R$, observe that ${\rm supp}\,K_t^j*\phi_{j-l}\subset \{2^{j-2}\leq|y|\leq 2^{j+2}\}$, and $\int_{|z|<R}\big|K_{t}^j*\phi_{j-l}(x-z)\big|{\rm d}z\not =0$ only if $2^j\approx |x|$. A trivial computation shows that $$\begin{aligned} \int_{|z|<R}\big|K_{t}^j*\phi_{j-l}(x-z)\big|{\rm d}z&\lesssim&\Big(\int_{|z|<R}\big|K_{t}^j*\phi_{j-l}(x-z)\big|^2{\rm d}z\Big)^{\frac{1}{2}}R^{\frac{n}{2}}\\ &\lesssim&\Big(\int_{\frac{|x|}{2}\leq |z|<2|x|}\big|K_{t}^j*\phi_{j-l}(z)\big|^2{\rm d}z\Big)^{\frac{1}{2}}R^{\frac{n}{2}}\\ &\lesssim&\|K_t^j\|_{L^1(S^{n-1})}\|\phi_{j-l}\|_{L^2(\mathbb{R}^n)}R^{\frac{n}{2}}\\ &\lesssim&2^{nl/2}|x|^{-\frac{n}{2}}R^{\frac{n}{2}}.\end{aligned}$$On the other hand, we have that $$\begin{aligned} &&\sum_{j\in\mathbb{Z}}\Big(\int_{|y|<R}|K_{t}^j*\phi_{j-l}(x-y)||f(y)|^s{\rm d}y\Big)^{\frac{1}{s}}\\ &&\quad=\sum_{j\in\mathbb{Z}:\, 2^j\approx |x|}\Big(\int_{|x|/2\leq |y-x|\leq 2|x|}|K_{t}^j*\phi_{j-l}(x-y)||f(y)|^s{\rm d}y\Big)^{\frac{1}{s}}\\ &&\quad\lesssim\Big(M_{\Omega}M(|f|^s)(x)\Big)^{1/s}.\end{aligned}$$ Another application of the Hölder inequality then yields $$\begin{aligned} \sum_{j\in\mathbb{Z}}|F_{j,\,b}^{l}f(x,\,t)\big|^2&\lesssim& \sum_{j\in\mathbb{Z}}\Big(\int_{|y|<R}|K_{t}^j*\phi_{j-l}(x-y)||f(y)|^s{\rm d}y\Big)^{2/s}\\ &&\qquad\times\Big(\int_{|y|<R}|K_{t}^j*\phi_{j-l}(x-y)|{\rm d}y\Big)^{2/s'}\nonumber\\ &\lesssim &2^{\frac{nl}{s'}}|x|^{-\frac{n}{s'}}R^{\frac{n}{s'}}\Big(M_{\Omega}M(|f|^s)(x)\Big)^{2/s}.\nonumber\end{aligned}$$ This, in turn leads to our claim (i). We turn our attention to claim (ii). Write $$\begin{aligned} |F_{j,\,b}^{l}f(x,\,t)-F_{j,\,b}^{l}f(x+h,\,t)|\leq|b(x)-b(x+h)||F_j^lf(x,\,t)|+{\rm J}^{l}_jf(x,\,t),\end{aligned}$$ with$${\rm J}_{j}^{l}f(x,t)=\Big|\int_{\mathbb{R}^n}\big(K_{t}^j*\phi_{j-l}(x-y)-K_{t}^j*\phi_{j-l}(x+h-y)\big)\big(b(x+h)-b(y)\big)f(y){\rm d}y\Big|.$$ It follows from Lemma 3.1 that $$\begin{aligned} \Big(\sum_{j>j_0}|{\rm J}_{j}^{l}f(x,t)|^2\Big)^{\frac{1}{2}} &\lesssim&\sum_{j>j_0}\int_{\mathbb{R}^n}\big|K_{t}^j*\phi_{j-l}(x-y)-K_{t}^j*\phi_{j-l}(x+h-y)\big||f(y)|{\rm d}y\\ &\lesssim&2^{l(n+1)}|h|2^{-j_0}M_sf(x).\end{aligned}$$ Therefore, $$\begin{aligned} && \Big(\int^2_1 \sum_{j>j_0}|F_{j,\,b}^{l}f(x,\,t)-F_{j,\,b}^{l}f(x+h,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\\ &&\quad\lesssim|h|\widetilde{\mathcal{M}}_{\Omega}^{l,\,j_0}f(x)+2^{l(n+1)}2^{-j_0}|h|M_sf(x).\nonumber\end{aligned}$$ We now verify claim (iii). For each fixed $\sigma\in (0,\,1/2)$ and $t\in [1,\,2]$, let $$U_{t,\,\sigma}^j(z)=\frac{1}{2^j}\frac{|\Omega(z)|}{|z|^{n-1}}\chi_{\{2^j(t-\sigma)\leq |z|\leq 2^jt\}}+\frac{1}{2^j}\frac{|\Omega(z)|}{|z|^{n-1}}\chi_{\{2^{j+1}t\leq |z|\leq 2^{j+1}(t+\sigma)\}},$$ and $$G_{l,\,t,\,\sigma}^jf(x)=\int_{\mathbb{R}^n}\big(U_{t,\,\sigma}^j*|\phi_{j-l}|\big)(x-y)|f(y)|{\rm d}y.$$ Note that for $t\in [1,\,2]$, $$\|U_{t,\,\sigma}^j*|\phi_{j-l}|\|_{L^1(\mathbb{R}^n)}\lesssim \sigma, \,\,\,\sup_{|j|\leq N}\sup_{t\in [1,\,2]}|G_{l,\,t,\,\sigma}^jf(x)|\lesssim MM_{\Omega}f(x).$$ Thus, $$\begin{aligned} \Big\|\sup_{|j|\leq N}\sup_{t\in [1,\,2]}|G_{l,\,t,\,\sigma}^jf|\Big\|_{L^{\infty}(\mathbb{R}^n)}\lesssim \sigma\|f\|_{L^{\infty}(\mathbb{R}^n)},\end{aligned}$$ and $$\begin{aligned} \qquad\Big\|\sup_{|j|\leq N}\sup_{t\in [1,\,2]}|G_{l,\,t,\sigma}^jf|\Big\|_{L^{p}(\mathbb{R}^n,w)}\lesssim\|MM_{\Omega}\|_{L^{p}(\mathbb{R}^n,w)}\lesssim \|f\|_{L^{p}(\mathbb{R}^n,\,w)}.\end{aligned}$$ Interpolating the estimates (3.8) and (3.9) shows that if $p_1\in (p,\,\infty)$, $$\begin{aligned} \Big\|\sup_{|j|\leq N}\sup_{t\in [1,\,2]}|G_{l,\,t,\,\sigma}^jf|\Big\|_{L^{p_1}(\mathbb{R}^n,\,w)}\lesssim \sigma^{1-p/p_1}\|f\|_{L^{p_1}(\mathbb{R}^n,\,w)}.\end{aligned}$$ On the other hand, if $p_0\in (1,\,p)$, it then follows from the weighted estimae $M$ and $M_{\Omega}$ that $$\begin{aligned} \int_{\mathbb{R}^n}\int^2_1\sum_{|j|\leq N}\big|G_{l,\,t,\,\sigma}^jf(x)\big|^{p_0}{\rm d}tw(x){\rm d}x \lesssim N\|f\|_{L^{p_0}(\mathbb{R}^n,\,w)}^{p_0}.\end{aligned}$$ Choosing $p_1\in (2,\,\infty)$ such that $1/p=1/2+(2-p_0)/(2p_1)$ in (3.10), we get from (3.10) and (3.11) that for $p\in (1,\,2)$, $$\begin{aligned} \Big\|\Big(\int^2_1\sum_{|j|\leq N}\big|G_{l,\,t,\,\sigma}^jf(x)\big|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^p(\mathbb{R}^n,\,w)} \lesssim N\sigma^{\tau_1}\|f\|_{L^{p}(\mathbb{R}^n,\,w)}.\end{aligned}$$ with $\tau_1\in (0,\,1)$ a constant. If $p\in [2,\,\infty)$, we obtain from the Minkowski inequality and the Young inequality that $$\begin{aligned} &&\Big\|\Big(\int^2_1\sum_{|j|\leq N}|G_{l,\,t,\,\sigma}^jf(x)|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^p(\mathbb{R}^n,\,w)}^2\\ &&\quad\lesssim\Big\{\int_{\mathbb{R}^n}\Big(\int^2_1\Big(\sum_{|j|\leq N}\int_{\mathbb{R}^n}\big(U_{l,t,\sigma}^j*|\phi_{j-l}|\big)(x-y)|f(y)|{\rm d}y\Big)^2{\rm d}t\Big)^{\frac{p}{2}}w(x){\rm d}x\Big\}^{\frac{2}{p}}\nonumber\\ &&\quad\lesssim\int^2_1\Big\{\sum_{|j|\leq N}\Big(\int_{\mathbb{R}^n}\Big(\int_{\mathbb{R}^n}\big(U_{l,\,t,\sigma}^j*|\phi_{j-l}|\big)(x-y)|f(y)|{\rm d}y\Big)^pw(x){\rm d}x\Big)^{\frac{1}{p}}\Big\}^2{\rm d}t\nonumber\\ &&\quad\lesssim N^2\|f\|_{L^p(\mathbb{R}^n,\,w)}^2.\nonumber\end{aligned}$$ Also, we have that $$\begin{aligned} &&\Big\{\int_{\mathbb{R}^n}\Big(\int^2_1\sum_{|j|\leq N}\Big(\int_{\mathbb{R}^n}\big(U_{l,\,t,\,\sigma}^j*|\phi_{j-l}|\big)(x-y)|f(y)|{\rm d}y\Big)^2{\rm d}t\Big)^{\frac{p}{2}}{\rm d}x\Big\}^{\frac{2}{p}}\\ &&\quad\lesssim\int^2_1\Big\{\sum_{|j|\leq N}\big\|\big(U_{l,\,t,\,\sigma}^j*|\phi_{j-l}|*|f|\big\|_{L^p(\mathbb{R}^n)}\Big\}^2{\rm d}t\nonumber\\ &&\quad\lesssim(2N\sigma)^2\|f\|_{L^p(\mathbb{R}^n)}^2,\,\,p\in [2,\,\infty).\nonumber\end{aligned}$$ The inequalities (3.13) and (3.14), via interpolation with changes of measures, give us that for $p\in [2,\,\infty)$, $$\begin{aligned} \qquad \Big\|\Big(\int^2_1\sum_{|j|\leq N}|G_{l,\,t,\,\sigma}^jf(x)|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^p(\mathbb{R}^n,\,w)}\lesssim N\sigma^{\tau_2}\|f\|_{L^p(\mathbb{R}^n,\,w)},\end{aligned}$$ with $\tau_2\in (0,\,1)$ a constant. Since $$\sup_{|s|\leq \sigma}|F_{j,\,b}^{l}f(x,\,t)-F_{j,\,b}^{l}f(x,\,t+s)|\leq G_{l,\,t,\,\sigma}^jf(x),$$ our claim (iii) now follow from (3.12) and (3.15) immediately if we choose $\sigma=\epsilon/(2N)$. It remains to prove (iv). Let $D>0$ and $N\in\mathbb{N}$ such that $2^{N-2}>D$. Then for $j> N$ and $x\in\mathbb{R}^n$ with $|x|\leq D$, $$\begin{aligned} \int_{\mathbb{R}^n}\big|K_{t}^j*\phi_{j-l}(x-y)f(y)\big|{\rm d}y&\leq&\int_{\mathbb{R}^n} \big|K_{t}^j*\phi_{j-l}(x-y)f(y)\big|\chi_{\{|y|\leq 2^{j+3}\}}{\rm d}y\\ &\lesssim&\int_{|y|\leq 2^{j+3}}|f(y)|{\rm d}y\|K_{t}^j\|_{L^1(\mathbb{R}^n)}\|\phi_{j-l}\|_{L^\infty(\mathbb{R}^n)}\nonumber\\ &\lesssim&2^{nl}2^{-nj/p}\|f\|_{L^p(\mathbb{R}^n)}.\nonumber\end{aligned}$$Therefore, $$\begin{aligned} \qquad\Big\|\Big(\int^2_1\sum_{j>N}\big|F_{j,\,b}^{l}f(\cdot,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\chi_{B(0,\,D)}\Big\|_{L^p(\mathbb{R}^n)} \lesssim 2^{nl}\big(\frac{D}{2^N}\big)^{n/p}\|f\|_{L^p(\mathbb{R}^n)}.\end{aligned}$$ It is obvious that $$\begin{aligned} \Big\|\Big(\int^2_1\sum_{j>N}\big|F_{j,\,b}^{l}f(\cdot,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\chi_{B(0,\,D)}\Big\|_{L^p(\mathbb{R}^n,\,w)} \lesssim l\|f\|_{L^p(\mathbb{R}^n,\,w)}.\end{aligned}$$ Interpolating the inequalities (3.16) and (3.17) yields $$\Big\|\Big(\int^2_1\sum_{j>N}\big|F_{j,\,b}^{l}f(\cdot,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\chi_{B(0,\,D)}\Big\|_{L^p(\mathbb{R}^n,\,w)} \lesssim 2^{\tau_3 nl}\big(\frac{D}{2^N}\big)^{\frac{\tau_3 n}{p}}\|f\|_{L^p(\mathbb{R}^n,\,w)}.$$ with $\tau_3\in (0,\,1)$ a constant depending only on $w$. The claim (iv) now follows directly. We can now conclude the proof of Theorem \[t1.2\]. Let $p\in (1,\,\infty)$. Note that $$\widetilde{\mathcal{M}}_{\Omega,\,b}^{l,\,j_0}f(x)\leq \widetilde{\mathcal{M}}_{\Omega,\,b}^{l}f(x),$$ and so $\widetilde{\mathcal{M}}_{\Omega,\,b}^{l,\,j_0}$ is bounded on $L^p(\mathbb{R}^n,\,w)$. Our claims (i)-(iv), via Lemma \[l3.4\], prove that for $b\in C^{\infty}_0(\mathbb{R}^n)$, $l\in\mathbb{N}$ and $j_0\in\mathbb{Z}_-$, the operator $\mathcal{F}^{l,\,}_{j_0}$ defined by $$\begin{aligned} \mathcal{F}^{l}_{j_0}:\, f(x)\rightarrow \{\dots,\,0,\,\dots,\,F_{j_0,\,b}^{l}f(x,\,t),\,F_{j_0+1,\,b}^lf(x,\,t),\,\dots\}\end{aligned}$$ is compact from $L^p(\mathbb{R}^n,w)$ to $L^p(L^2([1,\,2]),l^2;\,\mathbb{R}^n,w)$. Thus, $\widetilde{\mathcal{M}}_{\Omega,\,b}^{l,\,j_0}$ is completely continuous on $L^p(\mathbb{R}^n,\,w)$. This, via Lemma \[l3.2\] and Theorem \[t2.2\], shows that for $b\in C^{\infty}_0(\mathbb{R}^n)$, $\widetilde{\mathcal{M}}_{\Omega,\,b}$ is completely continuous on $L^p(\mathbb{R}^n,\,w)$. Note that $$\big|\mathcal{M}_{\Omega,\,b}f_k(x)-\mathcal{M}_{\Omega,\,b}f(x)\big|\lesssim \mathcal{M}_{\Omega,\,b}(f_k-f)(x)\lesssim \widetilde{\mathcal{M}}_{\Omega,\,b}(f_k-f)(x).$$ Thus, for $b\in C^{\infty}_0(\mathbb{R}^n)$, $\mathcal{M}_{\Omega,\,b}$ is completely continuous on $L^p(\mathbb{R}^n,\,w)$. Recalling that $\mathcal{M}_{\Omega,\,b}$ is bounded on $L^p(\mathbb{R}^n,\,w)$ with bound $C\|b\|_{{\rm BMO}(\mathbb{R}^n)}$, we obtain that for $b\in {\rm CMO}(\mathbb{R}^n)$, $\mathcal{M}_{\Omega,\,b}$ is completely continuous on $L^p(\mathbb{R}^n,\,w)$. Proof of Theorem \[t1.3\] ========================= The following lemma will be useful in the proof of Theorem \[t1.3\], and is of independent interest. \[l4.1\] Let $u\in (1,\,\infty)$, $m\in \mathbb{N}\cup\{0\}$, $S$ be a sublinear operator which satisfies that $$|Sf(x)|\leq \int_{\mathbb{R}^n}|b(x)-b(y)|^m|W(x-y)f(y)|{\rm d}y,$$ with $b\in {\rm BMO}(\mathbb{R}^n)$, and $$\begin{aligned} \sup_{R>0}R^{n/u}\Big(\int_{R\leq |x|\leq 2R}|W(x)|^{u'}{\rm d}x\Big)^{1/u'}\lesssim 1.\end{aligned}$$ - Let $p\in (u,\,\infty)$, $\lambda\in (0,\,1)$ and $w\in A_{p/u}(\mathbb{R}^n)$. If $S$ is bounded on $L^p(\mathbb{R}^n,\,w)$ with bound $D\|b\|_{{\rm BMO}(\mathbb{R}^n)}^m$, then for some $\varepsilon\in (0,\,1)$,$$\big\|Sf\big\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}\lesssim (D+D^{\varepsilon})\|b\|_{{\rm BMO}(\mathbb{R}^n)}^m\|f\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}.$$ - Let $p\in (1,\,u)$, $w^{r}\in A_1(\mathbb{R}^n)$ for some $r\in (u,\,\infty)$ and $\lambda\in (0,\,1-r'/u')$. If $S$ is bounded on $L^p(\mathbb{R}^n,\,w)$ with bound $D$, then for some $\varepsilon\in (0,\,1)$,$$\big\|Sf\big\|_{L^{p,\,\lambda}(\mathbb{R}^n)}\lesssim(D+ D^{\varepsilon})\|b\|_{{\rm BMO}(\mathbb{R}^n)}^m\|f\|_{L^{p,\,\lambda}(\mathbb{R}^n)}.$$ For simplicity, we only consider the case of $m=1$ and $\|b\|_{{\rm BMO}(\mathbb{R}^n)}=1$. For fixed ball $B$ and $f\in L^{p,\,\lambda}(\mathbb{R}^n,\,w)$, decompose $f$ as $$f(y)=f(y)\chi_{2B}(y)+\sum_{k=1}^{\infty}f(y)\chi_{2^{k+1}B\backslash 2^kB}(y)=\sum_{k=0}^{\infty}f_k(y).$$ It is obvious that $$\begin{aligned} \int_{B}|Sf_0(y)|^pw(y){\rm d}y\lesssim D^p\int_{2B}|f(y)|^pw(y)\,{\rm d}y\lesssim D^p\|f\|_{L^{p,\lambda}(\mathbb{R}^n,w)}^p\{w(B)\}^{\lambda}.\end{aligned}$$ Let $\theta\in (1,\,p/u)$ such that $w\in A_{p/(\theta u)}(\mathbb{R}^n)$. For each $k\in\mathbb{N}$, let $S_kf(x)=S\big(f\chi_{2^{k+1}B\backslash 2^kB}\big)(x)$. Then $S_k$ is also sublinear. We have by the Hölder inequality that for each $x\in B$, $$\begin{aligned} |S_kf(x)|&\lesssim&|b(x)-m_B(b)|\|f_k\|_{L^u(\mathbb{R}^n)}\Big(\int_{2^kB}|W(x-y)|^{u'}{\rm d}y\Big)^{1/u'}\\ &&+\big\|\big(b-m_B(b)\big)f_k\big\|_{L^u(\mathbb{R}^n)}\Big(\int_{2^kB}|W(x-y)|^{u'}{\rm d}y\Big)^{1/u'}\\ &\lesssim&|b(x)-m_B(b)|\|f_k\|_{L^p(\mathbb{R}^n,w)}\Big(\int_{2^kB}w^{-\frac{1}{p/u-1}}(y){\rm d}y\Big)^{\frac{1}{u(p/u)'}}|2^kB|^{-\frac{1}{u}}\\ &&+\Big(\int_{2^{k+1}B}|b(y)-m_B(b)|^{p\theta '}{\rm d}y\Big)^{1/(p\theta')}\|f_k\|_{L^p(\mathbb{R}^n,w)}\\ &&\quad\times\Big(\int_{2^kB}w^{-\frac{1}{p/(\theta u)-1}}(y){\rm d}y\Big)^{\frac{1}{u\big(p/(\theta u)\big)'}}|2^kB|^{-\frac{1}{u}},\end{aligned}$$ here, $m_B(b)$ denotes the mean value of $b$ on $B$. It follows from the John-Nirenberg inequality that $$\Big(\int_{2^{k+1}B}|b(y)-m_B(b)|^{p\theta '}{\rm d}y\Big)^{\frac{1}{p\theta '}}\lesssim k|2^kB|^{\frac{1}{p\theta'}}.$$Therefore, for $q\in (1,\,\infty)$ and $k\in\mathbb{N}$, we have $$\begin{aligned} \big\|S_kf\big\|_{L^q(B,\,w)}\lesssim k\{w(B)\}^{\frac{1}{q}-\frac{1}{p}}\Big(\frac{w(B)}{w(2^kB)}\Big)^{1/p}\|f_k\|_{L^p(\mathbb{R}^n,\,w)}.\end{aligned}$$ On the other hand, we deduce from the $L^p(\mathbb{R}^n,\,w)$ boundedness of $S$ that $$\begin{aligned} \int_{B}|S_kf(y)|^pw(y)\,{\rm d}y&\lesssim& D^p\int_{2^kB}|f(x)|^pw(x)\,{\rm d}x\end{aligned}$$ We then get from (4.2) (with $q=p$) and (4.3) that for $\sigma\in (0,\,1)$, $$\begin{aligned} \int_{B}|S_kf(y)|^pw(y)\,{\rm d}y&\lesssim&k^p D^{p(1-\sigma)}\big(\frac{w(B)}{w(2^kB)}\big)^{\sigma}\int_{2^kB}|f(x)|^pw(x)\,{\rm d}x\end{aligned}$$ Recall that $w\in A_{p/u}(\mathbb{R}^n)$. Thus, there exists a constant $\tau\in (0,\,1)$, $$\frac{w(B)}{w(2^kB)}\lesssim \big(\frac{|B|}{|2^kB|}\big)^{\tau},$$see [@gra2]. For fixed $\lambda\in (0,\,1)$, we choose $\sigma$ sufficiently close to $1$ such that $0<\lambda<\sigma$. It then follows from (4.4) that $$\begin{aligned} \sum_{k=0}^{\infty}\Big(\int_{B}|Sf_k(y)|^p{\rm d}y\Big)^{\frac{1}{p}} &\lesssim &D^{1-\sigma}\{w(B)\}^{\frac{\lambda}{p}}\sum_{k=0}^{\infty}(k+1)2^{-kn\tau(\sigma-\lambda)/p} \|f\|_{L^{p,\lambda}(\mathbb{R}^n,w)}\\ &\lesssim&D^{1-\sigma}\{w(B)\}^{\lambda/p}\|f\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}.\end{aligned}$$ THis leads to the conclusion (a). Now we turn our attention to conclusion (b). From (4.1), it is obvious that for $y\in 2^{k+1}B\backslash 2^kB$, $$\int_{B}|W(x-y)||b(x)-m_B(b)|w(x){\rm d}x\lesssim|2^{k}B|^{-1/u}|B|^{\frac{1}{u\vartheta'}}\Big(\int_{B}w^{u\vartheta}(x){\rm d}x\Big)^{\frac{1}{u\vartheta}},$$ with $\vartheta\in (1,\,\infty)$ small enough such that $w^{u\vartheta}\in A_1(\mathbb{R}^n)$. This, in turn implies that $$\begin{aligned} &&\int_{B}\int_{2^{k+1}B\backslash 2^kB}|W(x-y)h(y)|{\rm d}y|b(x)-m_B(b)|w(x){\rm d}x\\ &&\quad\lesssim 2^{kn/u'}\frac{w(B)}{w(2^kB)} \int_{2^{k}B}h(y)w(y)\,{\rm d}y.\end{aligned}$$ Therefore, for $s\in (1,\,\infty)$, $$\begin{aligned} \quad\int_{B}|S_kf(x)|w(x){\rm d}x&\lesssim&2^{kn/u'}\frac{w(B)}{w(2^kB)}\int_{2^kB}|f(x)|w(x)\,{\rm d}x\\ &&+2^{kn/u'}\frac{w(B)}{w(2^kB)}\int_{2^kB}|f(x)||b(x)-m_B(b)|w(x)\,{\rm d}x\nonumber\\. &\lesssim&k2^{\frac{kn}{u'}}\frac{w(B)}{w(2^{k+1}B)}\Big(\int_{2^kB}|f(x)|^sw(x){\rm d}x\Big)^{\frac{1}{s}}\big\{w(2^kB)\big\}^{\frac{1}{s'}}.\nonumber\end{aligned}$$ Also, we get by (4.2) that for $q\in (u,\,\infty)$ and $\theta\in (0,\,1)$ with $\theta q\in (u,\,\infty)$, $$\begin{aligned} \big\|S_kf\big\|_{L^q(B,\,w)}\lesssim k\{w(B)\}^{\frac{1}{q}-\frac{1}{\theta q}}\Big(\frac{w(B)}{w(2^kB)}\Big)^{\frac{1}{\theta q}}\|f\|_{L^{\theta q}(2^{k+1}B,\,w)}.\end{aligned}$$ For $p\in (1,\,\infty)$, we choose $q\in (u,\,\infty)$ and $\theta\in (0,\,1)$, $s\in (1,\,\infty)$ which is close to $1$ sufficiently such that $1/p=t+(1-t)/q$ and $1/p=t/s+(1-t)/(\theta q)$, with $t\in (0,\,1/p)$. By interpolating, we obtain from the inequalities (4.5) and (4.6) that $$\begin{aligned} \|S_kf\|_{L^p(\mathbb{R}^n,\,w)}\lesssim k2^{\frac{kn}{pu'}}\Big(\frac{w(B)}{w(2^kB)}\Big)^{1/p}\|f\|_{L^p(2^kB,\,w)}.\end{aligned}$$ The fact that $w^{r}\in A_1(\mathbb{R}^n)$ tells us that $$\frac{w(B)}{w(2^kB)}\lesssim 2^{-kn(r-1)/r},$$ see [@gra2 p. 306]. This, together with the fact that $S$ is bounded on $L^p(\mathbb{R}^n,\,w)$ with bound $D$, gives us that for any $\omega\in (0,\,1)$, $$\begin{aligned} \Big(\int_{B}|S_kf(x)|^pw(x){\rm d}x\Big)^{1/p}&\lesssim& D^{1-\omega}k2^{\frac{\omega kn}{pu'}}\Big(\frac{w(B)}{w(2^kB)}\Big)^{\omega/p}\|f\|_{L^p(2^kB,\,w)}\\ &\lesssim&\{w(B)\}^{\lambda/p} D^{1-\omega}k2^{\frac{kn}{p}\big(\frac{\omega}{u'}-\frac{\omega-\lambda}{r'}\big)}\|f\|_{L^{p,\lambda}(\mathbb{R}^n,w)}.\end{aligned}$$ For fixed $\lambda\in (0,\,1-r'/u')$, we choose $\omega\in (\lambda,\,1)$ sufficiently close to $1$ such that $\omega/u'-(\omega-\lambda)/r'<0$. Summing over the last inequality yields conclusion (b). Let $p,\,r\in[1,\,\infty)$, $\lambda\in (0,\,1)$, $q\in [1,\,\infty]$ and $w$ be a weight. Define the space $L^{p,\,\lambda}(L^q([1,\,2]),\,l^r;\,\mathbb{R}^n,\,w)$ by $$L^{p,\,\lambda}(L^q([1,\,2]),\,l^r;\,\mathbb{R}^n,\,w)=\big\{\vec{f}=\{f_k\}_{k\in \mathbb{Z}}:\, \|\vec{f}\|_ {L^{p,\,\lambda}(L^q([1,\,2]),\,l^r;\,\mathbb{R}^n,\,w)}<\infty\big\},$$ with $$\|\vec{f}\|_{L^{p,\,\lambda}(L^q([1,\,2]),\,l^r;\,\mathbb{R}^n,\,w)}=\Big\|\Big(\int^2_1\Big(\sum_{k\in \mathbb{Z}}|f_k(x,\,t)|^r\Big)^{\frac{q}{r}}{\rm d}t\Big)^{\frac{1}{q}}\Big\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}.$$ With usual addition and scalar multiplication, $L^{p,\,\lambda}(L^q([1,\,2]),\,l^{r};\,\mathbb{R}^n,\,w)$ is a Banach space. \[l4.2\] Let $p\in (1,\,\infty)$, $\lambda\in (0,\,1)$ and $w\in A_p(\mathbb{R}^n)$, $\mathcal{G}$ be a subset in $ L^{p,\,\lambda}(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)$. Suppose that $\mathcal{G}$ satisfies the following five conditions: - $\mathcal{G}$ is a bounded set in $L^{p,\,\lambda}(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)$; - for each fixed $\epsilon>0$, there exists a constant $A>0$, such that for all $\{f_k\}_{k\in\mathbb{Z}}\in \mathcal{G}$, $$\Big\|\Big(\int^2_1\sum_{k\in\mathbb{Z}}|f_k(\cdot,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\chi_{\{|\cdot|>A\}}(\cdot)\Big\|_{ L^{p,\,\lambda}(\mathbb{R}^n,\,w)}<\epsilon;$$ - for each fixed $\epsilon>0$ and $N\in\mathbb{N}$, there exists a constant $\varrho>0$, such that for all $\vec{f}=\{f_k\}_{k\in\mathbb{Z}}\in \mathcal{G}$, $$\Big\|\sup_{|h|\leq \varrho}\Big(\int^2_1 \sum_{|k|\leq N}|f_k(\cdot,\,t)-f_k(\cdot+h,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}< \epsilon;$$ - for each fixed $\epsilon>0$ and $N\in\mathbb{N}$, there exists a constant $\sigma\in (0,\,1/2)$ such that for all $\vec{f}=\{f_k\}_{k\in\mathbb{Z}}\in \mathcal{G}$, $$\Big\|\sup_{|s|\leq \sigma}\Big(\int^2_1 \sum_{|k|\leq N}|f_k(\cdot,\,t+s)-f_k(\cdot,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^{p,\lambda}(\mathbb{R}^n,\,w)}< \epsilon,$$ - for each fixed $D>0$ and $\epsilon>0$, there exists $N\in\mathbb{N}$ such that for all $\{f_k\}_{k\in\mathbb{Z}}\in \mathcal {G}$, $$\Big\|\Big(\int^2_1\sum_{|k|>N}|f_k(\cdot,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\chi_{B(0,\,D)}\Big\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}<\epsilon.$$ Then $\mathcal{G}$ is strongly pre-compact in $L^{p,\,\lambda}(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)$. The proof is similar to the proof of Lemma \[l3.4\], and so we only give the outline here. It suffices to prove that, for each fixed $\epsilon>0$, there exists a $\delta=\delta_{\epsilon}>0$ and a mapping $\Phi_{\epsilon}$ on $L^{p,\,\lambda}(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)$, such that $\Phi_{\epsilon}(\mathcal{G})=\{\Phi_{\epsilon}(\vec{f}):\,\vec{f}\in \mathcal G\}$ is a strongly pre-compact set in $L^{p,\,\lambda}(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)$, and for any $\vec{f}$, $\vec{g}\in \mathcal{G}$, $$\|\Phi_{\epsilon}(\vec{f})-\Phi_{\epsilon}(\vec{g})\|_{L^{p,\,\lambda}(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)}<\delta\Rightarrow \|\vec{f}-\vec{g}\|_{L^{p,\,\lambda}(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w))}<8\epsilon.$$ For fixed $\epsilon>0$, we choose $A>1$ large enough as in assumption (b), and $N\in\mathbb{N}$ such that for all $\{f_k\}_{k\in\mathbb{Z}}\in \mathcal {G}$, $$\Big\|\Big(\int^2_1\sum_{|k|>N}|f_k(\cdot,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\chi_{B(0,\,2A)}\Big\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}<\epsilon.$$ Let $Q$, $Q_1,\,\dots,\,Q_J$, $\mathcal{D}$, $I_1,\,\dots,\,I_L\subset [1,\,2]$, and $\Phi_{\epsilon}$ be the same as in the proof of Lemma \[l3.2\]. For such fixed $N$, let $\varrho$ and $\sigma\in (0,\,1/2)$ small enough such that for all $\vec{f}=\{f_k\}_{k\in\mathbb{Z}}\in \mathcal{G}$, $$\begin{aligned} \Big\|\sup_{|h|\leq \varrho}\Big(\int^2_1 \sum_{|k|\leq N}|f_k(\cdot,\,t)-f_k(\cdot+h,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}< \frac{\epsilon}{2J};\end{aligned}$$ $$\begin{aligned} \Big\|\sup_{|s|\leq \sigma}\Big(\int^2_1 \sum_{|k|\leq N}|f_k(\cdot,\,t+s)-f_k(\cdot,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^{p,\lambda}(\mathbb{R}^n,\,w)}< \frac{\epsilon}{2J},\end{aligned}$$ We can verify that $\Phi_{\epsilon}$ is bounded on $L^{p,\,\lambda}(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)$, and consequently, $\Phi_{\epsilon}(\mathcal{G})=\{\Phi_{\epsilon}(\vec{f}):\, \vec{f}\in \mathcal{G}\}$ is a strongly pre-compact set in $L^{p,\lambda}(L^2([1,\,2]),l^{2};\,\mathbb{R}^n,w)$. Recall that for $x\in Q_i$ with $1\leq i\leq J$, $$\begin{aligned} &&\Big\{\int^2_1\sum_{|k|\leq N}\big|f_k(x,\,t)-\sum_{v=1}^Lm_{Q_i\times I_v}(f_k)\chi_{I_v}(t)\big|^2{\rm d}t\Big\}^{\frac{1}{2}}\\ &&\quad\lesssim\sup_{|h|\leq \varrho}\Big(\int^2_1 \sum_{|k|\leq N}|f_k(x,\,t)-f_k(x+h,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\\ &&\qquad+\sup_{|s|\leq \sigma}\Big(\int^2_1 \sum_{|k|\leq N}|f_k(x,\,t+s)-f_k(x,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}.\end{aligned}$$ For a ball $B(y,\,r)$, a trivial computation involving (4.7) and (4.8), leads to that $$\begin{aligned} &&\int_{B(y,\,r)}\Big(\int^2_1\sum_{|k|\leq N}\big|f_k(x,\,t)\chi_{\mathcal{D}}-\sum_{i=1}^{J}\sum_{j=1}^{L}m_{Q_i\times I_j} (f_k)\chi_{Q_i\times I_j}(x,t)\big|^2{\rm d}t\Big)^{\frac{p}{2}}w(x){\rm d}x\\ &&\quad=\sum_{i=1}^J\int_{B(y,\,r)\cap Q_i}\Big(\int^2_1\sum_{|k|\leq N}\big|f_k(x,\,t)-\sum_{i=1}^{J}m_{Q_i\times I_j} (f_k)\chi_{I_j}(t)\big|^2{\rm d}t\Big)^{\frac{p}{2}}w(x){\rm d}x\\ &&\quad\lesssim \epsilon\{w(B(y,\,r))\}^{\lambda}.\end{aligned}$$ Therefore, $$\begin{aligned} &&\int_{B(y,\,r)}\big\|\vec{f}\chi_{\mathcal{D}}-\Phi_{\epsilon}(\vec{f}) \big\|_{L^2([1,\,2]),\,l^{2})}^pw(x) {\rm d}x\\ && \lesssim \int_{B(y,\,r)}\Big(\int^2_1\sum_{|k|\leq N}\big|f_k(x,\,t)\chi_{\mathcal{D}}-\sum_{i=1}^{J}\sum_{j=1}^{L}m_{Q_i\times I_j} (f_k)\chi_{Q_i\times I_j}(x,t)\big|^2{\rm d}t\Big)^{\frac{p}{2}}w(x){\rm d}x\\ &&\quad+\int_{B(y,\,r)}\Big(\int^2_1\sum_{|k|>N}\big|f_k(x,\,t)\big|^2\Big)^{p/2}\chi_{B(0,\,2A)}(x) w(x){\rm d}x\\ &&\lesssim 2\epsilon\{w(B(y,\,r))\}^{\lambda}.\end{aligned}$$ It then follows from the assumption (b) that for all $\vec{f}\in \mathcal{G}$, $$\begin{aligned} \|\vec{f}-\Phi_{\epsilon}(\vec{f})\|_{L^{p,\,\lambda}(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)}&\leq & \big\|\vec{f}\chi_{\mathcal{D}}-\Phi_{\epsilon}(\vec{f}) \big\|_{L^p(L^2([1,\,2]),\,l^{2};\,\mathbb{R}^n,\,w)}+\epsilon\\ &<&3\epsilon, \end{aligned}$$and $$\|\vec{f}-\vec{g}\|_{L^{p,\,\lambda}(\mathbb{R}^n)}< 6\epsilon+\|\Phi_{\epsilon}(f)-\Phi_{\epsilon}(\vec{g})\|_{L^{p,\,\lambda}(\mathbb{R}^n)}.$$ This completes the proof of Lemma \[l4.2\]. [*Proof of Theorem \[t1.3\]*]{}. We only consider the case of $p\in (q',\,\infty)$, $w\in A_{p/q'}(\mathbb{R}^n)$ and $\lambda\in (0,\,1)$. Recall that $\mathcal{M}_{\Omega,b}$ is bounded on $L^{p}(\mathbb{R}^n,\,w)$. Thus, by Lemma 4.2, we know that $\mathcal{M}_{\Omega,\,b}$ is bounded on $L^{p,\,\lambda}(\mathbb{R}^n,\,w)$. Thus, it suffices to prove that for $b\in C^{\infty}_0(\mathbb{R}^n)$, $\mathcal{M}_{\Omega,\,b}$ is completely continuous on $L^{p,\,\lambda}(\mathbb{R}^n,\,w)$. Let $j_0\in\mathbb{Z}_-$, $b\in C^{\infty}_0(\mathbb{R}^n)$ with ${\rm supp}\, b\subset B(0,\,R)$ and $\|b\|_{L^{\infty}(\mathbb{R}^n)}+\|\nabla b\|_{L^{\infty}(\mathbb{R}^n)}=1.$ Let $\widetilde{K^j}(z)=\frac{|\Omega(z)|}{|z|^n}\chi_{\{2^{j-1}\leq |z|\leq 2^{j+2}\}}(z).$ By the Minkowski inequality, $$\begin{aligned} \Big(\int^2_1\sum_{j\in\mathbb{Z}}|F_{j,\,b}^{l}f(x,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}&\leq & \Big(\sum_{j\in\mathbb{Z}}\int^2_1|F_{j,\,b}^{l}f(x,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\\ &\lesssim&\sum_{j\in\mathbb{Z}}\int_{\mathbb{R}^n}\widetilde{K^j}*\phi_{j-l}(x-y)|f(y)|\,{\rm d}y.\end{aligned}$$ It is obvious that ${\rm supp}\,\widetilde{K^j}*\phi_{j-l}\subset \{x:\, 2^{j-3}\leq |x|\leq 2^{j+3}\}$, and for any $R>0$, $$\int_{R\leq |x|\leq 2R}\Big|\sum_{j\in\mathbb{Z}}\widetilde{K^j}*\phi_{j-l}(x)\Big|^q{\rm d}x\leq \sum_{j:\, 2^j\approx R}\|\widetilde{K^j}*\phi_{j-l}\|_{L^q(\mathbb{R}^n)}^q\lesssim R^{-nq+n}.$$ Let $\epsilon>0$. We deduce from Lemma \[l4.1\] and the inequality (3.7) that, there exists a constant $A>0$, such that for all $f\in L^p(\mathbb{R}^n)$, $$\begin{aligned} \quad\Big\|\Big(\int^2_1\sum_{j\in\mathbb{Z}}|F_{j,\,b}^{l}f(x,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\chi_{\{|\cdot|>A\}}(\cdot)\Big\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}<\epsilon\|f\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}.\end{aligned}$$ Recall that $\widetilde{\mathcal{M}}_{\Omega}^{l,\,j_0}$ is bounded on $L^{p,\,\lambda}(\mathbb{R}^n,\,w)$. For $r>1$ small enough, $M_r$ is also bounded on $L^{p,\,\lambda}(\mathbb{R}^n,\,w)$ (see [@ks]). Thus by (3.4), we know that there exists a constant $\varrho>0$, such that $$\begin{aligned} && \Big\|\sup_{|h|\leq \varrho}\Big(\int^2_1 \sum_{j>j_0}|F_{j,\,b}^{l}f(\cdot,t)-F_{j,\,b}^{l}f(\cdot+h,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^{p,\lambda}(\mathbb{R}^n,\,w)}\\ &&\quad\lesssim \epsilon\|f\|_{L^{p,\lambda}(\mathbb{R}^n,\,w)}.\nonumber\end{aligned}$$ It follows from Lemma \[l4.1\], estimate (3.5) that for each $N\in\mathbb{N}$, there exists a constant $\sigma\in (0,\,1/2)$ such that $$\begin{aligned} &&\Big\|\sup_{|s|\leq \sigma}\Big(\int^2_1 \sum_{|j|\leq N}|F_{j,\,b}^{l}f(\cdot,\,s+t)-F_{j,\,b}^{l}f(\cdot,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\Big\|_{L^{p,\lambda}(\mathbb{R}^n,w)}\\ &&\quad< \epsilon\|f\|_{L^{p,\lambda}(\mathbb{R}^n,w)}.\nonumber\end{aligned}$$ We also obtain by Lemma \[l4.1\] and (3.6) that for each fixed $D>0$, there exists $N\in\mathbb{N}$ such that $$\begin{aligned} \Big\|\Big(\int^2_1\sum_{j>N}|F_{j,\,b}^{l}f(\cdot,\,t)|^2{\rm d}t\Big)^{\frac{1}{2}}\chi_{B(0,\,D)}\Big\|_{L^{p,\lambda}(\mathbb{R}^n,\,w)}<\epsilon\|f\|_{L^{p,\lambda}(\mathbb{R}^n,\,w)}.\end{aligned}$$ The inequalities (4.9)-(4.12), via Lemma 4.2, tell us for any $j_0\in \mathbb{Z}_-$, the operator $\mathcal{F}^{l}_{j_0}$ defined by (3.18) is compact from $L^{p,\,\lambda}(\mathbb{R}^n,\,w)$ to $L^{p,\,\lambda}(L^2([1,\,2]),\,l^2;\,\mathbb{R}^n,\,w)$. 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--- abstract: 'We present three-dimensional force-free electrodynamics simulations of magnetar magnetospheres that demonstrate the instability of certain degenerate, high energy equilibrium solutions of the Grad-Shafranov equation. This result indicates the existence of an unstable branch of twisted magnetospheric solutions and allows to formulate an instability criterion. The rearrangement of magnetic field lines as a consequence of this instability triggers the dissipation of up to 30% of the magnetospheric energy on a thin layer above the magnetar surface. During this process, we predict an increase of the mechanical stresses onto the stellar crust, which can potentially result in a global mechanical failure of a significant fraction of it. We find that the estimated energy release and the emission properties are compatible with the observed giant flare events. The newly identified instability is a candidate for recurrent energy dissipation, which could explain part of the phenomenology observed in magnetars.' author: - | J. F. Mahlmann$^1$[^1], T. Akgün$^2$, J. A. Pons$^2$, M.A. Aloy$^1$ and P. Cerdá-Durán$^1$[^2]\ $^1$Departament d’Astronomia i Astrofísica, Universitat de València, 46100, Burjassot, Spain\ $^2$Departament de Fisica Aplicada, Universitat d’Alacant, 03690, Alicante, Spain bibliography: - 'literature.bib' date: 'Accepted 2019 September 25. Received 2019 September 22; in original form 2019 July 31' title: Instability of twisted magnetar magnetospheres --- \[firstpage\] stars: magnetars – magnetic fields – methods: numerical – stars: neutron – X-rays: bursts Introduction {#sec:introduction} ============ Soft gamma-ray repeaters (SGRs) are neutron stars with recurrent X-ray activity in the form of short bursts with duration $\sim 0.1$s and luminosities in the range $10^{36}-10^{43}$ergs$^{-1}$. Over the last 40 years, three bursts have been uniquely energetic, the so-called [*giant flares*]{} (GFs) with luminosities of the order of $10^{44}-10^{47}$ergs$^{-1}$ [SGR0525-66, SGR1900+14 and SGR1806-20, see @Cline1980; @Hurley1999; @Hurley2005]. In the three referenced cases, a short initial peak was followed by a softer X-ray tail lasting for $50-400$s. The engine behind these extraordinary events are [*magnetars*]{}, neutron stars with the strongest known magnetic fields [$10^{14}-10^{16}$G; see comprehensive reviews of magnetar observations and physics, e.g. in @Woods2006; @Rea2011; @Turolla2015; @Mereghetti2015; @Kaspi2017]. The precise mechanism producing such energetic events is still unclear. Strong magnetic fields are a gigantic energy reservoir in magnetars, generally of the order $$\begin{aligned} {\mathcal{E}}_{\rm magnetar} \sim 1.6\times 10^{47} \,\text{erg}\,\left ( \frac{B}{10^{15}\,\text{G}}\right )^{2}\left ( \frac{R_*}{10\,\text{km}}\right )^3,\end{aligned}$$ where we consider a neutron star with radius $R_*$. The timescale on which the magnetar is evolving, mainly due to Hall drift and Ohmic dissipation in the crust, is of the order of $10^3-10^6$yrs [@Jones1988; @Goldreich1992; @Pons2007; @Pons2009; @Gourgouliatos2016], by itself too slow to explain this phenomenology. Two complementary models have tried to explain these observations. In the [*crustquake*]{} model [@Thompson1996; @Perna2011] the dynamical trigger is the mechanical failure of patches of the magnetar crust due to large stresses built during its magneto-thermal evolution. Numerical simulations of the Hall evolution of the crust [@Vigano2013] show that it is possible to recurrently reach the maximum stress supported by the very same [@Horowitz2009; @Baiko2018]. At this point, the crust likely becomes plastic [@Levin2012], i.e. the crust generates thermo-plastic waves emerging from such a localized trigger, or in other words *yields* [@Beloborodov2014; @Li2016]. The waves propagate into the magnetosphere, probably resulting in rapid dissipation through a turbulent cascade triggered by reconnection on slightly displaced flux surfaces [@Thompson1996; @Thompson2001; @Li2018]. The energy released in those events suffices to explain the observed luminosities, even for GFs [@Thompson1996; @Lander2015]. The burst duration ($\sim 0.1$s) is related to the crossing time of shear waves through the whole crust ($1-100$ ms). A limitation is that, if stressed for long periods of time ($\sim 1\,$yr) as it is the case due to the slow magneto-thermal evolution, the crust may yield at significantly lower breaking stresses [@Chugonov2010]. In that case, it would effectively deform as a plastic flow, and, depending on its (unknown) properties, cease to yield altogether [@Lyutikov2015; @Lander2019]. @Thompson2017 has argued that even in this case the crust could yield. The [*magnetospheric instability*]{} model requires a strongly twisted magnetosphere that becomes unstable and leads to a rapid reconnection event [@Lyutikov2003]. The existence of long-lived magnetospheric twists is supported by the observation of hard X-ray emission in persistent magnetars [@Beloborodov2013b; @Hascoet2014]. During the magneto-thermal evolution of the crust, the displacement of the magnetic field footprints can generate large twists in the magnetosphere [@Akgun2017; @Akgun2018b]. Above a critical twist, the magnetosphere becomes unstable and undergoes a rapid rearrangement where energy is dissipated by reconnection [@Lyutikov2003; @Gill2010; @Elenbaas2016] in a similar fashion as in the crustquake model. The main challenge of this scenario is the ability of the crust to produce significant twists in the magnetosphere. @Beloborodov2009 estimated that currents supporting magnetospheric twist are bound to dissipate on timescales of years, effectively leading to a progressive untwisting. Therefore, Hall evolution is required to proceed relatively fast in order to allow for significant twists. Plastic viscosity may also be a problem for similar reasons [@Lander2019]. The latter authors have also suggested that the dynamical crust fractures of the crustquake model could be substituted by sustained episodes of accelerated plastic flows which are able to generate large magnetospheric twists on times shorter than the untwisting timescale. Numerical simulations by @Parfrey2012 [@Parfrey2013] and @Carrasco2019 confirm the instability of the magnetosphere beyond a critical twist, accompanied by the formation of plasmoids. These results are an analogy to the context of eruption processes in the solar corona as found in numerical experiments by @Roumeliotis1994 [@Mikic1994]. The energy dissipated in the reconnection events is sufficient to explain the GF processes [@Parfrey2012]. A caveat to these simulations is that the applied twisting rate is larger than the one expected from the respective magneto-thermal evolution, although it would be fine if the trigger was a rapid plastic deformation. An alternative approach to the above is the study of stability properties of magnetospheres. A number of authors have constructed equilibrium solutions to the Grad-Shafranov equation (GSE) for neutron star magnetospheres [@Glampedakis2014; @Fujisawa2014; @Pili2015; @Akgun2016; @Kojima2017; @Kojima2018; @Kojima2018b; @Akgun2018a]. @Akgun2017 performed magneto-thermal evolutions coupling the crustal magnetic field at the stellar surface with an exterior equilibrium solution. The results showed that large twists grow in the magnetosphere up to a critical point beyond which no stable equilibrium solutions where found. A more detailed analysis by @Akgun2018a showed that, for sufficiently large twists, the solutions of the GSE are degenerate with several possible configurations of different energies but matching boundary conditions at the surface. This suggests the possibility of an unstable branch of the solutions and, thus, a possible explanation for the occurrence of bursts and GFs. In this work we explore this possibility by performing three-dimensional (3D) numerical simulations of the equilibrium models in @Akgun2018a. We asses their stability properties and their potential as candidates for transient magnetar phenomenology. This work is organised as follows. In section \[sec:physics\] we review and discuss the physics involved in magnetars relevant to the processes that we want to study. In section \[sec:ff\_equations\] we briefly review the equations of force-free electrodynamics (FFE) implemented for simulations conducted on the infrastructure of the `Einstein Toolkit` (supplemented by appendix \[sec:augmented\_system\]). A detailed description of the derivation of initial models according to @Akgun2018a is given in section \[sec:twisted\_models\]. In section \[sec:simulations\] we present the numerical setup of our simulations as well as the outcome of the conducted 3D force-free simulations of twisted magnetospheres (reviewing details on maintaining the force-free regime in appendix \[sec:ff\_breakdown\]). The observed rapid dissipation of electromagnetic energy through the magnetar crust is interpreted and related to observable quantities, such as luminosity estimates, shear stresses on the stellar crust, and opacity models, in section \[sec:discussion\]. Along this paper we use Gaussian units in CGS, except for section \[sec:ff\_equations\] in which we use Heaviside-Lorentz with geometrised units ($G=c=M_\odot=1$). For convenience we express current densities in A m$^{-2}$ and voltages in V, instead of the corresponding CGS units. Physics of magnetars {#sec:physics} ==================== The basic structure of the magnetar interior is a (likely) fluid core of $\sim 10$ km radius, amounting for most of the mass of the object, surrounded by a solid crust of about $1$ km size. Outside, there is a tenuous, co-rotating magnetosphere connected to the NS by magnetic field lines (threading the central object) that extend up to the light cylinder at distances larger than $10^5$ km. We start by discussing some basic properties of the different parts of the magnetosphere relevant for the interpretations and models presented later in this work. Currents supporting the magnetosphere {#sec:current_magnetsophere} ------------------------------------- For the typical rotation periods of magnetars ($P\sim 1$ - $10$s) the Goldreich-Julian particle density [@GJ1969] for a magnetar magnetosphere has the typical value $$\begin{aligned} n_{\textsc{gj}} = 7\times 10^{12} \,\text{cm}^{-3}\,\left ( \frac{B_{\rm pole}}{10^{15} \,{\rm G}}\right) \left( \frac{R_{*}}{r}\right )^3 \left ( \frac{10 \,{\rm s}}{P}\right),\end{aligned}$$ where $B_{\rm pole}$ is the magnetic field strength at the magnetar pole, $R_*$ the magnetar radius and $r$ the distance to the center of the star. This limits the magnetospheric current density close to the surface to $J<e\,c\,n_{\textsc{gj}} \approx 3\times 10^8$A/m$^2$, much below the typical values needed to support currents in strongly twisted magnetospheres of magnetars, of the order of $$J\sim \frac{B c}{4 \pi r}\sim 8.2\times 10^{12}\,\text{A}\,\text{m}^{-2}\,\left( \frac{B_{\rm pole}}{10^{15}\,\text{G}}\right ) \left( \frac{R_*}{10\,\text{km}}\right )^{-1}\label{eq:GJcurrent}.$$ In general, magnetospheric currents in magnetars cannot be supported neither by Goldreich-Julian charges nor by charges lifted from the surface. @Beloborodov2007 proposed that the currents are supported by $e^+$-$e^-$ pairs generated in the magnetosphere in an intermittent discharge process that can be sustained for voltages along magnetic field lines of about $10^8 -10^9$V. This voltage can be maintained by self-induction in untwisting magnetospheres [@Beloborodov2009]. This untwisting is driven by the effective resistivity of the magnetosphere; the thermal photons from the magnetar’s surface scatter resonantly off the charges supporting the magnetospheric currents, taking energy away, at the same time that pairs are produced. The untwisting timescale is $\sim 1$yr, and it may explain the spectral evolution of some magnetars [@Beloborodov2009]. Timescales {#sec:timecales} ---------- Changes in magnetars take place during two different timescales. On the one hand, there is a [*secular timescale*]{} of thousands of years during which the magnetar is essentially in equilibrium. On the other hand, there is a [*dynamical timescale*]{} associated to energetic events (burst, flares) that can produce observable variations on timescales as fast as $0.1$s. The latter are likely associated to out-of-equilbrium states. ### Secular timescales {#sec:secular} The [*secular timescale*]{} is set by the slow magnetothermal evolution of the cooling object. The interior magnetic field evolution is dominated by Hall drift and Ohmic diffusion at the crust [see, e.g. @Vigano2012; @Fujisawa2014 and references therein], which proceeds on typical timescales of $10^3-10^6$ yr. The long-term evolution of the magnetosphere is driven by the changes in the crustal magnetic field, which displaces the footprints of the magnetospheric magnetic field lines. Since this evolution is much slower than the dynamical timescale of the magnetosphere (see below), it can be considered that the magnetosphere evolves through a series of equilibrium states. This evolution creates a twist in the magnetosphere supported by currents - until a critical maximum twist is reached ($\varphi_{\rm crit} \sim 1$rad) beyond which no magnetospheric equilibrium solutions exist [@Akgun2017]. The stability of the magnetosphere close to this critical point is the subject of this paper. At the same time as the crustal magnetic field evolves, other processes in the magnetosphere can also contribute to the evolution. The untwisting of the magnetosphere on timescales of $\sim1$yr [@Beloborodov2009 and discussion in section \[sec:secular\]], may be a competing action to the twisting process described above. Although the velocity of the footprints is typically very slow, numerical simulations of the magnetothermal evolution of magnetars including the magnetosphere show that, close to the critical point, it can be as fast as $v_{\varphi}\sim 1$kmyr$^{-1}$ [see @Akgun2017] in the most optimistic scenario. Therefore, close to the critical twist, the magnetosphere twists slowly ($\dot\varphi_{\rm max,crit} \,\lesssim\, 0.1$radyr$^{-1}$), evolving on timescales $\,\gtrsim\, 10$yrs. In the best case scenario, this timescale is comparable to the untwisting timescale ($\sim1$yr) and, hence, parts of the magnetosphere could sustain a significant twist. This timescale is still much longer than the dynamical timescale of the system (see below). Therefore, in our study of the dynamical behavior we can neglect the secular evolution of the field. ### Dynamical timescales {#sec:dynamical} The [*dynamical timescale*]{} is set by the travel time of waves propagating in the different regions of the magnetar. In the magnetosphere, the mass density can be neglected in view of the dominating magnetic field energy density. Also, the velocity of Alfvén and fast magnetosonic waves is degenerated to the speed of light. Hence, within $\sim 100$km the whole magnetosphere is coupled through timescales smaller than $1$ms, which sets the scale for the dynamical evolution of the magnetosphere. In this region it is possible to neglect the inertia of the fluid in the evolution equations of so-called FFE, which is used in the numerical simulations of this work. In the outermost parts of the crust, the force-free condition still holds because of low densities. At sufficiently high densities, elastic forces of the solid crust and pressure gradients break this condition. To estimate the transition density one may consider the depth at which waves propagate at a velocity significantly different to the speed of light. Two possible waves can travel in the interior of the magnetised crust, the so-called magnetosonic (ms) waves, related to sound waves, and magneto-elastic (me) waves, a combination of Alfvén and shear waves. The complete eigenvalue structure of relativistic ideal MHD equations in the presence of an elastic solid is not known. To make a simple order of magnitude estimate of the different wave speeds, we use the expression of magneto-elastic torsional waves parallel to the magnetic field derived in [@Gabler2012] as well as the expression for fast magnetosonic waves perpendicular to the field[^3]: $$\begin{aligned} v_{\rm me} /c = \sqrt{\frac{\mu_s+B^2}{e + B^2}} \qquad \qquad v_{\rm ms} /c= \sqrt{\frac{e c_s^2 + B^2}{e+B^2}},\end{aligned}$$ where $e$ is the energy density and $\mu_s$ the shear modulus. Note that in the limit of low magnetic field ($B^2\ll \mu_s, \, B^2\ll e$) we recover the shear and sound speed, respectively. In the high magnetic field limit ($B^2\gg \mu_s,\, B^2\gg e$) both, $v_{\rm me}$ and $v_{\rm ms}$, coincide with the speed of light. Inside the fluid core ($\mu_s=0$) the magneto-elastic speed becomes the Alfvén speed. Figure \[fig:speeds\] shows the value of the characteristic speeds in the outer layers of a typical NS model for different magnetic fields in the magnetar range. Indeed, both fast magnetosonic waves and Alfvén waves have a degenerate speed equal to the speed of light in the magnetosphere. Inside the outer crust ($\rho < 4\times 10^{11}$gcm$^{-3}$), all characteristic speeds transition from the speed of light to a significantly lower value, in a region that can still be considered force-free. This transition depends on the magnetic field strength, happening deeper inside the star for larger values of $B_{\rm pole}$. Given these characteristic speeds, any global rearrangement of the magnetosphere can modify the entire structure of the crust (of size $\sim 2 \pi R_*$) on a timescale of $\sim 1$ ms for magnetosonic waves and $\sim 10$ms for magnetoelastic waves. One last aspect to consider is the ability of magnetospheric waves to transmit energy into the crust. The discussion should be limited to Alfvén waves, which become magnetoelastic waves once they penetrate the crust; the energy carried by fast magnetosonic waves in the magnetosphere can be neglected due to the small density, which renders the compressibility effects of fast-magnetosonic waves unimportant. Since the characteristic time in the magnetosphere is $\sim 1$ms, the typical frequency of the waves generated during its dynamics is in the kHz range. At this frequency, the crust can be considered as a thin layer because its thickness ($\sim 1$km) is much smaller than the typical wavelength in the magnetosphere ($\lambda \sim 100\,$km). In this case the energy transmission coefficient for waves perpendicular to the surface is approximately [cf. @Link2014; @Li2015] $$\mathcal{T} =\frac{4 v_{\rm me}/c}{(1+v_{\rm me}/c )^2}\approx 0.04 \left (\frac{v_{\rm me}/c}{0.01} \right), \label{eq:transmission}$$ for typical physical conditions in the magnetar crust. Given the low transmission coefficients of magnetospheric Alfvén waves hitting the crust as well as the differences on timescales between the crust and the magnetosphere (typically $\sim 10$ times shorter in the later) it is reasonable to consider that most of the crust remains rigid during any dynamical rearrangement of the magnetosphere. In our magnetar model we will consider two regions: A force-free region ([*exterior*]{}, hereafter) consisting of the magnetosphere and the force-free part of the outer crust as well as the magnetar [*interior*]{} for the remainder of the NS, which we will consider to be fixed during our simulations. The limit between both regions is a spherical surface below the NS surface, where magnetic field lines are anchored, and is located below the transition density between inner and outer crust at a density $\rho<4\times 10^{11}\,$gcm$^{-3}$. For the purpose of describing the simulations we will refer to this transition point simply as *surface*. ![Fast magnetosonic (solid lines) and magnetoelastic (dashed lines) speed in the outer layers of a magnetar, for different magnetic field strengths ranging from 0 to $10^{16}$G. The neutron star model corresponds to the $1.4 M_\odot$ mass APR+DH model of @Gabler2012. The magnetic field is considered to be constant for simplicity. []{data-label="fig:speeds"}](images/figure1.pdf){width="50.00000%"} Force-free Electrodynamics {#sec:ff_equations} ========================== In analogy to @Komissarov2004 and @Parfrey2017 we solve Maxwell’s equations in the force-free limit: $$\begin{aligned} \frac{\partial {\boldsymbol{{\tilde{B}}}}}{\partial t} = -{\boldsymbol{\nabla}}\times{\boldsymbol{{\tilde{E}}}}\qquad\text{and}\qquad\frac{\partial {\boldsymbol{{\tilde{E}}}}}{\partial t} ={\boldsymbol{\nabla}}\times{\boldsymbol{{\tilde{B}}}} - {\boldsymbol{{\tilde{J}}}}_{\rm \text{\tiny FF}}\:, \label{eq:maxwell}\end{aligned}$$ where ${\boldsymbol{{\tilde{E}}}}$, ${\boldsymbol{{\tilde{B}}}}$, and ${\boldsymbol{{\tilde{J}}}}_{\rm \text{\tiny FF}}$ are the electric field, the magnetic field, and the so-called force-free current, respectively. We place a tilde to distinguish quantities expressed in our Heaviside-Lorentz *geometrised* (HLG) system of units, while the same symbols without tilde express quantities in the Gaussian *non-geometrised* (GNG) system of units (see Table \[tab:unit\_conversions\]). We explicitly include the charge conservation equation $$\begin{aligned} \frac{\partial {\tilde{\rho}_{\rm e}}}{\partial t}+{\boldsymbol{\nabla}}\cdot{\boldsymbol{{\tilde{J}}}}_{\rm \text{\tiny FF}} = 0\,, \label{eq:chargecons}\end{aligned}$$ where ${\tilde{\rho}_{\rm e}}$ represents the charge density. Furthermore, we use a mixed hyperbolic/parabolic correction by the introduction of additional potentials (further discussed in appendix \[sec:augmented\_system\]) in order to numerically ensure the constraints ${\boldsymbol{\nabla}}\cdot{\boldsymbol{{\tilde{B}}}}=0$ and ${\boldsymbol{\nabla}}\cdot{\boldsymbol{{\tilde{E}}}}={\tilde{\rho}_{\rm e}}$ [@Dedner2002; @Palenzuela2009; @Mignone2010]. In the force-free limit it is necessary to guarantee that there are either no forces acting on the system or, more generally, that the forces of the system balance each other. This is equivalent to a vanishing net Lorentz force on the charges ${\tilde{\rho}_{\rm e}}$ [see, e.g. @Camenzind2007]: $$\begin{aligned} {\boldsymbol{{\tilde{E}}}}\cdot{\boldsymbol{{\tilde{J}}}}_{\rm \text{\tiny FF}}&=0\label{eq:electromagnetic_work}\\ {\tilde{\rho}_{\rm e}}{\boldsymbol{{\tilde{E}}}}+{\boldsymbol{{\tilde{J}}}}_{\rm \text{\tiny FF}}\times{\boldsymbol{{\tilde{B}}}}&=0\label{eq_lorentz_force}\end{aligned}$$ From equation (\[eq\_lorentz\_force\]) one readily obtains the degeneracy condition $$\begin{aligned} {\boldsymbol{{\tilde{E}}}}\cdot{\boldsymbol{{\tilde{B}}}}&=0.\label{eq:force_free_crossfield}\end{aligned}$$ Additionally, force-free fields are required to be magnetically dominant, the magnetic field being always stronger than the electric one, such that the following condition must hold: $$\begin{aligned} {\boldsymbol{{\tilde{B}}}}^2-{\boldsymbol{{\tilde{E}}}}^2&\geq 0.\label{eq:force_free_dominance}\end{aligned}$$ Conditions (\[eq:force\_free\_crossfield\]) and (\[eq:force\_free\_dominance\]), as well as the conservation condition $\partial_t\left({\boldsymbol{{\tilde{E}}}}\cdot{\boldsymbol{{\tilde{B}}}}\right)=0$ can be combined in order to obtain an explicit expression for ${\boldsymbol{{\tilde{J}}}}_{\rm \text{\tiny FF}}$ [cf. @Komissarov2011; @Parfrey2017]: $$\begin{aligned} {\boldsymbol{{\tilde{J}}}}_{\rm \text{\tiny FF}}=\left[{\boldsymbol{{\tilde{B}}}}\cdot{\boldsymbol{\nabla}}\times{\boldsymbol{{\tilde{B}}}}-{\boldsymbol{{\tilde{E}}}}\cdot{\boldsymbol{\nabla}}\times{\boldsymbol{{\tilde{E}}}}\right]\frac{{\boldsymbol{{\tilde{B}}}}}{{\tilde{B}}^2}+{\tilde{\rho}_{\rm e}}\frac{{\boldsymbol{{\tilde{E}}}}\times{\boldsymbol{{\tilde{B}}}}}{{\tilde{B}}^2}\label{eq:ff_current}\end{aligned}$$ Across the literature [e.g. @Komissarov2004; @Alic2012; @Parfrey2017] we find various modifications in the definition of ${\boldsymbol{{\tilde{J}}}}_{\rm \text{\tiny FF}}$ in order to drive the numerical solution of the system of partial differential equations (\[eq:maxwell\]) towards a state which fulfills equation (\[eq:force\_free\_crossfield\]) by introducing a suitable cross-field conductivity. In the numerical setup, we choose to combine the prescription of @Komissarov2004 with the force-free current given above. This strategy effectively minimises the violations of equations(\[eq:force\_free\_crossfield\]) and (\[eq:force\_free\_dominance\]) by exponentially damping the (numerically induced) components of the electric field parallel to ${\boldsymbol{{\tilde{B}}}}$ and suitably adjusting the electric field in magnetospheric current sheets in order to obtain ${\boldsymbol{{\tilde{B}}}}^2-{\boldsymbol{{\tilde{E}}}}^2\rightarrow 0$ at these locations. Throughout the literature, the magnetic dominance condition (\[eq:force\_free\_dominance\]) condensates to a necessary condition of FFE [e.g. @Uchida1997; @McKinney2006]. For some authors [e.g. @McKinney2006] the breakdown of the magnetic dominance implies the invalidity of the numerical model. Others [e.g. @Uchida1997] claim that some physical processes (e.g. radiation losses) taking place in the regions where condition (\[eq:force\_free\_dominance\]) is breached may restore the magnetic dominance condition. Indeed, [@Uchida1997] explicitly allows for transient phases violating condition (\[eq:force\_free\_dominance\]) - these regions are then interpreted as abandoning the freezing of magnetic flux onto the flux of matter, being necessarily accompanied by dissipation. Following @Uchida1997, the force-free regime continues to be a valid approximation as long as the dissipative effects are only a small fraction of the total energy. The violation of the perpendicularity condition (\[eq:force\_free\_crossfield\]) is an additional source of (Ohmic) dissipation [studied for example in the context of Alfvén waves in force-free electrodynamics by @Li2018]. In practice, this channel of dissipation occurs when ${\boldsymbol{{\tilde{E}}}}\cdot{\boldsymbol{{\tilde{B}}}}\neq 0$ such that ${\boldsymbol{{\tilde{J}}}}\cdot{\boldsymbol{{\tilde{E}}}}\neq 0$. Currently used force-free codes aim to avoid the transient into this regime by numerically cutting back all violations of condition (\[eq:force\_free\_dominance\]) [e.g. @Palenzuela2010; @Paschalidis2013; @Carrasco2016] or include a suitable Ohm’s law [e.g. @Komissarov2004; @Spitkovsky2006; @Alic2012; @Parfrey2017] in order to minimise these violations during a transient phase. Figure \[fig:energy\_dissipation\_channels\] shows the breakdown of condition (\[eq:force\_free\_dominance\]) during the simulation and hints towards the aforementioned dissipative processes. We refer to appendix \[sec:ff\_breakdown\] as well as, for example, @Lyutikov2003 for further details on the necessary constraint preservation and limitations of the highly magnetised regime (such as the lack of physical reconnection). We will give a thorough review of the procedures employed in our code in a subsequent technical paper. Twisted magnetar magnetosphere models {#sec:twisted_models} ===================================== Magnetospheres {#sec:initial_data} -------------- Quantity Nongeometrised unit Conversion factor ----------------- ---------------------------------------------- ------------------------------------------- Mass M $M_\odot$ Length L $M_\odot G c^{-2}$ Time T $M_\odot G c^{-3}$ Electric charge $\text{L}^{3/2}\text{M}^{1/2}\text{T}^{-1}$ $(4\pi)^{-1/2}M_\odot G^{1/2}$ Electric field $\text{L}^{-1/2}\text{M}^{1/2}\text{T}^{-1}$ $(4\pi)^{1/2}M_\odot^{-1} G^{-3/2}c^{4}$ Magnetic field $\text{L}^{-1/2}\text{M}^{1/2}\text{T}^{-1}$ $(4\pi)^{1/2}M_\odot^{-1} G^{-3/2}c^{4}$ Current density $\text{L}^{-1/2}\text{M}^{1/2}\text{T}^{-2}$ $(4\pi)^{-1/2}M_\odot^{-2} G^{-5/2}c^{7}$ (EM) Energy $\text{L}^{2}\text{M}\:\text{T}^{-2}$ $ M_\odot\:c^{2}$ (EM) Stress $\text{L}^{-1}\text{M}\:\text{T}^{-2}$ $ M_\odot^{-2} G^{-3}c^{8}$ : Conversion table between code output in Heaviside-Lorentz *geometrised* units ($M_\odot=G=c=1$) and *non-geometrised* Gaussian units. In order to transform the respective quantities from code quantities to the *non-geometrised* system, one has to multiply the *geometrised* quantity by its conversion factor expressed in CGS.[]{data-label="tab:unit_conversions"} ![Magnetospheric energy normalised to the vacuum dipole energy (equation \[eq:edipole\]) of the initial equilibrium models, for different values of the parameter ${P_{\rm c}}$ (in units of $P$ at the equator). The solid and dashed lines correspond to a series of models with constant $s$ and $\sigma$. The colored dots correspond to the initial data models used in our simulations.[]{data-label="fig:initial_model"}](images/figure2.pdf){width="45.00000%"} Due to the long rotational period of observed magnetars pushing the location of the light cylinder to great distances, it is possible to neglect the rotation of the neutron star when building numerical models of magnetospheres in the near zone. The equilibrium structure of a non-rotating axisymmetric force-free magnetosphere is given through the well-known GSE [@Luest1954; @Grad1958; @Shafranov1966]. This approach has been followed in several recent papers [e.g. @Spitkovsky2006; @Beskin2010; @Vigano2011; @Glampedakis2014; @Fujisawa2014; @Pili2015; @Akgun2016; @Akgun2018a; @Kojima2017; @Kojima2018; @Kojima2018b]. In most of these works, the toroidal field is confined within a magnetic surface near the equator, smoothly transitioning to vacuum at large distances. In stationary, non-rotating, axisymmetric magnetosphere models, the toroidal field cannot extend to the poles. Otherwise, the toroidal field would extend all the way to infinity, thus, violating the requirements of finite magnetic energy. Following the notation of @Akgun2016 [@Akgun2018a], we write the axisymmetric magnetic field in terms of its poloidal and toroidal components: $$\begin{aligned} {\boldsymbol{B}} = {\boldsymbol{\nabla}} P \times {\boldsymbol{\nabla}}\varphi + T {\boldsymbol{\nabla}}\varphi \ ,\end{aligned}$$ where $\varphi$ is the azimuthal angle in spherical coordinates. Here, $P$ and $T$ are the poloidal and toroidal stream functions. Expressed in the orthonormal spherical basis corresponding to the coordinates $(r,\theta,\varphi)$, the magnetic field can be explicitly computed from the potentials $P$ and $T$ as $$\begin{aligned} B^r &= \frac{1}{r^2\sin\theta}\partial_\theta P, \label{eq:Br}\\ B^\theta &= - \frac{1}{r \sin\theta} \partial_r P, \label{eq:Btheta}\\ B^\varphi &= \frac{T}{r \sin\theta}. \label{eq:Bphi}\end{aligned}$$ For an axially symmetric force-free field, the functions $T$ and $P$ may be expressed in terms of each other and appear as solutions of the force-free GSE: $$\begin{aligned} \left[\partial_r^2 + \frac{1 - \mu^2}{r^2}\partial_\mu^2\right] P + T \frac{dT}{dP} = 0,\end{aligned}$$ where $\mu=\cos\theta$. $P$ and $T$ are constant on magnetic surfaces or, equivalently, along magnetic field lines. $P$ is related to the magnetic flux passing through the area centered on the axis and delineated by the magnetic surface. Therefore, its value at the poles is zero and increases towards the equator. The function $T$ is related to the current passing through the same area. Its functional dependence on $P$ can be chosen freely (consistently with any continuity and convergence requirements, particularly for the currents), which is equivalent to setting boundary conditions for $T$ at the surface of the star. Here, we invoke the same functional form for $T(P)$ as in @Akgun2016 [@Akgun2018a]. Thus, the toroidal field is confined within some [*critical*]{} magnetic surface ($P = {P_{\rm c}}$), $$\begin{aligned} T\left(P\right)=\left\{\begin{array}{ccc} s\times (P-{P_{\rm c}})^\sigma& : & P \geqslant {P_{\rm c}}\\ 0 & : & \text{else} \end{array}\right., \label{eq:toroidal}\end{aligned}$$ $s$ being a parameter determining the relative strength of the toroidal field with respect to the poloidal field. In order to avoid divergences in the currents we must demand that the power index satisfies $\sigma \geqslant 1$. For a pure dipolar field, the poloidal stream function in the magnetosphere is $$\begin{aligned} P = \frac{1}{2}B_{\rm pole}\frac{R_*^3}{r}\sin^2\theta,\end{aligned}$$ while the toroidal stream function is $T=0$ everywhere. We will consider the simplest cases where the boundary value of $P$ at the surface of the magnetar coincides with that of a dipolar field, and, therefore, the initial data is symmetric with respect to the equator. For different choices of the functional relation $T(P)$ given by equation (\[eq:toroidal\]) we solve the GSE and obtain a twisted magnetospheric initial model. We would like to note that all equations can be rescaled with $B_{\rm pole}$, hence, the results of our numerical simulations can be normalised to the field strength of interest. The energy stored in the magnetosphere can be computed as a volume integral $$\begin{aligned} {\mathcal{E}}= \frac{1}{8\pi}\int ({\boldsymbol{B}}^2 + {\boldsymbol{E}}^2)\, {\rm d}V . \end{aligned}$$ For later reference and in order to normalize the energetic content of our models, we provide the energy stored in the magnetosphere of a pure dipolar magnetic field ($\vec{E}=0$, $B^r=B_{\rm pole}(R_*/r)^3\cos{\theta}$, $B^\theta=(B_{\rm pole}/2)(R_*/r)^3\sin{\theta}$, $B^\varphi=0$): $$\begin{aligned} \begin{split} {\mathcal{E}}_{\rm d} = \frac{1}{12}B_{\rm pole}^2 R_*^3 =8.3\times 10^{46}\,\text{erg}\,\left(\frac{B_{\rm pole}}{10^{15}\,\text{G}}\right)^2\left(\frac{R_*}{10\,\text{km}}\right)^3. \end{split} \label{eq:edipole} \end{aligned}$$ Once the surface value of $P$ and the functional relation $T(P)$ are defined, one can solve the GSE iteratively (as it is a non-linear equation), while imposing vacuum boundary conditions at large distances. We use the numerical code described in @Akgun2018a to build our initial data. Using this parametrisation, the boundary condition at the surface of the neutron star for the GSE (values of $P$ and $T$) is fully determined by four parameters $B_{\rm pole}$, $s$, ${P_{\rm c}}$ and $\sigma$. However, the solution of the GSE with this fixed boundary condition is not necessarily unique. @Akgun2018a showed that for sufficiently large magnetospheric twists, there exist degeneracies, i.e. different solutions of the GSE for the same boundary conditions (the same set of four parameters). These solutions differ in their energy, twist and the radial extent of the toroidal currents. Table \[tab:model\_overview\] shows the parameters used to construct the initial data for our numerical simulations. Each of the series A, B and C of initial models were chosen to have identical parameters but [*different*]{} magnetospheric energies and, hence, represent degenerate magnetospheric models. We would like to point out that the value of ${P_{\rm c}}$ is only equal, within each series, up to the second significant digit, due to numerical reasons. Figure \[fig:initial\_model\] shows the energy of the initial models as a function of the parameter ${P_{\rm c}}$. Models within each spiral curve (constant $s$ and $\sigma$) and with the same value of ${P_{\rm c}}$ have identical boundary conditions but different energies. In the interpretation made by @Akgun2018a, the lower energy state for each series of degenerate models (i.e., A1, B1 and C1) corresponds to stable configurations, while high energy states (i.e., A2, B2, C2 and C3) may be unstable and would evolve towards the stable configuration releasing the respective energy difference. This instability is a possible scenario for the flare activity observed in magnetars. The lowest energy solutions are the ones that are most similar to the vacuum solutions, with all field lines connected to the surface, while the higher energy solutions are more radially extended, and can contain disconnected field lines. {width="80.00000%"} Magnetar interior {#sec:interior} ----------------- The initial models described above provide solutions only for the magnetosphere. For each possible magnetospheric model one can build infinite solutions to describe the neutron star interior. The magnetospheric (exterior) values of $P$ and $T$ determine the magnetic field ${\boldsymbol{B}}$ at the exterior side of the surface (equations \[eq:Br\] to \[eq:Bphi\]). To match this solution to the interior, one has to ensure the continuity of $B^r$ at the surface. This is valid if $P$ is continuous and, hence, $T$ and $B^\varphi$ are continuous as well. However, $B^\theta$ does not necessarily match continuously to the neutron star interior because current sheets (thin current-carrying layers across which the magnetic field changes either direction or magnitude) in the $\varphi$ direction may occur. Even if all components of ${\boldsymbol{B}}$ are continuous at the surface, the magnetic field structure in the interior depends completely on how currents are internally distributed. In the astrophysical scenario we are considering, the magnetar reaches the initial state in which we start our numerical simulation after a slow magnetothermal evolution that proceeds in a long timescale compared to the dynamical timescales (cf. section \[sec:timecales\]) of the magnetosphere ($\sim 1$ms) or the crust ($\sim 10$ms). On such long timescales, any current close to the surface of the NS is expected to be dissipated by Ohmic diffusion. Therefore, we consider that initially all fields are continuous across the surface. We build our interior solution by extrapolating the exterior magnetic field towards the stellar interior across a number of grid cells as needed by the reconstruction algorithm used for the magnetospheric evolution in our simulations. Since the neutron star is basically a perfect conductor, the initial charge density and electric field in the interior (and the magnetosphere) are set to zero. The surface values of $B^r$ and $B^\varphi$ are coincident for degenerate models (e.g. within the series C1, C2 and C3 in Figure \[fig:initial\_model\]) because $P$ and $T$ at the surface are identical. However, since $P$ and $T$ may have a different radial dependence outside of the magnetar, and $B^\theta$ depends on the radial derivative of $P$ (equation \[eq:Btheta\]), it is different for every model of the same series. Simulations {#sec:simulations} =========== We have performed numerical simulations of the neutron star magnetosphere using the initial models in Table \[tab:model\_overview\]. For all the simulations we employ our own implementation of a General Relativistic FFE code in the framework of the `Einstein Toolkit`[^4] [@Loeffler2012]. The `Einstein Toolkit` is an open-source software package utilizing the modularity of the `Cactus`[^5] code [@Goodale2002a] which enables the user to specify so-called `thorns` in order to set up customary simulations. There exist other code packages such as `GiRaFFE` [@Etienne2017], which integrate the equations of force-free electrodynamics employing an evolution scheme based on the Poynting flux as a conserved quantity [cf. @McKinney2006; @Paschalidis2013] rather than the electric field and its current sources [as formulated in, e.g. @Komissarov2004; @Parfrey2017]. The `Einstein Toolkit` employs units where $M_\odot=G=c=1$, which sets the respective time and length scales to be $1M_\odot\equiv 4.93\times 10^{-6}\text{ s}\equiv 1477.98\text{ m}$. This unit system is a variation of the so-called system of *geometrised units* [as introduced in appendix F of @Wald2010], with the additional normalisation of the mass to $1M_\odot$ (i.e. our HLG units, as introduced in section \[sec:ff\_equations\]). For easy reference, we provide a set of conversion factors for relevant physical quantities in Table \[tab:unit\_conversions\]. Numerical setup {#sec:numerical_setup} --------------- All shown simulations are conducted on a 3D box with dimensions $\left[4741.12 M_\odot\times4741.12 M_\odot\times4741.12 M_\odot\right]$ with a grid spacing of $\Delta_{x,y,z}=74.08M_\odot$ on the coarsest grid level. For the chosen magnetar model of radius $R_*=9.26M_\odot$ ($\simeq {13.7\,\text{km}}$km) this corresponds to a $\left[512R_*\times 512R_*\times 512R_*\right]$ box with a grid spacing of $\Delta_{x,y,z}=8R_*$. For the low and high resolution tests we employ seven and eight additional levels of mesh refinement, each increasing the resolution by a factor of two and encompassing the central object, respectively. This means that the finest resolution of our models (close to the magnetar surface) are $\Delta_{x,y,z}^{\rm min}=0.0625\times R_* = 0.5787M_\odot$ and $\Delta_{x,y,z}^{\rm min}=0.03125\times R_*= 0.2894M_\odot$ for the low and high resolution models, or in other words 16 and 32 points per $R_*$, respectively. The initial data is evolved for a period of $t=1185.28M_\odot\simeq 5.84\,$ms, which is chosen to be well below the dynamical timescale of the magnetar crust, which can be considered as a fixed boundary (see section \[sec:timecales\]). In order to ensure the conservation properties of the algorithm, it is critical to employ [*refluxing*]{} techniques correcting numerical fluxes across different levels of mesh refinement [see, e.g. @Collins2010]. Specifically, we make use of the thorn `Refluxing`[^6] in combination with a cell-centered refinement structure [cf. @Shibata2015]. We highlight the fact that employing the refluxing algorithm makes the numerical code $2-4$ times slower for the benefit of enforcing the conservation properties of the numerical method (specially of the charge). Refluxing also reduces the numerical instabilities, which tend to develop at mesh refinement boundaries. In conservative schemes, numerical reconstruction algorithms [we employ an MP7 scheme, cf. @Suresh1997] derive inter-cell approximations of the conservative variables by making use of their values at several adjacent grid-points (for MP7, one requires seven points). As a result of the numerical coupling between the magnetosphere and the magnetar crust introduced by the inter-cell reconstruction at the stellar surface, the field dynamics induce a mismatch in the current flowing through the surface and effectively trigger a (numerical) flow of charges leaving or entering the domain. In order to avoid this artifact, we replace the reconstructed values of the radial current ${\tilde{J}}^r_{\rm FFE}$ at interfaces between the stellar interior and exterior by the cell-centered value in the stellar interior. This procedure ensures a conservation of magnetospheric charge. The (3D) initial data is imported from the (2D) initial models (see se. \[sec:initial\_data\]) by bicubic spline interpolation. Throughout the numerical evolution, all quantities on grid-points inside of the magnetar radius are fixed to their initial values. Instability onset and magnetospheric energy balance {#sec:instability_relax} --------------------------------------------------- ![Energy balance during the evolution of the high resolution model C2 (Table \[tab:model\_overview\]). *Top:* Comparison of the change in total magnetospheric energy, normalized to the energy of a magnetosphere equiped with a pure dipolar magnetic field, $\Delta {\mathcal{E}}/{\mathcal{E}}_{\rm d}$, as well as the Poynting flux through the magnetar surface. Up to a simulation time of $t~\sim 3.33\,$ms the energy change is dominated by Poynting flux onto the magnetar crust. *Middle:* Maximum violation of the ${\boldsymbol{{\tilde{B}}}}^2-{\boldsymbol{{\tilde{E}}}}^2\ge 0$ condition throughout the numerical grid. *Bottom:* Maximum violation of the ${\boldsymbol{{\tilde{B}}}}\cdot{\boldsymbol{{\tilde{E}}}}= 0$ constraint throughout the numerical grid. At the time of the breakdown of conditions (\[eq:force\_free\_crossfield\]) and (\[eq:force\_free\_dominance\]), the energy change is dominated by secondary (possibly numerical) effects. []{data-label="fig:energy_dissipation_channels"}](images/figure5.pdf){width="45.00000%"} We have performed simulations with initial models in the low energy branch (A1, B1 and C1) and in the high energy branch (A2, B2, C2, C3). We observe a differentiated behavior in the evolution of the system depending on the class of initial model. For models in the low energy branch we find that the magnetosphere is stable and that the system remains essentially unchanged. The energy of the system remains constant throughout the simulation (see blue lines in Figure \[fig:absolute\_energy\_evolution\]), confirming the stability of these configurations, at least on dynamical timescales. This is specially true in the high resolution models, which exhibit a smaller numerical dissipation. The slightly larger numerical dissipation of the low resolution models explains the small drift in time with respect to the initial energy displayed by the blue dashed lines in Figure \[fig:absolute\_energy\_evolution\]. On the other hand, models in the high energy branch become unstable on a timescale of a few milliseconds and the magnetosphere changes its shape roughly at the same time as the energy of the magnetosphere decreases (see red and green lines in Figure \[fig:absolute\_energy\_evolution\]). This numerical experiment confirms the hypothesis of @Akgun2018a that, for degenerate initial models, only the lowest energy state is stable, and that all corresponding degenerate cases of high energy are unstable. In addition, we note that the lower energy states are closer to a purely dipolar magnetosphere, hence, the minimised circumference of the magnetic surfaces minimise the magnetospheric energy content [cf. @Thompson1996]. For configurations in the unstable branch, the onset of the instability proceeds earlier for lower numerical resolution. This is expected because a coarser grid contains larger numerical discretisation errors acting as a seed for the instability onset. However, the rapid drop in energy during the instability proceeds in a similar fashion for both numerical resolutions, indicating that the instability has a physical origin and is not a numerical artifact. In the case of the high energy initial model C2 we observe a rearrangement of the lobes of magnetic twist towards a dipolar structure (see Figure \[fig:high\_energy\_relaxation\]) prior to a significant drop of magnetospheric energy (by approximately $30\%$ of its initial value). During the phase of full validity of the force-free condition (see equation \[eq:force\_free\_dominance\]) the loss of magnetospheric energy is dominated by an outgoing Poynting flux at the innermost boundary (see Figure \[fig:energy\_dissipation\_channels\]). For our boundary condition it can be interpreted as the formation of a strong current on a thin layer below the surface, where energy can be efficiently dissipated. Following @Parfrey2013 in the context of twisted magnetar fields and @Li2018 in a study of energy dissipation in collisions of force-free Alfvén waves, the onset of the (topological) relaxation is likely to be linked to Ohmic heating ${\boldsymbol{J}}\cdot{\boldsymbol{E}}\neq 0$, which occurs as a result of (minor) violations of the force-free condition (\[eq:force\_free\_crossfield\]), as can be seen in the bottom panel of Fig. \[fig:energy\_dissipation\_channels\] (note the much smaller scale of that panel compared to the middle one). We give a more detailed review of the treatment of these violations in our code and throughout the literature in appendix \[sec:ff\_breakdown\]. Surface currents and long-time evolution {#sec:surface_currents} ---------------------------------------- Following the initial instability and subsequent rapid rearrangement of the magnetar magnetosphere (section \[sec:instability\_relax\]), thin currents form at the magnetar surface (see Figures \[fig:surface\_currents\_xz\] and \[fig:surface\_currents\_avg\]). These currents are expected to appear as the initial model in the high energy state tries to relax to the lowest energy magnetospheric configuration, while keeping the interior field fixed (see the discussion in section \[sec:interior\]). There are two possible fates for these currents: i) they could propagate inwards, inside the magnetar crust, deforming the magnetic field inside and creating a mechanical stress in the crust, on a timescale of several $10 \text{ ms}$, or ii) they could form a thin surface current dissipating on a timescale shorter that the time it takes to deform the crust. These two possibilities are not mutually exclusive and a combination of both is possible. In none of the cases our simulations can give a conclusive answer because i) we are not evolving the magnetar interior as we are considering only timescales smaller than the dynamical timescale of the crust, ii) the formation of thin surface currents is numerically challenging (would require a computationally prohibitively high resolution near the magnetar surface), and iii) it would eventually violate the FF conditions (\[eq:force\_free\_crossfield\]) and (\[eq:force\_free\_dominance\]), hence invalidating our current numerical approach. ![*xz*-cross-sections of the toroidal current in geometrised units showing the development of strong surface currents during the evolution, in addition to other currents extended on larger magnetospheric volumes. *Top:* Low resolution model C2 (16 points per $R_*$). *Bottom:* High resolution model C2 (32 points per $R_*$). The high resolution evolution shows currents located around the magnetar surface with more detailed structures, emphasizing their interpretation as surface currents. The spatial coincidence of the currents in both resolutions reinforce the argument that the observed currents are of physical origin (in spite of the - relatively small - differences among different resolutions).[]{data-label="fig:surface_currents_xz"}](images/figure6.pdf){width="49.00000%"} The aforementioned current layers are expected to be regions of strong energy dissipation and the breakdown of the force-free conditions [see, e.g. @Uchida1997; @McKinney2006; @Palenzuela2010; @Parfrey2013]. Figures \[fig:energy\_dissipation\_channels\] and \[fig:surface\_currents\_avg\] link the breakdown of the force-free condition (\[eq:force\_free\_dominance\]) and the occurrence of surface currents with the opening of dissipation channels different to the energy flow through the magnetar surface (see appendix \[sec:ff\_breakdown\] for a short review of the force-free breakdown). We find the violation of condition (\[eq:force\_free\_crossfield\]) to be continuously occurring with peaks at the instance of rapid energy dissipation. Condition (\[eq:force\_free\_dominance\]) starts to fail on longer timescales at the moment of fastest transfer of magnetic energy through the surface. At this time, further dissipation mechanisms (see Figure \[fig:energy\_dissipation\_channels\]) come into play, as is expected throughout the literature [@Uchida1997; @McKinney2006; @Li2018]. It should be noted that the total magnetospheric energy for the models B2, C2, and C3 drops below the energy of their respective low energy equilibrium solutions, and even below the magnetospheric energy of the vacuum dipole (equation \[eq:edipole\]). However, this energy drop is (slightly) smaller for the high resolution simulations, and shows some dependence on the chosen setup of the hyperbolic/parabolic cleaning procedures (see appendix \[sec:augmented\_system\]) at the magnetar surface. The sensitivity of this behavior to the numerical details at the location of the (3D Cartesian) crust may be attributed to the numerical dissipation of the employed code. ![Azimuthal angular averages of the toroidal current (normalised to its initial value at the stellar surface) in the equatorial plane showing the development of surface currents during the evolution of the C2 initial model. We display the current evolution for both low resolution (16 points per $R_*$, denoted by dashed lines), and high resolution (32 points per $R_*$, denoted by solid lines) models. The increase of the toroidal current during the transient of energy dissipation (see Figure \[fig:absolute\_energy\_evolution\]) at the lower resolution (compare the two blue lines) may be attributed to a faster onset of the twist instability for this model.[]{data-label="fig:surface_currents_avg"}](images/figure7.pdf){width="45.00000%"} -------------- -------------------------------- ---------------------------------------------- --------------------- -------------------------------- ------------------------------- Model $\Delta t_{\rm r} \text{(ms)}$ $\Delta {\mathcal{E}}/{\mathcal{E}}_{\rm d}$ $\Delta_{\rm mx} J$ $\Delta_{\rm mx} T^{r\varphi}$ $\Delta_{\rm mx} T^{r\theta}$ \*\[0pt\] A1 5.8400 0.0033 0.0159 0.0012 0.0010 A2 1.4162 0.0963 1.6350 0.0295 0.0150 B1 5.8400 0.0042 0.0363 0.0012 0.0014 B2 3.0427 0.1002 0.9805 0.0358 0.0232 C1 5.8400 0.0009 0.0640 0.0008 0.0013 C2 2.1604 **0.2808** **3.5400** 0.0851 0.0414 C3 1.0490 0.1962 3.1720 **0.1008** **0.0811** -------------- -------------------------------- ---------------------------------------------- --------------------- -------------------------------- ------------------------------- : Selection of electromagnetic quantities monitored throughout the (high resolution, 32 points per $R_*$) simulation of the models of Table \[tab:model\_overview\]. The total change in energy $\Delta {\mathcal{E}}$ (displayed as a fraction of the vacuum dipole energy; equation \[eq:edipole\]) corresponds to the maximum drop of electromagnetic energy during the total runtime (see section \[sec:energy\_release\]). The operator $\Delta_{\rm mx}$ acting on any quantity $A(t,{\boldsymbol{x}})$ is defined as $\Delta_{\rm mx}A:=\max_{\{t,|{\boldsymbol{x}}|=R_*\}}{ \{ A(t,{\boldsymbol{x}})-A(0,{\boldsymbol{x}}) \}/\max_{\{|{\boldsymbol{x}}|=R_*\}}A(0,{\boldsymbol{x}}) }$. Hence, $\Delta_{\rm mx} J$ is the maximum increase in current density in the magnetosphere during the relaxation relative to the initial values (see section \[sec:current\_magnetsophere\]). In the right columns, $\Delta_{\rm mx} T^{r\varphi}$ and $\Delta_{\rm mx} T^{r\theta}$ denote the maximum increase of electromagnetic stresses relative to their corresponding initial values (see section \[sec:crust\_stresses\]) on the stellar surface compared to its initial value. We highlight with bold face the maximum values of each of the last four columns.[]{data-label="tab:model_results"} Discussion {#sec:discussion} ========== Conclusions {#sec:conclusion} =========== In this work, we explore the stability properties of force-free equilibrium configurations of magnetar magnetospheres by performing numerical simulations of a selection of the models computed in @Akgun2018a. For the case of degenerate magnetospheres (i.e. the same boundary conditions but different energies) we validate the hypothesis of @Akgun2018a that configurations in the high-energy branches are unstable while those in the lowest energy branch are stable. This confirms the existence of an unstable branch of twisted magnetospheres. It also allows to formulate an instability criterion for the sequences of models computed in @Akgun2018a. Our results are consistent with an interesting scenario where bursts and GFs in magnetars are triggered without involving crustal failures. The twist that is naturally produced in the magnetosphere by the Hall evolution of the crust [@Akgun2017] can lead to unstable configurations that will release up to a 10% of the energy stored in the magnetosphere, sufficient to explain the observations. [@Akgun2017] have shown that the magneto-thermal evolution of the crust leads naturally to configurations close to the instability threshold. However, the amount of energy released depends on how far away from the stable branch can the evolution drive the configuration. This is essentially a problem of comparing the evolution timescale and the instability timescale. For the models studied in this work the instability timescale is of the order of milliseconds, much shorter that the magneto-thermal evolution timescales of the object (see Sect. \[sec:secular\]). However, close to the critical point, the growth rate of the instability could be significantly smaller (actually, it should be zero at the critical point) which would allow to overshoot the instability threshold. Note that, since the energy reservoir is large ($\sim 10^{46}$ erg), even a very small fraction of energy release could explain many of the phenomenology of magnetars. Alternatively, there could be phenomena leading to fast dynamics in the crust such as sustained episodes of accelerated plastic flows triggered by the magnetic stresses in the crust [@Lander2019]. For the unstable models, we observe the development of almost axisymmetric instabilities on a timescale of a few ms rearranging the magnetic field to a configuration similar to those in the (stable) lower energy branch. The energy of the magnetosphere also decreases towards the value of the stable configuration. Differences with respect to the corresponding stable configuration can be attributed to the influence of the non-preservation of the force-free constraints (\[eq:force\_free\_crossfield\]) and (\[eq:force\_free\_dominance\]). Using (much) larger numerical resolution (beyond the scope of our computational resources) we envision that the violation of the force-free constraints would be significantly reduced and the expected (low-energy) states would be the endpoint of the evolution after a full relaxation of the magnetosphere takes place. The energy decrease is explained, mainly, by a flow of energy towards the surface of the star, where it is dissipated efficiently. A large fraction of this energy is also dissipated in the magnetosphere at locations where the force-free conditions break. This contrasts with the work of @Beloborodov2011, @Parfrey2013 and [@Carrasco2019] in which most of the energy is dissipated by the formation and ejection of plasmoids. The different setup used in these workst (dynamically twisting vs. unstable equilibrium configurations) makes a direct comparison difficult. A possible source for the qualitative discrepancy may be differences in the boundary condition at the surface of the star. While we use a boundary condition that dissipates very efficiently any strong currents formed at the surface, in their work, their use of essentially non-dissipative boundary conditions make the surface perfectly reflective. For the future it would be interesting to compare more closely the differences in the boundary condition and to develop a better physical model for dissipation at the NS surface. The magnetic field remains nearly axisymmetric throughout the simulation indicating that the instability is mostly an m=0 instability. A complete theoretical analysis of the origin of the instability and its properties is beyond the scope of this paper. However, we anticipate that such analysis has to be carried out on a global scale either by calculating the eigenmodes or by using the so-called energy principle of @Bernstein1958 and is not trivial due to the presence of both poloidal and toroidal components [@Akguen2013 and references therein]. However, we note that, since the poloidal field structure changes somewhat less than the toroidal field, this instability could be compared to the interchange instability discussed by @Tayler1973, where displacing the toroidal field radially decreases the energy (even in the absence of a fluid). We have made a crude estimation of the observational properties of the energy liberated in the magnetosphere as a result of the instability. The fact that large amounts of energy (in excess of $10^{46}\,$erg) are released on milliseconds timescales results in dynamical luminosities significantly larger than $10^{48}\,$ergs$^{-1}$ (reaching in some models $4\times 10^{49}\,$ergs$^{-1}$). This should trigger the expansion of a pair-photon fireball polluted with baryons unbound from the magnetar crust. The bolometric signature of these fireballs seems incompatible with the observations of the initial spikes observed in GFs. With our simple analytic model, most of the unstable magnetospheres produce over-luminous, too cool and excessively short flashes. However, this problem can be solved if the energy can be liberated on longer timescales, of the order of the observed GF spikes ($\Delta t_{\rm spike}\sim 0.1\,$s). This could be possible in a scenario of slow energy dissipation as the one proposed by [@Li2018], which we plan to explore in the future. The currents produced during the instability increase significantly the amount of pairs in the magnetosphere, a large fraction of which, of size $\sim 10 R_*$, becomes optically thick. The hot plasma magnetically confined in this region could be responsible for the extended thermal X-ray emission lasting for $50-300$s after GFs. Our force-free numerical method cannot properly deal with the evolution of extremely thin surface currents. Therefore, the dynamical millisecond timescales computed in our models should be taken as a lower bound for the physical timescales. The magnetic dissipation taking place at these locations can be due to, e.g. Ohmic processes or to non-linear Alfvén wave interactions. Assuming that energy is released on $\sim\Delta t_{\rm spike}$, our estimate of the electromagnetic signature yields photospheric luminosities and temperatures compatible with observational data. Since this is a sound physical assumption, we conclude that observed GFs in SGRs are broadly compatible with the development of instabilities in twisted magnetospheres. Acknowledgements ================ We thank Amir Levinson for his support in challenging our numerical code prior to the production of the presented results. We also thank Oscar Reula, Federico Carrasco and Carlos Palenzuela for their valuable discussions on the boundary conditions. We acknowledge the support from the grants AYA2015-66899-C2-1-P and PROMETEO-II-2014-069. JM acknowledges a Ph.D. grant of the *Studienstiftung des Deutschen Volkes*. PC acknowledges the Ramon y Cajal funding (RYC-2015-19074) supporting his research. We acknowledge the partial suport of the PHAROS COST Action CA16214 and GWverse COST Action CA16104. The shown numerical simulations have been conducted on *MareNostrum 4* of the *Red Española de Supercomputación* (AECT-2019-1-0004) as well as on the computational infrastructure of the *University of Valencia*. We thank the *EWASS 2019* and *GR22* conferences for the possibility to disseminate the results of this work. Numerical details {#sec:num_details} ================= The augmented system {#sec:augmented_system} -------------------- ![Energy evolution of the high energy initial data models A2, B2, and C2 using different damping constants $\kappa_\psi$ (divergence cleaning) in a low resolution study (16 points per $R_*$). While one observes a converging evolution for the lower cleaning potentials $\kappa_\psi=0.03125$ and $\kappa_\psi=0.125$, the energy evolution shows a strong (non-physical) dependence on $\kappa_\psi$ for larger damping constants. This effect is amplified in the high resolution (32 points per $R_*$).[]{data-label="fig:psi_optimization"}](images/figureA1.pdf){width="49.00000%"} In order to preserve the physical conditions $\text{div}{\boldsymbol{{\tilde{B}}}}=0$ and $\text{div}{\boldsymbol{{\tilde{E}}}}={\tilde{\rho}_{\rm e}}$ we make use of hyperbolic/parabolic cleaning potentials [@Dedner2002; @Palenzuela2009; @Mignone2010]. Specifically, we implement an augmented system of Maxwell’s equations as follows [@Palenzuela2009; @Miranda-Aranguren2018]: $$\begin{aligned} \partial_t\phi-\partial_i {\tilde{E}}^i&=-\tilde{\rho}_{\rm e}-\kappa_\phi\phi\label{eq:scalar_phi}\\ \partial_t {\tilde{E}}^i-\partial_j\left(\epsilon^{ijk}{\tilde{B}}_k+\delta^{ij}\phi\right)&=-{\tilde{J}}^i_{\rm \textsc{FF}}\label{eq:vector_phi}\\ \partial_t\psi+\partial_i {\tilde{B}}^i&=-\kappa_\psi\psi\label{eq:scalar_psi}\\ \partial_t {\tilde{B}}^i+\partial_j\left(\epsilon^{ijk}{\tilde{E}}_k+\delta^{ij}\psi\right)&=0\label{eq:vector_psi}\end{aligned}$$ Here, $\psi$ (divergence cleaning) and $\phi$ (charge conservation) are the scalar potentials, $\kappa_\phi$ and $\kappa_\psi$ the respective damping constants and $\delta^{ij}$ denotes the Kronecker delta. As for the practical implementation, we follow a Strang splitting approach [as employed, e.g. in @Komissarov2004], effectively solving part of the scalar equations (\[eq:scalar\_phi\]) and (\[eq:scalar\_psi\]) analytically. Prior (before `MoL_Step`) and after (before `MoL_PostStep`) the time integration of the Einstein Toolkit `thorn` `MoL` we evolve in time the equations $$\begin{aligned} \phi\left(t\right)&=\phi_0\exp\left[-\kappa_\phi t\right],\\ \psi\left(t\right)&=\psi_0\exp\left[-\kappa_\psi t\right],\end{aligned}$$ for a time $t=\Delta t/2$. The coefficients $\kappa_\phi$ and $\kappa_\psi$ have to be chosen by optimisation in accordance with the grid properties. We find it beneficial to choose a large value for $\kappa_\phi$, effectively dissipating charge conservation errors on very short timescales. As for the divergence cleaning, we conducted a series of tests, optimizing $\kappa_\psi$ to yield stable and converging evolution for all shown resolutions, ultimately resorting to $\kappa_\psi = 0.125$ (see Figure \[fig:psi\_optimization\] for a review of the optimisation process). It should be noted at this point that @Mignone2010 present a promising scheme of choosing $\kappa_\psi$ according to the grid resolution that has also been used in [@Miranda-Aranguren2018]. In the framework of mesh refinement of the Einstein Toolkit, this would result in a different damping of the cleaning potentials across the refinement levels. We have found that the optimisation of the hyperbolic/parabolic cleaning becomes a very subtle issue and may experience strong numerical effects when increasing the overall resolution. This observation may, however, be an artifact of the fixed boundary of the magnetar surface - which on a Cartesian grid, resembles an accumulation of boxes rather than a perfectly aligned spherical boundary. The exploration of these effects and the transition to a fully spherical version of this force-free `thorn` [as introduced in @Baumgarte2013; @Montero2014] will be a subject of future efforts. Conservation of force-free constraints {#sec:ff_breakdown} -------------------------------------- FFE codes are valid in the limit of high electromagnetic energy compared to the rest mass and thermal energy of the respective plasma. The dynamics of force-free fields is described entirely without the plasma four-velocity. However, demanding the existence of a physical, timelike velocity field ${\boldsymbol{u}}$ with $F_{\mu\nu}u^\nu = 0$, as well as the degeneracy condition $F_{\mu\nu}J^\nu=0$ [see @Uchida1997 for a detailed algebraic review] one is left with the aforementioned constraints: $$\begin{aligned} {\boldsymbol{{\tilde{E}}}}\cdot{\boldsymbol{{\tilde{B}}}}&=0 \tag{\ref{eq:force_free_crossfield}}\\ {\boldsymbol{{\tilde{B}}}}^2-{\boldsymbol{{\tilde{E}}}}^2&\geq 0 \tag{\ref{eq:force_free_dominance}}\end{aligned}$$ Within the shown simulations we find it beneficial to employ an approach presented in @Komissarov2011 and @Parfrey2017 in order to archive $\partial_t\left({\boldsymbol{{\tilde{E}}}}\cdot{\boldsymbol{{\tilde{B}}}}\right)=0$ throughout the evolution (by making use of the force-free current as in equation \[eq:ff\_current\]) without the employment of target currents [as discussed in @Parfrey2017]. Additionally, we include a suitable Ohm’s law [@Komissarov2004 section C3] into our Strang splitting approach aiming towards an evolution minimizing the violation of conditions (\[eq:force\_free\_crossfield\]), and (\[eq:force\_free\_dominance\]). In order to build up a force-free current, @Komissarov2004 introduces a generalised Ohm’s law in the context of FFE: $$\begin{aligned} \begin{split} {\boldsymbol{{\tilde{J}}}}=\sigma_\parallel{\boldsymbol{{\tilde{E}}}}_\parallel+\sigma_\perp{\boldsymbol{{\tilde{E}}}}_\perp+{\boldsymbol{\tilde{j}}}_d\label{eq:GeneralizedOhmsLaw}, \end{split}\end{aligned}$$ where the subscripts $\parallel$ and $\perp$ denote the components parallel and perpendicular to the magnetic field, ${\boldsymbol{{\tilde{B}}}}$. A to be specified model for $\sigma$ introduces a suitable resistivity into the force-free system [see also @Lyutikov2003 for further comments on resistive FFE], while ${\boldsymbol{\tilde{j}}}_d$ is the drift current perpendicular to the electric and magnetic fields. In its general form, (\[eq:GeneralizedOhmsLaw\]) plays the central role in ensuring the force-free conditions (\[eq:force\_free\_crossfield\]) and (\[eq:force\_free\_dominance\]). @Komissarov2004 suggests a resistivity model that depends on the time-step of the evolution $\Delta t$ (throughout the presented simulations we employ CFL $=0.2$), where $$\begin{aligned} \begin{split} \sigma_\parallel = \frac{d}{\Delta t}\label{eq:sigmapar}. \end{split}\end{aligned}$$ The cross-field resistivity $\sigma_\perp$ is strongly linked to the violation of condition (\[eq:force\_free\_dominance\]), $$\begin{aligned} \arraycolsep=1.4pt\def{1.5}{1.5} \begin{split} \sigma_\perp=\left\{\begin{array}{ccc} 0 & \qquad :\qquad & {\boldsymbol{B}}^2\geq{\boldsymbol{E}}^2\\ b\displaystyle \frac{\left({\tilde{E}}_\perp-{\tilde{E}}_\perp^*\right)}{{\tilde{E}}_\perp^*} & \qquad :\qquad & {\boldsymbol{{\tilde{B}}}}^2<{\boldsymbol{{\tilde{E}}}}^2 \end{array}\right.\label{eq:sigmaperp}, \end{split}\end{aligned}$$ where ${\tilde{E}}_\perp=\left|{\boldsymbol{{\tilde{E}}}}_\perp\right|$ and $\left({\tilde{E}}_\perp^*\right)^2=\left({\boldsymbol{{\tilde{B}}}}-{\boldsymbol{{\tilde{E}}}}_\parallel\right)^2$ and $b$ is an scalar parameter controlling the magnitude of $\sigma_\perp$. Equations (\[eq:sigmapar\]) and (\[eq:sigmaperp\]) have a pair of analytic solutions: $$\begin{aligned} \begin{split} {\boldsymbol{{\tilde{E}}}}_\parallel\left(t\right)=\:&{\boldsymbol{{\tilde{E}}}}_\parallel\left(0\right)\times e^{-\sigma_\parallel t} \end{split}\label{eq:StrangCurrentPar}\\ \begin{split} {\boldsymbol{{\tilde{E}}}}_\perp\hspace{-3pt}\left(t\right)=\:&\left[{\tilde{E}}_\perp^*\hspace{-3pt}\left(0\right)+\frac{{\tilde{E}}_\perp^*\hspace{-3pt}\left(0\right)\left[{\tilde{E}}_\perp\hspace{-3pt}\left(0\right)-{\tilde{E}}_\perp^*\hspace{-3pt}\left(0\right)\right]\times e^{-b\sigma_\parallel t}}{{\tilde{E}}_\perp\hspace{-3pt}\left(0\right)-\left[{\tilde{E}}_\perp\hspace{-3pt}\left(0\right)-{\tilde{E}}_\perp^*\hspace{-3pt}\left(0\right)\right]\times e^{-b\sigma_\parallel t}}\right]\\ &\times\frac{{\boldsymbol{{\tilde{E}}}}_\perp\hspace{-3pt}\left(0\right)}{{\tilde{E}}_\perp\hspace{-3pt}\left(0\right)}. \end{split}\label{eq:StrangCurrentPerp}\end{aligned}$$ During our numerical simulations), we usually choose $d=5.0$, and $b=0.1$, and solve equation (\[eq:StrangCurrentPar\]) prior to equation (\[eq:StrangCurrentPerp\]) in a Strang splitting scheme in direct analogy to the implementation described in section \[sec:augmented\_system\]. This resistivity model ensures the validity of the force-free regime throughout time, in other words, the evolution is driven towards a force-free state $$\begin{aligned} \begin{split} {\boldsymbol{{\tilde{E}}}}\cdot{\boldsymbol{{\tilde{B}}}}&\rightarrow0 \\ {\boldsymbol{{\tilde{B}}}}^2-{\boldsymbol{{\tilde{E}}}}^2&\rightarrow 0 \qquad\text{ : }{\boldsymbol{{\tilde{B}}}}^2<{\boldsymbol{{\tilde{E}}}}^2. \end{split}\end{aligned}$$ Optical depth to resonant cyclotron scattering {#sec:optical_thickness} ============================================== For the presented modeling of the optical thickness of highly magnetised force-free plasmas around magnetars (see section \[sec:emission\_processes\]), we adapt the techniques describing resonant scattering as presented by @Beloborodov2013 (from now on Be13). In the following, we will give a short review of the underlying equations. In order to derive the optical thickness $\tau$, we integrate equation (Be13/A15), $$\begin{aligned} \frac{\text{d}\tau}{\text{d} s}=2\pi^2 r_e\frac{c}{\omega}\frac{\xi}{\left|\tilde{\mu}\right|}n_e\left[f_e\left(p_1\right)+f_e\left(p_2\right)\right]. \label{eq:optical_thickness_integrand}\end{aligned}$$ Here, $r_e=e^2/m_e c^2$ denotes the photon wavelength, $\omega$ the frequency of the seed photon (we consider 1keV photons), and $\xi=1$ or $\xi=\tilde{\mu}^2$ depending on the photon polarisation ($\perp$ or $\parallel$, respectively). The relativistic particles require the specification of the quantities $\mu=\cos\vartheta$ and $\tilde{\mu}=\cos\tilde{\vartheta}$, where $\vartheta$ is the angle between the photon path and the magnetic field ${\boldsymbol{B}}$ in the lab frame and $\tilde{\vartheta}$ in the rest frame of the electron. The dimensionless momenta $p_{1,2}$ correspond to the electron (or positron) velocities favored by the resonant scattering model. As both polarisations yield similar results, we only consider the slightly dominant $\perp$ orientation for our model. @Beloborodov2013 estimated that the contribution of non-resonant scattering to the optical depth is negligible and will not be considered in our calculations (see, however, footnote \[foot:one\]). Following Be13, we employ the so-called waterbag model as a distribution function for electron (or positron) momenta. In analogy to a two-fluid model, the distribution function is characterised by the two parameters (dimensionless momenta) $p_+$ and $p_-$, with the overall shape $$\begin{aligned} f_e\left(p\right)=\left\{\begin{array}{ccc} (p_+ - p_-)^{-1}& : & p_-<p<p_+ \\ 0 & : & \text{else} \end{array}\right.. \label{eq:waterbag}\end{aligned}$$ Applying the waterbag model (\[eq:waterbag\]) in equation (\[eq:optical\_thickness\_integrand\]) selects the relevant electron (or positron) momenta for the scattering process. The distribution of this normalisation factor throughout the magnetosphere especially depends on the flow direction of charges along ${\boldsymbol{B}}$. As described in section 5.2 of Be13, we adjust their model according to a flow of electrons (or positrons) which turns back to the central object when field lines cross the equator. We apply this to all field lines crossing regions with $B<10^{13}$G (this holds everywhere except in the inner coronal region of strong closed magnetic field lines). \[lastpage\] [^1]: jens.mahlmann@uv.es [^2]: pablo.cerda@uv.es [^3]: Slow magnetosonic waves are also possible but their velocity is much smaller and not relevant for this work, in fact, for the case of waves perpendicular to the magnetic field their speed is zero. [^4]: <http://www.einsteintoolkit.org> [^5]: <http://www.cactuscode.org> [^6]: Refluxing at mesh refinement interfaces by Erik Schnetter: <https://svn.cct.lsu.edu/repos/numrel/LSUThorns/Refluxing/trunk>

--- abstract: 'Hierarchical assembly models predict a population of supermassive black hole (SMBH) binaries. These are not resolvable by direct imaging but may be detectable via periodic variability (or nanohertz frequency gravitational waves). Following our detection of a 5.2 year periodic signal in the quasar PG 1302-102 [@graham15], we present a novel analysis of the optical variability of 243,500 known spectroscopically confirmed quasars using data from the Catalina Real-time Transient Survey (CRTS) to look for close ($< 0.1$ pc) SMBH systems. Looking for a strong Keplerian periodic signal with at least 1.5 cycles over a baseline of nine years, we find a sample of 111 candidate objects. This is in conservative agreement with theoretical predictions from models of binary SMBH populations. Simulated data sets, assuming stochastic variability, also produce no equivalent candidates implying a low likelihood of spurious detections. The periodicity seen is likely attributable to either jet precession, warped accretion disks or periodic accretion associated with a close SMBH binary system. We also consider how other SMBH binary candidates in the literature appear in CRTS data and show that none of these are equivalent to the identified objects. Finally, the distribution of objects found is consistent with that expected from a gravitational wave-driven population. This implies that circumbinary gas is present at small orbital radii and is being perturbed by the black holes. None of the sources is expected to merge within at least the next century. This study opens a new unique window to study a population of close SMBH binaries that must exist according to our current understanding of galaxy and SMBH evolution.' author: - | Matthew J. Graham,$^1$[^1] S. G. Djorgovski,$^1$ Daniel Stern,$^2$ Andrew J. Drake,$^1$ Ashish A. Mahabal,$^1$ Ciro Donalek,$^1$ Eilat Glikman$^3$, Steve Larson$^4$, Eric Christensen$^4$\ $^{1}$California Institute of Technology, 1200 E. California Blvd, Pasadena, CA 91125, USA\ $^{2}$Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA\ $^{3}$Department of Physics, Middlebury College, Middlebury, VT 05753, USA\ $^{4}$University of Arizona, Department of Planetary Sciences, Lunar and Planetary Lab, Tucson, AZ 85721, USA date: 'Accepted . Received ; in original form' title: 'A systematic search for close supermassive black hole binaries in the Catalina Real-Time Transient Survey' --- \[firstpage\] methods: data analysis — quasars: general — quasars: supermassive black holes — techniques: photometric — surveys Introduction ============ Supermassive black hole (SMBH) binary systems are an expected consequence of hierarchical models of galaxy formation [@haehnelt02; @volonteri03] and an important sources of nanohertz frequency gravitational waves. Theoretically, they are seen to evolve through three stages (for a recent review, see [@colpi14] and references therein): in the first, the SMBH pair sinks towards the center of the newly merged system via dynamical friction (the SMBHs feel the collective effect of the stellar distribution). The binary continues to decay due to interactions with stars in the nuclear region on intersecting orbits with the binary. These stars carry away energy and angular momentum. The merger process may stall in this phase (at separations of 0.01 – 1 pc) owing to the depletion of stars in the nuclear region. However, other factors such as the binary mass, the presence of cold gas in the nuclear region, or deviations from axial symmetry in the galaxy remnant can affect the length of the potential stall. Depending on these factors as well as the stellar distribution, the binary will generally spend most of its lifetime in this stage. Finally, if the binary separation decreases sufficiently through these processes, the remaining orbital angular momentum of the pair is efficiently dissipated by the emission of gravitational radiation, leading to an inevitable merger. At coalescence, the merged SMBH receives a kick velocity and may recoil and oscillate about the core of the galaxy or even be ejected. SMBH binary pairs have been detected at kiloparsec separations, e.g., NGC 6240 at a separation of 1.4 kpc by [*Chandra*]{} [@komossa03], but the observational evidence for close (subparsec) pairs is tentative. The highest milliarcsecond angular resolution imaging with VLBA/VLBI radio observations is only feasible for the very nearest systems – a 1 pc separation pair at a distance of 100 Mpc ($z \sim 0.024$) subtends 2 milliarcsecs. [@rodriguez06] report the discovery of two resolved compact, variable, flat-spectrum radio sources with a projected separation of 7.3 pc in the radio galaxy 0402+379 ($z = 0.055$). The best known close SMBH binary candidate, OJ 287 [@valtonen08], shows a pair of outburst peaks in its optical light curve every 12.2 years for at least the last century, which has been interpreted as evidence for a secondary SMBH perturbing the accretion disk of the primary SBMH at regular intervals [@valtonen11]. Most other claims of photometric [(quasi)]{}[periodicity]{}, e.g., [@fan02; @rieger00; @depaolis02; @liu14], however, tend to have far shorter temporal baselines and do not show the same level of regularity. To date the search for SMBH binary systems on [(sub)]{}[parsec]{} scales has therefore focused on detection by spectroscopic discovery, almost solely within the SDSS data set. It should be noted, though, that there is no unique spectral characteristic signature of a SMBH binary, despite several theoretical efforts to identify one, e.g., [@shen10; @montuori11]. Initial searches [@bogdanovich08; @dotti09; @boroson09] looked for double broad-line emission systems – thought to originate in gas associated with the two SMBHs, where the velocity separation between the two emission line systems traces the projected orbital velocity of the binary – or single line systems with a systematic velocity offset of the broad component from the narrow component (e.g., [@tsalmantza11; @tsai13]), under the assumption that only one SMBH is active and Keplerian rotation about this produces the shift. However, it is possible to explain such phenomena via a disk emitter model with a single SMBH rather than requiring a binary system. More recent searches have been based on multi-epoch spectroscopy, looking for temporal changes in the velocity shifts consistent with binary motion (acceleration effects). This can either be from dedicated follow-up spectroscopy of initial epoch candidates (e.g., [@eracleous12; @decarli13]) or more general searches using emerging collections of multi-epoch data, e.g., [@shen13; @liu14; @ju13]. With typical velocity offsets of $\sim1000$ km s$^{-1}$, searches are typically sensitive to systems with mass $M \sim 10^{8} M_\odot$ separated by $\sim0.1$ pc and with orbital periods of $\sim300$ years. They will therefore only capture a fraction of an orbital cycle and other astrophysical processes, such as an orbiting hotspot produced by a local instability in the broad line region (BLR) of a galaxy or outflows associated with the accretion disk, may still be responsible for any effect seen [@bogdanovich14; @barth15]. At small separations ($\sim$ 0.01 pc), the size of any BLR is significantly larger than the semimajor axis of the binary. The velocity dispersion of the BLR gas bound to a single SMBH is higher than the velocity difference between the two SMBHs in a binary. It is very unlikely, therefore, that emission lines from the two BLRs associated with the SBMHSs will be resolved since their separation is likely to be smaller than their widths. The optical and UV spectral features are therefore no longer directly related to the period of the binary [@shen10]. However, variability related to the periodicity of the accretion flows onto the binary may be directly detectable [@artymowicz96; @hayasaki08]. As noted earlier, there have been reports of [(quasi)]{}[periodic]{} behavior detected in the long-term multiwavelength monitoring of particular blazars, e.g., PKS 2155-304, AO 0235+164 [@rani09; @wang14]. These can be separated into short timescale high-energy (X-ray, gamma ray) phenomena, typically with periods of a few hours or days, and longer timescale radio and optical behaviors with timescales of a fraction of a year to decades. Given the nature of these objects, though, these are generally explained as related to jet-disk interactions around a single SMBH rather than a SMBH binary. The expected period for a pair of $10^8 M_{\odot}$ black holes at small separations ($\sim0.01$ pc) is $\sim$10 years and there are now a number of synoptic sky surveys with sufficient temporal baselines and sky coverage for a large scale search for such objects based on their photometry [@pojmanski02; @udalski93; @rau09; @sesar11]. In fact, [@haiman09] proposed systematic searches for periodic quasars in large optical or X-ray surveys and calculated the likely detection rates for various observing strategies. The expectation is that such objects are rare – [@volonteri09] estimate that there should be $\sim$10 with $z < 0.7$. In this work, we present a systematic search for periodic behavior in quasars covered by the Catalina Real-time Transient Survey (CRTS[^2]; [@drake09; @mahabal11; @djorgovski12]). This is the largest open (publicly accessible) time domain survey currently operating, covering $\sim 33000$ deg$^2$ between $-75^{\circ} < {\mathrm Dec} < 70^{\circ}$ (but avoiding regions within $\sim 10^{\circ} - 15^{\circ}$ of the Galactic plane) to a depth of $V \sim 19$ to 21.5. Time series exist[^3] for approximately 500 million objects with an average of $\sim$250 observations over a 9-year baseline. This paper is structured as follows: in section 2, we present the selection techniques for identifying periodic quasars and in section 3, the data sets we have applied them to. We discuss our results in section 4. We consider the CRTS time series of other periodic candidates in the literature in section 5. We discuss our findings in section 6 and present our conclusions in section 7. We assume a standard WMAP 9-year cosmology ($\Omega_\Lambda = 0.728$, $\Omega_M = 0.272$, $H_0 = 70.4$ km s$^{-1}$ Mpc$^{-1}$; [@jarosik]) and our magnitudes are approximately on the Vega system. Identifying periodic quasars ============================ The CRTS time series are typically noisy, gappy and irregularly sampled and this precludes any standard Fourier-based analysis of periodicity which assumes regular sampling. Variant techniques that seek to regularize the data, such as Lomb-Scargle, may be $\sim 30$% ineffective for this type of data [@graham13] and so are not suitable for this analysis. Instead we have developed a joint wavelet and autocorrelation function based approach to select our periodic quasar candidates. Wavelet analysis ---------------- Wavelets are an increasingly popular tool in analyses of time series and are particularly attractive since they allow both localized time and frequency analysis. In particular, the power spectrum of a time series can be evaluated as a function of time and the time evolution of parameters associated with possible [(quasi)]{}[periodic]{} behavior determined, i.e., period, amplitude and phase. Although conventional wavelet analysis via the discrete wavelet transform requires regularly sampled data, a number of techniques have been developed to deal with irregularly sampled data. The discrete wavelet transform (DWT) of a time series, $X_t$, is a convolution with the complex conjugate of the particular wavelet function, $f$, being used: $$W (\omega, \tau, X_t) = \sqrt{\omega} \sum_{\alpha = 1}^{N} X_{t_\alpha} f^{\ast}(\omega(t_\alpha - \tau))$$ where $\omega$ is the test frequency (scale factor) and $\tau$ is an offset relative to the start of the time series (time shift). A popular choice for the convolution function (wavelet) is the (abbreviated) Morlet wavelet, a plane wave modulated by a Gaussian decay profile (determined by the decay constant, $c$): $$f(z) = \exp(i \omega(t - \tau) - c\omega^{2}(t - \tau)^2)$$ With unevenly sampled data, however, the DWT is prone to spurious high-frequency spikes and fake structures in the time-frequency plane. [@foster96] proposed that the DWT can be reconsidered as a projection of the time series onto a trial function $\phi(t) = \exp(i\omega(t - \tau))$ with statistical weighting $w_a = \exp(-c\omega^2(t - \tau)^2)$, which compensates for irregularities in the data coverage. Furthermore, a bias at lower frequencies due to effectively sampling more data points through a wider window can be corrected by using $Z$-statistics to give the weighted wavelet $Z$-transform (WWZ). The peak of the WWZ (for constant $\tau$) can be used to determine the period of any signal in a time series within a given time window and the tracks of peaks in the time-frequency plane will show any changes of period or amplitude with time. This also allows a check on whether any periodicity is present for the whole observed time or is transient. The resolution of WWZ is dependent on the decay constant $c$ used to determine the width of the wavelet window. If the value is too high then the WWZ power decays too rapidly and small amplitude periodic signals will tend to be missed whilst if it is too low then the WWZ only has broad frequency resolution. A general recommendation is to use a value such that the exponential term decreases significantly in a single cycle $2\pi / \omega$. Finally, one limitation of the WWZ is that it cannot be used for data with gaps larger than the period in question. An alternate approach finds the wavelets as solutions to an eigenvalue problem where they maximize the energy in a frequency band $|f - f_c| \le f_w$ about a central frequency $\pm f_c$. These Slepian wavelets are related to discrete prolate spheroidal functions and have the advantage that they can be applied to irregular and gappy time series [@mondal11]. This type of wavelet analysis makes use of the variance of the wavelet coefficients to identify those scales making the largest contribution to the total wavelet variance of a time series. [@graham14] used this to identify a characteristic $\sim$50 day restframe timescale in quasars that marks an apparent deviation from the expected behavior for a first-order continuous time autoregressive process (CAR(1), also known as a damped random walk or Ornstein-Uhlenbeck process). This restframe timescale also strongly (anti)correlates with absolute magnitude (see Fig. \[slepmag\]). Periodic behavior is not expected as a result of a CAR(1) process (although see the discussion in Sec. \[carma\] about higher order processes). Periodic objects should therefore show a strong deviation in their wavelet variance from that expected for a CAR(1) model with a shorter characteristic timescale than regular quasars. A periodic candidate should therefore show as a statistically significant outlier from the timescale - absolute magnitude relationship, appearing below the median curve in Fig. \[slepmag\]. ![The characteristic restframe timescale determined from the Slepian wavelet variance as a function of absolute magnitude for the 243,500 quasars in the data set. The black points indicate the median value in each magnitude bin.[]{data-label="slepmag"}](slepmag){width="3.5in"} Autocorrelation function (ACF) ------------------------------ The autocorrelation function (ACF) is a measure of how closely a quantity observed at a given time is related to the same quantity at another time (the time difference between the two is the [*lag time*]{}). In the case of a [(quasi)]{}[periodic]{} signal, peaks in the ACF will indicate lag times at which the signal is perfectly (highly) correlated with itself and these can be interpreted as periods at which the signal [(quasi)]{}[repeats]{}. The peak at zero lag $(\tau = 0)$ is the strongest peak in the ACF of a [(quasi)]{}[periodic]{} signal. Successive peaks at increasing multiples of the period are increasingly weaker, and as the lag value approaches the coherence time – the time required for a signal to change its frequency and phase information, the peaks in the autocorrelation function approach the “noise floor” generated by the non-periodic components of the signal and the noise. [@mcquillan] discuss the robustness of the ACF for period detection in time series data. Since the ACF measures only the degree of self-similarity of the light curve at a given lag time, the period remains detectable even when the amplitude and phase of the photometric modulation evolve significantly during the timespan of the observations. The ACF is also not susceptible to residual instrumental systematics since correlated noise, long-term trends and discontinuities manifest as monotonic trends in the ACF, on top of which local maxima may still be identified. In fact, the ACF is preferable as a period-finding method to the Lomb-Scargle periodogram in terms of clarity and robustness. For irregularly sampled data, the standard ACF estimator of [@edelson] can be defined as: $$ACF(k \Delta\tau) = \frac{\sum_{i=1}^{n}\sum_{j>i}x_i x_j b_k(t_j - t_i)}{\sum_{i=1}^{n}\sum_{j=1}^{n} b_k(t_j - t_i)}$$ where the observations have been normalized to zero mean and unit variance before the analysis. The kernel $b_k(t_j - t_i)$ selects the observations whose time lag is not further than half the bin width from $k \Delta \tau$: $$\begin{aligned} b_k(t_j - t_i) & = & 1 \,\, \mathrm{for} \mid (t_j - t_i) / \Delta \tau - k \mid \,\, < 1/2 \\ & & 0 \,\, \mathrm{otherwise} \end{aligned}$$ However, this estimator has a number of disadvantages including a high variance and not always producing positive semidefinite covariance matrix estimates (necessary to ensure the non-negativity of its Fourier transform; [@rehfeld]). Whilst a Gaussian kernel addresses some of these, [@alexander13] recommends Fisher’s $z$-transform and equal population binning (at least 11 points in each $\Delta \tau$ bin) for a more effective estimator. Fig. \[acfexample\] shows the ACF for a sample quasar light curve calculated using both the z-transform method and Scargle’s alternative method [@scargle89]. {width="7.0in"} [@mcquillan] identify the rotation periods of [*Kepler*]{} field M-dwarfs from the largest peaks (relative to neighbouring local minima) in their ACFs smoothed with a fixed width Gaussian kernel. Our data, however, have variable sampling in terms of cadence, temporal baseline and number of observations, which makes defining a single smoothing width value difficult. We have found that a better procedure to determine the period of any signal in the data is to fit the ACF of a light curve with a Gaussian process model (also known as kriging). Assuming the process has a squared-exponential covariance, this provides the best linear unbiased prediction of the underlying function from noisy data. We define the period as the time-lag value associated with the largest peak in the ACF between the second and third zero-crossings of the fitted model. This shows very good agreement with values determined from ACFs smoothed with a range of different width Gaussian kernels. We have verified the period determined by this method with that obtained with the conditional entropy algorithm of [@graham13]. We would expect broad agreement between the period determined from the ACF and that identified by the wavelet technique. Periodically driven stochastic systems are expected to have an ACF with the form of an exponentially decaying cosine function [@jung93]. Alternatively, if the light curve is the result of a continuous autoregressive moving average CARMA process (CARMA) (see Sec. \[carma\]) then its ACF is a sum of exponentially damped sinusoids and exponential decays, depending on whether the roots of the characteristic polynomial of the CARMA process are real or imaginary [@kelly14]. To test either case, we have also determined the best fitting exponentially damped sinusoid to the ACF of the form: $ACF(\tau) = A \cos (\omega \tau) \exp(-\lambda \tau)$, where the amplitude $A$, period $p = 2 \pi / \omega$, and decay $\lambda$ are determined using maximum likelihood estimation. Shape and coverage ------------------ In the simplest case, periodicity associated with a Keplerian orbit would show as a pure sinusoidal signal in the light curve. Although the projective geometry of a system – orbital inclination, eccentricity and argument of periastron – will distort this, the variation will remain a smooth periodic function, essentially a multiharmonic sine function. Regular flaring activity and similar phenomena not necessarily associated with a close SMBH binary pair would show as [(quasi)]{}[periodic]{} discontinuities. We can therefore use the scatter around an expected sinusoidal waveform to identify those objects most closely exhibiting the desired behavior. The best-fit truncated Fourier series (up to 6 terms) is determined for a given light curve phased using the period calculated by the ACF method (see above): $$\phi(t) = m_0 + \sum_{n=1}^{6} a_n sin (n \omega t + \phi_n)$$ where $\phi(t)$ is the phased light curve, $m_0$ is the median magnitude and $\omega = 2 \pi / period_{ACF}$. An F-test is used to determine the exact number of harmonics to include. The scatter about the best fit will be a combination of the intrinsic variability of the quasar and the quality of the fit. We therefore scale the rms scatter of the phased data about the fit by the median absolute deviation (MAD) of the light curve and only consider objects with scatter below some cutoff value as binary candidates. We note, however, that the MAD is a function of magnitude: fainter objects show large MAD values as there is a larger noise contribution for low S/N (faint) sources (see Fig. \[mad\]). We therefore employ a MAD value normalized by the median value for a given magnitude to ensure that objects with equivalent variability strength (irrespective of magnitude) can be compared. ![The median absolute deviation - magnitude relationship for the 243,500 quasars in the data set. The black points indicate the median value in each magnitude bin. The contours indicate the 1$\sigma$, 2$\sigma$ and 3$\sigma$ levels.[]{data-label="mad"}](madmag){width="3.5in"} The temporal coverage of the data set used will determine the range of SMBH masses and binary separations to which it is sensitive (see Fig. \[sensitive\]). A decade-long baseline will only sample a fraction of an orbital period for either a low mass SMBH pair (where the primary BH mass $< 10^7 M_\odot$) or a more massive pair at separations $> 1$pc. Since we want objects with well-sampled periodic behavior, we limit our search to objects with coverage of at least 1.5 cycles, assuming their ACF determined period. We also only consider objects with more than 50 observations in their light curves to minimize any systematic effects in the statistics from poor sampling. ![The range of SMBH masses and binary separations to which a synoptic data set is sensitive. The solid upper line for each separation indicates a $z=5$ track and the solid lower line a $z=0.05$ track whilst the two internal dotted lines show $z=1.0$ (lower) and $z = 2.0$ (upper) tracks respectively. The hatched region indicates the range over which CRTS has temporal coverage of 1.5 cycles or more of a periodic signal. The points indicate the locations of the CRTS candidates and the best known SMBH binary candidate, OJ 287 (solid black star).[]{data-label="sensitive"}](mbhperiod3){width="3.5in"} Data sets ========= There are few data sets with sufficient sky and/or temporal coverage and sampling to support an extensive search for quasars exhibiting periodic behavior. Most large studies of long-term quasar variability, e.g., SDSS with POSS [@mac12] or Pan-STARRS1 [@morganson14], consist of relatively few epochs of data spread over a roughly decadal baseline, which is sufficient to model ensemble behavior but not to identify specific patterns in individual objects. CRTS represents the best data currently available with which to systematically define sets of quasars with particular temporal characteristics. Catalina Real-time Transient Survey (CRTS) {#crts} ------------------------------------------ CRTS leverages the Catalina Sky Survey data streams from three telescopes – the 0.7 m Catalina Sky Survey Schmidt and 1.5 m Mount Lemmon Survey telescopes in Arizona and the 0.5 m Siding Springs Survey Schmidt in Australia – used in a search for Near-Earth Objects, operated by Lunar and Planetary Laboratory at University of Arizona. CRTS covers up to $\sim$2500 deg$^2$ per night, with 4 exposures per visit, separated by 10 min, over 21 nights per lunation. All data are automatically processed in real-time, and optical transients are immediately distributed using a variety of electronic mechanisms[^4]. The data are broadly calibrated to Johnson $V$ (see [@drake13] for details) and the full CRTS data set[^5] contains time series for approximately 500 million sources. The Million Quasars (MQ) catalogue[^6] v3.7 contains all spectroscopically confirmed type 1 QSOs (309,525), AGN (21,728) and BL Lacs (1,573) in the literature up to 2013 November 26 and formed the basis for the results of [@graham15].[^7] We have extended this with 297,301 spectroscopically identified quasars in the SDSS Data Release 12 [@paris15]. We crossmatched this combined quasar list against the CRTS data set with a 3 matching radius and find that 334,446 confirmed quasars are covered by the full CRTS. Of these, 83,782 do not pass our sampling criterion (i.e., they have less than 50 observations in their light curves), leaving a data set of 250,664 quasars. Note that the majority of rejected objects are faint ($V > 20$). The magnitude and redshift distributions for our sample are shown in Figs. \[magnitude\] and \[redshift\]. ![The $V$-band magnitude distribution for the CRTS known quasar sample (red). The tail of sources fainter than $V \sim 20.3$ are from the Mount Lemmon 1.5m telescope in the Catalina surveys. The stepped distribution (black) shows the rejected objects (less than 50 data points) from the sample which are primarily fainter sources.[]{data-label="magnitude"}](magnitude){width="3.5in"} ![The redshift distribution for the CRTS known quasar sample (red), omitting the tail of the distribution which contains 164 quasars with $z > 5.$ The stepped distribution (black) again shows the rejected objects.[]{data-label="redshift"}](redshift){width="3.5in"} Preprocessing ------------- It is common to preprocess data to remove spurious outlier points which may be caused by technical or photometric error. The danger, of course, is removing real signal, although a robust method should be unaffected by the presence of noisy data. We have created a cleaned version of each data set following the procedure of [@palanque11] (PD11): a 3-point median filter was first applied to each time series, followed by a clipping of all points that still deviated significantly from a quintic polynomial fit to the data. To ensure that not too many points are removed, the clipping threshold was initially set to 0.25 mag and then iteratively increased (if necessary) until no more than 10% of the points were rejected. Note that PD11 used a 5$\sigma$ threshold but we found that this removed too few points and so used the limit set by [@schmidt10] instead. There is always a chance that the light curve for a particular object may be contaminated by stray light (e.g., diffraction spikes, reflections) from nearby bright sources (up to $\sim$2 away for CRTS), low surface brightness galaxies or genuine blends, which can give a false seeing-dependent result in the analysis. If an object has a nearby counterpart (within $\sim$5) which is unresolved by CRTS ($\sim$2 pixels), the light curve will be the combined signal from both, dominated by the brighter of the two. We have defined a distance and magnitude-based criterion to exclude quasars near bright stars by crossmatching the quasar list against the Tycho catalog of bright stars $(V < 14)$ and reject 5,319 sources. We reject all objects within a $350''$ radius of any star brighter than $V = 6.67$ and within a radius of $200'' / (V - 6)$ for $V > 6.67$. To identify blends, we crossmatched our list against SDSS and flagged those objects where there is a significant difference (taking into account the variability of the source) between the magnitude of the CRTS source and the SDSS $r$ of its match, typically indicating a faint close SDSS companion contributing to the CRTS photometry: $|V - r| > 5 \sigma_V$. We also flagged those CRTS quasars outside the SDSS footprint (or without a SDSS companion) where the distribution of the spatial positions of individual photometry points in the light curve show significant scatter, e.g., a bimodal distribution indicating two sources: $\sigma_{pos} > 5 \sigma_pos(V)$ where $\sigma_pos(V)$ is the expected positional scatter for all quasar of magnitude $V$. Together these reject another 4.560 sources. The final size of the data set is 243.486 quasars. Mock data --------- To first approximation, quasar variability is well-described by a Gaussian first-order continuous autoregressive model (CAR(1)), also known as a damped random walk or Ornstein-Uhlenbeck process (see [@graham14; @kelly14] for details of where this description breaks down). Formally, the temporal behavior of the quasar flux $X(t)$ is given by: $$dX(t) = -\frac{1}{\tau}X(t)dt + \sigma \sqrt{dt}\epsilon(t) + b dt, \tau, \sigma, t > 0$$ where $\tau$ is the relaxation time of the process, $\sigma$ is the variability of the time series on timescales short compared to $\tau$, $b\tau$ is the mean magnitude, and $\epsilon(t)$ is a white noise process with zero mean and variance equal to 1. For each quasar in our data set, we have generated a simulated light curve assuming that it follows a CAR(1) model. Using the actual observation times, $t_i$, we replace the observed magnitudes with those that would be expected under a CAR(1) model. The magnitude $X(t)$ at a given timestep $\Delta t$ from a previous value $X(t - \Delta t)$ is drawn from a Gaussian distribution with mean and variance given by, e.g., [@kelly09]: $$E(X(t) | X(t - \Delta t)) = e^{-\Delta t / \tau} X(t - \Delta t) + b\tau(1 - e^{-\Delta t / \tau})$$ $$\mathrm{Var}(X(t) | X(t - \Delta t)) = \frac{\tau \sigma^2}{2} [1 - e^{-2\Delta t / \tau}]$$ We add a Gaussian deviate normalized by the photometric error associated with the magnitude to be replaced at each time $t$ to incorporate measurement uncertainties into the mock light curves. For each light curve, we set $b\tau$ to its median value and use the rest frame CAR(1) fitting functions determined by [@mac10]: $$\log f = A + B \log \left(\frac{\lambda_{RF}}{4000 {\mathrm \AA}} \right) + C (M_i + 23) + D \log \left(\frac{M_{BH}}{10^9 M_{\odot}}\right)$$ where $(A,B,C,D) = (-0.51, -0.479, 0.131, 0.18)$ for $f = SF_\infty$ and $(A,B,C,D) = (2.4, 0.17, 0.03, 0.21)$ for $f = \tau$. Note that a value of $C = 0.113$ is used for $f = SF_\infty$ for non K-corrected magnitudes. $M_i$ is the absolute magnitude of the quasar and $\lambda_{RF}$ is the restframe wavelength of the filter – here taken to be SDSS $r$, $\lambda_{RF} = 6250 {\mathrm \AA} / (1 + z)$. The mass of the black hole given the absolute magnitude is drawn from a Gaussian distribution: $$p(\log M_{BH} | M_i) = \frac{1}{\sqrt{2\pi}\sigma} \exp \left[ - \frac{(\log M_{BH} - \mu)^2}{2\sigma^2}\right]$$ where $\mu = 2.0 - 0.27 M_i$ and $\sigma = 0.58 + 0.011 M_i$. This is based on [@shen08] results. Differences in cosmologies used in estimating the best-fit parameters for CAR(1) and black hole mass should only have a 1% effect (MacLeod thesis, 2012). Results ======= We have applied our technique which identifies strong periodic behavior to both the real CRTS quasar data set and its simulated counterpart. From the real data set, we define an initial candidate sample according to the following criteria (see Table \[selection\] for a summary). For the wavelet component, we set the decay constant $c = 0.001$ and use the frequency range 0.0003 – 0.05 day$^{-1}$. As we are interested in well-defined periodicity, we only consider those objects whose wavelet peak significance places them in the top quartile of the data set (in terms of significance), which translates to a WWZ peak value cutoff of $wwzpk > 50$. We also only consider objects with a characteristic timescale from their Slepian wavelet variance at least $1\sigma$ below the expected value for their absolute magnitude. This reflects that periodic objects are not expected to be well-described by a CAR(1) model. Component Constraint Real Mock --------------------------- --------------------------------------------- -------- -------- Wavelet peak value $wwzpk > 50$ 24437 32078 Slepian wavelet deviation $\tau_{slep} < \tau(M_{V}) - \sigma_{\tau}$ 37828 27746 WWZ-ACF period $ 0.9 < p_{ACF}/p_{WWZ} < 1.1$ 30330 63637 ACF amplitude $A > 0.3$ 108625 143447 ACF decay $\lambda < 10^{-3}$ 172049 182592 Shape $rms / MAD < 0.67$ 11794 3944 Temporal coverage $\tau / p_{ACF} > 1.5$ 245234 182257 Number of points $n > 50$ 243486 243486 Combined - 101 0 Final - 111 0 The first ACF-based constraint is that the periods determined by the WWZ and the ACF, respectively, should agree to within 10 per cent. The shape of the ACF should also be reasonably described by an exponentially damped cosine with not too small an amplitude $(A > 0.3)$ and a decay constant such that the ACF drops by at most a factor of $1/e$ over the temporal baseline of the time series. This corresponds to $\lambda < 10^{-3}$ for the shortest time coverage in the data set and so we use this as a fiducial value. In terms of the shape constraint, we limit the selection to those objects whose rms scatter about the best-fit truncated Fourier series to their phased light curve is less than the $1\sigma$ lower limit on the magnitude-normalized median absolute deviation, i.e., $rms / mad < 0.67$. Finally, as previously described, we also limit the allowed temporal coverage to 1.5 cycles or greater, i.e. $\tau / period_{ACF} > 1.5$, where $\tau$ is the timespan of a light curve, and only consider light curves with 50 or more observations. Ideally, we want to determine the discriminating hyperplane in the multidimensional feature space that provides a clear separation between periodic and non-periodic objects. The feature cuts that we have defined provide a first approximation to this but likely candidates close to a selection value may have been missed due to error, poor sampling, etc. We therefore use the initial sample defined by the feature cuts as part of the training set for a support vector machine (SVM) with a radial basis function kernel. A number of similar machine learning algorithms could also be employed here but we found from simulations that the SVM was the best in terms of speed and accuracy. The other part of the training set comprises a set of randomly selected quasars that do not meet the periodicity selection criteria. From simulations with known numbers of periodic objects and periodic/non-periodic ratios, we have determined that the most adequate training set mix is $\sim$100:1, given our expected detection rate (see Section \[detection\]). 10000 randomly selected “non-periodic" quasars are thus added to the training set. The trained SVM is then used to classify each quasar in the full data set as periodic or not periodic based on their feature values. We repeat this process 50 times and identify those objects in the full data set that are selected each time as periodic as a final binary candidate. Our final sample consists of 111 quasars (see Table \[candidates\] for details and Fig. \[lcs\] for their light curves). We have also applied the same selection process to the simulated data set of objects with CAR(1) model light curves and find that none of them satisfy our criteria. If we apply the SVM trained on the real data to the simulated data set, we also find that that no objects are selected in all iterations as periodic. There are, however, three mock sources which are selected in almost all iterations (see Fig. \[mock\]). Though these objects are the closest to our selection criteria, they look qualitatively different from the real sample with no clear periodicity. The range of variability they exhibit is also significantly more than in the real sample: their mock light curves have MAD values two or three times the values seen in the real data. -------------------------- ------------ --------------- ------- -------------- -------- ----------------------------------------------- ------- ----------------------- ----------------- Id RA Dec $z$ $V_{median}$ Period $\log\left( \frac{M_{BH}}{M_{\odot}} \right)$ $r$ $t_{insp}$ $\Delta t_{GW}$ (days) (pc) (yrs) (ns) UM 211 00 12 10.9 $-$01 22 07.6 1.998 17.38 1886 - - - - UM 234 00 23 03.2 +01 15 33.9 0.729 17.82 1818 9.19 0.013 $1.2 \times 10^4$ 3.0 SDSS J014350.13+141453.0 01 43 50.0 +14 14 54.9 1.438 17.68 1538 9.21 0.009 $2.8 \times 10^3$ 2.4 PKS 0157+011 02 00 03.9 +01 25 12.6 1.170 18.04 1052 - - - - RX J024252.3-232633 02 42 51.9 $-$23 26 34.0 0.680 19.01 1818 - - - - US 3204 02 49 28.9 +01 09 25.0 0.954 18.09 1666 8.95$^{1}$ 0.009 $1.7 \times 10^4$ 1.0 CT 638 03 18 06.5 $-$34 26 37.4 0.265 16.68 1515 - - - - RXS J04117+1324 04 11 46.9 +13 24 16.5 0.277 16.20 1851 8.16 0.006 $1.4 \times 10^6$ 0.1 HS 0423+0658 04 26 30.2 +07 05 30.3 0.170 15.73 1123 - - - - 2MASS J04352649-1643460 04 35 26.5 $-$16 43 45.7 0.098 16.96 1369 7.78 0.004 $4.1 \times 10^6$ 0.1 SDSS J072908.71+400836.6 07 29 08.6 +40 08 37.0 0.074 16.08 1612 5.71 0.001 $1.9 \times 10^{10} $ 0.0 SDSS J080237.60+340446.3 08 02 37.6 +34 04 46.6 1.119 18.40 1428 8.96 0.007 $8.8 \times 10^3 $ 0.9 SDSS J080648.65+184037.0 08 06 48.6 +18 40 37.3 0.745 20.61 0892 7.99 0.003 $1.7 \times 10^5 $ 0.0 SDSS J080809.56+311519.1 08 08 09.5 +31 15 18.9 2.642 18.91 1162 8.36 0.003 $1.2 \times 10^4 $ 0.1 SDSS J081133.43+065558.1 08 11 33.4 +06 55 58.3 1.266 18.48 1587 9.39 0.010 $1.8 \times 10^3 $ 5.0 SDSS J081617.73+293639.6 08 16 17.8 +29 36 40.7 0.768 17.80 1162 9.77 0.013 $3.6 \times 10^2 $ 23.7 FBQS J081740.1+232731 08 17 40.2 +23 27 32.0 0.891 16.58 1190 9.55 0.011 $7.4 \times 10^2 $ 9.6 SDSS J082121.88+250817.5 08 21 22.0 +25 08 16.2 1.906 17.74 1886 9.53 0.011 $8.8 \times 10^2 $ 8.2 SDSS J082716.85+490534.0 08 27 16.9 +49 05 34.9 0.682 18.38 1612 8.96 0.009 $2.2 \times 10^4 $ 1.3 SDSS J082827.84+400333.9 08 28 27.8 +40 03 34.1 0.968 17.79 1886 8.87 0.008 $3.1 \times 10^4 $ 0.8 SDSS J082926.01+180020.7 08 29 26.0 +18 00 20.7 0.810 19.81 1449 8.42 0.005 $1.1 \times 10^5 $ 0.1 SDSS J083349.55+232809.0 08 33 49.6 +23 28 09.2 1.155 17.58 1086 9.40 0.008 $7.4 \times 10^2 $ 4.7 SDSS J084146.19+503601.1 08 41 46.3 +50 36 00.5 0.555 18.49 1694 7.44 0.003 $1.1 \times 10^7 $ 0.0 BZQJ0842+4525 08 42 15.3 +45 25 45.0 1.408 17.20 1886 9.48 0.012 $1.7 \times 10^3 $ 7.3 SDSS J091554.50+352949.6 09 15 54.5 +35 29 49.9 0.896 17.92 1369 9.05 0.008 $7.3 \times 10^3 $ 1.5 SBS 0920+590 09 23 58.7 +58 49 06.3 0.709 16.76 0649 8.76 0.004 $4.1 \times 10^3 $ 0.4 SDSS J092911.35+203708.5 09 29 11.3 +20 37 09.2 1.845 18.51 1785 9.92 0.014 $1.8 \times 10^2 $ 36.2 HS 0926+3608 09 29 52.1 +35 54 49.6 2.150 16.99 1562 9.95 0.012 $8.4 \times 10^1 $ 38.8 SDSS J093819.25+361858.7 09 38 19.3 +36 18 58.9 1.677 18.80 1265 9.32 0.007 $8.3 \times 10^2 $ 3.3 SDSS J094450.76+151236.9 09 44 50.7 +15 12 37.5 2.118 17.72 1428 9.61 0.009 $2.6 \times 10^2 $ 10.0 SDSS J094715.56+631716.4 09 47 15.6 +63 17 17.3 0.487 16.10 1724 9.22 0.012 $1.3 \times 10^4 $ 4.3 HS 0946+4845 09 50 00.7 +48 31 29.9 0.589 16.87 1587 8.59 0.007 $1.0 \times 10^5 $ 0.3 KUV 09484+3557 09 51 23.9 +35 42 49.2 0.398 17.77 1162 8.31 0.005 $1.8 \times 10^5 $ 0.1 SDSS J102255.21+172155.7 10 22 55.2 +17 21 56.0 1.062 18.48 1666 8.69 0.006 $3.9 \times 10^4 $ 0.4 SDSS J102349.38+522151.2 10 23 49.5 +52 21 51.8 0.955 16.96 1785 9.59 0.014 $1.8 \times 10^3 $ 12.1 RXS J10304+5516 10 30 25.0 +55 16 23.4 0.435 16.74 1515 8.43 0.006 $2.2 \times 10^5 $ 0.2 SDSS J103111.52+491926.5 10 31 11.5 +49 19 27.2 1.203 17.64 1612 9.04 0.008 $7.8 \times 10^3 $ 1.3 SDSS J104430.25+051857.2 10 44 30.3 +05 18 56.8 0.905 17.33 1333 9.24 0.009 $3.3 \times 10^3 $ 3.0 SDSS J104758.34+284555.8 10 47 58.3 +28 45 56.2 0.792 19.09 1851 8.72 0.008 $6.9 \times 10^4 $ 0.5 SDSS J104941.01+085548.4 10 49 41.0 +08 55 48.5 1.185 17.87 1428 9.37 0.009 $1.7 \times 10^3 $ 4.5 MS 10548-0335 10 57 22.3 –3 51 31.3 0.555 17.67 0892 - - - - CSO 67 11 03 27.5 +29 48 11.2 0.909 17.52 2083 9.35 0.013 $7.1 \times 10^3 $ 5.2 SDSS J110554.78+322953.7 11 05 54.8 +32 29 54.1 0.151 17.45 1724 8.24 0.007 $1.2 \times 10^6 $ 0.2 SDSS J113050.21+261211.4 11 30 50.2 +26 12 11.8 1.012 17.69 2173 9.32 0.013 $7.6 \times 10^3 $ 4.6 SDSS J113916.47+254412.6 11 39 16.4 +25 44 13.0 1.012 17.61 2439 9.16 0.012 $1.9 \times 10^4 $ 2.6 SDSS J114438.34+262609.4 11 44 38.3 +26 26 10.1 0.974 18.39 1315 9.38 0.010 $1.7 \times 10^3 $ 4.9 SDSS J114749.70+163106.7 11 47 49.7 +16 31 06.8 0.554 18.78 1449 8.22 0.005 $3.5 \times 10^5 $ 0.1 SDSS J114857.33+160023.1 11 48 57.4 +16 00 22.7 1.224 18.14 1851 9.90 0.017 $4.1 \times 10^2 $ 37.6 SDSS J115141.81+142156.6 11 51 41.8 +14 21 57.0 1.002 18.11 1492 9.11 0.009 $6.3 \times 10^3 $ 1.8 SDSS J115346.39+241829.4 11 53 46.4 +24 18 29.8 0.753 18.41 1666 8.97 0.009 $2.1 \times 10^4 $ 1.3 SDSS J121018.34+015405.9 12 10 18.3 +01 54 06.2 0.216 16.90 1612 8.54 0.008 $2.6 \times 10^5 $ 0.6 SDSS J121018.66+185726.0 12 10 18.7 +18 57 27.0 1.516 17.42 1754 9.53 0.011 $1.1 \times 10^3 $ 8.4 SDSS J121056.83+231912.5 12 10 56.8 +23 19 13.0 1.260 18.47 1785 8.78 0.007 $2.7 \times 10^4 $ 0.5 -------------------------- ------------ --------------- ------- -------------- -------- ----------------------------------------------- ------- ----------------------- ----------------- -------------------------- ------------ ------------- ------- -------------- -------- ----------------------------------------------- ------- -------------------- ----------------- Id RA Dec $z$ $V_{median}$ Period $\log\left( \frac{M_{BH}}{M_{\odot}} \right)$ $r$ $t_{insp}$ $\Delta t_{GW}$ (days) (pc) (yrs) (ns) SDSS J121457.39+132024.3 12 14 57.4 +13 20 24.5 1.494 18.59 1923 9.46 0.011 $1.8 \times 10^3 $ 6.6 SDSS J123147.27+101705.3 12 31 47.3 +10 17 05.4 1.733 18.83 1851 9.20 0.009 $3.5 \times 10^3 $ 2.3 SDSS J123821.84+030024.2 12 38 21.8 +03 00 24.6 0.380 18.46 1250 8.92 0.008 $2.2 \times 10^4 $ 1.4 SDSS J124044.49+231045.8 12 40 44.5 +23 10 46.1 0.722 18.40 1428 8.94 0.008 $1.6 \times 10^4 $ 1.1 SDSS J124119.04+203452.7 12 41 19.0 +20 34 53.4 1.492 18.44 1219 9.40 0.008 $6.8 \times 10^2 $ 4.5 SDSS J124157.90+130104.1 12 41 57.9 +13 01 04.7 1.227 18.83 1538 8.95 0.007 $9.4 \times 10^3 $ 0.9 PGC 3096192 12 50 29.0 +06 36 11.1 0.133 16.93 1562 7.06 0.003 $8.5 \times 10^7 $ 0.00 SDSS J125414.23+131348.1 12 54 14.2 +13 13 48.4 0.655 18.11 1754 8.94 0.009 $3.2 \times 10^4 $ 1.2 SDSS J130040.62+172758.4 13 00 40.6 +17 27 58.5 0.863 19.26 1818 8.88 0.008 $3.2 \times 10^4 $ 0.8 BZQJ1305-1033 13 05 33.0 –10 33 19.1 0.286 14.99 1694 8.50 0.008 $3.0 \times 10^5 $ 0.4 SNU J13120+0641 13 12 04.7 +06 41 07.6 0.242 15.82 1492 9.14 0.012 $2.0 \times 10^4 $ 5.1 SDSS J131706.19+271416.7 13 17 06.2 +27 14 16.7 2.672 17.83 1666 9.92 0.011 $7.7 \times 10^1 $ 33.9 SDSS J131909.08+090814.7 13 19 09.1 +09 08 15.1 0.882 18.01 1298 8.67 0.006 $2.8 \times 10^4 $ 0.3 SDSS J132103.41+123748.2 13 21 03.4 +12 37 48.1 0.687 18.62 1538 8.91 0.008 $2.4 \times 10^4 $ 1.0 SDSS J133127.31+182416.9 13 31 27.3 +18 24 17.1 0.938 17.44 1639 9.39 0.011 $3.0 \times 10^3 $ 5.6 SDSS J133516.17+183341.4 13 35 16.1 +18 33 41.8 1.192 17.75 1724 9.76 0.015 $6.0 \times 10^2 $ 21.7 SDSS J133631.45+175613.8 13 36 31.4 +17 56 14.1 0.421 18.20 1562 9.03 0.010 $2.4 \times 10^4 $ 2.2 SDSS J133654.44+171040.3 13 36 54.4 +17 10 40.8 1.231 17.96 1408 9.24 0.008 $2.5 \times 10^3 $ 2.7 SDSS J133807.69+360220.3 13 38 07.7 +36 02 20.3 1.196 18.44 1960 9.14 0.010 $9.1 \times 10^3 $ 2.1 SDSS J134820.42+194831.5 13 48 20.4 +19 48 31.9 0.594 18.34 1388 7.63 0.003 $2.8 \times 10^6 $ 0.0 SDSS J134855.27-032141.4 13 48 55.3 –3 21 41.4 2.099 17.19 1428 9.89 0.011 $8.7 \times 10^1 $ 30.0 SDSS J135225.80+132853.2 13 52 25.8 +13 28 53.3 0.402 16.44 1754 8.75 0.009 $1.0 \times 10^5 $ 0.8 SDSS J140600.26+013252.2 14 06 00.3 +01 32 52.4 0.454 18.00 2000 8.41 0.007 $4.7 \times 10^5 $ 0.2 SDSS J140704.43+273556.6 14 07 04.5 +27 35 56.3 2.222 17.32 1562 9.94 0.012 $8.4 \times 10^1 $ 36.3 HE 1408-1003 14 10 40.3 –10 17 29.7 0.882 17.01 1923 - - - - SDSS J141425.92+171811.2 14 14 25.9 +17 18 11.6 0.409 18.33 1785 8.59 0.008 $1.9 \times 10^5 $ 0.4 3C 298.0 14 19 08.2 +06 28 35.1 1.437 16.53 1960 9.57 0.013 $1.3 \times 10^3 $ 10.5 SDSS J142301.96+101500.1 14 23 02.0 +10 15 00.0 1.052 17.96 1234 9.46 0.010 $9. \times 10^2 $ 6.3 SDSS J143621.29+072720.8 14 36 21.3 +07 27 21.1 0.889 17.07 1886 9.03 0.010 $1.9 \times 10^4 $ 1.5 SDSS J143820.60+055447.9 14 38 20.6 +05 54 48.1 0.614 18.71 1851 8.19 0.006 $6.9 \times 10^5 $ 0.1 SDSS J144754.62+132610.0 14 47 54.6 +13 26 10.4 0.572 18.39 1960 8.58 0.008 $1.9 \times 10^5 $ 0.4 SDSS J144755.57+100040.0 14 47 55.6 +10 00 40.4 0.678 17.65 0862 8.93 0.006 $4.7 \times 10^3 $ 0.9 SDSS J150450.16+012215.5 15 04 50.2 +01 22 15.8 0.967 17.75 1724 9.20 0.010 $7.0 \times 10^3 $ 2.7 QNZ3:54 15 18 06.6 +01 31 34.9 1.402 18.64 1724 9.27 0.009 $3.1 \times 10^3 $ 3.2 SDSS J152035.23+095925.2 15 20 35.2 +09 59 25.7 1.049 18.88 1204 9.12 0.007 $3.3 \times 10^3 $ 1.7 SDSS J152157.02+181018.6 15 21 57.0 +18 10 19.2 0.731 18.23 1960 7.95 0.005 $1.6 \times 10^6 $ 0.0 SDSS J153636.22+044127.0 15 36 36.2 +04 41 26.9 0.379 16.75 1111 8.82 0.007 $2.4 \times 10^4 $ 0.9 SDSS J154409.61+024040.0 15 44 09.6 +02 40 39.8 0.964 18.72 2000 8.76 0.008 $5.5 \times 10^4 $ 0.5 SDSS J155449.11+084204.8 15 54 49.1 +08 42 05.4 0.786 17.03 1562 8.85 0.008 $2.6 \times 10^4 $ 0.8 SDSS J155647.78+181531.5 15 56 47.8 +18 15 32.1 1.502 18.79 1428 9.51 0.010 $6.7 \times 10^2 $ 7.2 SDSS J160730.33+144904.3 16 07 30.3 +14 49 04.2 1.800 17.57 1724 9.82 0.013 $2.5 \times 10^2 $ 24.6 SDSS J161013.67+311756.4 16 10 13.7 +31 17 56.6 0.245 17.19 1724 7.92 0.005 $3.2 \times 10^6 $ 0.1 SDSS J161854.64+230859.1 16 18 54.6 +23 08 59.3 0.561 18.22 1666 8.37 0.006 $2.9 \times 10^5 $ 0.1 163107.34+560905.3 16 31 07.4 +56 09 05.1 0.731 19.13 1724 8.38 0.006 $2.2 \times 10^5 $ 0.1 HS 1630+2355 16 33 02.7 +23 49 28.8 0.821 15.23 2040 9.86 0.020 $1.1 \times 10^3 $ 38.7 SDSS J164452.71+430752.2 16 44 52.7 +43 07 52.9 1.715 17.06 2000 10.15 0.019 $1.1 \times 10^2 $ 91.6 SDSS J165136.76+434741.3 16 51 36.8 +43 47 41.9 1.604 18.13 1923 9.34 0.010 $2.6 \times 10^3 $ 4.1 MCG 5-40-026 17 01 07.8 +29 24 24.6 0.036 14.65 1408 - - - - SDSS J170616.24+370927.0 17 06 16.2 +37 09 27.0 1.267 18.19 1388 9.10 0.007 $3.9 \times 10^3 $ 1.6 HS 1715+2131 17 17 20.1 +21 28 15.0 0.590 16.19 1250 - - - - FBQS J17239+3748 17 23 54.3 +37 48 41.7 0.828 17.55 1960 9.38 0.013 $6.0 \times 10^3 $ 6.1 SDSS J172656.96+600348.5 17 26 56.9 +60 03 49.1 0.991 18.50 1923 9.15 0.010 $1.1 \times 10^4 $ 2.3 4C 50.43 17 31 03.7 +50 07 35.7 1.111 16.74 1075 - - - - BZQJ2156-2012 21 56 33.7 –20 12 30.2 1.309 17.01 1333 - - - - SDSS J221016.97+122213.9 22 10 17.0 +12 22 14.0 0.717 18.32 1333 9.00 0.008 $1.1 \times 10^4 $ 1.4 6QZ J221925.1-305408 22 19 25.2 –30 54 08.1 0.579 19.15 1408 - - - - HS 2219+1944 22 22 21.1 +19 59 48.1 0.211 16.33 1724 - - - - SDSS J224829.47+144418.0 22 48 29.4 +14 44 18.4 0.424 18.82 1219 8.86 0.008 $2.4 \times 10^4 $ 1.0 -------------------------- ------------ ------------- ------- -------------- -------- ----------------------------------------------- ------- -------------------- ----------------- {width="7.0in"} If we take the simulated data results as an indication of the expected number of quasars to show comparable periodic behavior by chance then the number of real objects selected is statistically significant, particularly with the higher temporal coverage constraint. It also suggests strong periodicity (on an approximately decadal timescale) is not common in quasars nor is it expected as an artifact of a CAR(1) process. We note, however, that our selection criteria will select against longer periodic as well as more general quasi-periodic behavior as may be expected from SMBH binaries at greater than 1 pc separation. Binary Candidates in the Literature =================================== Spectroscopic ------------- As previously mentioned, the availability of multiepoch spectroscopy enables searches for SMBH binaries on the basis of time-dependent velocity shifts in the broad lines, either with respect to each other or narrow lines associated with the host galaxy. It is interesting to see whether any spectroscopically selected candidates in the literature show periodic photometric variability in CRTS data. We have examined the light curves of 30 spectroscopic SMBH binary candidates from the literature (see Table \[litcands\]). We only consider here the most likely binary candidates reported, e.g., classified as such from the shape of their Balmer lines [@tsalmantza11; @decarli13], and not secondary candidates also reported with asymmetric line profiles, double-peaked emitters, or other features since these do not sufficiently indicate a SMBH binary. Although the predicted separation of these candidates is greater than in our candidates, it is still typically less than 1 pc and we would expect there to be some detectable behavior in the photometric data. -------------- ------------ --------------- ------------ ------ ------------------ ---------------------- -------- ------ -------------------- Id RA Dec Shape Poly $\log_{10} \tau$ $\log_{10} \sigma^2$ CARMA MAD Source J0049+0026 00 49 19.0 +00 26 09.4 Hump 3 2.999 -4.579 (2, 0) 0.10 Optical$^{10}$ J0301+0004 03 01 00.2 +00 04 29.3 Periodic 1 -1.957 0.753 (4, 0) 0.21 Optical$^{9}$ J0322+0055 03 22 13.9 +00 55 13.4 Polynomial 5 2.720 -4.327 (6, 0) 0.07 Optical$^{9}$ J0757+4248 07 57 00.7 +42 48 14.5 Logistic 3 3.235 -5.031 (2, 0) 0.11 Optical$^{10}$ J0829+2728 08 29 30.6 +27 28 22.7 Polynomial 4 3.095 -3.827 (4, 0) 0.23 Optical$^{8,12}$ J0847+3732 08 47 16.0 +37 32 18.1 Polynomial 4 2.844 -4.001 (4, 0) 0.12 Optical$^{8}$ J0852+2004 08 52 37.0 +20 04 11.0 Hump 4 2.594 -3.871 (4, 0) 0.15 Optical$^{8}$ OJ 287 08 54 48.9 +20 06 30.6 Periodic 5 1.678 -2.021 (7, 0) 0.34 Optical$^{1}$ J0927+2943 09 27 04.4 +29 44 01.3 Periodic 2 2.882 -4.639 (1, 0) 0.09 Optical$^{2,7,11}$ J0928+6025 09 28 38.0 +60 25 21.0 Periodic 1 2.650 -4.105 (6, 0) 0.10 Optical$^{8}$ J0932+0318 09 32 01.6 +03 18 58.7 Polynomial 5 2.921 -4.119 (4, 0) 0.13 Optical$^{6,7,11}$ J0935+4331 09 35 02.5 +43 31 10.7 Hump 5 2.946 -4.541 (6, 0) 0.05 Optical$^{10}$ 4C 40.24 09 48 55.3 +40 39 44.6 Polynomial 5 2.900 -4.500 (2, 0) 0.09 Radio$^{13}$ J0956+5350 09 56 56.4 +53 50 23.2 Periodic 1 2.713 -4.392 (4, 0) 0.10 Optical$^{10}$ J1000+2233 10 00 21.8 +22 33 18.6 Polynomial 7 2.142 -3.593 (7, 0) 0.12 Optical$^{5,7,11}$ J1012+2613 10 12 26.9 +26 13 27.2 Logistic 2 2.936 -4.557 (5, 0) 0.09 Optical$^{7,11}$ J1030+3102 10 30 59.1 +31 02 55.8 Hump 3 2.780 -4.134 (5, 0) 0.08 Optical$^{8}$ J1050+3456 10 50 41.4 +34 56 31.3 Polynomial 10 2.689 -4.154 (2, 0) 0.12 Optical$^{4,7,11}$ J1100+1709 11 00 51.0 +17 09 34.3 Hump 1 2.906 -4.232 (6, 0) 0.12 Optical$^{8,12}$ J1112+1813 11 12 30.9 +18 31 11.4 Polynomial 6 2.784 -4.264 (3, 0) 0.09 Optical$^{8}$ J1154+0134 11 54 49.4 +01 34 43.6 Logistic 1 2.906 -3.853 (5, 2) 0.15 Optical$^{7,11}$ J1229-0035 12 29 09.5 $-$00 35 30.0 Polynomial 4 2.834 -3.947 (6, 5) 0.13 Optical$^{9}$ J1229+0203 12 29 06.7 +02 03 08.7 Polynomial 5 1.397 -3.057 (7, 2) 0.07 Radio$^{13}$ J1305+1819 13 05 34.5 +18 19 32.9 Polynomial 8 2.728 -4.899 (5, 0) 0.04 Optical$^{8,12}$ J1345+1144 13 45 48.5 +11 44 43.5 Hump 4 2.440 -4.039 (6, 0) 0.08 Optical$^{8}$ J1410+3643 14 10 20.6 +36 43 22.7 Logistic 3 3.240 -4.434 (5, 0) 0.15 Optical$^{9}$ J1536+0441 15 36 36.2 +04 41 27.1 Polynomial 10 2.732 -4.474 (6, 0) 0.07 Optical$^{3,7,11}$ J1537+0055 15 37 06.0 +00 55 22.8 Polynomial 2 3.105 -5.987 (2, 0) 0.03 Optical$^{9}$ J1539+3333 15 39 08.1 +33 33 27.6 Periodic 1 3.042 -5.591 (2, 0) 0.06 Optical$^{7,11}$ J1550+0521 15 50 53.2 +05 21 12.1 Hump 1 2.804 -4.817 (2, 0) 0.03 Optical$^{9}$ J1616+4341 16 16 09.5 +43 41 46.8 Logistic 1 -1.383 -0.033 (4, 0) 0.18 Optical$^{10}$ J1714+3327 17 14 48.5 +33 27 38.3 Polynomial 7 3.060 -4.865 (1, 0) 0.06 Optical$^{7,11}$ PKS 2155-304 21 58 52.1 $-$30 13 32.1 Logistic 1 2.483 -2.648 (3, 1) 0.44 Radio$^{13}$ 3C 454.3 22 53 57.7 +16 08 53.6 Polynomial 4 1.978 -1.817 (6, 0) 0.55 Radio$^{13}$ 1ES 2321+419 23 23 52.1 +42 10 59 Polynomial 5 2.531 -2.974 (6, 0) 0.29 X-ray$^{14}$ J2349-0036 23 49 32.8 $-$00 36 45.8 Periodic 1 -0.337 -1.153 (7, 3) 0.10 Optical$^{9}$ -------------- ------------ --------------- ------------ ------ ------------------ ---------------------- -------- ------ -------------------- References: \[1\] Valtonen et al. (2008); \[2\] Dotti et al. (2009); \[3\] Boroson & Lauer 2009; \[4\] Shields, Bonning & Salviander (2009) ; \[5\] Decarli et al. (2010); \[6\] Barrows et al. (2011); \[7\] Tsalmantza et al. (2011); \[8\] Liu et al. (2014); \[9\] Shen et al. (2013); \[10\] [@ju13]; \[11\] [@decarli13]; \[12\] Eracleous et al. (2012); \[13\] Ciaramella et al. (2004); \[14\] Rani et al. (2009) There seem to be a number of morphologically distinct behaviors seen with these candidates. To quantify this, we have defined a number of template functions that capture different shape profiles: [*Linear*]{}: a line with gradient given by the Theil-Sen estimator for the light curve which passes through the median data point $(t_{med}, m_{med})$; [*Polynomial*]{}: the best fitting polynomial up to 10$^{th}$ order determined by a F-test between successive orders; [*Sinusoidal*]{}: the best fitting sinusoid with period determined by the ACF method (see section 2.2); [*Logistic*]{}: a generalized logistic function fit to the light curve: $$m(t) = A + \frac{K - A}{(1 + Q e^{-B(t-M)}) ^{1/\nu}}$$ [*Hump/Dip*]{}: a linear least-squares fit to the light curve plus a quadratic to the largest contiguous amount of data above or below the linear fit. For each light curve of interest, we have determined which template function provides the best chi-squared fit. The shape of the light curve may also just be a random pattern arising from the stochastic nature of the variability and so we have also determined both the best-fit CAR(1) and CARMA(p,q) model parameters for each light curve and checked for correlations between these and the best-fitting template. We note that none of the spectroscopic candidates shows the type of strong periodicity exhibited by the candidates identified in this paper. The division of template shapes is fairly uniform, although objects from [@liu14] tend to be best-fit by the hump template. These candidates specifically show significant radial accelerations in their broad H$\beta$ lines from dual epoch spectroscopy whereas other studies are based on more or other lines. There is also some suggestion of preferential localization in the CAR(1) $\tau-\sigma^2$ plane with objects best fit by a sinusoid function lying outside the $1\sigma$ contour. Those best fit by CARMA models with $p=7$ also show the same effect but to a higher degree. However, the sample size here is too small to determine whether these effects are statistically significant but they certainly warrant further study. [@fliu14] proposed SDSS J120136.02+300305.5 as the first milliparsec SMBH binary in a quiescent galaxy on the basis of a candidate tidal disruption event in its X-ray light curve. Its optical light curve shows no significant variation – it has a MAD value of 0.07 mag which is the median value for its magnitude – or evidence of an associated flare and there is no structure to it. This suggests that this population of SMBH binaries will only be detected in the optical as a result of serendipity (assuming that the candidate is an actual binary). Blazars ------- Similarly we have examined the CRTS light curves of the few blazars reported in the literature as exhibiting [(quasi)]{}[periodic]{} behavior, e.g., PKS 2155-304 and 1ES 2321+419 [@rani09]. We note that these do not show the consistent periodicity that we see in the candidates in this analysis. However, their CAR(1) and CARMA fits are comparable with those seen for optical candidates in the literature as in the previous section. Photometric ----------- [@liu15] report the discovery of a periodic quasar candidate, FBQS J221648.7+012427, out of a sample of 168 quasars in the Pan-STARRS1 Medium Deep Survey (PSMDS) with an average of 350 detections in four filters. It has an observed period of 542 days, a redshift of 2.06, and an inferred separation of 0.006 pc. Whilst this object is part of the data we have considered, it does not appear as a candidate in our analysis. Assuming the quoted period, its CRTS light curve covers 5.8 cycles but no significant or consistent periodic signal is identified by either the WWZ or ACF algorithms and its Slepian wavelet characteristic timescale is not a statistically significant outlier. The candidate was identified using a Lomb-Scargle (LS) periodogram to look for evidence of periodicity. A similar approach was used by [@mac10] in a search of 8863 SDSS Stripe 82 quasar light curves (with an average of over 60 observations) with 88 candidates identified. None were considered significant, however, as the standard likelihood calculation assumes white noise as the null hypothesis rather than colored noise as appropriate for an underlying CAR(1) process. The technique is regarded as not very powerful for poorly sampled light curves. It is also interesting to note that both analyses have a detection rate of about 1 in 100 rather than the far more conservative 1 in 10000 that we find, which is much more in line with expected rates from population models (see section \[detection\]). The PSMDS candidate FBQS J221648.7+012427 is also reported to have a (restframe) inspiral time of $\sim$ 7 years thus putting its final coalescence within the timeframe of pulsar timing array experiments within the next decade or so. Given that the merger timescale is typically 10$^7$ years, it seems statistically unlikely that an object would be found in its final phase in such a small sample size. The inspiral time is, however, very dependent on the black hole mass and so it is possible that the authors have overestimated the mass of the system, particularly as the line and continuum fluxes were not determined from an electronic format spectrum. Discussion ========== Quasiperiodic behavior in stochastic models {#carma} ------------------------------------------- Although CAR(1) models have been used as popular statistical descriptors of quasar optical variability, e.g., [@kelly09; @mac12], there is a growing body of evidence that a more sophisticated approach is required [@mushotzky11; @zu13; @graham14; @kasliwal15]. [@kelly14] describe the use of CARMA models as a better method to quantify stochastic variability in a light curve. (Note that a CAR(1) model is the same as a CARMA(1, 0) model). A zero-mean CARMA(p,q) process $y(t)$ is defined to be the solution to the stochastic differential equation: $$\begin{aligned} \frac{d^py(t)}{dt^p} + \alpha_{p-1} \frac{d^{p-1}y(t)}{dt^{p-1}} + \ldots + \alpha_0y(t) = \\ \beta_q \frac{d^q\epsilon(t)}{dt^q} + \beta_{q-1} \frac{d^{q-1}\epsilon(t)}{dt^{q-1}} + \ldots + \epsilon(t)\end{aligned}$$ where $\epsilon(t)$ is a continuous time white noise process with zero mean and variance $\sigma^2$. This is associated with a characteristic polynomial: $$A(z) = \sum_{k=0}^p \alpha_k z^k$$ with roots $r_1, \ldots, r_p$. The power spectral density (PSD) for a CARMA(p, q) process is given by: $$P(\omega) = \sigma^{2} \frac{\mid \sum_{j=0}^q \beta_j(i \omega)^j \mid^2} {\mid \sum_{k=0}^p \alpha_k(i \omega)^k \mid^2}$$ Now $A(z)$ can be written in terms of its roots: $$A(z) = \prod_{j=1}^p (z - r_j) = 0$$ where $r = a + ib$ and so the PSD is proportional to: $$P(\omega) \propto \frac{1}{\prod_{j = 1}^p a_j^2 + (w - b_j)^2}$$ For complex roots (and $p > 1$), local maxima will exist at $w = b_j$ which correspond to quasi-periodicities in the data. If the roots have negative real parts and $q < p$ then the CARMA process is stationary. By analogy with underdamped second order systems, the quasi-periodicities can be characterized by the damping ratio, $\zeta_j = -a_j / \sqrt{a_j^2 + b_j^2}$, for a given frequency, $b_j$. Smaller values of $\zeta$ indicate a stronger oscillating signal (less damping) with $\zeta = 0$ defining a pure sinusoidal signal. The bandwidth of the signal, i.e., the range of frequencies associated with it, is given by $\Delta f = \zeta b / \pi$. For each of our periodic candidates, we have determined the best-fitting CARMA(p,q) model using the [carma\_pack]{} code[^8] of [@kelly14] and checked whether there is any quasi-periodic behavior expected at its identified period from peaks in the PSD. Out of the 111 candidates, we find that five show a detected periodicity which overlaps with that expected from its best-fit CARMA model. However, the bandwidth of all of these is such that any period within the temporal baseline of the object would fit. None of the periodic behavior is therefore consistent with that potentially expected from stochastic variability. We have also generated 100 mock light curves of the appropriate CARMA(p,q) type for each periodic candidate with the same time sampling as the CRTS data and determined the expected wavelet and ACF parameters. The median differences between the real value and CARMA model-based values are statistically significant in all cases. From this we conclude that CARMA(p,q) models do not provide an adequate statistical description for our periodic candidates as they cannot reproduce the same range of features values seen. Physical interpretation ----------------------- Although we have searched for optical periodicity in quasars on the assumption that this is associated with a close SMBH binary system, the exact physical mechanism responsible for this behavior is uncertain. The sinusoidal nature of the signal suggests that it is kinematic in origin and there are a number of potential explanations that should be considered. ### A jet with a varying viewing angle The optical flux could be the superposition of thermal emission from the accretion disk and a differential Doppler boosted non-thermal contribution associated with a jet [@rieger04]. The observed periodicity would be the result of a periodically varying viewing angle, driven either by the orbital motion of a SMBH binary system, a precessing jet or an internally rotating jet flow. In the latter case, the expected observed period is typically $P_{obs} \lesssim 10$ days for massive quasars and even less for objects with significant bulk Lorentz factors, i.e., blazars. Given the range of values we have identified, this seems an unlikely model. In a single SMBH system, precession of the jet can arise if the accretion disk is tilted (or warped) with respect to the plane of symmetry (although the origin of the tilt would still need to be accounted for). Using data in the literature, [@lu05] find that the expected precession period for a jet with a single SMBH [@lu05] is: $$\log(\tau_{prec} / \mathrm{yr}) \sim 0.48M_{abs} + 0.14 \log \left( \frac{M}{10^8 M_\odot} \right) + A$$ where $A = 15.18 - 18.86$, depending on the values of constant parameters relating to the viscosity and angular momentum of the SMBH system. For a representative quasar with $M_{abs} = -25$ with a $10^8 M_{\odot}$ SMBH, this gives a period between $10^{2.2 - 6.9}$ years which is much longer than the observed periods reported here. We note as well that jets in non-blazar systems are predominantly associated with geometrically thick accretion flows in low-luminosity AGN ($L_{bol} \le 10^{42}$ erg s$^{-1}$) whereas the candidate objects considered here span the range $10^{45} \le L_{bol} \le 10^{48}$ erg s$^{-1}$. Reported jetted systems in the high luminosity range tend to be blazars and radio galaxies, e.g., [@ghisellini10], [@sbarrato14], but only a few of the candidates fit that category. It is still possible, though, that the central part of the accretion disk in the highest luminosity systems could be geometrically thick with a slim disk associated to accretion rates close to or above the Eddington limit, e.g. [@sadowski15] and references therein. Shorter periods are possible, however, for jet procession in a SMBH binary system. In this case, the jet precesses as a result of inner disk precession due to the tidal interaction of an inclined secondary SMBH. ### Warped accretion disk In addition to precessing any potential jet, a warped accretion disk might be responsible for the exhibited periodicity by obscuring the continuum-emitting region or at least modulating its luminosity as it precesses. Warped features are known to exist when the data is of sufficient quality to identify them, e.g. [@herrnstein05] for NGC 4258, and may be common to most AGN, according to high resolution simulations of gas inflow in active galaxies [@hopkins10]. In general, the warping may be produced by a number of different mechanisms: the Bardeen-Peterson effect, a quadrupole potential or tidal interaction in a binary system, self-gravity, angular momentum transport by viscous disk stresses, and radiation- and magnetic-driven instabilities. Simulations suggest that the Bardeen-Peterson effect may not typically occur in AGN disks [@fragile07]. Radiation- and magnetic-driven instabilities are also not feasible for AGNs as they predict behaviors inconsistent with observation [@caproni06]. [@tremaine14] argue that, in the absence of any source of external torque, self-gravity is expected to play a prominent role in the dynamics of AGN accretion disks. The precession rate of a self-gravitating warped disk around a SMBH is [@ulubay09]: $$\omega \simeq C \left( \frac{G M_{d}^2}{M_{BH} r_{w}^3} \right) ^ {\frac{1}{2}}$$ where $r_w$ is the radius of the warp, $M_{BH}$ is the mass of the central object, $M_{d}$ is the mass of the accretion disk, and $C$ is a constant of order unity accounting for the disk configuration (inclination, etc). For a fiducial value of $M_{BH}$ = 10$^8 M_\odot$, [@tremaine14] find a warp radius of $\sim 300 R_g$ ($R_g = G M_{BH} / c^2$) and, assuming $M_{disk} = 0.01 M_{BH}$, the precession timescale is 51 years. This is only an order of magnitude larger than the typical periods reported here. The typical lifetime for a warp is $t_{align} = 1.3 \times 10^5 (r_w/ 300 R_g)^4$ yr [@tremaine14], much shorter than the typical AGN lifetime, and so it would also be a relatively rare phenomenon to encounter. A common explanation for a warped accretion disk is a binary system [@macfadyen08; @hayasaki14; @tremaine14], particularly in stellar systems where all components are resolvable. In the case of a SMBH binary, a warp will occur in the accretion disk of one of the SMBHs if its spin axis is misaligned with the orbital axis of the binary. It is also possible that there is a circumbinary accretion disk which is warped. ### Periodic accretion [@farris14] describe how periodic mass accretion rates in a SMBH binary system can give rise to an overdense lump in the inner circumbinary accretion disk (CAD) and that the observation of periodicity in quasar emission would be associated with this. Periodic accretion would not be expected for a single SMBH system. [@hayasaki08] propose that small temperature changes associated with accretion variations would produce large-amplitude variations in the UV region, where the spectral energy distribution (SED) of the quasar decays exponentially, but would have little effect on the Rayleigh-Jeans part of the spectrum. However, [@rafikov13] argues that the SED of a circumbinary disk exhibits a power law segment with $\nu F_\nu \propto \nu^{12/7}$ rather than $\nu^{4/3}$. Accretion variations would therefore have a more noticeable effect at UV wavelengths in a binary system. We note as well that close pre-main-sequence binary stars show periodic changes in their observed optical luminosity which is attributed to periodic accretion from the circumbinary disk, e.g., [@jensen07]. ### Accretion disk gap From hydrodynamical simulations of SBMH binaries, [@dorazio15] have proposed that the detected periodicity in PG1302-102 could be associated with a lopsided central cavity in the CAD for $q > 0.3$, where $q$ is the ratio of the two SMBH masses: $q = M_2 / M_1$. Significantly, in this scenario, the strongest period detected is a factor of $\chi = 3 - 8$ times greater than the true period of the binary and so the true binary separation is a factor of $\chi^{-2/3}$ smaller. This model predicts additional smaller amplitude periodic variability on timescales of $t_{bin}$ and $0.5 t_{bin}$ ($t_{bin}$ is the true binary periodicity), and periodic spectral variations in broad line widths and narrow line offsets. There are also measurable relativistic effects on the Fe K$\alpha$ line and an observable decrement in the spectral energy distribution of the source at a wavelength determined by the width of the gap, e.g., [@gultekin12]. ### Relativistic beaming [@dorazio15] have also advanced an alternate suggestion for the periodicity of PG1302-102 – the emission seen is from a mini-disk around the secondary black hole which is in orbital motion around the system barycenter. Doppler boosting of the emission as the disk moves along the line-of-sight is sufficient to account for the periodic variation seen. With the right choice of parameters, the variability of PG 1302-102 can explained by this if the broad lines do not originate from any circumbinary disk. The model also predicts different amplitude variability at different frequencies, assuming that the spectral slope is different, with no phase difference, i.e, it should track the optical variability. ### Quasi-periodic oscillation Quasi-periodic oscillations (QPOs) are a common phenomenon in the X-ray emission of stellar mass black hole binaries. Although the underlying physical cause is not understood, high frequency QPOs are consistent with the dynamical time of the system whilst low frequency ones are associated with Lense-Thirring precession of a geometrically thick accretion flow near the primary black hole, e.g., [@ingram09]. If the frequencies involved are scaled up by mass (many black hole timescales depend approximately on the inverse of the black hole mass), they are roughly consistent with the periodic behavior reported in this work. For example, the microquasar GRS 1915+105 has a mass of $\sim12 M_\odot$ and exhibits QPOs at $\sim$1 Hz (e.g., [@yan13] and references therein) with slowly varying frequency. Scaling this to the observed mass range of PG 1302-102 ($10^{8.3 - 9.4} M_\odot$) gives QPOs with a period between $\sim$200 and 2400 days (the restframe period is $\sim$1440 days). [@king13] reported the first AGN analog of a low-frequency QPO in the 15 GHz light curve of the blazar J1359+4011 with a timescale varying between 120 and 150 days over a $\sim$4 year timespan. Stellar black hole binary QPOs are predicted to show a decreasing rms amplitude with longer wavelength [@veledina15]. The degree and angle of linear polarization of the precessing accretion flow are also predicted to modulate on the QPO frequency [@ingram15]. Theoretical detection rates {#detection} --------------------------- Given our large data set of 250,000 quasars, it is worthwhile considering how many SMBH binary systems we could expect to find. [@haiman09] used simple disk models for circumbinary gas and the binary-disk interaction to determine the number of SMBH binaries that may be expected in a variety of surveys, assuming that such objects are in the final gravitational wave-dominated phase of coalescence (which equates to separations less than $\sim$0.01 pc for a $10^8 M_\odot$ system). [@volonteri09] combined this with merger tree assembly models to similarly predict the number of expected SMBH binaries at wider separations where spectral line variations may be seen (equating to separations greater than $\sim 0.2$ pc for a $10^8 M_\odot$ SMBH binary). The latter shows that in a sample of 10000 quasars with $z < 0.7$, there should be $\sim10$ such objects and this number increases by a factor of $\sim$5–10 for $z < 1$. We note that our sample has $\sim 75000$ quasars with $z < 1$. Using these approaches, we can estimate our predicted sample size. Assuming a limiting magnitude of $V \sim 20$, a detectable range of orbital periods from 20–300 weeks (thus spanning both gravitational-wave and gas-dominated regimes), a survey sky coverage of $2\pi$ steradians (there are far fewer known spectroscopically confirmed quasars in the regions where CRTS does not overlap with SDSS), and a redshift range of 0.0 – 4.5, we should identify $\sim 450$ SMBH binaries. Our detection rate of $10^{-4}$ is therefore conservative compared to the predicted rate of $2.3 \times 10^{-3}$. Virial black hole mass estimates [@shen11] are available for 88,000 quasars in our sample (of which 23 per cent have $z > 2$). If we assume that each of these is a SMBH binary with a separation of 0.01 pc then the CRTS temporal baseline is sufficient to detect 1.5 cycles or more of periodic behavior in 63 per cent of them (including 55 per cent of the $z > 2$ population). Our search should therefore be sensitive to a large fraction of the close SMBH population. We note, however, these theoretical arguments are still subject to considerable uncertainties, for example, if the final parsec problem cannot be resolved then there will not be any binaries in the $\sim$0.01 pc regime. [@haiman09] also predict that in the gravitational wave (GW)-dominated regime of the merger process, the number of expected binaries is: $$n_{\mathrm var} = \left( \frac{10^7 \mathrm{yr}}{t_Q} \right) \left[ \frac{t_{\mathrm{orb}}}{50.2 \,\, \mathrm{week}} \right]^{8/3} M_7^{-5/3} q_s^{-1}$$ where $t_{orb}$ is the restframe period, $t_Q$ is the timescale, $M_7$ is the black hole mass in units of $10^7 M_{\odot}$ and $q_s$ is the black hole mass ratio. Fig. \[bhnum\] shows the observed distributions for the binary candidates with $\log(M_{BH}/M_{\odot}) < 9$ and $\log(M_{BH}/M_{\odot}) > 9$ together with $n_{\mathrm var} \propto t_{orb}^{8/3}$ relationships fit to these ($\log(M_{BH}/M_{\odot}) \sim 9$ is the median value for the data set and provides a natural division point). The results suggest that there is a transition timescale below which objects follow the power law expected from orbital decay driven by GWs. This timescale also anticorrelates with object mass such that higher mass objects (here $\log(M_{BH}/M_{\odot}) > 9$) change behavior at about $t_{orb} \sim 750$ days and lower mass objects at $t_{orb} \sim 1000$ days. At longer timescales (and lower masses), the number of objects is expected to follow a power law distribution $n_{var} \propto t^{\alpha}$, where the scaling index $\alpha$ is dependent on the physics of the circumbinary accretion disk and viscous orbital decay. However, since one of our selection criteria is that the temporal coverage of a light curve should cover at least 1.5 cycles of the periodic signal in the observed frame, we are incomplete at restframe timescales larger than the transition timescale. We can therefore not say anything about the true value of $\alpha$ and the physics of the gas-driven phase from our sample due to insufficient coverage. Nevertheless the statistical detection of GW-driven binaries can also be seen to confirm that circumbinary gas is present at small orbital radii and is being perturbed by the black holes [@haiman09]. Gravitational wave detection ============================ Given that some of the candidates appear to be in the gravitational-wave driven phase of merging, it is worth determining whether any of them might be resolvable as individual sources for current or upcoming nanohertz GW detectors (pulsar timing arrays (PTA)) rather than just be contributors to the stochastic GW background. For each candidate, we have determined the maximum signal that could be detected (see Table \[candidates\]). The GW frequency of a binary with a circular orbit is $f_{GW} = 2 / t_{per}$, where $t_{per}$ is the orbital period. The intrinsic GW strain amplitude is: $$h_s = \left( \frac{128}{15} \right)^{1/2} \frac{(GM_c)^{5/3}}{c^4 d_L} (\pi f)^{2/3}$$ where $M_c = (M_1M_2)^{3/5}(M_1 + M_2)^{-1/5}$ is the chirp mass and $d_L$ is the comoving distance to the source. The maximum induced timing residual amplitude is then $h_s / (2 \pi f_{GW})$. We have also calculated the inspiral time for each candidate (see Table \[candidates\]) to check that there is no significant frequency evolution over the lifetime of a detection experiment, i.e., $t_{expt} << t_{insp}$, where $t_{insp} = 5 (1+q)^2 a^4 / 2 q R_S^3$ and $R_S = 2GM/c^2$. The median inspiral time is $\sim$8,000 years, although there are four candidates predicted to merge within the next century and , with decadal baselines, we should be able to detect period changes in these objects. Using PTA noise estimates for Nanograv and the Parkes PTA ([@arzoumanian14; @zhu14]), we calculate that none of the candidates will be resolvable as sources by the current generation of experiments (see Fig. \[gw\]). However, some of the sources will be viable by the next generation of detectors, e.g., SKA. Conclusions =========== We have detected strong periodicity in the optical photometry of 111 quasars which we ascribe to the presence of a close SMBH binary in these systems. The Keplerian nature of the signal suggests that it is kinematic in origin and may be produced by a precessing jet, warped accretion disk or periodic accretion driven by the SMBH binary. We note our detection methodology may not be particularly sensitive to other types of periodic behavior exhibited by SMBH binary candidates, such as OJ 287. This would then suggest that our sample of objects represents a small fraction of a much larger close binary SMBH population which is yet to be detected. We are therefore planning further studies of the [(quasi)]{}[periodic]{} quasar population with less stringent constraints than those applied in this analysis. This will also address our incomplete coverage of the gas-driven merger regime. Further observations of our sample can test the different interpretations, as well as the primary SMBH binary attribution. [@shen10] suggest that reverberation mapping is a particularly useful diagnostic since the behaviour of emission line response to continuum variations is expected to be different for alternate explanations. We plan to see whether stacked reverberation employing time series and pairs of multiepoch spectra [@fine13] is sufficient. Obviously continued monitoring by CRTS and other synoptic surveys will track future cycles of periodicity and historical photometric data from photographic plate collections, such as DASCH[^9], may provide more data for previous cycles. For example, DASCH data for PG 1302-102 is consistent with the reported period (J. Grindlay, private communication). With such decadal baselines, any change in the period of the system expected from general relativity may be detectable. Future spectroscopic observations can also test whether there is any spectral variation in the sample consistent with binary orbital timescales, although this is not necessarily present in such close systems. The theoretical expectation for pairs at the close separations that we are probing is that they should not show any such effect – the size of any broad line region (BLR) dwarfs the orbital dimensions of the binary and at separations closer than 0.03 pc in some scenarios, the BLR is truncated or destroyed [@ju13]. However, [@dorazio15] suggest that there may be broadened emission due to recombination in orbiting circumbinary gas which would show as periodic variation in the FWHM of associated lines and smaller shifts in their centroids. Continued spectral monitoring, particularly coincident with an extremum in the photometric light curve, will help to confirm the origin of any spectral variability detected. With a binary, the variation should follow a regular evolution whereas a bright spot (non-axisymmetric perturbation in the BLR emissivity) in the BLR, say, should just be a transient feature. One caveat with any detection of spectral variability is that asymmetric reverberation can act as a major source of confusion noise in multiepoch spectroscopic data [@barth15]. Multiwavelength observations, particularly of the radio quiet objects, will also provide more information about them. In particular, the specific predictions that [@dorazio15] make regarding the Fe K$\alpha$ line, broad line widths and narrow line offsets for the accretion disc cavity model can be tested. Finally, we note that these objects are strong candidates for any gravitational wave experiment sensitive to nanohertz frequency waves such as those using pulsar timing arrays as well as any future space-borne gravitational wave detection mission. Acknowledgments {#acknowledgments .unnumbered} =============== We thank Zoltan Haiman for useful discussions. This work was supported in part by the NSF grants AST-0909182, IIS-1118041 and AST-1313422, and by the W. M. Keck Institute for Space Studies. The work of DS was carried out at Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. This work made use of the Million Quasars Catalogue. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University. 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--- author: - | Jiaguo Zhang$^a$[^1] , Eckhart Fretwurst$^a$, Robert Klanner$^a$, Ioana Pintilie$^b$, Joern Schwandt$^a$, and Monica Turcato$^{c}$\ Institute for Experimental Physics, Hamburg University,\ Luruper Chaussee 149, D-22761 Hamburg, Germany\ National Institute of Materials Physics,\ P.O.Box MG-7, Bucharest-Magurele, Romania\ European XFEL GmbH,\ Albert-Einstein-Ring 19, D-22761 Hamburg, Germany\ E-mail: title: 'Investigation of X-ray induced radiation damage at the Si-SiO$_{2}$ interface of silicon sensors for the European XFEL' --- Introduction ============ At DESY, Hamburg, the European X-ray Free Electron Laser (XFEL) [@bib1] is under construction. Starting in late 2015, it will open up completely new research opportunities for science as well as for industrial applications. Examples are the structural analysis of single complex organic molecules, the investigation of chemical reactions at the femtosecond scale, and the study of processes that occur in the interior of planets. At the European XFEL, imaging experiments require silicon pixel detectors. One of the X-ray pixel detectors under development is the Adaptive Gain Integrating Pixel Detector (AGIPD) [@bib2; @bib3], which has to satisfy extraordinary performance specifications [@bib4]: Doses of up to 1 GGy in three years of operation, up to 10$^{5}$ 12 keV photons per pixel of 200 $\mu$m $\times$ 200 $\mu$m arriving within less than 100 fs, and a time interval between XFEL pulses of 220 ns. To address these challenges, radiation-hard silicon pixel sensors need to be developed, which requires good understanding of X-ray induced radiation damage. The aim of this work is to (1) understand the radiation damage induced by X-rays, (2) extract the damage-related parameters, i.e. the surface density of oxide charges and surface-current density, which are the main inputs for sensor optimization with TCAD simulation [@bib5], (3) investigate the effects due to the voltage applied to the gates of the MOS capacitor and the gate-controlled diode during irradiation, and (4) verify the long term stability and performance of silicon sensors with the help of annealing studies. The following sections will discuss the results for the above topics separately. Radiation damage at the European XFEL environment ================================================= There are two kinds of radiation damage: bulk damage and surface damage. The former is due to the non-ionization energy loss (NIEL) [@bib6; @bib61] of incident particles, i.e. protons, neutrons, electrons and gamma-rays, which cause silicon crystal damage; the latter is due to the ionization energy loss of charged particles or X-ray photons, which cause positive charges and traps to build up in the SiO$_{2}$ and at the Si-SiO$_{2}$ interface. The threshold energy for X-rays to cause bulk damage is . Therefore, the main damage in silicon sensors at the European XFEL with a typical energy of is the surface damage. The mechanisms of surface damage have been described extensively in [@bib7; @bib8; @bib9; @bib10; @bib11]. We shortly summarize them as follow: X-rays (or charged particles) produce electron-hole pairs in the SiO$_{2}$. Depending on the strength of the electric field in the SiO$_{2}$ and the type of incident particles, as seen in figure \[Figure1\](a), a fraction of electrons and holes recombine. The remaining electrons and holes escaping from the initial recombination either drift to the electrode or to the Si-SiO$_{2}$ interface, depending on the direction of the electric field in the SiO$_{2}$. Some of the holes drift close to the interface, are captured by oxygen vacancies (most of the vacancies locate in the SiO$_{2}$ close to the Si-SiO$_{2}$ interface) and form trapped positive charges in the oxide, called oxide charges. During the transport of holes, some react with hydrogenated oxygen vacancies and result in protons. Those protons, which drift to the interface, break the hydrogenated silicon bonds at the interface and produce dangling silicon bonds, namely interface traps, with energy levels distributed throughout the band gap of silicon. Figure \[Figure1\](b) shows the mechanisms of formation of oxide charges (with density $N_{ox}$) and interface traps (with density $N_{it}$) in a MOS capacitor biased with positive voltage. The values of $N_{ox}$ and $N_{it}$ induced by X-ray ionizing radiation mainly depend on dose, electric field in the SiO$_{2}$, annealing time and temperature, crystal orientation, and quality of the oxide. The influence of the above factors has been investigated and results will be discussed in Section $5$. ![(a) Fraction of electrons and holes escaping from initial recombination. (b) Mechanisms of formation of oxide charges and interface traps, shown in band diagrams of SiO$_{2}$, Si-SiO$_{2}$ interface and Si, [@bib11].[]{data-label="Figure1"}](Initial_recombination.eps "fig:") ![(a) Fraction of electrons and holes escaping from initial recombination. (b) Mechanisms of formation of oxide charges and interface traps, shown in band diagrams of SiO$_{2}$, Si-SiO$_{2}$ interface and Si, [@bib11].[]{data-label="Figure1"}](Mechanism.eps "fig:") Investigated structures and irradiation ======================================= To study the surface damage, test fields have been designed. Each test field includes a MOS capacitor and a gate-controlled diode. The MOS capacitor has a circular shape with a diameter $\phi$ of 1.5 mm. The gate-controlled diode contains a circular diode ($\phi$ = 1.0 mm) in the center and 5 concentric surrounding gate rings. The width of the gate rings is 50 $\mu$m and neighbouring gate rings are separated by 5 $\mu$m spacing. The investigated test fields fabricated on high resistivity n-type silicon with different orientations produced by two vendors, used for the study in Section $5.1$, are listed in table \[Table1\]. For the studies in Section $5.2$ and $5.3$, the test fields with orientation &lt;100&gt; and &lt;111&gt; produced by CiS have been used, respectively. Label CE2250 CB0450 6336-01-03 HAMA-MOS-04 ----------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- Producer CiS [@bib111] CiS CiS Hamamatsu [@bib112] Material FZ DOFZ Epi FZ Orientation &lt;100&gt; &lt;111&gt; &lt;111&gt; &lt;100&gt; Doping $7.6\times 10^{11}$ cm$^{-3}$ $1.1\times 10^{12}$ cm$^{-3}$ $7.8\times 10^{13}$ cm$^{-3}$ $9.0\times 10^{11}$ cm$^{-3}$ \*[Insulator]{} 330 nm SiO$_{2}$ 360 nm SiO$_{2}$ \*[335 nm SiO$_{2}$]{} \*[700 nm SiO$_{2}$]{} + + 50 nm Si$_{3}$N$_{4}$ 50 nm Si$_{3}$N$_{4}$ : List of investigated test fields in the study of Section $5.1$. Each contains a MOS capacitor and a gate-controlled diode. FZ - float zone; DOFZ - diffusion oxygenated float zone; Epi - epitaxial.[]{data-label="Table1"} The irradiation of the test fields have been done at the DESY DORIS III beamline F4 with a “white” photon beam. The typical energy of the X-rays is with a full width at half maximum of $\sim$10 keV. The flux at the beam center is approximately $1.1 \times 10^{14}$ photons/(mm$^{2}\cdot$s), which corresponds to a dose rate of 180 kGy/s in SiO$_{2}$ . A chopper has been used to reduce the dose rate in order to avoid heating up the silicon during X-ray irradiation. The typical temperature during irradiations was in the range from 25 $^{\circ}$C to 36 $^{\circ}$C [@bib121]. Details on the X-ray beam profile and dose calibration can be found in [@bib121; @bib12]. The following procedures are used for the study in Section $5.1$: (1) solid state measurements were performed on MOS capacitors and gate-controlled diodes before irradiation; (2) the test fields, each consisting of a MOS capacitor and a gate-controlled diode, were irradiated to a specific dose (the first irradiation dose is 10 kGy); (3) after irradiation, the solid state measurements were performed before and after annealing at for 10 minutes[^2]; (4) the previously irradiated test fields were irradiated to higher doses and measured afterwards. The steps (2)-(4) were repeated till the 1 GGy dose had been reached. For the studies in Section $5.2$ and $5.3$, only steps (1), (2) and (3) were done without repeat. Methods to extract the parameters related to surface damage =========================================================== Extraction of $N_{ox}$ and $N_{it}$ ----------------------------------- The capacitance/conductance-voltage (C/G-V) and thermal dielectric relaxation current (TDRC) measurements have been performed on the MOS capacitors, which are used to extract the surface density of oxide charges, $N_{ox}$, and the density of interface traps, $N_{it}$. During the measurements, the voltage is applied to the gate of the MOS capacitor while the backside is grounded. The measurement procedures have been described in [@bib12; @bib13]. The measured TDRC signal, $I_{tdrc}$, as function of temperature, $T$, allows to determine the distribution of the density of interface states, $D_{it}(E_{it})$, in the silicon band gap [@bib131; @bib132; @bib133]. The total density of interface traps, $N_{it}$, can be estimated by integrating the TDRC signal divided by the heating rate $\beta$ used in the measurement and the . To extract the surface density of oxide charges $N_{ox}$, a model describing the irradiated MOS capacitor by an RC-network is employed. The model allows to calculate the capacitance and conductance of a MOS capacitor as function of gate voltage for different frequencies using the measured TDRC signal and $N_{ox}$ as input. $N_{ox}$, which just shifts the C/G-V curves along the voltage-axis, is obtained by adjusting its value till the calculated C/G-V curves describe the measurements. Figure \[Figure2\](a) shows the comparison of the measured to the calculated C/G-V curves (parallel mode) of a MOS capacitor irradiated to 5 MGy after annealing for 120 minutes at 80 $^{\circ}$C for frequencies between 1 kHz and 100 kHz. The model has been described in [@bib13]. In the model calculations, all the interface traps within the silicon band gap are assumed to be acceptors, which gives the maximal estimate for the surface density of oxide charges introduced by X-rays. ![The C/G-V and I-V curves of irradiated MOS capacitor and gate-controlled diode. (a) Comparison of the measured and calculated C/G-V curves (parallel mode) of a MOS capacitor irradiated to 5 MGy for frequencies between 1 kHz and 100 kHz using the model described in [@bib13]. The extracted values of $N_{ox}$ and $N_{it}$ are both $2.5 \times 10^{12}$ cm$^{-2}$. (b) I-V curve of a gate-controlled diode irradiated to 5 MGy annealed at for 10 minutes.[]{data-label="Figure2"}](CGV.eps "fig:") ![The C/G-V and I-V curves of irradiated MOS capacitor and gate-controlled diode. (a) Comparison of the measured and calculated C/G-V curves (parallel mode) of a MOS capacitor irradiated to 5 MGy for frequencies between 1 kHz and 100 kHz using the model described in [@bib13]. The extracted values of $N_{ox}$ and $N_{it}$ are both $2.5 \times 10^{12}$ cm$^{-2}$. (b) I-V curve of a gate-controlled diode irradiated to 5 MGy annealed at for 10 minutes.[]{data-label="Figure2"}](IV_GCD_acc.eps "fig:") Extraction of $J_{surf}$ ------------------------ To determine the surface-current density, $J_{surf}$, current-voltage (I-V) measurements have been done at room temperature for the gate-controlled diodes. For the I-V measurement, a constant DC voltage (-12 V in our case) is applied to the p$^{+}$ electrode through a voltage source to partially deplete the central diode, and the current flow from the rear side n$^{+}$ electrode is recorded as function of voltage on the 1$^{st}$ gate ring while keeping the 2$^{nd}$ gate ring grounded. The DC voltage applied to the p$^{+}$ electrode should be enough so that the depletion regions below the diode and the 1$^{st}$ gate merge when the 1$^{st}$ gate is biased to depletion. Figure \[Figure2\](b) shows the I-V curve of a gate-controlled diode irradiated to 5 MGy after annealing for 10 minutes at 80 $^{\circ}$C. The surface current, $I_{surf}$, is extracted from the “maximum” current measured in depletion of the 1$^{st}$ gate ring and the average value of the currents obtained in accumulation. It should be noted that the measured surface current $I_{surf}(T_{meas})$ is very sensitive to the temperature $T_{meas}$ during the measurement, i.e. at room temperature its value changes by $\sim$8% if the temperature changes by 1 $^{\circ}$C. In figure \[Figure21\](a), the temperature dependence of the I-V curves of an irradiated gate-controlled diode measured at different temperatures from 213 K to 295 K is shown. The following scaling formula allows to describe the data $$\label{eq:scale} I_{surf}(T) = I_{surf}(T_{meas}) \cdot \left( \frac{T}{T_{meas}} \right) ^{2} \cdot \textrm{exp} \left[ \frac{0.605 \textrm{eV}}{k_{B}} \cdot \left( \frac{1}{T_{meas}} - \frac{1}{T} \right) \right]$$ with $k_{B}$ the Boltzmann constant. ![(a) I-V curves of a gate-controlled diode irradiated to $\sim$100 kGy after annealing at 80 $^{\circ}$C for 60 minutes, measured in the temperature range from 213 K to 295 K. (b) Comparison of the measured values of surface currents at different temperatures with the scaling formula (4.1).[]{data-label="Figure21"}](Isurface_temperature.eps "fig:") ![(a) I-V curves of a gate-controlled diode irradiated to $\sim$100 kGy after annealing at 80 $^{\circ}$C for 60 minutes, measured in the temperature range from 213 K to 295 K. (b) Comparison of the measured values of surface currents at different temperatures with the scaling formula (4.1).[]{data-label="Figure21"}](Isurface_scaling.eps "fig:") Figure \[Figure21\](b) shows the comparison between the measured surface currents and the calculated ones according to the scaling formula \[eq:scale\] in the temperature range from 203 K to , which shows good agreement. Thus, all surface currents extracted from the measurements in our study have been scaled to the values at 20 $^{\circ}$C. The surface-current density at 20 $^{\circ}$C, $J_{surf}(T=293\textrm{ K})$, is calculated from the surface current scaled to 20 $^{\circ}$C and the area of the 1$^{st}$ gate ring: . Results ======= Dose dependence --------------- We first present the results from MOS capacitors and gate-controlled diodes irradiated to eight doses in the range between 10 kGy to 1 GGy. During the irradiations, the electrodes of the MOS capacitors or the gate-controlled diodes were kept floating. The first C/G-V and I-V measurements were performed within 1 hour after each irradiation. As mentioned above, due to significant annealing of the radiation induced defects already at room temperature, measurements were also done after annealing at 80 $^{\circ}$C for 10 minutes in order to obtain reproducible results. Hence, only the results after annealing will be shown in the following. Figure \[Figure3\] shows the surface density of oxide charges, $N_{ox}$, and the surface-current density, $J_{surf}$, as function of dose. A comparison of the measurements between cyan stars (CS: CiS-&lt;100&gt;-350 nm SiO$_{2}$ + 50 nm Si$_{3}$N$_{4}$) and green dots (GD: CiS-&lt;100&gt;-330 nm SiO$_{2}$ + 50 nm Si$_{3}$N$_{4}$) shows that the results for structures with the same technology and crystal orientation are compatible[^3]. The results for CS, measured in 2011, are obtained from eight different MOS capacitors each one directly irradiated to the dose shown in figure \[Figure3\](a). GD, measured in 2012, are the results from one MOS capacitor irradiated in steps to the doses given in the figure and annealed for 10 minutes at 80 $^{\circ}$C after each step. Thus, it can be concluded that, under the same irradiation environment and annealing condition, the densities of defects introduced by X-ray ionizing radiation are independent of the way the irradiation is performed but depend on the “accumulated dose”. ![Dose dependence of the surface density of oxide charges and the surface-current density scaled to after annealing at 80 $^{\circ}$C for 10 minutes. (a) $N_{ox}$ vs. dose. (b) $J_{surf}$ vs. dose.[]{data-label="Figure3"}](Dose_dependence_of_oxide_charge_density.eps "fig:") ![Dose dependence of the surface density of oxide charges and the surface-current density scaled to after annealing at 80 $^{\circ}$C for 10 minutes. (a) $N_{ox}$ vs. dose. (b) $J_{surf}$ vs. dose.[]{data-label="Figure3"}](Dose_dependence_of_surface_current_density.eps "fig:") From the dose dependence of the surface density of oxide charges $N_{ox}$ and the surface-current density $J_{surf}$, it is found that: (1) $N_{ox}$ and $J_{surf}$ for &lt;100&gt; silicon is lower than that for &lt;111&gt; silicon. (2) Little difference in $N_{ox}$ is observed for the MOS capacitors with an insulating layer made of SiO$_{2}$ and that made of SiO$_{2}$ and Si$_{3}$N$_{4}$. (3) The values found for samples fabricated by CiS and Hamamatsu are different, agree however within a factor of 2, which indicates a dependence of radiation-induced defects on technology. (4) $N_{ox}$ and $J_{surf}$ either saturate or decrease at high irradiation doses. The significant decrease of $J_{surf}$ at high doses is not understood and still under study. Gate-voltage dependence ----------------------- The presence of an electric field in the SiO$_{2}$ during the irradiation plays an important role in the formation of oxide charges and interface traps: it determines not only the fraction of electrons and holes escaping from the initial recombination, but also the direction the electrons and holes drift, which impacts on the amount of oxide charges and interface traps that form in the Si-SiO$_{2}$ interface region. ![Gate voltage dependence of the surface density of oxide charges and the surface-current density scaled to 20 $^{\circ}$C for doses of 100 kGy and 100 MGy. Results are obtained after annealing at 80 $^{\circ}$C for 10 minutes from test fields with crystal orientation &lt;100&gt; produced by CiS. For positive gate voltages, the electric field points towards the Si-SiO$_{2}$ interface.[]{data-label="Figure4"}](Bias_voltage_dependence_of_oxide_charge_density.eps "fig:") ![Gate voltage dependence of the surface density of oxide charges and the surface-current density scaled to 20 $^{\circ}$C for doses of 100 kGy and 100 MGy. Results are obtained after annealing at 80 $^{\circ}$C for 10 minutes from test fields with crystal orientation &lt;100&gt; produced by CiS. For positive gate voltages, the electric field points towards the Si-SiO$_{2}$ interface.[]{data-label="Figure4"}](Bias_voltage_dependence_of_surface_current_density.eps "fig:") Figure \[Figure4\] shows the gate-voltage dependence of the surface density of oxide charges, $N_{ox}$, and the surface-current density, $J_{surf}$, for doses of 100 kGy and 100 MGy. The maximum gate voltage used, 25 V, corresponds to an electric field of $\sim$0.7 MV/cm in the SiO$_{2}$. The results will be discussed for three cases: \(i) For 0 V, the initial electric field in the SiO$_{2}$ is zero and thus the situation is similar to the case without gate voltage applied. In both cases, the electric field in the SiO$_{2}$ during irradiation is dominated by the field created by the oxide charges, which points from the Si-SiO$_{2}$ interface to the aluminium gate. Hence, the values of $N_{ox}$ and $J_{surf}$ obtained under 0 V are similar to the values obtained without gate voltage (refer to the values of $N_{ox}$ and $J_{surf}$ at 100 kGy and 100 MGy in figure \[Figure3\]). \(ii) For negative gate voltages, the direction of the electric field in the SiO$_{2}$ points from the Si-SiO$_{2}$ interface to the aluminium gate, as for 0 V. The electric field in the SiO$_{2}$ is the sum of the electric field due to external voltage and that created by the positive oxide charges close to the Si-SiO$_{2}$ interface. With increasing “accumulated dose”, the electric field due to the oxide charges increases; thus, the field due to the gate voltage is reduced. For example, the electric field created by the oxide charges with a density of $N_{ox}=2 \times 10^{12}$ cm$^{-2}$ is $\sim$0.9 MV/cm if it is assumed that the oxide charges are located at the interface, compared to a field of 0.7 MV/cm for a gate voltage of . Therefore, no big difference is observed for the values of $N_{ox}$ and $J_{surf}$ obtained under negative gate voltage and under 0 V. However, the situation may change for a different spatial distribution of oxygen vacancies in the SiO$_{2}$. \(iii) For positive gate voltages, the direction of the electric field in the SiO$_{2}$ points from the aluminium gate to the Si-SiO$_{2}$ interface. The number of holes drifting to the interface is larger than for negative or zero gate voltage. Thus, more holes are able to be captured by the oxygen vacancies located close to the interface and produce oxide charges. As the fraction of holes escaping from the initial recombination process increases with the electric field in the SiO$_{2}$, the number of holes drifting to the interface increases with the positive gate voltage applied to the aluminium gate. Thus, a strong voltage dependence of $N_{ox}$ and $J_{surf}$ is observed in this case. Time and temperature dependence of annealing -------------------------------------------- In order to understand the long term behaviour of the oxide charges and interface traps, annealing studies have been performed on MOS capacitors and gate-controlled diodes irradiated to 5 MGy. Different processes are responsible for the annealing of the oxide charges and interface traps. The annealing of oxide charges is due to two different hole-removal processes [@bib8]. Below $\sim$125 $^{\circ}$C, the removal of oxide charges is mainly due to the recombination of trapped holes with electrons tunnelling into the SiO$_{2}$ from the silicon, which has been described by a tunnelling : $$\label{eq:Nox} N_{ox}(t) = N_{ox}^{0} \cdot (1+t/t_{0})^{-\frac{\lambda}{2\beta}}$$ with $t_{0}(T) = t_{0}^{*} \cdot \textrm{exp} (\frac{\Delta E}{k_{B}T})$. $N_{ox}^{0}$ is the surface density of oxide charges at $t=0$, $t_{0}$ the effective tunnelling time constant, $1/\lambda$ the characteristic depth of the spatial distribution of oxide charges in the SiO$_{2}$ and $\beta$ a parameter related to the barrier height, $t_{0}^{*}$ the tunnelling time constant, and $\Delta E$ the difference between the energy level of the defects in the SiO$_{2}$ and the Fermi level in the silicon. Above $\sim$150 $^{\circ}$C, a rapid removal or recombination of trapped holes in the SiO$_{2}$ is observed, which is described by a thermal detrapping model [@bib141]. The annealing mechanism of interface traps (dangling silicon bonds) is not well understood, but the kinetics can be described by the “two reaction model” according to [@bib15; @bib16]. The first reaction is the passivation of dangling silicon bonds by hydrogen at the Si-SiO$_{2}$ interface. The second reaction is the binding of two hydrogen atoms to a hydrogen molecule. The “two reaction model” predicts that the time dependence of the density of interface traps follows a power law. As the surface current is mainly generated by the interface states at the mid-gap of silicon, the annealing behaviour of the surface-current density is also described by a similar expression as for the annealing of $N_{it}$: $$\label{eq:Jsurf} J_{surf}(t) = J_{surf}^{0} \cdot (1+t/t_{1})^{-\eta}$$ with $t_{1}(T) = t_{1}^{*} \cdot \textrm{exp} (\frac{E_{\alpha}}{k_{B}T})$. $J_{surf}^{0}$ is the surface-current density at $t=0$, $\eta$ a parameter related to the ratio of the two reaction rates, $1/t_{1}^{*}$ the frequency factor and $E_{\alpha}$ the activation energy. ![Annealing of the surface density of oxide charges and the surface-current density at elevated temperatures 60 $^{\circ}$C and 80 $^{\circ}$C. Results are obtained from test fields with crystal orientation &lt;111&gt; produced by CiS. Measurements fit by functions given in eq(5.1) and (5.2) and an exponential function shown.[]{data-label="Figure5"}](Annealing_data_oxide_charge_density.eps "fig:") ![Annealing of the surface density of oxide charges and the surface-current density at elevated temperatures 60 $^{\circ}$C and 80 $^{\circ}$C. Results are obtained from test fields with crystal orientation &lt;111&gt; produced by CiS. Measurements fit by functions given in eq(5.1) and (5.2) and an exponential function shown.[]{data-label="Figure5"}](Annealing_data_surface_current_density.eps "fig:") The annealing of oxide charges and interface traps in this study has been performed at 60 $^{\circ}$C and 80 $^{\circ}$C. Figure \[Figure5\] shows $N_{ox}$ and $J_{surf}$ as function of annealing time. We fit to the measurements by the functions given in eq(\[eq:Nox\]) and (\[eq:Jsurf\]) and an exponential function, expected for a constant annealing probability. It can be seen that eq(\[eq:Nox\]) and (\[eq:Jsurf\]) provide a good description of the data supporting the tunnelling model and the “two reaction model”. Table \[Table2\] and \[Table3\] show the parameters found from the fits for $N_{ox}$ and $J_{surf}$ by the functions given in eq(\[eq:Nox\]) and (\[eq:Jsurf\]). The data can be used to calculate the annealing behaviour of the oxide charges and the surface current at other temperatures. It is found that the annealing of $N_{ox}$ is a slow process whereas the annealing of $J_{surf}$ is relative fast. For example, using the parameters found it is predicted that it takes three years to remove 50% of the oxide charges at 20 $^{\circ}$C but only 5 days to reduce the surface-current density by 50%. $N_{ox}^{0}$ \[cm$^{-2}$\] $\frac{\lambda}{2 \beta}$ $t_{0}^{*}$ \[s\] $\Delta E$ \[eV\] ---------------------------- --------------------------- ----------------------- ------------------- $3.6 \times 10^{12}$ $0.070$ $5.4 \times 10^{-12}$ $0.91$ : Parameters found from the fit of the annealing data for $N_{ox}$ by the function given in eq(5.1) described by the tunnelling model.[]{data-label="Table2"} $J_{surf}^{0}$ \[$\mu$A$\cdot$cm$^{-2}$\] $\eta$ $t_{1}^{*}$ \[s\] $E_{\alpha}$ \[eV\] ------------------------------------------- -------- ---------------------- --------------------- $8.1$ $0.21$ $1.4 \times 10^{-8}$ $0.70$ : Parameters found from the fit of the annealing data for $J_{surf}$ by the function given in eq(5.2) described by the “two reaction model”.[]{data-label="Table3"} Summary and outlook =================== Results on the surface densities of oxide charges and the surface-current densities from MOS capacitors and gate-controlled diodes built on high resistivity n-type silicon with orientations &lt;100&gt; and &lt;111&gt; produced by two vendors, CiS and Hamamatsu, as function of 12 keV X-ray doses up to 1 GGy have been presented. The influence of the electric field in the oxide on the formation of oxide charges and interface traps has been investigated. Finally, first results on the annealing of the X-ray induced oxide charges and the surface current due to interface traps are presented. Additional annealing studies will be performed for MOS capacitors and gate-controlled diodes irradiated to a dose of 100 MGy. The extracted parameters, extrapolated annealing behaviour at room temperature and detailed discussions will be reported in a separated paper. The work was performed within the AGIPD Project. J. Zhang would like to thank the Marie Curie Initial Training Network “MC-PAD” for his PhD funding. I. Pintilie gratefully acknowledges the financial support from the Romanian National Authority for Scientific Research through the Project PCE 72/5.10.2011. We would like to thank the colleagues within AGIPD collaboration for their helpful discussions on the results. 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The webpage of CiS Forschungsinstitut fuer Mikrosensorik und Photovoltaik GmbH: *http://www.cismst.org*. The webpage of Hamamatsu Photonics: *http://www.hamamatsu.com*. H. Perrey, *Jets at Low Q$^{2}$ at HERA and Radiation Damage Studies for Silicon Sensors for the XFEL*, [*PhD thesis*, [**DESY-THESIS-2011-021**]{} (2011)]{}. J. Zhang et al., *Study of X-ray Radiation Damage in Silicon Sensors*, [*JINST* [**6**]{} (2011) C11013]{}, . J. Zhang et al., *Study of radiation damage induced by 12 keV X-rays in MOS structures built on high-resistivity n-type silicon*, [*J. Synchrotron Rad.* [**19**]{} (2012) 340-346]{}, . J.G. Simmons and G.W. Taylor, *Dielectric Relaxation and Its Effects on the Thermal Electric Characteristics of Insulators*, [*Phys. Rev.* B [**6**]{} (1972) 4804-4814]{}. J.S. Uranwala et al., *Nonsteady-state techniques for determining the energy distribution of interface traps in MNOS (memory) devices*, [*Appl. Phys. Lett.* [**26**]{} (1975) 697-699]{}. H.A. Mar and J.G. Simmons, *A Review of the Techniques Used to Determine Trap Parameters in the MNOS Structure*, [*IEEE Trans. Electron Devices* [**24**]{} (1977) 540-546]{}. A.J. Lelis et al., *The Nature of the Trapped Hole Annealing Process*, [*IEEE Trans. Nucl. Sci.* [**36**]{} (1989) 1808-1815]{}. M. Schmidt and H. Koester, *Hole Trap Analysis in SiO$_{2}$/Si Structures by Electron Tunneling*, [*Phys. Stat. Sol.* (b) [**174**]{} (1992) 53-66]{}. T.R. Oldham et al., *Spatial Dependence of Trapped Holes Determined from Tunnelling Analysis and Measured Annealing*, [*IEEE Trans. Nucl. Sci.* [**33**]{} (1986) 1203-1209]{}. M.L. Reed and J.D. Plummer, *Chemistry of Si-SiO$_{2}$ Interface Trap Reactions*, [*J. Appl. Phys.* [**63**]{} (1988) 5776-5793]{}. M.L. Reed, *Models of Si-SiO$_{2}$ Interface Reactions*, [*Semicond. Sci. Technol.* [**4**]{} (1989) 980-985]{}. [^1]: Corresponding author. [^2]: Annealing needs to be done in order to obtain reproducible results [@bib13]. [^3]: The thickness of the SiO$_{2}$ has been estimated from the capacitance of the MOS capacitor biased to accumulation assuming a Si$_{3}$N$_{4}$ thickness of 50 nm.

--- abstract: 'A bound on the maximum information transmission rate through a cascade of Gaussian links is presented. The network model consists of a source node attempting to send a message drawn from a finite alphabet to a sink, through a cascade of Additive White Gaussian Noise links each having an input power constraint. Intermediate nodes are allowed to perform arbitrary encoding/decoding operations, but the block length and the encoding rate are fixed. The bound presented in this paper is fundamental and depends only on the design parameters namely, the network size, block length, transmission rate, and signal-to-noise ratio.' author: - '\' bibliography: - 'References.bib' title: 'Information Loss due to Finite Block Length in a Gaussian Line Network: An Improved Bound' --- Introduction {#sec:intro} ============ Transmission of messages through a series of links corrupted by noise is a situation that occurs frequently in communication networks. When the transmission block length is allowed to be arbitrarily large, it is quite simple to show (using the data-processing inequality) that the maximum information transfer rate is equal to the capacity of the weakest link. The possibilities in the finite block length regime are far less clear. Past work by Niesen *et al.* in [@Niesen07] and by us in [@IZS12-Paper] have addressed this question for the Discrete Memoryless Channel (DMC) case and the Additive White Gaussian Noise (AWGN) case, respectively. These results are *asymptotic* and provide scaling laws for the block length in terms of the number of nodes. In this paper, we provide a universal *non-asymptotic* bound on the maximum rate of information transfer for a line network consisting of a cascade of AWGN links. This complements and improves the asymptotic scaling results derived in [@IZS12-Paper]. The bound derived here is universal in the following sense: 1. While we assume that the block length and encoding rates are constant for all the nodes, we do not assume any particular structure for the channel codes and decision rules employed at any of the nodes. 2. In addition, no assumption is made on the absolute/relative magnitudes of the network size and the block length. It is to be noted that the analysis in [@IZS12-Paper] was found to be unsuitable to our requirement that the bound be non-asymptotic, and hence we take a totally new approach here. The rest of the paper is organized as follows. In Section \[sec:defs\], we introduce the notations and definitions used in the rest of the paper. In Section \[sec:network-model\], we introduce the network and the signal transmission models. We then provide our main result followed by its derivation in Section \[sec:analysis\], followed by a short discussion in Section \[sec:discussion\] that includes a comparison of our current results in relation to our previous results in [@IZS12-Paper]. Notations and Definitions {#sec:defs} ========================= Let $\mathds{R}$ be the set of all real numbers and $\mathds{N}$ be the set of all natural numbers. Natural logarithms are assumed unless the base is specified. The notation $\|\cdot\|$ represents $\mathcal{L}^2$ norm throughout. $\mathscr{S}_{M \times M}$ denotes the set of all $M \times M$ row-stochastic matrices, and $\mathscr{S}^*_{M \times M}$ denotes the set of all $M \times M$ row-stochastic matrices whose rows are identical. Let $N \in \mathds{N}$ denote the *code length* or *block length* of the transmission scheme. A *code rate* $R > 0$ is a real number such that $2^{NR}$ is an integer. Let $\mathscr{M} \triangleq \{1,2,3,\ldots,2^{NR}\}$ be the *message alphabet*. For a certain $P_0 \geq 0$, a rate $R$ length $N$ *code* $\mathscr{C}$ with power constraint $P_0$ is an ordering of $M = 2^{NR}$ elements from $\mathds{R}^N$, called *codewords*, such that the power of any codeword is lower than $P_0$: $$\mathscr{C} = \left(\mathbf{c}_1,\mathbf{c}_2,\mathbf{c}_3,\ldots,\mathbf{c}_{M}\right) \text{ s.t. } \forall {w \in \mathscr{M}}, \frac{1}{N}\|\mathbf{c}_w\|^2 \leq P_0.\nonumber$$ A rate $R$ length $N$ *decision rule* $\mathscr{R} = \left(\mathcal{R}_1,\mathcal{R}_2,\ldots,\mathcal{R}_M\right)$ is an ordered partition of $\mathds{R}^N$ of size $M = 2^{NR}$. The *encoding function* $\operatorname{ENC}_\mathscr{C}:\mathscr{M} \rightarrow \mathds{R}^N$ for a code $\mathscr{C}$ is defined by $\operatorname{ENC}_\mathscr{C}(w) = \mathbf{c}_w$, where $\mathbf{c}_w$ is the $w^{\text{th}}$ codeword in $\mathscr{C}$. The *decoding function* $\operatorname{DEC}_\mathscr{R}:\mathds{R}^N \rightarrow \mathscr{M}$ for a decision rule $\mathscr{R}$ is defined by: $$\operatorname{DEC}_\mathscr{R}(\mathbf{y}) = w \text{ iff } \mathbf{y} \in \mathcal{R}_w,\nonumber$$ where $\mathcal{R}_w$ is the $w^{\text{th}}$ partition in $\mathscr{R}$. Let $\Omega_0 = \frac{ 2\pi^{\frac{N}{2}} }{\Gamma \left(\frac{N}{2}\right)}$, the solid angle of a $N$-sphere. Here, $\Gamma(\cdot)$ is the standard gamma function given by $\Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\,{\rm d}t$. We also define the following functions: \[def:functions\] Let $Z_1,\ldots,Z_N$ be i.i.d. zero-mean unit-variance Gaussian random variables. Let for any $\gamma \geq 0$, $$\Phi_\gamma \triangleq \begin{cases}\cot^{-1} \left( \frac{ \sqrt{N\gamma} + z_1 }{ \sqrt{ \sum_{l=2}^{N} z_l^2 } } \right), & \sum_{l=2}^{N} z_l^2 > 0 \\ 0, \text{otherwise}.\end{cases}$$ Also, let $\forall v \in [0,\pi]$ $$g(v) \triangleq \frac{ (N-1) \pi^{ \frac{N-1}{2} } }{ \Gamma\left( \frac{N+1}{2} \right) } \int_0^{v} \left( \sin \theta \right)^{N-2} \mathrm{d}\theta$$ We then define the following function for $x \in \left[0,\Omega_0\right]$: $$\begin{aligned} \mathcal{Q}\left(x,N,\gamma\right) \triangleq \Pr\left[ g\left(\Phi_{\gamma}\right)\geq x \right]. \end{aligned}$$ The above function is the same as $Q^*\left(\cdot\right)$ defined and used by Shannon in [@Shannon-Papers]. In other words, computing $\mathcal{Q}\left(x,N,\gamma\right)$ gives the probability that a signal point on the power-constraint sphere $\| \mathbf{x} \|^2 = N P_0$ is displaced by a noise vector consisting of i.i.d. zero-mean unit-variance Gaussian random variables in each dimension outside an infinite right-circular cone of solid angle $x$ whose apex is at the origin and axis runs through the original signal point. Note that $\Phi_{\gamma}$ is a random variable whose probability distribution function has $N$ and $\gamma$ as parameters. The inverse cotangent function is assumed to have $[0, \pi]$ for its range so that it is continuous. Noting that $\cot^{-1}x$ will then be a decreasing function of $x$, we have the following remark about the monotonicity of the $\mathcal{Q}$-function w.r.t. $\gamma$: \[rem:Q-is-monotonic\] For any $\gamma_1, \gamma_2 > 0$ s.t. $\gamma_1 \geq \gamma_2$ and $x \in \left[0,\Omega_0\right]$, $$\mathcal{Q}\left(x,N,\gamma_2\right) \geq \mathcal{Q}\left(x,N,\gamma_1\right).$$ The term $g \left( \Phi_{\gamma} \right)$ is equal to the solid angle of the cone formed by rotating the line joining the origin and the displaced signal point about the line joining the origin and the original signal point as the axis. Network Model {#sec:network-model} ============= The line network model to be considered is given in Fig. \[fig:netwk-model\]. There are $n+1$ nodes in the network identified by the indices $\left\{0,1,2,\ldots,n\right\}$. The $n$ hops in the network are each associated with noise variances ${\sigma_i}^2 \geq \sigma_0^2 > 0, 1 \leq i \leq n$. In other words, the noise variances can be different for each link, but they are equal to or greater than a certain minimum $\sigma_0^2$ that is strictly positive. Nodes $0, 1,\ldots,n-1$ choose codes $\mathscr{C}_0,\mathscr{C}_1,\ldots,\mathscr{C}_{n-1}$ respectively to transmit, and Nodes $1,2,\ldots,n$ choose decision rules $\mathscr{R}_1,\mathscr{R}_2,\ldots,\mathscr{R}_{n}$ for reception. From here on, for the sake of simplicity, we let $\operatorname{ENC}_i$ and $\operatorname{DEC}_i$ to denote $\operatorname{ENC}_{\mathscr{C}_i}$ and $\operatorname{DEC}_{\mathscr{R}_i}$, respectively. All the codes and decision rules have the same rate $R$ and block length $N$. Node 0 generates a random message $W \in \mathscr{M}$ with probability distribution $p_W(w)$ and intends to convey the same to Node $n$ through the noisy multihop path in the network. Each node estimates the message sent by the node in the previous hop from its noisy observation, encodes the message as a codeword, and transmits the resulting codeword to the next hop. The codeword transmitted by Node $i$, for any $0 \leq i \leq n-1$ is given by $\mathbf{X}_i = \operatorname{ENC}_i(\hat{W}_i)$, where $\hat{W}_i$ is the estimate of the message at Node $i$ after decoding (Note that $\hat{W}_0 = W$ in this notation). The observation received by Node $i$, for any $1 \leq i \leq n$ is given by $\mathbf{Y}_i$, which follows a conditional density function that depends on the codeword $\mathbf{X}_{i-1}$ sent by the previous node: $$p_{\mathbf{Y}_i|\mathbf{X}_{i-1}}\left(\mathbf{y}|\mathbf{x}\right) = \frac{1}{\left(2\pi{\sigma_i}^2\right)^\frac{N}{2}}e^{-\frac{\left\|\mathbf{y}-\mathbf{x}\right\|^2}{2\sigma_i^2}}.\label{eqn:conditional:distrib}$$ The above density function follows from the assumptions of AWGN noise and memorylessness of the channel. The message $\hat{W}_i$ decoded by Node $i$ is given by $\hat{W}_i = \operatorname{DEC}_i(\mathbf{Y}_i)$. Note that the random variable $\hat{W}_n$ represents the message decoded by the final sink. {width="\textwidth"} The main result and analysis {#sec:analysis} ============================ The following theorem summarizes our main result. \[thm:unif-full\] In a line network employing any choice of rate $R$ length $N$ codes $\mathscr{C}_0,\mathscr{C}_1,\ldots,\mathscr{C}_{n-1}$ and rate $R$ dimension $N$ decision rules $\mathscr{R}_1,\mathscr{R}_2,\ldots,\mathscr{R}_n$, $$\mathcal{I}\left(W;\hat{W}_n\right) \leq NR \left[1 - M \mathcal{Q} \left( \frac{M-1}{M}\Omega_0, N, \frac{P_0}{\sigma_0^2} \right) \right]^n.$$ We now delve into the proof of Theorem \[thm:unif-full\]. Let $p_{\hat{W}_i \mid \hat{W}_{i-1}}\left(k \mid j\right), \forall j,k \in \mathscr{M}$ denote the conditional probabilities induced by channel encoding, noisy reception, and decoding at the $i^{\text{th}}$ hop. For each hop $i$, let $\mathbf{P}_i$ be the $M \times M$ row-stochastic matrix whose entry in row $j$ and column $k$ is $p_{\hat{W}_i \mid \hat{W}_{i-1}}\left(k \mid j\right)$. Note that the $j^{\text{th}}$ row in $\mathbf{P}_i$ gives the conditional probability mass function on the estimate $\hat{W}_i$ of the original message $W$ at hop $i$, given that the message sent by Node $i - 1$ is $j$. Let $$\mathbf{P} \triangleq \prod_{i = 1}^{n} \mathbf{P}_i.$$ Then, $\mathbf{P}$ clearly represents the row-stochastic probability transition matrix between the original message $W$ and the message decoded at the sink $\hat{W}_n$. The transition matrix $\mathbf{P}$ along with $p_W$ (the probability mass function of the original message $W$) together induce a joint distribution between $W$ and $\hat{W}_n$. Our goal is to find an upper bound on $\mathcal{I}\left(W; \hat{W}_n\right)$, with the constraints given in Section \[sec:network-model\]. For any $M \times M$ row-stochastic matrix $\mathbf{Q}$, define $\psi\left(\mathbf{Q} \mid p_W \right) \triangleq \mathcal{I}\left(W; \tilde{W}\right)$, where $\tilde{W}$ is a random variable conditionally dependent on $W$ according to the probability transition matrix $\mathbf{Q}$ and $W$ is drawn according to the distribution $p_W$ (which is the distribution of the message at Node 0). For simplicity, we just write $\psi\left(\mathbf{Q}\right)$ instead of $\psi\left(\mathbf{Q} \mid p_W \right)$ for the rest of the paper, assuming throughout that the specific distribution $p_W$ is used. Ultimately, our final bound is independent of $p_W$. We now have $$\begin{aligned} \mathcal{I}\left(W;\hat{W}_n\right) &=& \psi \left( \prod_{i = 1}^n \mathbf{P}_i \right).\label{eqn:psi-prod}\end{aligned}$$ Before proceeding further to bound $\mathcal{I}\left(W;\hat{W}_n\right)$, we introduce the following useful lemma: \[lem:ss-stoch-prod\] For any $\mathbf{Q}_1 \in \mathscr{S}^*_{M \times M}$ and any $\mathbf{Q}_2 \in \mathscr{S}_{M \times M}, \psi\left(\mathbf{Q}_1\mathbf{Q_2}\right) = 0$. The result follows from noting that for any $\mathbf{Q} \in \mathscr{S}^*_{M \times M}$, $\psi\left(\mathbf{Q}\right) = 0$ and $\mathbf{Q}_1 \mathbf{Q}_2 \in \mathscr{S}^*_{M \times M}$ for $\mathbf{Q}_1 \in \mathscr{S}^*_{M \times M}$ and $\mathbf{Q}_2 \in \mathscr{S}_{M \times M}$. Now for each $i$, consider $\beta_i \in \left[0,1\right]$ such that $$\begin{aligned} \mathbf{P}_i = \beta_i\mathbf{P}_{\beta_i} + \bar{\beta_i}\mathbf{P}_{\bar{\beta_i}},\label{eqn:stoch-matrix-convex-combo}\end{aligned}$$ where $\bar{\beta_i} = 1 - \beta_i$, $\mathbf{P}_{\beta_i} \in \mathscr{S}_{M \times M}$ and $\mathbf{P}_{\bar{\beta_i}} \in \mathscr{S}^*_{M \times M}$. In other words for each $i$, $\mathbf{P}_i$ be expressed as a convex combination of two row-stochastic matrices, one of them being a steady-state matrix. From (\[eqn:psi-prod\]), $$\begin{aligned} \mathcal{I}\left(W;\hat{W}_n\right) &= \psi\Big(\prod_{i = 1}^n \mathbf{P}_i\Big)\hspace{-0.5mm}=\hspace{-0.5mm} \psi\Big(\hspace{-0.5mm} \big( \beta_1 \mathbf{P}_{\beta_1} \hspace{-0.5mm}+\hspace{-0.5mm} \bar{\beta_1}\mathbf{P}_{\bar{\beta_1}} \big)\hspace{-0.5mm} \prod_{i=2}^n\hspace{-0.5mm} \mathbf{P}_i \Big)\nonumber\\ &\stackrel{(a)}{\leq} \beta_1 \psi \Big( \mathbf{P}_{\beta_1} \prod_{i=2}^n \mathbf{P}_i \Big) + \bar{\beta_1} \psi \Big( \mathbf{P}_{\bar{\beta_1}} \prod_{i=2}^n \mathbf{P}_i \Big) \nonumber\\ &\stackrel{(b)}{\leq}\beta_1 \psi \Big( \prod_{i=2}^n \mathbf{P}_i\Big),\nonumber\end{aligned}$$ where (a) follows from the convexity property of mutual information w.r.t. the probability transition function, and (b) follows from applying the data processing inequality to the first term and Lemma \[lem:ss-stoch-prod\] to the second term. By induction, we have: $$\begin{aligned} \mathcal{I}\left(W; \hat{W}_n\right) &\leq& \left( \prod_{i=1}^n \beta_i \right) \psi\left( I_{M} \right) = NR \prod_{i=1}^n \beta_i, \label{eqn:mutual-info-beta-product}\end{aligned}$$ where $I_M$ is the $M \times M$ identity matrix. The above procedure is based on a key idea developed in the proof of Theorem V.1 in [@Niesen07] in a different context. We have applied the same to facilitate a useful intermediate result given by (\[eqn:mutual-info-beta-product\]). The remaining portion of the analysis that enables us to obtain the final bound involves novel steps. We now need to determine how each $\mathbf{P}_i$ is to be split in the form given by (\[eqn:stoch-matrix-convex-combo\]) in an optimal manner, to obtain the best possible bound using this approach. Specifically, we need $\beta_i$ to be as small as possible for each $i$. Consider the following choice: $$\begin{aligned} \beta_i &=& 1 - \sum_{k = 1}^M \min_j p_{\hat{W}_i \mid \hat{W}_{i-1}} \left( k \mid j \right)\nonumber\\ P_{\bar{\beta_i}; j,k} &=& \frac{1}{1 - \beta_i} \min_{j'} p_{\hat{W}_i \mid \hat{W}_{i-1}} \left( k \mid j' \right),\label{eqn:convex-combo-params-choice}\end{aligned}$$ where $P_{\bar{\beta_i}; j,k}$ denotes the element on $j^{\text{th}}$ row and $k^{\text{th}}$ column of the matrix $\mathbf{P}_{\bar{\beta_i}}$. Note that this matrix consists of identical rows, where each entry in any row is equal to the smallest element in the corresponding column of $\mathbf{P}_i$ scaled by a normalizing factor. The other matrix $\mathbf{P}_{\beta_i}$ is determined by substituting these expressions for $\beta_i$ and $\mathbf{P}_{\bar{\beta_i}}$ into (\[eqn:stoch-matrix-convex-combo\]). Note that the two matrices $\mathbf{P}_{\beta_i}$ and $\mathbf{P}_{\bar{\beta}_i}$ determined thus will be stochastic for any $i$, and that $\beta_i \in [0,1]$. Hence, we can obtain $\mathbf{P}_i$ as a convex combination of two stochastic matrices in this manner for any $i$. The following lemma shows that the value of $\beta_i$ provided in (\[eqn:convex-combo-params-choice\]) is the best possible value for the purpose of the bound in (\[eqn:mutual-info-beta-product\]). \[lem:best-beta\] Let $\mathbf{Q} = \left[Q_{jk}\right] \in \mathscr{S}_{M \times M}$, $\mathbf{Q}_1 \in \mathscr{S}_{M \times M}, \mathbf{Q}_2 \in \mathscr{S}^*_{M \times M}$, and let $\beta \in [0,1]$ be chosen such that $\mathbf{Q} = \beta \mathbf{Q}_1 + (1-\beta)\mathbf{Q}_2$. Then, $\beta \geq 1 - \sum_{k = 1}^M \min_j Q_{jk}$. Since $\mathbf{Q} = \beta \mathbf{Q}_1 + (1-\beta)\mathbf{Q}_2$, every element of the matrix $(1-\beta) \mathbf{Q}_2$ must be smaller than the corresponding element in $\mathbf{Q}$. Consider any column $k$ of $\mathbf{Q}_2$. All the elements in that column are equal to, say, $q_k$. It then follows that $(1-\beta)q_k \leq Q_{jk}$ for every $j$, and hence $(1-\beta)q_k \leq \min_j Q_{jk}$. Summing over all $k$ and noting that $\sum_{k=1}^M q_k = 1$, we obtain the desired result. Let $\mathscr{C}_{i-1}$ be the code used by Node $i - 1$ and let $\mathscr{R}_i$ be the decision rule used by Node $i$. As per the argument above, the optimal choice of $\beta_i$ for this link will be: $$\begin{aligned} 1 - \beta_i &=& \sum_{k = 1}^M \min_j p_{\hat{W}_i \mid \hat{W}_{i-1}} \left( k \mid j \right) \nonumber \\ &=& \sum_{\mathcal{R} \in \mathscr{R}_i} \min_{\mathbf{c} \in \mathscr{C}_{i-1}} \int_{\mathcal{R}} \frac{e^{ -{\left\|\mathbf{y} - \mathbf{c}\right\|^2}/{2 \sigma_i^2 }}}{ \left( 2 \pi \sigma_i^2 \right)^{{N}/{2}} } \mathrm{d} \mathbf{y}.\end{aligned}$$ In other words, we can write $$\begin{aligned} \beta_i &=& 1 - \mu_{\sigma_i}\left(\mathscr{C}_{i-1},\mathscr{R}_i\right),\end{aligned}$$ where for any $\sigma > 0$, rate $R$ length $N$ code $\mathscr{C}$, and rate $R$ dimension $N$ decision rule $\mathscr{R}$, $$\begin{aligned} \mu_{\sigma}\left(\mathscr{C},\mathscr{R}\right) &\triangleq& \sum_{\mathcal{R} \in \mathscr{R}} \min_{\mathbf{c} \in \mathscr{C}} \int_{\mathcal{R}} \frac{e^{ -\left\|\mathbf{y} - \mathbf{c}\right\|^2 /2 \sigma^2 } }{ \left( 2 \pi \sigma^2 \right)^{N/2} } \mathrm{d}\mathbf{y}. \label{eqn:mu-defined}\end{aligned}$$ We would like to find a lower bound on $\mu_{\sigma_i} \left(\mathscr{C}_{i-1}, \mathscr{R}_i\right)$ that depends only on the parameters $N, R, P_0$ and $\sigma_0$. To do so, we need the following three lemmas. Lemma \[lem:opt-R-given-C\] removes the dependency of the bound on the choice of the decision rule. Lemma \[lem:ball-to-sphere\] shows that we can restrict our choice of codes to those having all codewords that satisfy the power constraint with equality. For this class of codes, Lemma \[lem:opt-code-config\] gives a bound in the desired form, depending solely on $N, R, P_0$ and $\sigma_0$. From now on, we denote $(2 \pi \sigma_0^2)^{\frac{N}{2}}$ by $\eta$ for brevity. \[lem:opt-R-given-C\] Let $\mathscr{C} = \left( \mathbf{c}_1, \mathbf{c}_2, \ldots, \mathbf{c}_M \right)$ be a given rate $R$ length $N$ code. Further, let $\mathscr{R}^*\left(\mathscr{C}\right)$ be the decision rule given by $\left( \mathcal{R}^*_1,\mathcal{R}^*_2,\ldots,\mathcal{R}^*_M \right)$ where for $1 \leq i \leq M$, $$\begin{aligned} \mathcal{R}^*_i &=& \left\{ \mathbf{y} \in \mathds{R}^N \mid i = \operatorname*{arg\,max}_{i'} \left\|\mathbf{y} - \mathbf{c}_{i'}\right\|\right\}.\nonumber \end{aligned}$$ Then, for any rate $R$ dimension $N$ decision rule $\mathscr{R}$, $$\begin{aligned} \mu_{\sigma}\left(\mathscr{C},\mathscr{R}\right) \geq \mu_{\sigma}\left(\mathscr{C},\mathscr{R}^*\left(\mathscr{C}\right)\right).\nonumber \end{aligned}$$ For any code $\mathscr{C}$ and decision rule $\mathscr{R}$, we have: $$\begin{aligned} \eta \mu_{\sigma}\left(\mathscr{C},\mathscr{R}\right) &= \sum_{\mathcal{R} \in \mathscr{R}} \min_j \int_{\mathcal{R}} e^{ -\frac{ \left\|\mathbf{y} - \mathbf{c}_j\right\|^2 }{ 2 \sigma^2 } } \mathrm{d}\mathbf{y} \nonumber\\ &\geq \sum_{\mathcal{R} \in \mathscr{R}} \int_{\mathcal{R}} \min_j e^{ -\frac{ \left\|\mathbf{y} - \mathbf{c}_j\right\|^2 }{ 2 \sigma^2 } } \mathrm{d}\mathbf{y}\nonumber\\ &= \int_{\mathds{R}^N} \min_j e^{ -\frac{ \left\|\mathbf{y} - \mathbf{c}_j\right\|^2 }{ 2 \sigma^2 } } \mathrm{d}\mathbf{y}\nonumber\\ &= \sum_{k = 1}^{M}\int_{\mathcal{R}^*_k} \min_{j} e^{ -\frac{ \left\|\mathbf{y} - \mathbf{c}_j\right\|^2 }{ 2 \sigma^2 } } \mathrm{d}\mathbf{y}\nonumber\\ &\stackrel{(a)}{=} \sum_{k = 1}^{M} \int_{\mathcal{R}^*_k} e^{ -\frac{ \left\|\mathbf{y} - \mathbf{c}_k\right\|^2 }{ 2 \sigma^2 } } \mathrm{d}\mathbf{y}\nonumber\\ &\stackrel{(b)}{=} \sum_{k = 1}^{M} \min_{j} \int_{\mathcal{R}^*_k} e^{ -\frac{ \left\|\mathbf{y} - \mathbf{c}_j\right\|^2 }{ 2 \sigma^2 } } \mathrm{d}\mathbf{y}\nonumber\\ &= \eta \mu_{\sigma} \left( \mathscr{C} , \mathscr{R}^* \left( \mathscr{C} \right) \right). \end{aligned}$$ Here, (a) and (b) follow from the definition of $\mathscr{R}^* \left( \mathscr{C} \right)$: for any $\mathbf{y} \in \mathcal{R}^*_k, \operatorname*{arg\,min}_{j} e^{ -\frac{ \left\|\mathbf{y} - \mathbf{c}_j\right\|^2 }{ 2 \sigma^2 } } = k$. It is useful to note that the decision rule $\mathscr{R}^* \left( \mathscr{C} \right)$ given by the above lemma is the same as the $(M - 1)^\text{th}$-order Voronoi partitioning (called “farthest-point Voronoi partitioning”, see Section 3.3 in [@Voronoi-Book]) of $\mathds{R}^N$ w.r.t. $\mathscr{C}$. \[lem:ball-to-sphere\] Let $\mathscr{C}$ be a code satisfying $\| \mathbf{c} \|^2 \leq N P_0, \forall \mathbf{c} \in \mathscr{C}$. Then, there exists a code $\mathscr{C}'$ such that $\forall \mathbf{c'} \in \mathscr{C}', \| \mathbf{c'} \|^2 = NP_0$ and $ \mu_{\sigma}\left(\mathscr{C},\mathscr{R}^*\left(\mathscr{C}\right)\right) \geq \mu_{\sigma}\left(\mathscr{C}',\mathscr{R}^*\left(\mathscr{C}'\right)\right) $. Let $\mathscr{C} = \left( \mathbf{c}_1, \ldots, \mathbf{c}_M \right)$ and let $\mathscr{R}^*\left( \mathscr{C} \right) = \left( \mathcal{R}^*_1,\ldots,\mathcal{R}^*_M\right)$. Consider a codeword that lies strictly inside the ball $\| \mathbf{x} \|^2 < NP_0$. If no such codeword exists, the statement of the lemma is trivially true with $\mathscr{C}' = \mathscr{C}$. For the non-trivial case, we can assume that such a codeword exists. Let $\mathbf{c}_{k_0}$ be that codeword. Consider the decision region $\mathcal{R}^*_{k_0} = \left\{ \mathbf{y} \in \mathds{R}^N \mid k_0 = \operatorname*{arg\,max}_{i'} \left\|\mathbf{y} - \mathbf{c}_{i'}\right\|\right\} \in \mathscr{R}^*\left( \mathscr{C} \right)$. The region $\mathcal{R}^*_{k_0}$ (if non-empty) is convex since $\mathscr{R}^*\left( \mathscr{C} \right)$ is a Voronoi tesselation of order $M-1$ and since Voronoi cells of any order are convex regions (see Property OK.1 in Section 3.2 of [@Voronoi-Book]). Hence, there exists a unique point $\mathbf{z}_{k_0}$ in $\mathcal{R}^*_{k_0}$ nearest to $\mathbf{c}_{k_0}$. By moving the codeword at $\mathbf{c}_{k_0}$ along the line joining $\mathbf{c}_{k_0}$ and $\mathbf{z}_{k_0}$ away from the latter, the distance from the codeword to every point in $\mathcal{R}^*_{k_0}$ is increased. We continue thus until the codeword is moved to the surface of the power-constraint sphere, at say $\mathbf{c'}_{k_0}$. Let us call the resulting code $\mathscr{C}_1$. Note that $\mathscr{C} \backslash \left\{\mathbf{c}_{k_0}\right\} = \mathscr{C} \backslash \left\{\mathbf{c'}_{k_0}\right\}$. Now consider $$\begin{aligned} \eta \mu_{\sigma}\left(\mathscr{C},\mathscr{R}^*\left(\mathscr{C}\right)\right) &= \sum_{k = 1}^{M} \min_{j} \int_{\mathcal{R}^*_k} {e^{ -\frac{ \left\|\mathbf{y} - \mathbf{c}_j\right\|^2 }{ 2 \sigma^2 } }}\mathrm{d}\mathbf{y}\nonumber\\ &\stackrel{(c)}{=} \sum_{k = 1}^{M}\int_{\mathcal{R}^*_k} {e^{ -\frac{ \left\|\mathbf{y} - \mathbf{c}_k\right\|^2 }{ 2 \sigma^2 } } } \mathrm{d}\mathbf{y}\nonumber\\ \begin{split}&= \sum_{\substack{k = 1 \\ k \neq k_0}}^{M}\int_{\mathcal{R}^*_k} {e^{ -\frac{ \left\|\mathbf{y} - \mathbf{c}_k\right\|^2 }{ 2 \sigma^2 } }} \mathrm{d}\mathbf{y} \\ &\quad \quad + \int_{\mathcal{R}^*_{k_0}} {e^{ -\frac{ \left\|\mathbf{y} - \mathbf{c}_{k_0}\right\|^2 }{ 2 \sigma^2 } }} \mathrm{d}\mathbf{y}\end{split}\nonumber\\ \begin{split}& \stackrel{(d)}{\geq} \sum_{\substack{k = 1 \\ k \neq k_0}}^{M}\int_{\mathcal{R}^*_k} {e^{ -\frac{ \left\|\mathbf{y} - \mathbf{c}_k\right\|^2 }{ 2 \sigma^2 } }} \mathrm{d}\mathbf{y}\\ &\quad \quad + \int_{\mathcal{R}^*_{k_0}} {e^{ -\frac{ \left\|\mathbf{y} - \mathbf{c'}_{k_0}\right\|^2 }{ 2 \sigma^2 } }} \mathrm{d}\mathbf{y}\end{split}\nonumber\\ \begin{split}&\geq \sum_{\substack{k = 1 \\ k \neq k_0}}^{M} \min_{\mathbf{c} \in \mathscr{C}_1} \int_{\mathcal{R}^*_k} {e^{ -\frac{ \left\|\mathbf{y} - \mathbf{c}\right\|^2 }{ 2 \sigma^2 } }} \mathrm{d}\mathbf{y}\\ & \quad \quad + \min_{\mathbf{c} \in \mathscr{C}_1} \int_{\mathcal{R}^*_{k_0}} {e^{ -\frac{ \left\|\mathbf{y} - \mathbf{c}\right\|^2 }{ 2 \sigma^2 } }}\mathrm{d}\mathbf{y}\end{split}\nonumber\\ &= \eta \mu_{\sigma}\left(\mathscr{C}_1,\mathscr{R}^*\left(\mathscr{C}\right)\right)\nonumber\\ &\stackrel{(e)}{\geq} \eta \mu_{\sigma}\left(\mathscr{C}_1,\mathscr{R}^*\left(\mathscr{C}_1\right)\right). \end{aligned}$$ Here, (c) follows from (b) in the proof of Lemma \[lem:opt-R-given-C\], (d) follows from the construction of $\mathbf{c'}_{k_0}$ so that for every $\mathbf{y} \in \mathcal{R}^*_{k_0}, \| \mathbf{y} - \mathbf{c}_{k_0} \| \leq \| \mathbf{y} - \mathbf{c'}_{k_0} \|$, and (e) follows from Lemma \[lem:opt-R-given-C\]. Note also that we have only treated the case where $\mathcal{R}^*_{k_0}$ is non-empty. If on the other hand, that decision region was empty, we can move the codeword at $\mathbf{c}_{k_0}$ along any arbitrary direction. For such a case inequality (b) becomes an equality since the integrals over $\mathcal{R}^*_{k_0}$ would be zero. From a given code $\mathscr{C}$, we can thus obtain a code $\mathscr{C}_1$ having one more codeword on the surface of the power-constraint sphere, also satisfying $\mu_{\sigma}\left(\mathscr{C},\mathscr{R}^*\left(\mathscr{C}\right)\right) \geq \mu_{\sigma}\left(\mathscr{C}_1,\mathscr{R}^*\left(\mathscr{C}_1\right)\right)$. We can repeat this process several times to eventually obtain a code $\mathscr{C}'$ with all codewords on the power constraint sphere. \[lem:opt-code-config\] For any rate $R$ length $N$ code $\mathscr{C}$ satisfying the power constraint $\forall \mathbf{c} \in \mathscr{C}, \| \mathbf{c} \|^2 = N P_0$, $$\mu_{\sigma}\left(\mathscr{C},\mathscr{R}^*\left(\mathscr{C}\right)\right) \geq \mathcal{Q}\left( \frac{M-1}{M} \Omega_0, N, \frac{P_0}{\sigma^2} \right).$$ For any code $\mathscr{C}$ that satisfies the requirements of the lemma, the decision regions in $\mathscr{R}^*\left( \mathscr{C} \right)$ consists of pyramids with their apex at the origin and extending out to infinity (see Appendix \[appdx:voronoi-sphere-pyramids\] for a proof). Assume that each of these regions $\left\{\mathcal{R}^*_k\right\}_{k=1}^M$ cut out a surface of area $\Omega_k$ on the unit $N$-sphere centered at the origin. Note that for each codeword $\mathbf{c}_k \in \mathscr{C}$ corresponding to message $k$, the decision region $\mathcal{R}^*_k$ contains the point $-\mathbf{c}_k$. Consider any term in the summation of the expression for $\mu_{\sigma} \left( \mathscr{C}, \mathscr{R}^*\left( \mathscr{C} \right) \right)$: $$\begin{aligned} \min_{j} \int_{\mathcal{R}^*_k} \frac{e^{ -{ \left\|\mathbf{y} - \mathbf{c}_j\right\|^2 }/{ 2 \sigma^2 } } }{ \left( 2 \pi \sigma^2 \right)^{\frac{N}{2}} }\mathrm{d}\mathbf{y} &=& \int_{\mathcal{R}^*_k} \frac{e^{ -{ \left\|\mathbf{y} - \mathbf{c}_k\right\|^2 }/{ 2 \sigma^2 } } }{ \left( 2 \pi \sigma^2 \right)^{\frac{N}{2}} }\mathrm{d}\mathbf{y}.\nonumber \end{aligned}$$ The right hand side of the above equation is equal to the probability of the event $E_1$ that the transmitted codeword in $\mathds{R}^N$ located at $\mathbf{c}_k$ on the sphere $ \| \mathbf{x} \|^2 = N P_0$ is displaced by the noise vector into a specific region $\mathcal{R}^*_k$ that contains the point $-\mathbf{c}_k$. Now consider the probability of the event $E_2$ that the same transmitted codeword is displaced into the $N$-dimensional circular cone $\mathcal{C}^*_k$ that has its apex at the origin, axis running through $-\mathbf{c}_k$, and cutting out a surface of area $\Omega_k$ on the unit sphere centered at the origin (i.e., the solid angle of the $N$-dimensional circular cone is $\Omega_k$). We claim that the probability of $E_1$ cannot be smaller than the probability of $E_2$. A proof of this claim is provided in Appendix \[appdx:pyramid-to-cone\]. The probability of the event $E_2$ is equal to $\mathcal{Q} \left( \Omega_0-\Omega_k, N, P_0/\sigma^2 \right)$, from the definition of the $\mathcal{Q}$-function in Definition \[def:functions\]. Hence, $$\begin{aligned} \mu_{\sigma}\left(\mathscr{C},\mathscr{R}^*\left(\mathscr{C}\right)\right) &=& \sum_{k = 1}^M \min_{j} \int_{\mathcal{R}^*_k} \frac{e^{{ \left\|\mathbf{y} - \mathbf{c}_j\right\|^2 }/{ 2 \sigma^2 } } }{ \left( 2 \pi \sigma^2 \right)^{N/2} }\mathrm{d}\mathbf{y} \nonumber\\ &\geq& \sum_{k = 1}^M \mathcal{Q} \left( \Omega_0-\Omega_k, N, \frac{P_0}{\sigma^2} \right).\label{eqn:q-before-jensen} \end{aligned}$$ Noting that $\mathcal{Q}$ is a convex function of $\Omega_0 - \Omega_k$ (See Section III in [@Shannon-Papers]), we apply Jensen’s inequality to (\[eqn:q-before-jensen\]): $$\begin{aligned} \mu_{\sigma}\left(\mathscr{C},\mathscr{R}^*\left(\mathscr{C}\right)\right) &\geq& M \frac{1}{M} \sum_{k = 1}^M \mathcal{Q} \left( \Omega_0-\Omega_k, N, \frac{P_0}{\sigma^2} \right)\nonumber\\ &\geq& M \mathcal{Q} \left( \frac{ \sum_{k = 1}^M \left( \Omega_0-\Omega_k \right) }{M}, N, \frac{P_0}{\sigma^2} \right) \nonumber\\ &=& M \mathcal{Q} \left( \frac{M-1}{M}\Omega_0, N, \frac{P_0}{\sigma^2}\right), \end{aligned}$$ since $\sum_{k=1}^M \Omega_k = \Omega_0$. We are now ready to prove Theorem \[thm:unif-full\]. Consider any link $i$. For the code $\mathscr{C}_{i-1}$ satisfying $\| \mathbf{c} \|^2 \leq N P_0, \forall \mathbf{c} \in \mathscr{C}_{i-1}$, we can apply Lemma \[lem:ball-to-sphere\] to construct another code $\mathscr{C}'_{i-1}$ such that $\forall \mathbf{c} \in \mathscr{C}'_{i-1}, \| \mathbf{c} \|^2 = N P_0$ and $ \mu_{\sigma_i}\left(\mathscr{C}_{i-1},\mathscr{R}^*\left(\mathscr{C}_{i-1}\right)\right) \geq \mu_{\sigma_i}\left(\mathscr{C}'_{i-1},\mathscr{R}^*\left(\mathscr{C}'_{i-1}\right)\right) $. We then have: $$\begin{aligned} \mu_{\sigma_i}\left(\mathscr{C}_{i-1},\mathscr{R}_i\right) &\stackrel{(a)}{\geq}& \mu_{\sigma_i}\left(\mathscr{C}_{i-1},\mathscr{R}^*\left(\mathscr{C}_{i-1}\right)\right)\nonumber\\ &\stackrel{(b)}{\geq}& \mu_{\sigma_i}\left(\mathscr{C}'_{i-1},\mathscr{R}^*\left(\mathscr{C}'_{i-1}\right)\right)\nonumber\\ &\stackrel{(c)}{\geq}& M \mathcal{Q} \left( \frac{M-1}{M}\Omega_0, N, \frac{P_0}{\sigma_i^2}\right)\nonumber\\ &\stackrel{(d)}{\geq}& M \mathcal{Q} \left( \frac{M-1}{M}\Omega_0, N, \frac{P_0}{\sigma_0^2}\right).\label{eqn:link-i:mu-bound-Q} \end{aligned}$$ In the above chain of equations (a), (b), and (c) follow from Lemma \[lem:opt-R-given-C\], Lemma \[lem:ball-to-sphere\] (as discussed above), and Lemma \[lem:opt-code-config\] respectively. Inequality (d) follows from Remark \[rem:Q-is-monotonic\], since $\sigma_i^2 \geq \sigma_0^2$. Recalling that $\beta_i = 1 - \mu_{\sigma_i}\left(\mathscr{C}_{i-1},\mathscr{R}_i\right)$ and applying (\[eqn:link-i:mu-bound-Q\]) to (\[eqn:mutual-info-beta-product\]), we have the desired result: $$\mathcal{I}\left(W;\hat{W}_n\right) \leq NR \left[1 - M \mathcal{Q} \left( \frac{M-1}{M}\Omega_0, N, \frac{P_0}{\sigma_0^2} \right) \right]^n.\nonumber$$ Discussion {#sec:discussion} ========== We had mentioned in Section \[sec:intro\] that the bound presented in the current paper improves and complements the bound provided by [@IZS12-Paper]. In this section, we demonstrate this fact with a comparison plot. The bound given by [@IZS12-Paper] is: $$\begin{aligned} \mathcal{I}\left( W; \hat{W}_n \right) &\leq& 2^{NR} \left( 1 - e^{-N E(P_0 / \sigma_0^2) } \right) ^{n \left(1-\epsilon\right)},\label{eqn:expo-old}\end{aligned}$$ where for any $S \geq 0$, $$\begin{aligned} E\left(S\right) &\triangleq& \frac{(S+2) + \sqrt{(S+2)^2 - 4}}{4}\nonumber\\ & & + \frac{1}{2}\log\left\{ (S+2)+\sqrt{(S+2)^2-4} \right\}.\nonumber\end{aligned}$$ for asymptotically large $n$. The above bound decays with $n$ as $\left( 1 - e^{-N E(P_0 / \sigma_0^2) } \right)$ for any code rate $R$, while the bound given by Theorem \[thm:unif-full\] decays as $\left( 1 - M\mathcal{Q}\left(\frac{M-1}{M}\Omega_0, N, \frac{P_0}{\sigma_0^2} \right) \right)$. Now, let $$E_{as}\left(R,S\right) \triangleq - \lim_{N \rightarrow \infty} \frac{\log\left(M\mathcal{Q}\left(\frac{M-1}{M}\Omega_0, N, S\right)\right)} {N}.$$ We now investigate how these two bounds compare when $N$ is asymptotically large, by comparing the values of $E(S)$ and $E_{as}\left(R,S\right)$ where $S = P_0/\sigma_0^2$. To do so we obtained $E_{as}\left(R,S\right)$ as a function of $R$ and $S$ using the asymptotic analysis in [@Shannon-Papers]. The expression for $E_{as}\left(R,S\right)$ is given below, with a justification in Appendix \[appdx:asymptotics\]: $$E_{as}\left(R,S\right) = S - \frac{1}{2}\sqrt{S} G \cos \theta - \log\left(G \sin \theta \right),$$ where $$G = \frac{1}{2} \left( \sqrt{S} \cos \theta + \sqrt{ 4 + S \cos^2 \theta }\right),$$ and $\theta = \pi - \sin^{-1} 2^{-R}$. Shown in Fig. \[fig:plot-exponent-compare\] is a plot of $E_{as}(R,S)/E(S)$ as a function of $S$, repeated for various $R$. As can be seen from the plot $E_{as}(R,S)$ is always smaller than $E(S)$ for any $R$ and $S$, thus showing that the bound obtained in Theorem \[thm:unif-full\] is tighter than the one given by (\[eqn:expo-old\]). The current bound is also seen to be better when the SNR $S$ is not very high. The Farthest-point Voronoi Tessellation for Points on a Sphere {#appdx:voronoi-sphere-pyramids} ============================================================== \[lem:voronoi-sphere-pyramids\] Given any $M, N \in \mathds{N}$, the non-empty cells in the farthest-point Voronoi tessellation of $\mathds{R}^N$ w.r.t. any set of $M$ points on an $N$-sphere of radius $A > 0$ are all semi-infinite pyramids. Consider the farthest-point Voronoi tessellation of $\mathds{R}^N$ w.r.t. a code $\mathscr{C} = \left( \mathbf{c}_1,\ldots,\mathbf{c}_M \right) \text{s.t.} \|\mathbf{c}_i\|^2 = A^2, 1 \leq i \leq M$. Consider any $\mathbf{x} \in \mathds{R}^N$ and assume without loss of generality that it is contained in $\mathcal{R}^*_1$, the farthest-point Voronoi cell for $\mathbf{c}'_1$. In that case, we have for all $i$ s.t. $ 1 \leq i \leq M$, $$\begin{aligned} \|\mathbf{c}_1 - \mathbf{x}\|^2 &\geq& \|\mathbf{c}_i - \mathbf{x}\|^2 \nonumber\\ \Rightarrow \| \mathbf{c}_1 \|^2 + \| \mathbf{x} \|^2 - 2 \langle \mathbf{c}_1,\mathbf{x} \rangle &\geq& \| \mathbf{c}_i \|^2 + \| \mathbf{x} \|^2 - 2 \langle \mathbf{c}_i,\mathbf{x} \rangle \nonumber\\ \Rightarrow - \langle \mathbf{c}_1,\mathbf{x} \rangle &\geq& - \langle \mathbf{c}_i,\mathbf{x} \rangle.\label{eqn:pyramid-proof-inner-product} \end{aligned}$$ Consider any $\alpha \geq 0$. We claim that $\alpha \mathbf{x}$ is also contained in $\mathcal{R}^*_1$. This can be shown to be true by applying (\[eqn:pyramid-proof-inner-product\]) to the expansion of $\| \mathbf{c}_1 - \alpha \mathbf{x}\|^2$: $$\begin{aligned} \| \mathbf{c}_1 - \alpha \mathbf{x}\|^2 &=& \| \mathbf{c}_1 \|^2 + \alpha^2 \| \mathbf{x} \|^2 - 2 \alpha \langle \mathbf{c}_1,\mathbf{x} \rangle \nonumber\\ &\geq& \| \mathbf{c}_i \|^2 + \alpha^2 \| \mathbf{x} \|^2 - 2 \alpha \langle \mathbf{c}_i,\mathbf{x} \rangle \nonumber\\ &=& \| \mathbf{c}_i - \alpha \mathbf{x}\|^2,\nonumber \end{aligned}$$ for all $i$ s.t. $1 \leq i \leq M$. Hence, $\alpha \mathbf{x}$ is also contained in $\mathcal{R}^*_1$. Generalizing this, we have shown that any non-empty Voronoi cell that contains a point $\mathbf{x}$ also contains the point $\alpha \mathbf{x}$ for any $\alpha \geq 0$. Such a region is a semi-infinite pyramid by definition. Proof of the claim in Lemma \[lem:opt-code-config\] {#appdx:pyramid-to-cone} =================================================== Consider the $N-1$ dimensional cross-section of the pyramid $\mathcal{R}^*_k$ cut out by a sphere of radius $R$ centered at the origin. This will be an arbitrary spherical polygon. The cross-section of the cone $\mathcal{C}^*_k$ by the same sphere will be a spherical cap with its center at $-\mathbf{c}_k$. The axis of the cone cuts through the spherical cap at its center. The non-overlapping regions of such a spherical cap and a polygon are illustrated in Fig. \[fig:pyramid-to-cone\]. Since the both the cross sections have the same surface area $R^N \Omega_k$, the surface areas of the non-overlapping parts of both the cross-sections (indicated as $A_1$ and $A_2$ and by two different shadings in Fig. \[fig:pyramid-to-cone\]) are equal. Now, every point in the shaded region $A_1$ on the polygon is nearer to $\mathbf{c}_k$ than any point in $A_2$ is to $\mathbf{c}_k$. This is because the former lies outside the spherical cap centered at $-\mathbf{c}_k$ while the latter is inside the same. This in turn implies that the angle $\theta_2 \in [0, \pi]$ between the axis of the cone and the line joining any point $\mathbf{y}_2$ on $A_2$ and the origin is smaller than the angle $\theta_1 \in [0, \pi]$ between the axis and the line joining any point $\mathbf{y}_1$ on $A_1$ and the origin, as shown in the right hand side of Fig. \[fig:pyramid-to-cone\]. This in turn implies that $\mathbf{y}_1$ is closer to $\mathbf{c}_k$ than $\mathbf{y}_2$ is to $\mathbf{c}_k$, as shown below: $$\begin{aligned} \| \mathbf{c}_k - \mathbf{y}_1 \|^2 &=& N P_0 + R^2 + 2 R \sqrt{N P_0} \cos{\theta_1}\nonumber\\ & \leq & N P_0 + R^2 + 2 R \sqrt{N P_0} \cos{\theta_2} \nonumber\\ &=& \| \mathbf{c}_k - \mathbf{y}_2 \|^2.\nonumber\end{aligned}$$ This in turn means that the integral of the density function of the Gaussian noise vector with center at $\mathbf{c}_k$ over the volume $\mathcal{R}^*_k$ is greater than the integral over the volume $\mathcal{C}^*_k$. The former is the probability of the event $E_1$ and the latter is the probability of the event $E_2$. {width="66.00000%"} Asymptotic exponential decay of the $\mathcal{Q}$ function with $N$ {#appdx:asymptotics} =================================================================== Though it is hard to express $\mathcal{Q}$ in terms of elementary functions, it is easy to obtain asymptotic approximations when the block length $N$ is very large. The idea is to use Shannon’s computation of the sphere-packing exponent. Shannon derives a bound on $\mathcal{Q}\left(.\right)$ as a function of the cone angle $\theta \in [0,\pi]$ instead of the solid angle $\Omega$ since this makes asymptotic analysis easier. This results in a bound for $\mathcal{Q}\left(.\right)$ that decays exponentially in $N$, with the exponent being $$E_L\left(\theta\right) = \frac{P_0}{2\sigma_0^2} - \frac{1}{2}\sqrt{ \frac{P_0}{\sigma_0^2} } G \cos \theta - \log\left(G \sin \theta \right),$$ where $$G = \frac{1}{2} \left( \sqrt{ \frac{P_0}{\sigma_0^2} } \cos \theta + \sqrt{ 4 + \frac{P_0}{\sigma_0^2} \cos^2 \theta }\right).$$ The bound on $\mathcal{Q}\left(.\right)$ for a given $\Omega$ can then be evaluated numerically or by any other means, since there is a one-to-one correspondence between the cone angle $\theta_0$ and the solid angle $\Omega$ (see Fig. \[fig:cone-solid\]): $$\Omega\left(\theta_0\right) = \frac{ (N-1) \pi^{ \frac{N-1}{2} } }{ \Gamma\left( \frac{N+1}{2} \right) } \int_0^{\theta_0} \left( \sin \theta_0 \right)^{N-2} \mathrm{d} \theta_0.$$ The particular case of interest in [@Shannon-Papers] is $\mathcal{Q} \left( \frac{1}{M}\Omega_0, N, \frac{P_0}{\sigma_0^2} \right)$, which corresponds to the cone angle $\theta = \sin^{-1} 2^{-R}$ and the sphere-packing lower bound is obtained thus (see pages 620 and 625 in [@Shannon-Papers]). Our bound involves $\mathcal{Q} \left( \frac{M-1}{M}\Omega_0, N, \frac{P_0}{\sigma_0^2} \right)$ instead, and hence we will have to evaluate the exponent $E_L\left(\theta\right)$ with $\theta = \pi - \sin^{-1} 2^{-R}$ instead, giving us the following result: ![Relation between solid angle and cone angle.[]{data-label="fig:cone-solid"}](cone-solid.eps){width="0.40\columnwidth"} $$\begin{aligned} \mathcal{I}\left(W;\hat{W}_n\right) \lessapprox \left( 1 - e^{- N E_{as}\left(R,P_0/\sigma_0^2\right)} \right)^n,\end{aligned}$$ where $E_{as}\left(R,S\right)$ is as shown in (\[eqn:E\_as-appdx\]). $$\begin{aligned} E_{as}\left(R,S\right) &=& \frac{S}{2^{2R + 2}} \left( (2^{2R} + 1) + (2^{2R} - 1) \sqrt{ 1 + \frac{2^{2R+2}}{\left(2^{2R} - 1\right) S} }\right)\nonumber\\ & & \ \ \ \ \ + \frac{1}{2} \log \left[ 2^{2R} + \frac{S}{2} (2^{2R} - 1) \left( \sqrt{ 1 + \frac{2^{2R+2}}{\left(2^{2R} - 1\right) S} } + 1\right) \right] - R \log 2.\label{eqn:E_as-appdx}\end{aligned}$$

-0.5cm -0.5cm INR-0904/95\ October 1995\ hep-ph/9510247 [**BFKL QCD Pomeron in High Energy Hadron Collisions\ and Inclusive Dijet Production**]{} [^1]\ \ [^2]\ [^3] [**Abstract**]{} We calculate inclusive dijet production cross section in high energy hadron collisions within the BFKL resummation formalism for the QCD Pomeron. We take into account the Pomerons which are adjacent to the hadrons. With these adjacent Pomerons we define a new object — the BFKL structure function of hadron — which enables one to calculate the inclusive dijet production for any rapidity intervals. We present predictions for the dijet K-factor and azimuthal angle decorrelation. Estimations for some NLO BFKL corrections are also given. At present, much attention is being paid to the perturbative QCD Pomeron obtained by Balitsky, Fadin, Kuraev and Lipatov (BFKL) [@Lip76]. One of the reasons is that it relates hard processes ($ -t = Q^2 \gg {\Lambda^2_{QCD}}$) and semi-hard ones ($s \gg -t = Q^2 \gg {\Lambda^2_{QCD}}$): It sums up leading energy logarithms of perturbative QCD into a singularity in the complex angular momentum plane. Several proposals to find direct manifestations of the BFKL Pomeron are available in the literature, see, e.g., Refs. \[2-4\], but it is still difficult to get the necessary experimental data. In this presentation we outline, within the BFKL approach, the inclusive dijet cross section in high energy hadron collisions without any restrictions on untagging jets [@Kim95]. Our goal is to push further towards the existing experimental conditions the BFKL Pomeron predictions. Removing the restriction on tagging jets to be most forward/backward, which was imposed in the previous studies, one should take into account additional contributions to the cross section with jets more rapid than the tagging ones. There are three such contributions: two with a couple of Pomerons (Figs. 1(b),1(c)) and one with three (Fig. 1(d)). We will call the Pomerons developing between colliding hadrons and their descendant jets the adjacent Pomerons and the Pomeron developing between the tagging jets the inner Pomeron. These additional contributions contain extra power of $\alpha_S$ per extra Pomeron but hardly could they be regarded as corrections since they are also proportional to a kinematically dependent factor which one can loosely treat as the number of partons in the hadron moving faster than the descendant tagging jet. Mueller and Navelet result [@Mue87] for contribution to the cross section of Fig. 1(a) could be recast [@Kim95] as $$\begin{aligned} &&\frac{x_1x_2d\sigma_{\{P\}}}{dx_1dx_2d^{2}k_{1 \perp}d^{2}k_{2 \perp}} = \frac{\alpha_{S} C_A}{k^2_{1 \perp}}\frac{\alpha_{S} C_A}{k^2_{2 \perp}} \times \nonumber \\ &&\sum_n\int d\nu x_1 F_A(x_1,\mu^2_1) \left[ \chi_{n,\nu}(k_{1 \perp}) e^{y\omega(n,\nu)} \chi_{n,\nu}^*(k_{2 \perp}) \right] x_2 F_B(x_2,\mu^2_2). \label{cast}\end{aligned}$$ where the subscript on $\sigma_{\{P\}}$ labels the contribution to the cross section as a single inner Pomeron; $C_A=3$ is a color group factor; $x_i$ are the longitudinal momentum fractions of the tagging jets; $k_{i\perp}$ are the transverse momenta; $xF_{A,B}$ are the effective structure functions of colliding hadrons; $y = \ln(x_1x_2s/k_{1\perp}k_{2\perp})$ is the relative rapidity of tagging jets; $\chi_{n,\nu}(k_{\perp})=\frac{(k_{\perp}^2)^{-\frac{1}{2}+i\nu} e^{in\phi}}{2\pi} $ are Lipatov’s eigenfunctions and $\omega(n,\nu) = \frac{2 \alpha_{S} C_A}{\pi} \biggl[ \psi(1) - Re \, \psi \biggl( \frac{|n|+1}{2} + i\nu \biggr) \biggr]$ are Lipatov’s eigenvalues. Here $\psi$ is the logarithmic derivative of Euler Gamma-function. As we have shown [@Kim95], subprocesses of Fig. 1(b)-1(d) with the adjacent Pomerons contribute to the effective structure functions, i.e., one can account for them by just adding some “radiation corrections” to the structure functions of Eq.(\[cast\]): $$\begin{aligned} x F_A(x_1,\mu^2_1) & \Rightarrow &x \Phi_{A}(x_1,\mu^2_1,n,\nu,k_{1\perp}) \equiv x F_A(x_1,\mu^2_1) + x D_{A}(x_1,\mu^2_1,n,\nu,k_{1\perp}), \nonumber \\ x F_B(x_2,\mu^2_2)&\Rightarrow& x \Phi_{B}^{\ast}(x_2,\mu^2_2,n,\nu,k_{2\perp}) \equiv x F_B(x_2,\mu^2_2) + x D_{B}^{\ast}(x_2,\mu^2_2,n,\nu,k_{2\perp}), \nonumber \\ && \label{subs}\end{aligned}$$ where $x \Phi_{A,B}$ are the new structure functions that depend on Lipatov’s quantum numbers $(n,\nu)$ — we call them BFKL structure functions; the complex conjugation on $\Phi_B$ could be understood if one look at rhs of Eq. (\[cast\]) as a matrix element of a $t$-channel evolution operator with the relative rapidity, $y$, as an evolution parameter and $F_B$ as a final state; $(n,\nu)$ are then “good quantum numbers” conserved under the evolution—this makes room for $(n,\nu)$-dependence of the corrected structure functions. We note also that the corrected structure functions may depend on the transverse momenta of the tagging jets. An explicit expression for the radiation correction $D_{A,B}$ to the effective hadron structure functions see in Ref. [@Kim95]. Eq. (\[cast\]) with the substitution (\[subs\]), makes possible to get updated predictions for the $K$-factor and the azimuthal angle decorrelation of $x$-symmetric ($x_1=x_2$) dijets on an effective relative rapidity $y^{\ast}\equiv\ln({x_1x_2s}/{k_{\perp min}^2})$ (see Figs. 2,3, where the LO CTEQ3L structure functions [@Lai94] have been used). A look at the plots brings a conclusion that the adjacent Pomerons may play a decisive role in the high energy hadron collisions. We note also that one should not stick anymore to the large dijet relative rapidity region in the BFKL Pomeron manifestations hunting, since, from the one hand, we include the region of the moderate rapidity intervals into our consideration and, from the other hand, the resummation effects are quite pronounced at the moderate rapidity region. We present also in Figs. 2,3 estimations for NLO BFKL effects using the results of Ref. [@Cor95], where conformal NLO contributions to the Lipatov’s eigenvalues were calculated. The estimations incorporate the NLO conformal corrections to the Lipatov’s eigenvalues (see Fig. 4) and the NLO CTEQ3M structure functions [@Lai94]. We should note here that the extraction of data on high-$k_{\perp}$ jets from the event samples in order to compare them with the BFKL Pomeron predictions should be different from the algorithms directed to a comparison with perturbative QCD predictions for the hard processes. These algorithms, motivated by the strong $k_{\perp}$-ordering of the hard QCD regime, employ hardest-$k_{\perp}$ jet selection (see, e.g., Ref. [@Alg94]). It is doubtful that one can reconcile these algorithms with the weak $k_{\perp}$-diffusion and the strong rapidity ordering of the semi-hard QCD regime, described by the BFKL resummation. We also note that our predictions should not be compared with the preliminary data [@Heu94] extracted by the most forward/backward jet selection criterion. Obviously, one should include for tagging all the registered pairs of jets (not only the most forward–backward pair) to compare with our predictions. In particular, to make a comparison with Figs. 2,3, one should sum up all the registered $x$-symmetric dijets ($x_1=x_2$) with transverse momenta harder than $k_{\perp min}$. We thank E.A.Kuraev and L.N.Lipatov for stimulating discussions. We are grateful to A.J.Sommerer, J.P.Vary, and B.-L.Young for their kind hospitality at the IITAP, Ames, Iowa and support. V.T.K. is indebted to S.Ahn, C.L.Kim, T.Lee, A.Petridis, J.Qiu, C.R.Schmidt, S.I.Troyan, and C.P.Yuan for helpful conversations. V.T.K. also thanks the Fermilab Theory Division for hospitality. G.B.P. wishes to thank F.Paccanoni for fruitful discussions and hospitality at the Padova University. [10]{} L.N.Lipatov, Yad. Fiz. [**23**]{}, 642 (1976) \[Sov. J. Nucl. Phys. [**23**]{}, 338 (1976)\];\ E.A.Kuraev, L.N.Lipatov and V.S.Fadin, Zh. Eksp. Teor. Fiz. [**71**]{}, 840 (1976) \[Sov. JETP [**44**]{}, 443 (1976)\]; [*ibid.*]{} [**72**]{}, 377 (1977) \[[**45**]{}, 199 (1977)\];\ Ya.Ya.Balitsky and L.N.Lipatov, Yad. Fiz. [**28**]{}, 1597 (1978) \[Sov. J. Nucl. Phys. [**28**]{}, 822 (1978)\];\ L.N.Lipatov, Zh. Eksp. Teor. Fiz. [**90**]{}, 1536 (1986) \[Sov. JETP [**63**]{}, 904 (1986)\] M.G.Ryskin, Yad. Fiz. [**32**]{}, 259 (1980) \[Sov. J. Nucl. Phys. [**32**]{}, 133 (1980)\];\ E.M.Levin and M.G.Ryskin, Yad. Fiz. [**32**]{}, 802 (1980) \[Sov. J. Nucl. Phys. [**32**]{}, 413 (1980)\];\ L.L.Frankfurt and M.I.Strikman, Phys. Rev. Lett. [**63**]{}, 1914 (1989);\ J.Bartels, A.De Roeck and M.Loewe, Z. Phys. C [**54**]{}, 635 (1992);\ J.Kwiecinski, A.D.Martin and P.J.Sutton, Phys. Rev. D [**46**]{}, 921 (1992);\ A.H.Mueller and W.K.Tang, Phys. Lett. [**B284**]{}, 123 (1992);\ V.Del Duca, M.E.Peskin and W.K.Tang, Phys. Lett. [**B306**]{}, 151 (1993);\ N.N.Nikolaev, B.G.Zakharov and V.R.Zoller, Zh. Eksp. Teor. Fiz. [**105**]{}, 1498 (1994) \[Sov. JETP [**78**]{}, 806 (1994)\];\ R.Ragazzon and D.Treleani, UTS-DFT-95-06, Trieste (1995);\ I.Bojak and M.Ernst, DO-TH 95/14, Dortmund (1995) A.H.Mueller and H.Navelet, Nucl. Phys. [**B282**]{}, 727 (1987) V.Del Duca and C.R.Schmidt, Phys. Rev. D [**49**]{}, 4510 (1994);\ V.Del Duca and C.R.Schmidt, Phys. Rev. D [**51**]{}, 4510 (1995);\ DESY 94-128, SCIPP 94/20 (1994); DESY 94-163, SCIPP 94/27 (1994);\ W.J.Stirling, Nucl. Phys. [**B423**]{}, 56 (1994) V.T.Kim and G.B.Pivovarov, IITAP-95-01, Ames (1995); INR-0891/95, Moscow (1995); hep-ph/9506381 H.L.Lai et al., Phys. Rev. D [**51**]{}, 4763 (1995) C.Coriano and A.R.White, Phys. Rev. Lett. [**74**]{}, 4980 (1995); ANL-HEP-PR-95-12, Argonne (1995) A.Biatti for CDF Collaboration, [*XX ICHEP*]{} paper GLS 0368, Glasgow, Aug. 1994;\ F.Nang for D$\emptyset$ Collaboration, Proceedings of APS Division of Particle and Fields, Albuquerqe, N.M. 1994 D$\emptyset$ Collaboration, A.Brandt, [*Jet Production at Large Rapidity Intervals*]{}, talk presented at [*XX ICHEP*]{}, Glasgow, Aug. 1994;\ T.Heuring, [*Jets with Large Rapidity Separation*]{}, talk presented at APS DPF Meeting, Albuquerqe 1994;\ C.L.Kim, [*Jet Production at Large Rapidity Intervals*]{}, talk presented at Small-x Workshop, Fermilab, Sept. 1994 Figure Captions {#figure-captions .unnumbered} =============== Fig. 1: Subprocesses for the dijet production in hadron collision. Fig. 2: The $y^{\ast}$-dependence of the dijet K-factor. Fig. 3: The $y^{\ast}$-dependence of the average azimuthal angle cosine between the\ tagging jets. Fig. 4: The Lipatov’s eigenvalues at $n=0$. [^1]: To appear in [*the Proceedings of the Workshop on Particle Theory and Phenomenology*]{}, the International Institute of Theoretical and Applied Physics, Ames, Iowa, May 1995 [^2]: *e-mail: $kim@fnpnpi.pnpi.spb.ru$* [^3]: *e-mail: $gbpivo@ms2.inr.ac.ru$*

--- abstract: 'We study finite dimensional *almost* and *quasi-effective* prolongations of nilpotent $\operatorname{\mathbb{Z}}$-graded Lie algebras, especially focusing on those having a decomposable reductive structural subalgebra. Our assumptions generalize *effectiveness* and *algebraicity* and are appropriate to obtain Levi-Malčev and Levi-Chevalley decompositions and precisions on the heigth and other properties of the prolongations in a very natural way. In a last section we systematically present examples in which simple Lie algebras are obtained as prolongations, for reductive structural algebras of type A, B, C and D, of nilpotent $\operatorname{\mathbb{Z}}$-graded Lie algebras arising as their linear representations.' address: - | Stefano Marini: Dipartimento di Scienze Matematiche, Fisiche e Informatiche\ Università di Parma\ Parco Area delle Scienze 53/A (Campus), 43124 Parma (Italy) - | Costantino Medori: Dipartimento di Scienze Matematiche, Fisiche e Informatiche\ Università di Parma\ Parco Area delle Scienze 53/A (Campus), 43124 Parma (Italy) - | Mauro Nacinovich: Dipartimento di Matematica\ II Università di Roma “Tor Vergata”\ Via della Ricerca Scientifica\ 00133 Roma (Italy) author: - 'S.Marini, C.Medori, M.Nacinovich' bibliography: - 'homog.bib' title: 'On some classes of $\operatorname{\mathbb{Z}}$-graded Lie algebras' --- Introduction {#introduction .unnumbered} ============ Cartan’s method for the study of equivalence and symmetries of differential $\operatorname{\mathbf{G}}$-structures naturally leads to consider $\operatorname{\mathbb{Z}}$-graded Lie algebras and their prolongations. This approach is clearly explained in the classical books [@Sternberg] of Sternberg and [@Kob] of Kobayashi. The work of N. Tanaka (see e.g. [@Tan67; @Tan70]), extending the scope to general contact and $CR$ structures, set the path for further developments of the subject (see e.g. [@AlSp; @MMN2018; @Ottazzi2011]). An additional motivation is the fact that filtered Lie algebras are the core of the algebraic model for transitive differential geometry (see [@GS]). Their $\operatorname{\mathbb{Z}}$-graded associated objects have therefore an essential role in the study of several differential geometrical structures. Our interest in this topic was fostered by our previous work on homogeneous $CR$ manifolds (se e.g. [@AMN06; @AMN06b; @AMN10b; @AMN2013; @LN05; @LN08; @MaNa1; @MaNa2]). The more recent [@NMSM] showed us that some of the $\operatorname{\mathbb{Z}}$-graded Lie algebra naturally arising in this context do not satisfy all standard requirements of Tanaka’s theory under which e.g. are usually discussed effective maximal prolongations (cf. [@Ottazzi2011; @Warhurst2007]). This motivates our consideration of fairly general classes of $\operatorname{\mathbb{Z}}$-graded Lie algebras. Our main concern here is not on the maximality of prolongations, for which we refer throughout to our [@MMN2018]. Instead, we often restrict to prolongations that are *assumed* to be finite dimensional. Since in this case homogeneous terms of degree different from zero are $\operatorname{\mathrm{ad}}$-nilpotent, we keep most of the structure of a $\operatorname{\mathbb{Z}}$-graded Lie algebra while passing, following [@Bou82 Ch.VII,§[5]{}], to its *decomposable envelope*. This observation was very useful to clarify and simplify several points of the theory. We found also convenient to introduce weaker assumptions of effectiveness which are nevertheless sufficient to get informations on the positive degree homogeneous summands. Let us briefly summarise the contents of the paper. In the first section we collect some general structure property of $\operatorname{\mathbb{Z}}$-graded Lie algebras. As explained above, a key role is played by the concept of decomposability, which generalizes algebraicity and which is used here also to obtain a shorter proof of the existence of $\operatorname{\mathbb{Z}}$-graded Levi-Malčev and Levi-Chevalley decompositions (cf. [@CC2017; @MN02]). In §\[sec-effect\] we consider various effectiveness conditions, weakening those in [@Tan67], and some of their consequences. We call the subalgebra ${\mathfrak{g}}_{0}$ of $0$-degree homogeneous terms of a $\operatorname{\mathbb{Z}}$-graded Lie algebra ${\mathfrak{g}}$ its *type*, or *structural subalgebra*. We look at ${\mathfrak{g}}_{0}$ as infinitesimally describing the basic symmetries of the structure of a differential geometrical object under consideration. Starting from §\[sec-reduc\], we study the consequences of assuming that the adjoint action of ${\mathfrak{g}}_{0}$ on ${\mathfrak{g}}$ is reductive. In §\[sec-quasi\] we come back to the effectiveness conditions, showing that, together with assumptions on the type, they bring some further precision on the features of the $\operatorname{\mathbb{Z}}$-graded Levi-Malčev decomposition. In the last §\[sect9\] we deal with semisimple prolongations. After some general considerations, we systematically develop a series of examples, also relating to the exceptional Lie algebras and in which spin representations play an important role. We believe that some of them could also be of some interest in physics, where graded Lie algebras may contribute to better understand basic symmetries of nature. For the nilpotent depth $2$ case some similar descriptions were obtained in [@Mor2018]. $\operatorname{\mathbb{Z}}$-graded Lie alebras {#struct} ============================================== In this preliminary section we fix some notation and definitions that will be used throughout the paper. We consider $\operatorname{\mathbb{Z}}$-graded Lie algebras $$\label{e8.1} {\mathfrak{g}}\,{=}\,{\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}{\mathfrak{g}}_{{\mathpzc{p}}}$$ over a field ${\mathbb{K}}$ of characteristic $0.$ The Lie subalgebra ${\mathfrak{g}}_{0}$ of the $\operatorname{\mathbb{Z}}$-graded Lie algebra is called its *structure subalgebra*. The *depth* of ${\mathfrak{g}}$ is $\muup{=}\sup\{{\mathpzc{p}}\,{\in}\,\operatorname{\mathbb{Z}}\,{\mid}\, {\mathfrak{g}}_{-{\mathpzc{p}}}{\neq}0\}$ end its *heigth* is $\nuup{=}\sup\{{\mathpzc{p}}\,{\in}\,\operatorname{\mathbb{Z}}\,{\mid}\, {\mathfrak{g}}_{{\mathpzc{p}}}{\neq}0\}.$ The map $$\label{char} D_{E}:{\mathfrak{g}}\to{\mathfrak{g}},\;\;\text{with}\;\; D_{E}(X_{{\mathpzc{p}}}) ={\mathpzc{p}}\cdot{X}_{{\mathpzc{p}}},\;\;\forall{\mathpzc{p}}\in\operatorname{\mathbb{Z}},\;\;\forall X_{{\mathpzc{p}}}\in{\mathfrak{g}}_{{\mathpzc{p}}}$$ is a degree $0$ derivation of ${\mathfrak{g}},$ that we call *characteristic*. We call *characteristic* a $\operatorname{\mathbb{Z}}$-graded Lie algebra ${\mathfrak{g}}$ which contains a homogeneous element $E$ of degree $0$ (its *characteristic element*) such that $[E,X]=D_{E}(X)$ for all $X\,{\in}\,{\mathfrak{g}}.$ A $\operatorname{\mathbb{Z}}$-graded Lie algebra ${\mathfrak{g}}'{=}{\sum}_{{{\mathpzc{p}}}{\in}\operatorname{\mathbb{Z}}}{\mathfrak{g}}'_{{\mathpzc{p}}}$ is called a *prolongation* of the $\operatorname{\mathbb{Z}}$-graded Lie algebra ${\mathfrak{g}}$ of if ${\mathfrak{g}}$ is a Lie subalgebra of ${\mathfrak{g}}'$ and $${\mathfrak{g}}'_{{\mathpzc{p}}}={\mathfrak{g}}_{{\mathpzc{p}}},\;\forall{\mathpzc{p}}\,{<}\,0,\;\;\; {\mathfrak{g}}'_{{\mathpzc{p}}}\supseteqq{\mathfrak{g}}_{{\mathpzc{p}}},\;\forall{\mathpzc{p}}\,{\geq}\,0.$$ We say that a $\operatorname{\mathbb{Z}}$-graded prolongation ${\mathfrak{g}}'$ is *of type ${\mathfrak{g}}_{0}$* if ${\mathfrak{g}}'_{0}\,{=}\,{\mathfrak{g}}_{0}.$ For a thorough discussion of prolongations of $\operatorname{\mathbb{Z}}$-graded Lie algebras we refer the reader to [@MMN2018]. \[lm8.1\] A $\operatorname{\mathbb{Z}}$-graded Lie algebra ${\mathfrak{g}}$ admits a prolongation ${\mathfrak{g}}\,{\hookrightarrow}\,{\mathfrak{g}^{c}},$ with ${\mathfrak{g}^{c}}{=}{\mathfrak{g}}$ if ${\mathfrak{g}}$ is characteristic and $\dim_{{\mathbb{K}}}({\mathfrak{g}^{c}}/{\mathfrak{g}})\,{=}\,1$ otherwise. The adjoint representation of ${\mathfrak{g}^{c}}$ restricts to a representation of ${\mathfrak{g}}$ on ${\mathfrak{g}^{c}},$ whose kernel is the centralizer of ${\mathfrak{g}}$ in ${\mathfrak{g}}_{0}$: $$\label{8-3eq} \operatorname{\mathfrak{c}}_0=\{A\in{\mathfrak{g}}_0\mid [A,{\mathfrak{g}}]\,{=}\,\{0\}\}.$$ If ${\mathfrak{g}}$ is not characteristic, we may consider ${\mathfrak{g}^{c}}={\mathfrak{g}}\,{\oplus}\,\langle{E}\rangle$, with the gradation ${\mathfrak{g}^{c}}{=}{\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}{\mathfrak{g}^{c}}_{{\mathpzc{p}}}$ in which ${\mathfrak{g}^{c}}_{{\mathpzc{p}}}{=}{\mathfrak{g}}_{{\mathpzc{p}}}$ when ${\mathpzc{p}}{\neq}0,$ ${\mathfrak{g}^{c}}_{0}\,{=}\,{\mathfrak{g}}_{0}\,{\oplus}\,\langle{E}\rangle$ and the Lie algebra structure on ${\mathfrak{g}^{c}}$ is defined by requiring that ${\mathfrak{g}}$ is an ideal in ${\mathfrak{g}^{c}}$ and that $[E,X]\,{=}\,D_{E}(X)$ for all $Z\,{\in}\,{\mathfrak{g}}.$ Formula follows because the elements of the center of a characteristic $\operatorname{\mathbb{Z}}$-graded Lie algebra are homogeneous of degree zero. The $\operatorname{\mathbb{Z}}$-graded Lie algebra ${\mathfrak{g}^{c}}$ of Lemma \[lm8.1\] will be called the *characteristic prolongation* of ${\mathfrak{g}}.$ Finite dimensional $\operatorname{\mathbb{Z}}$-graded Lie algebras {#secnot} ------------------------------------------------------------------ Assuming that ${\mathfrak{g}}$ is finite dimensional, we denote by ${\mathfrak{r}}$ its solvable radical and by ${\mathfrak{n}}$ its maximal nilpotent ideal. Both ${\mathfrak{r}}$ and ${\mathfrak{n}}$ are graded ideals of ${\mathfrak{g}}$: $$\label{8.2} {\mathfrak{n}}={\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}{\mathfrak{n}}_{{\mathpzc{p}}} \subseteq{\mathfrak{r}}={\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}{\mathfrak{r}}_{{\mathpzc{p}}}$$ and ${\mathfrak{n}}$ consists of the $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}$-nilpotent elements of ${\mathfrak{r}}.$ In particular, $$\label{8.3} \begin{cases} {\mathfrak{n}}_{{\mathpzc{p}}}={\mathfrak{r}}_{{\mathpzc{p}}}, & \forall {\mathpzc{p}}\neq{0},\\ {\mathfrak{n}}_{0}\subseteq{\mathfrak{r}}_{0} \end{cases}$$ and any derivation $D$ of ${\mathfrak{g}}$ maps ${\mathfrak{r}}$ into ${\mathfrak{n}}.$ Set $${\textswab{m}}={\sum}_{{\mathpzc{p}}<0}{\mathfrak{g}}_{{\mathpzc{p}}}, \quad {\mathfrak{g}}_{+}={\sum}_{{\mathpzc{p}}{\geq}0}{\mathfrak{g}}_{{\mathpzc{p}}},$$ and denote by ${\textsf{k}}_{{\mathfrak{g}}}$ the Killing form of ${\mathfrak{g}}.$ We recall, from [@n1998lie Ch.I,§[5]{}.5,Prop.5], that the radical ${\mathfrak{r}}$ is the orthogonal of $[{\mathfrak{g}},{\mathfrak{g}}]$ with respect to ${\textsf{k}}_{{\mathfrak{g}}}.$ $\operatorname{\mathbb{Z}}$-graded linear representations --------------------------------------------------------- A $\operatorname{\mathbb{Z}}$-graded ${\mathbb{K}}$-vector space is the datum of a ${\mathbb{K}}$-vector space ${\textsf{V}}$ and of its decomposition $$\label{eq8.12} {\textsf{V}}={\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}{\textsf{V}}_{{\mathpzc{p}}},$$ into a direct sum of vector subspaces, indexed by the integers. A linear map $\phiup\,{\in}\,{\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})$ is called *homogeneous of degree ${\mathpzc{p}}$* if $$\phiup(V_{{\mathpzc{q}}})\subseteq{\textsf{V}}_{{\mathpzc{q}}+{\mathpzc{p}}},\;\;\forall{\mathpzc{q}}\in\operatorname{\mathbb{Z}}.$$ The homogeneous elements of ${\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})$ generate a Lie subalgebra ${\mathfrak{gl}^{\vee}}_{{\mathbb{K}}}({\textsf{V}})$ of ${\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}}),$ which has the natural $\operatorname{\mathbb{Z}}$-gradation $$\label{eqgrad} \begin{cases} {\mathfrak{gl}^{\vee}}_{{\mathbb{K}}}({\textsf{V}})={\sum}_{{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}}[{\mathfrak{gl}^{\vee}}_{{\mathbb{K}}}({\textsf{V}})]_{{\mathpzc{p}}}, \;\;\text{with}\;\;\\ [{\mathfrak{gl}^{\vee}}_{{\mathbb{K}}}({\textsf{V}})]_{{\mathpzc{p}}}=\{A\in{\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})\mid A({\textsf{V}}_{{\mathpzc{q}}}) \subseteq{\textsf{V}}_{{\mathpzc{q}}{+}{\mathpzc{p}}},\;\forall {\mathpzc{q}}\in\operatorname{\mathbb{Z}}\}. \end{cases}$$ If the $\operatorname{\mathbb{Z}}$-gradation of ${\textsf{V}}$ is finite, i.e. when the set of ${\mathpzc{p}}\,{\in}\operatorname{\mathbb{Z}}$ with ${\textsf{V}}_{{\mathpzc{p}}}\,{\neq}\,\{0\}$ is finite, then ${\mathfrak{gl}^{\vee}}_{{\mathbb{K}}}({\textsf{V}})\,{=}\,{\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}}).$ We keep however also in this case the notation ${\mathfrak{gl}^{\vee}}_{{\mathbb{K}}}({\textsf{V}})$ to specify that we are considering on ${\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})$ the $\operatorname{\mathbb{Z}}$-gradation , related to a given $\operatorname{\mathbb{Z}}$-gradation of ${\textsf{V}}.$ We denote by $E_{{\textsf{V}}}$ the map in ${\mathfrak{gl}^{\vee}}_{{\mathbb{K}}}({\textsf{V}})$ with $E_{{\textsf{V}}}({\mathpzc{v}}_{{\mathpzc{p}}})={\mathpzc{p}}\,{\cdot}\,{\mathpzc{v}}_{{\mathpzc{p}}}$ for ${\mathpzc{p}}\,{\in}\operatorname{\mathbb{Z}}$ and ${\mathpzc{v}}_{{\mathpzc{p}}}\,{\in}\,{\textsf{V}}_{{\mathpzc{p}}}.$ A $\operatorname{\mathbb{Z}}$-graded *linear representation* of ${\mathfrak{g}}$ is a linear representation $\rhoup\,{:}\,{\mathfrak{g}}\,{\to}\,{\mathfrak{gl}^{\vee}}_{{\mathbb{K}}}({\textsf{V}})$ of ${\mathfrak{g}}$ on a $\operatorname{\mathbb{Z}}$-graded vector space ${\textsf{V}},$ such that $$\label{qq8.11} \rhoup(X_{\mathpzc{p}})({\textsf{V}}_{{\mathpzc{q}}})\subseteq{\textsf{V}}_{{\mathpzc{p}}+{\mathpzc{q}}}, \;\;\;\forall {\mathpzc{p}},{\mathpzc{q}}\in\operatorname{\mathbb{Z}},\;X_{{\mathpzc{p}}}\in{\mathfrak{g}}_{{\mathpzc{p}}}.$$ \[propfinrep\] Let ${\mathfrak{g}}$ be a $\operatorname{\mathbb{Z}}$-graded characteristic Lie algebra. If ${\textsf{V}}$ is a finite dimensional ${\mathbb{K}}$-vector space and $\rhoup\,{:}\,{\mathfrak{g}}\,{\to}\,{\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})$ a linear Lie algebra representation of ${\mathfrak{g}},$ then we can find a gradation of ${\textsf{V}}$ for which $\rhoup$ satisfies . Let $E$ be the characteristic element of ${\mathfrak{g}}.$ For polynomials $\psiup({\texttt{t}})\,{\in}\,{\mathbb{K}}[{\texttt{t}}],$ we set $$V_{\psiup({\texttt{t}})}=\ker(\psiup(\rhoup(E)))= \{{\mathpzc{v}}\in{\textsf{V}}\mid \psiup(\rhoup(E))({\mathpzc{v}})=0\}.$$ Take the spectral decomposition of ${\textsf{V}}$ with respect to $\rhoup(E)$: $${\textsf{V}}={\textsf{V}}_{\psiup_{1}({\texttt{t}})}\oplus\cdots\oplus{\textsf{V}}_{\psiup_{m}({\texttt{t}})}.$$ Here $\psiup_{1}({\texttt{t}}),\hdots,\psiup_{m}({\texttt{t}})$ are powers of distinct irreducible monic polynomials in ${\mathbb{K}}[{\texttt{t}}].$ For every ${\mathpzc{p}}\,{\in}\,\operatorname{\mathbb{Z}}$ and $X_{{\mathpzc{p}}}\,{\in}\,{\mathfrak{g}}_{{\mathpzc{p}}},$ we have $\rhoup(X_{{\mathpzc{p}}})\,{\circ}(\rhoup(E){+}{\mathpzc{p}}{\cdot}{\mathrm{I}}_{{\textsf{V}}}) =\rhoup(E)\,{\circ}\,\rhoup(X_{{\mathpzc{p}}})$ and hence, for every positive integer $k,$ $$\rhoup(X_{{\mathpzc{p}}}) \circ(\rhoup(E)\,{+}\,{\mathpzc{p}}{\cdot}{\mathrm{I}}_{V})^{k} =\rhoup(E)^{k}\circ\rhoup(X_{{\mathpzc{p}}}).$$ This shows that $$\rhoup(X_{{\mathpzc{p}}})({\textsf{V}}_{\psiup({\texttt{t}})}) \subseteq{\textsf{V}}_{\psiup({\texttt{t}}+{\mathpzc{p}})},\;\;\forall\psiup({\texttt{t}})\in{\mathbb{K}}[{\texttt{t}}].$$ Fix $\phiup_{1}({\texttt{t}}),\hdots,\phiup_{r}({\texttt{t}})\in \{\psiup_{1}({\texttt{t}}),\hdots,\psiup_{m}({\texttt{t}})\}$ such that $$\begin{cases} \phiup_{i}({\texttt{t}})\neq\phiup_{j}({\texttt{t}}{+}{\mathpzc{p}}),\;\;\forall j\,{\neq}\,i,\;\forall{\mathpzc{p}}\,{\in}\,\operatorname{\mathbb{Z}},\\ \{\psiup_{1},\hdots,\psiup_{m}\} \subset\{\phiup_{i}({\texttt{t}}\,{+}\,{\mathpzc{p}}) \mid 1{\leq}i{\leq}{\mathpzc{r}},\; {\mathpzc{p}}\in\operatorname{\mathbb{Z}}\}. \end{cases}$$ Then the gradation $${\textsf{V}}={\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}{\textsf{V}}_{{\mathpzc{p}}},\;\;\text{with}\;\; {\textsf{V}}_{{\mathpzc{p}}}={\sum}_{i=1}^{r}{\textsf{V}}_{\phi_{i}({\texttt{t}}+{\mathpzc{p}})}$$ satisfies the requirements of the Theorem. Any $\operatorname{\mathbb{Z}}$-graded linear representation of a $\operatorname{\mathbb{Z}}$-graded Lie algebra ${\mathfrak{g}}$ extends to a linear representation of its characteristic prolongation ${\mathfrak{g}^{c}}.$ Thus Prop. \[propfinrep\] tells us that actually all finite dimensional $\operatorname{\mathbb{Z}}$-graded linear representations of ${\mathfrak{g}}$ are restrictions to ${\mathfrak{g}}$ of linear representations of ${\mathfrak{g}^{c}}.$ The gradation on ${\textsf{V}}$ is not uniquely determined: indeed we can always change its grading by shifting the indices by an integral constant and this can be done independently on each subspace $V^{i}={\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}V_{\phiup_{i}({\texttt{t}}+{\mathpzc{p}})}.$ Note also that $\rhoup(E)$ may not be a characteristic element of ${\mathfrak{gl}^{\vee}}_{{\mathbb{K}}}({\textsf{V}}).$ In the finite dimensional case we can normalise the grading of ${\textsf{V}}$ by requiring that $E_{{\textsf{V}}}\,{\in}\,{\mathfrak{sl}}_{{\mathbb{K}}}({\textsf{V}}),$ i.e. that ${\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}\,({\mathpzc{p}}\,{\cdot}\dim_{{\mathbb{K}}}({\textsf{V}}_{{\mathpzc{p}}}))\,{=}\,0.$ We obtain a graded version of Ado’s theorem. \[ado\] Every finite dimensional $\operatorname{\mathbb{Z}}$-graded Lie algebra admits a $\operatorname{\mathbb{Z}}$-graded faithful representation such that the linear maps corresponding to elements of its maximal nilpotent ideal are nilpotent. Let ${\mathfrak{g}}$ be a finite dimensional $\operatorname{\mathbb{Z}}$-graded Lie algebra and ${\mathfrak{g}^{c}}$ its characteristic prolongation. By Ado’s theorem (see e.g. [@n1998lie Ch.I,§[7.3]{},Theorem 3]) ${\mathfrak{g}^{c}}$ has a faithful finite dimensional linear representation $\rhoup{:}{\mathfrak{g}^{c}}{\to}{\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})$ such that $\rhoup(X)$ is nilpotent for all $X$ in the maximal nilpotent ideal of ${\mathfrak{g}^{c}},$ which coincides with the maximal nilpotent ideal ${\mathfrak{n}}$ of ${\mathfrak{g}}.$ By Prop. \[propfinrep\] we can find a $\operatorname{\mathbb{Z}}$-gradation of ${\textsf{V}}$ to make $\rhoup$ graded. If $\operatorname{\mathfrak{c}}_{0}\,{=}\,0,$ i.e. if no nonzero element of the center is homogeneous of degree $0,$ then the restriction to ${\mathfrak{g}}$ of the adjoint representation of its characteristic prolongation ${\mathfrak{g}^{c}}$ is a faithful $\operatorname{\mathbb{Z}}$-graded finite dimensional linear representation of ${\mathfrak{g}},$ mapping the elements of ${\mathfrak{n}}$ into nilpotent endomorphisms. Let ${\mathfrak{g}}$ be a finite dimensional $\operatorname{\mathbb{Z}}$-graded Lie algebra and ${\textsf{V}}$ a $\operatorname{\mathbb{Z}}$-graded finite dimensional vector space over ${\mathbb{K}}.$ A finite dimensional faithful representation $\rhoup\,{:}\,{\mathfrak{g}}\,{\to}\,{\mathfrak{gl}^{\vee}}_{{\mathbb{K}}}({\textsf{V}})$ for which $\rhoup({\mathfrak{n}})$ is a Lie algebra of nilpotent endomorphisms of ${\textsf{V}}$ will be called a *realisation* of ${\mathfrak{g}}.$ Decomposable Lie algebras ------------------------- In this subsection we will not assume that the Lie algebras we consider are $\operatorname{\mathbb{Z}}$-graded. Let ${\textsf{V}}$ be a finite dimensional vector space over a field ${\mathbb{K}}$ of characteristic $0.$ Every $A$ in ${\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})$ admits a Jordan-Chevalley decomposition: there are $A_{s},A_{n}\,{\in}\,{\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})$ such that $$\begin{aligned} A=A_{s}+A_{n},\;[A_{s},A_{n}]=0,\; \; \text{with $A_{s}$ semisimple and $A_{n}$ nilpotent on ${\textsf{V}}.$}\end{aligned}$$ The summands $A_{s},$ $A_{n}$ are uniquely determined and $A_{s}\,{=}\,\psiup_{s}(A),$ $A_{n}\,{=}\,\psiup_{n}(A)$ with $\psiup_{s}({\texttt{t}}),\psiup_{n}({\texttt{t}})\,{\in}\,{\mathbb{K}}[{\texttt{t}}]$ polynomials with no constant term. We recall some notion and results from [@Bou82 Ch.VII,§[5]{}]. The *decomposable envelope* ${\Tilde{\mathfrak{g}}}$ of a Lie subalgebra ${\mathfrak{g}}$ of ${\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})$ is the Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})$ generated by the semisimple and nilpotent components of its elements. Note that ${\mathfrak{g}}$ is an ideal in ${\Tilde{\mathfrak{g}}}.$ We say that ${\mathfrak{g}}$ is *decomposable in ${\textsf{V}}$* when ${\Tilde{\mathfrak{g}}}\,{=}\,{\mathfrak{g}}.$ A necessary and sufficient condition for ${\mathfrak{g}}$ to be decomposable in ${\textsf{V}}$ is that its solvable radical ${\mathfrak{r}}$ is decomposable in ${\textsf{V}}.$ We denote by $${\mathfrak{n}}_{{\textsf{V}}}=\{X\in{\mathfrak{r}}\mid X\;\text{is nilpotent on ${\textsf{V}}$}\}$$ the maximal ideal in ${\mathfrak{g}}$ consisting of nilpotent endomorphisms of ${\textsf{V}}.$ We have $$[{\mathfrak{g}},{\mathfrak{g}}]\cap{\mathfrak{r}}\subseteq{\mathfrak{n}}_{{\textsf{V}}}\subseteq{\mathfrak{n}}.$$ We say that ${\mathfrak{g}}$ is *${\textsf{V}}$-reductive* if the ${\mathfrak{g}}$-module ${\textsf{V}}$ is completely reducible. A ${\textsf{V}}$-reductive ${\mathfrak{g}}$ is also decomposable in ${\textsf{V}}.$ Let ${\mathfrak{g}}$ be a Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}}).$ A *Levi-Chevalley decomposition* of ${\mathfrak{g}}$ in ${\textsf{V}}$ is a direct sum decomposition $$\label{LCdec} {\mathfrak{g}}={\mathfrak{l}}_{{\textsf{V}}}\oplus{\mathfrak{n}}_{{\textsf{V}}},$$ where ${\mathfrak{l}}_{{\textsf{V}}}$ is a ${\textsf{V}}$-reductive Lie subagebra of ${\mathfrak{g}}.$ We call such an ${\mathfrak{l}}_{{\textsf{V}}}$ a of ${\mathfrak{g}}.$ The condition of being decomposable in ${\textsf{V}}$ is necessary and sufficient to ensure that ${\mathfrak{g}}$ admits a Levi-Chevalley decomposition in ${\textsf{V}}$ and the group of elementary automorphisms of ${\mathfrak{g}}$ is transitive on the set of ${\textsf{V}}$-reductive Levi factors of ${\mathfrak{g}}.$ This is the contents of [@Bou82 Ch.VII,§[5]{}, Proposition 7], that we further precise by giving here the analogue of a theorem that Mostow ([@Most56]) proved for algebraic groups. \[prop8.7\] Let ${\mathfrak{g}}$ be a decomposable Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})$ and ${\mathfrak{t}}$ a commutative Lie subalgebra of ${\mathfrak{g}},$ consisting of semisimple endomorphisms. Then ${\mathfrak{g}}$ has a ${\textsf{V}}$-reductive Levi factor ${\mathfrak{l}}_{{\textsf{V}}}$ containing ${\mathfrak{t}}.$ Take a ${\textsf{V}}$-reductive Levi factor ${\mathfrak{l}}_{{\textsf{V}}}$ of ${\mathfrak{g}}$ for which ${\mathfrak{l}}_{{\textsf{V}}}\,{\cap}\,{\mathfrak{t}}$ is maximal. We claim that ${\mathfrak{t}}\,{\subseteq}\,{\mathfrak{l}}_{{\textsf{V}}}.$ To prove this fact we argue by contradiction. Assume that there is an $A\,{\in}\,{\mathfrak{t}}{\backslash}{\mathfrak{l}}_{{\textsf{V}}}.$ This $A$ uniquely decomposes into a sum $A\,{=}\,A'{+}N,$ with $A'\,{\in}{\mathfrak{l}}_{{\textsf{V}}}$ and $N\,{\in}\,{\mathfrak{n}}_{{\textsf{V}}}.$ If $B\,{\in}\,{\mathfrak{t}}\,{\cap}{\mathfrak{l}}_{{\textsf{V}}},$ from $$0=[A,B]=[A',B]+[N,B]$$ we obtain that both $[A',B]\,{=}\,0$ and $[N,B]\,{=}\,0,$ because the first summand is in ${\mathfrak{l}}_{{\textsf{V}}}$ and the second in ${\mathfrak{n}}_{{\textsf{V}}}.$ Let $X$ and $Y$ be the semisimple and nilpotent summands in the Jordan-Chevalley decomposition of $A'.$ Being polynomials in $A',$ they both commute with the elements of ${\mathfrak{t}}\,{\cap}\,{\mathfrak{l}}_{{\textsf{V}}}.$ Let us consider now the Lie subalgebra $\operatorname{{\kappaup}}$ of ${\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})$ generated by $X,Y,N.$ It is solvable and the ideal ${\mathfrak{n}}_{{\textsf{V}}}'$ of its nilpotent endomorphisms, which is generated by $Y,N,[X,N],$ has codimension $1$ in $\operatorname{{\kappaup}}$ and its elements commute with those in ${\mathfrak{t}}\,{\cap}\,{\mathfrak{l}}_{{\textsf{V}}}.$ We note that both $\langle{X}\rangle$ and $\langle{A}\rangle$ are ${\textsf{V}}$-reductive Levi factors of $\operatorname{{\kappaup}}.$ Then there is an elementary automorphism $\Psi$ of $\operatorname{{\kappaup}}$ mapping $\langle{X}\rangle$ onto $\langle{A}\rangle.$ This $\Psi$ is a composition of automorphisms of the form $\exp(\operatorname{\mathrm{ad}}_{\operatorname{{\kappaup}}}(T)),$ with $T\,{\in}\,{\mathfrak{n}}_{{\textsf{V}}}'.$ Its extension to an automorphism $\tilde{\Psi}$ of ${\mathfrak{g}}$ is a composition of $\exp(\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}(T)),$ with $T\,{\in}\,{\mathfrak{n}}_{{\textsf{V}}}'$ and hence leaves invariant all elements of ${\mathfrak{t}}{\cap}{\mathfrak{l}}_{{\textsf{V}}}.$ Therefore $\tilde{\Psi}({\mathfrak{l}}_{{\textsf{V}}})$ is a ${\textsf{V}}$-reductive Levi factor with $${\mathfrak{t}}\cap{\mathfrak{l}}_{{\textsf{V}}}\oplus\langle{A}\rangle\subseteq\tilde{\Psi}({\mathfrak{l}}_{{\textsf{V}}}),$$ contradicting the choice of ${\mathfrak{l}}_{{\textsf{V}}}$ and showing therefore that in fact ${\mathfrak{t}}\,{\subseteq}\,{\mathfrak{l}}_{{\textsf{V}}}.$ The proof is complete. Decomposable prolongations -------------------------- We specialise the general notions of the previous subsection to the case of $\operatorname{\mathbb{Z}}$-graded Lie algebras. Let ${\textsf{V}}$ be a finite dimensional $\operatorname{\mathbb{Z}}$-graded ${\mathbb{K}}$-vector space and ${\mathfrak{g}}$ a $\operatorname{\mathbb{Z}}$-graded Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}}).$ Then its decomposable envelope ${\Tilde{\mathfrak{g}}}$ in ${\textsf{V}}$ is the $\operatorname{\mathbb{Z}}$-graded Lie subalgebra $$\label{e8.16} {\Tilde{\mathfrak{g}}}={\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}{\Tilde{\mathfrak{g}}}_{{\mathpzc{p}}}$$ of ${\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}}),$ where ${\Tilde{\mathfrak{g}}}_{{\mathpzc{p}}}\,{=}\,{\mathfrak{g}}_{{\mathpzc{p}}}$ for all ${\mathpzc{p}}{\neq}0,$ while ${\Tilde{\mathfrak{g}}}_{0}$ equals the decomposable envelope of ${\mathfrak{g}}_{0}$ in ${\textsf{V}}.$ The Lie algebra defined by is decomposable by [@Bou82 Ch.VII, §[5.5]{}, Theorem 1], being generated by the elements of ${\bigcup}_{{\mathpzc{p}}{\neq}0}{\mathfrak{g}}_{{\mathpzc{p}}},$ which are nilpotent, and by the semisimple and nilpotent components of the elements of ${\mathfrak{g}}_{0}.$ Moreover, ${\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})$ contains a characteristic element $E_{{\textsf{V}}}$ and $$[{\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})]_{0}\,{=}\,\{A\,{\in}\,{\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}}) \,{\mid}\,E_{{\textsf{V}}}{\circ}\,A\,{=}\,A\,{\circ}\,E_{{\textsf{V}}}\}.$$ Since the semisimple and nilpotent summands $A_{s}$ and $A_{n}$ of an element $A$ of ${\mathfrak{g}}_{0}$ are polynomials of $A,$ they also commute with $E_{{\textsf{V}}}$ and therefore belong to $[{\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})]_{0}.$ $\operatorname{\mathbb{Z}}$-graded Levi-Malčev and Levi-Chevalley decompositions -------------------------------------------------------------------------------- Finite dimensional $\operatorname{\mathbb{Z}}$-graded Lie algebras admit $\operatorname{\mathbb{Z}}$-graded Levi-Malčev and Levi-Chevalley decompositions. The Levi-Malčev decompositions was stated and proved in [@CC2017; @MN02] for real and complex finite dimensional graded Lie algebras. We provide here a short proof for general fields of characteristic zero. \[lmc-dec\] Every finite dimensional $\operatorname{\mathbb{Z}}$-graded Lie algebra admits a $\operatorname{\mathbb{Z}}$-graded Levi-Malčev decomposition. Let ${\textsf{V}}$ be a finite dimensional $\operatorname{\mathbb{Z}}$-graded ${\mathbb{K}}$-vector space. Then every decomposable $\operatorname{\mathbb{Z}}$-graded Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})$ admits a Levi-Chevalley decomposition in ${\textsf{V}}.$ We begin by proving the last statement under an additional assumption. Let ${\textsf{V}}$ be a finite dimensional $\operatorname{\mathbb{Z}}$-graded vector space and assume that ${\mathfrak{g}}$ is a decomposable $\operatorname{\mathbb{Z}}$-graded Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}(V),$ containing the characteristic element $E_{{\textsf{V}}}$ of ${\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}}).$ Since $E_{{\textsf{V}}}$ is semisimple, by Proposition \[prop8.7\], there is a ${\textsf{V}}$-reductive Levi factor ${\mathfrak{l}}_{{\textsf{V}}}$ of ${\mathfrak{g}}$ containing $E_{{\textsf{V}}}$ and hence $\operatorname{\mathbb{Z}}$-graded. Consider now any $\operatorname{\mathbb{Z}}$-graded Lie algebra ${\mathfrak{g}}.$ By using Theorem \[ado\], we identify ${\mathfrak{g}}$ with a $\operatorname{\mathbb{Z}}$-graded Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}}),$ for a suitable finite dimensional $\operatorname{\mathbb{Z}}$-graded ${\mathbb{K}}$-vector space ${\textsf{V}}.$ Then ${\mathfrak{g}^{c}}{\coloneqq}{\mathfrak{g}}{+}\langle{E_{{\textsf{V}}}}\rangle$ is a $\operatorname{\mathbb{Z}}$-graded Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})$ which, by the argument above, has a $\operatorname{\mathbb{Z}}$-graded ${\textsf{V}}$-reductive Levi factor ${\mathfrak{l}}_{{\textsf{V}}}.$ Then $\operatorname{\mathfrak{s}}{\coloneqq}[{\mathfrak{l}}_{{\textsf{V}}},{\mathfrak{l}}_{{\textsf{V}}}]$ is a semisimple $\operatorname{\mathbb{Z}}$-graded Levi factor of ${\mathfrak{g}}$. It remains to prove the existence a $\operatorname{\mathbb{Z}}$-graded Levi-Chevalley decomposition in ${\textsf{V}}$ for a decomposable $\operatorname{\mathbb{Z}}$-graded Lie subalgebra ${\mathfrak{g}}$ of ${\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})$ without assuming that $E_{{\textsf{V}}}\,{\in}\,{\mathfrak{g}}.$ We begin by taking a Levi-Malčev decomposition ${\mathfrak{g}}\,{=}\,\operatorname{\mathfrak{s}}\,{\oplus}\,{\mathfrak{r}}$ with a $\operatorname{\mathbb{Z}}$-graded semisimple Levi factor $\operatorname{\mathfrak{s}}.$ Since ${\sum}_{{\mathpzc{p}}\neq{0}}{\mathfrak{r}}_{{\mathpzc{p}}}\,{\subseteq}\,{\mathfrak{n}}_{{\textsf{V}}},$ we obtain that ${\mathfrak{r}}\,{=}\,{\mathfrak{n}}_{{\textsf{V}}}{+}{\mathfrak{r}}_{0}.$ The subalgebra ${\mathfrak{r}}_{0}$ is solvable and decomposable in ${\textsf{V}}.$ Therefore, if ${\mathfrak{t}}_{0}$ is a maximal abelian subalgebra of ${\mathfrak{r}}_{0}$ consisting of semisimple endomorphisms of ${\textsf{V}},$ then ${\mathfrak{r}}_{0}\,{=}\,{\mathfrak{t}}_{0}\,{\oplus}\,({\mathfrak{n}}_{{\textsf{V}}}{\cap}{\mathfrak{r}}_{0})$ (see e.g. [@Bou82 Ch.VII,§[5]{}, Corollary 2]) and hence ${\mathfrak{l}}_{{\textsf{V}}}\,{=}\,\operatorname{\mathfrak{s}}\,{\oplus}\,{\mathfrak{t}}_{0}$ is a ${\textsf{V}}$-reductive Levi factor of ${\mathfrak{g}}.$ We recall (see e.g. [@n1998lie Ch.I,§[6.4]{}]) that a Lie algebra $\operatorname{{\kappaup}}$ is *reductive* if its adjoint representation is semisimple. This is equivalent to the fact that its derived algebra $[\operatorname{{\kappaup}},\operatorname{{\kappaup}}]$ is semisimple. If $\operatorname{{\kappaup}}$ is reductive, then its center is a direct sum complement of $[\operatorname{{\kappaup}},\operatorname{{\kappaup}}]$ in $\operatorname{{\kappaup}}.$ A finite dimensional Lie algebra ${\mathfrak{g}}$ is *decomposable* if the semisimple and nilpotent components of its inner derivations are still inner derivations: this means that $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}({\mathfrak{g}})$ is a ${\mathfrak{g}}$-decomposable Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}({\mathfrak{g}}).$ A *linear realisation* of ${\mathfrak{g}}$ is a faithful finite dimensional linear representation $\rhoup\,{:}\,{\mathfrak{g}}\,{\to}\,{\mathfrak{gl}}_{{\mathbb{K}}}({\textsf{V}})$ such that $\rhoup(X)$ is nilpotent on ${\textsf{V}}$ for each $X$ in the maximal nilpotent ideal ${\mathfrak{n}}$ of ${\mathfrak{g}}.$ If ${\mathfrak{g}}$ is decomposable, by using its linear realisation, we obtain a direct sum decomposition $${\mathfrak{g}}\,{=}\,{\mathfrak{l}}\,{\oplus}\,{\mathfrak{n}},$$ where ${\mathfrak{l}}$ is a reductive Lie algebra of ${\mathfrak{g}}.$ We call *reductive Levi factors* of ${\mathfrak{g}}$ the reductive Lie algebras which are complements of ${\mathfrak{n}}$ in ${\mathfrak{g}}.$ The elementary automorphisms act transitively on the set of reductive Levi factors of ${\mathfrak{g}}.$ From Theorems \[ado\] and \[lmc-dec\] we obtain A decomposable finite dimensional $\operatorname{\mathbb{Z}}$-graded Lie algebra ${\mathfrak{g}}$ contains $\operatorname{\mathbb{Z}}$-graded reductive factors. \[lem8.2\] Let ${\mathfrak{g}}\,{=}\,{\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}{\mathfrak{g}}_{{\mathpzc{p}}}$ be a characteristic finite dimensional $\operatorname{\mathbb{Z}}$-graded Lie algebra. Then the Cartan subalgebras of ${\mathfrak{g}}_{0}$ are Cartan subalgebras of ${\mathfrak{g}}.$ Every Cartan subalgebra ${\mathfrak{h}}$ of ${\mathfrak{g}}_{0}$ contains the characteristic element $E,$ because it belongs to the center of ${\mathfrak{g}}_{0}.$ Let ${\mathfrak{h}}$ be a Cartan subalgebra of ${\mathfrak{g}}_{0}.$ If $X\,{=}\,{\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}X_{{\mathpzc{p}}},$ with $X_{{\mathpzc{p}}}\,{\in}\,{\mathfrak{g}}_{{\mathpzc{p}}},$ is in the normaliser of ${\mathfrak{h}}$ in ${\mathfrak{g}},$ then the condition $$[E,X]={\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}{\mathpzc{p}}\cdot{X}_{{\mathpzc{p}}}\in{\mathfrak{h}}\subseteq {\mathfrak{g}}_{0}$$ implies that $X$ belongs to ${\mathfrak{g}}_{0}$ and therefore to ${\mathfrak{h}},$ which is its own normaliser in ${\mathfrak{g}}_{0}.$ Thus ${\mathfrak{h}}$ is also its own normaliser in ${\mathfrak{g}}$ and thus is Cartan in ${\mathfrak{g}}.$ In a $\operatorname{\mathbb{Z}}$-graded finite dimensional Lie algebra ${\mathfrak{g}}{=}{\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}{\mathfrak{g}}_{{\mathpzc{p}}},$ when ${\mathpzc{p}}{+}{\mathpzc{q}}{\neq}0$ the subspaces ${\mathfrak{g}}_{{\mathpzc{p}}}$ and ${\mathfrak{g}}_{{\mathpzc{q}}}$ are orthogonal for the Killing form ${\textsf{k}}_{{\mathfrak{g}}}.$ Since the restriction of ${\textsf{k}}_{{\mathfrak{g}}}$ to a semisimple Levi factor of ${\mathfrak{g}}$ is nondegenerate, Theorem \[lmc-dec\] yields Let ${\mathfrak{g}}$ be a $\operatorname{\mathbb{Z}}$-graded finite dimensional Lie algebra. Then: 1. if ${\textswab{m}}\,{\subseteq}\,{\mathfrak{r}},$ then ${\sum}_{{\mathpzc{p}}{\neq}0}{\mathfrak{g}}_{{\mathpzc{p}}}\subseteq{\mathfrak{n}}$; 2. if ${\textswab{m}}\,{\cap}\,{\mathfrak{r}}\,{=}\,\{0\},$ then $\dim_{{\mathbb{K}}}({\mathfrak{g}}_{{\mathpzc{p}}})=\dim_{{\mathbb{K}}}({\mathfrak{g}}_{-{\mathpzc{p}}})$ for all ${\mathpzc{p}}\,{\in}\,\operatorname{\mathbb{Z}}.$ Effectiveness conditions {#sec-effect} ======================== Let ${\mathfrak{g}}\,{=}\,{\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}{\mathfrak{g}}_{{\mathpzc{p}}}$ be a $\operatorname{\mathbb{Z}}$-graded Lie algebra and set ${\mathfrak{g}}_{+}\,{=}\,{\sum}_{{\mathpzc{p}}{\geq}0}{\mathfrak{g}}_{{\mathpzc{p}}},$ ${\textswab{m}}{=}{\sum}_{{\mathpzc{p}}<0}{\mathfrak{g}}_{{\mathpzc{p}}}.$ We say that ${\mathfrak{g}}$ is - *effective* if $$\label{eq8.17} \{X\in{\mathfrak{g}}_{+}\mid [X,{\mathfrak{g}}_{-1}]=\{0\}\}=\{0\},$$ - *quasi-effective* if the following two conditions are fulfilled: $$\label{eq8.18} \{X\in{\mathfrak{g}}_{+}\mid [X,{\textswab{m}}]\subseteq{\textswab{m}}\}\subseteq{\mathfrak{g}}_{0}, \;\; \{X\in{\mathfrak{g}}_{0}\mid [X,{\textswab{m}}]=\{0\}\}=\{0\},$$ - *almost effective* if $$\label{eq8.19} \{X\in{\mathfrak{g}}_{+}\mid [X,{\textswab{m}}]=\{0\}\}=\{0\}.$$ We clearly have $\Rightarrow$$\Rightarrow$ and the three notions are equivalent when ${\textswab{m}}$ is *fundamental*, i.e. generated by ${\mathfrak{g}}_{-1}.$ \[ntz-a.e.\] Let us indicate by ${\mathfrak{P}}_{0}({\textswab{m}},{\mathfrak{g}}_{0})$ the collection of all finite dimensional $\operatorname{\mathbb{Z}}$-graded *almost effective* prolongations of type ${\mathfrak{g}}_{0}$ of ${\textswab{m}},$ where ${\mathfrak{g}}_{0}$ is a subalgebra of ${\mathpzc{Der}}_{\!0}({\textswab{m}})$ and, for $D\,{\in}\,{\mathfrak{g}}_{0}$ and $X\in{\textswab{m}}$ we have $[D,X]\,{=}\,D(X).$ We denote by ${\textswab{m}}\,{\oplus}\,{\mathfrak{g}}_{0}$ the object of ${\mathfrak{P}}_{0}({\textswab{m}},{\mathfrak{g}}_{0})$ which is the semidirect product of ${\textswab{m}}$ and ${\mathfrak{g}}_{0}.$ An almost effective ${\mathfrak{g}}$ contains at most one characteristic element. If ${\mathfrak{g}}$ is almost effective, then the restriction to ${\mathfrak{g}}$ of the adjoint representation of its characteristic prolongation is faithful. \[lm8.7\] Let ${\mathfrak{g}}$ be an almost effective $\operatorname{\mathbb{Z}}$-graded Lie algebra. Then a nonnegative degree homogeneous derivation of ${\mathfrak{g}}$ is zero if and only if its restriction to ${\textswab{m}}$ is zero. Let $D\,{\in}\,{\mathpzc{Der}}_{\!{\mathpzc{q}}}({\mathfrak{g}})$ be a homogeneous derivation of degree ${\mathpzc{q}}{\geq}0$ which vanishes on ${\textswab{m}}.$ We prove by recurrence that $D({\mathfrak{g}}_{{\mathpzc{p}}})\,{=}\,\{0\}$ for all ${\mathpzc{p}}\,{\in}\,\operatorname{\mathbb{Z}}.$ By assumption this is true for ${\mathpzc{p}}{<}0.$ Assume that ${\mathpzc{p}}\,{\geq}\,0$ and $D({\mathfrak{g}}_{{\mathpzc{s}}})\,{=}\,0$ for ${\mathpzc{s}}{<}{\mathpzc{p}}.$ If $X_{{\mathpzc{p}}}\in{\mathfrak{g}}_{{\mathpzc{p}}}$ and $Y_{{\mathpzc{r}}}\,{\in}\,{\mathfrak{g}}_{{\mathpzc{r}}}$ with ${\mathpzc{r}}{<}0,$ then $$[D(X_{{\mathpzc{p}}}),Y_{{\mathpzc{r}}}]=D([X_{{\mathpzc{p}}},Y_{{\mathpzc{r}}}])-[X_{{\mathpzc{p}}},D(Y_{{\mathpzc{r}}})]=0,$$ because the first summand in the right hand side is zero, being $[X_{{\mathpzc{p}}},Y_{{\mathpzc{r}}}]{\in}{\mathfrak{g}}_{{\mathpzc{p}}{+}{\mathpzc{r}}}$ and ${\mathpzc{p}}{+}{\mathpzc{r}}{<}{\mathpzc{p}}$; the second is zero because $D(Y_{{\mathpzc{r}}})\,{=}\,0$ since ${\mathpzc{r}}{<}0.$ Then $D(X_{{\mathpzc{p}}})$ is an element of ${\mathfrak{g}}_{{\mathpzc{p}}+{\mathpzc{q}}}$ with $[D(X_{{\mathpzc{p}}}),{\textswab{m}}]\,{=}\,\{0\}$ and hence is $0$ by the almost effectiveness assumption, because ${\mathpzc{p}}{+}{\mathpzc{q}}{\geq}0.$ This completes the proof. \[lemma8.11\] Let ${\mathfrak{g}}$ be a finite dimensional almost effective $\operatorname{\mathbb{Z}}$-graded Lie algebra. Then a degree $0$-homogeneous derivation of ${\mathfrak{g}}$ is nilpotent if and only if its restriction to ${\textswab{m}}$ is nilpotent. For a derivation $D$ of ${\mathfrak{g}}$ one can easily prove by recurrence that $$[D^{k}(X),Y]={\sum}_{h=0}^{k}c_{k,h}D^{h}[X,D^{k-h}(Y)],$$ for suitable constants $c_{k,h}.$ Then, if $D\,{\in}\,{\mathpzc{Der}}_{0}({\mathfrak{g}})$ is nilpotent on ${\textswab{m}},$ we can prove recursively that it is nilpotent on ${\mathfrak{g}}.$ Indeed, if ${\mathpzc{p}}\,{\geq}\,0$ and we know that $D^{m}$ is zero on ${\mathfrak{g}}_{{\mathpzc{q}}}$ for ${\mathpzc{q}}{<}{\mathpzc{p}},$ then we obtain that, for every $X_{{\mathpzc{p}}}\in{\mathfrak{g}}_{{\mathpzc{p}}}$ $$[D^{2m}(X_{{\mathpzc{p}}}),Y]={\sum}_{h=0}^{2m}c_{2m,h}D^{h}[X_{{\mathpzc{p}}},D^{2m-h}(Y)]=0,\;\;\forall Y\in{\textswab{m}}$$ by the inductive assumption, because $[X_{{\mathpzc{p}}},D^{2m-h}(Y)]\,{\in}\,{\sum}_{{\mathpzc{q}}<{\mathpzc{p}}}{\mathfrak{g}}_{{\mathpzc{q}}}$ and either $h$ or $2m{-}h$ is ${\geq}m.$ Since ${\mathfrak{g}}$ is almost effective, this implies that $D^{2m}(X_{{\mathpzc{p}}})\,{=}\,0.$ The proof is complete. A consequence of Lemma \[lemma8.11\] is that for an almost effective ${\mathfrak{g}}$ the Lie subalgebra of the homogeneous elements of degree $0$ of its maximal nilpotent ideal ${\mathfrak{n}}$ only depends on ${\textswab{m}}$ and the structure subalgebra ${\mathfrak{g}}_{0}.$ \[rapp\_notation\] Let $\rhoup\,{:}\,{\mathfrak{g}}_0\,{\to}\,{\mathfrak{gl}}_{{\mathbb{K}}}({\textswab{m}})$ be the linear representation obtained by restricting the adjoint representation to ${\textswab{m}}.$ Likewise, for all ${\mathpzc{p}}\,{\in}\,\operatorname{\mathbb{Z}},$ we denote by $\rhoup_{{\mathpzc{p}}}\,{:}\,{\mathfrak{g}}_0\,{\to}\,{\mathfrak{gl}}_{{\mathbb{K}}}({\mathfrak{g}}_{{\mathpzc{p}}})$ the restrictions of $\rhoup$ to the homogeneous subspaces ${\mathfrak{g}}_{{\mathpzc{p}}}.$ \[thm8.12\] Let ${\mathfrak{g}}_{0}$ be a Lie algebra of degree $0$ derivations of a finite dimensional $\operatorname{\mathbb{Z}}$-graded nilpotent Lie algebra ${\textswab{m}}{=}{\sum}_{{\mathpzc{p}}<0}\,{\mathfrak{g}}_{{\mathpzc{p}}}$. If ${\mathfrak{n}}({\mathfrak{g}})$ is the maximal nilpotent ideal of an almost effective prolongation ${\mathfrak{g}}$ of type ${\mathfrak{g}}_{0}$ of ${\textswab{m}},$ then $$\label{qq8.17} {\mathfrak{n}}({\mathfrak{g}})\cap{\mathfrak{g}}_{0}= {\mathfrak{n}}_{0}\coloneqq\{X\in{\mathfrak{r}}_0 \mid \rhoup(X) \;\;\text{is nilpotent on ${\textswab{m}}$}\}.$$ Let us fix a $\operatorname{\mathbb{Z}}$-graded Levi-Malčev decomposition ${\mathfrak{g}}\,{=}\,\operatorname{\mathfrak{s}}\,{\oplus}\,{\mathfrak{r}}$ of ${\mathfrak{g}}.$ The subalgebra $\operatorname{\mathfrak{s}}_{0}{\coloneqq}\operatorname{\mathfrak{s}}\,{\cap}\,{\mathfrak{g}}_{0}$ of its $\operatorname{\mathbb{Z}}$-graded semisimple Levi factor is reductive and decomposes into the direct sum $\operatorname{\mathfrak{s}}_{0}'\,{\oplus}\,\operatorname{\mathfrak{z}}_{0}$ of its semisimple ideal $\operatorname{\mathfrak{s}}'_{0}{=}\,[\operatorname{\mathfrak{s}}_{0},\operatorname{\mathfrak{s}}_{0}]$ and its center $\operatorname{\mathfrak{z}}_{0}{=}\,\{X\,{\in}\, \operatorname{\mathfrak{s}}_{0}\,{\mid}\,[X,\operatorname{\mathfrak{s}}_{0}]=\{0\}\}.$ We note that $\operatorname{\mathfrak{s}}$ contains a characteristic element $E_{\operatorname{\mathfrak{s}}}$ and therefore $\operatorname{\mathfrak{z}}_{0}$ is contained in a Cartan subalgebra of $\operatorname{\mathfrak{s}}$ contained in $\operatorname{\mathfrak{s}}_{0}.$ The elements of $\operatorname{\mathfrak{z}}_{0}$ are $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}$-semisimple on $\operatorname{\mathfrak{s}}$ and on ${\mathfrak{g}}.$ Therefore, if $A$ is a nonzero element of $\operatorname{\mathfrak{z}}_{0}$ and $B\,{\in}\,{\mathfrak{r}},$ then $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}(A{+}B)$ is not nilpotent on ${\mathfrak{g}}.$ Therefore the $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}$-nilpotent elements of $\operatorname{\mathrm{rad}}({\mathfrak{g}}_{0})$ are contained in ${\mathfrak{r}}_{0}$ and therefore belong to ${\mathfrak{n}}_{0}.$ The claim of the Theorem follows from Lemma \[lemma8.11\], because, by the assumption that ${\mathfrak{g}}$ is almost effective, an $X\,{\in}\,{\mathfrak{g}}_{0}$ is $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}$-nilpotent if and only if $\rhoup(X)$ is nilpotent on ${\textswab{m}}.$ \[lemma8.13\] For an almost effective finite dimensional $\operatorname{\mathbb{Z}}$-graded Lie algebra ${\mathfrak{g}}$ the following are equivalent: - ${\mathfrak{g}}$ is decomposable; - $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}({\mathfrak{g}}_{0})$ is decomposable in ${\mathfrak{g}}$; - $\rhoup({\mathfrak{g}}_{0})$ is decomposable in ${\textswab{m}}$. If ${\mathpzc{p}}\,{\in}\,\operatorname{\mathbb{Z}}{\backslash}\{0\}$ and $X_{{\mathpzc{p}}}\,{\in}{\mathfrak{g}}_{{\mathpzc{p}}},$ then the inner derivation $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}(X_{{\mathpzc{p}}})$ is nilpotent. Hence ${\mathfrak{g}}$ is decomposable if and only if the semisimple and nilpotent parts of the $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}(A),$ with $A\,{\in}\,{\mathfrak{g}}_{0},$ are still inner derivations. This shows that $(i){\Leftrightarrow}(ii).$ Let us prove the equivalence $(ii){\Leftrightarrow}(iii).$ If $A\,{\in}\,{\mathfrak{g}}_{0},$ then $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}(A)\,{\in}\,{\mathpzc{Der}}_{0}({\mathfrak{g}})$ has a Jordan-Chevalley decomposition $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}(A)=D_{s}{+}D_{n}$ with $D_{s}$ semisimple and $D_{n}$ nilpotent, both belonging to ${\mathbb{K}}[\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}(A)].$ If $A\,{=}\,A_{s}{+}A_{n}$ with $A_{s},A_{n}\,{\in}\,{\mathfrak{g}}_{0},$ then, by Lemma\[lm8.7\], we have $D_{s}\,{=}\,\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}(A_{s})$ and $D_{n}\,{=}\,\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}(A_{n})$ if and only if $D_{s}|_{{\textswab{m}}}\,{=}\,\rhoup(A_{s})$ and $D_{n}|_{{\textswab{m}}}\,{=}\,\rhoup(A_{n}).$ This yields the last equivalence. Lemma \[lemma8.13\] tells us that for an almost effective ${\mathfrak{g}}$ *being decomposable* is a property of its *type* ${\mathfrak{g}}_{0}.$ Let ${\textswab{m}}{=}{\sum}_{{{\mathpzc{p}}}<0}{\mathfrak{g}}_{{\mathpzc{p}}}$ be a finite dimensional $\operatorname{\mathbb{Z}}$-graded nilpotent Lie algebra and ${\mathfrak{g}}_{0}$ a Lie algebra of degree $0$ homogeneous derivations of ${\textswab{m}}.$ Then the following are equivalent: - ${\mathfrak{g}}_0\,{\oplus}\,{\textswab{m}}$ is decomposable; - there is a decomposable prolongation of type ${\mathfrak{g}}_{0}$ of ${\textswab{m}}$; - all ${\mathfrak{g}}$ in ${\mathfrak{P}}_{0}({\textswab{m}},{\mathfrak{g}}_{0})$ are decomposable. Reductive type {#sec-reduc} ============== Let ${\mathfrak{g}}{=}{\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}\,{\mathfrak{g}}_{{\mathpzc{p}}}$ be a finite dimensional $\operatorname{\mathbb{Z}}$-graded Lie algebra, ${\mathfrak{n}}$ its maximal nilpotent ideal and ${\mathfrak{n}}_{0}{\coloneqq}{\mathfrak{n}}\,{\cap}\,{\mathfrak{g}}_{0}.$ We say that ${\mathfrak{g}}$ is of *reductive type* if ${\mathfrak{n}}_{0}\,{=}\,\{0\}.$ By Theorem \[thm8.12\] we obtain Let ${\textswab{m}}{=}{\sum}_{{\mathpzc{p}}<0}\,{\mathfrak{g}}_{{\mathpzc{p}}}$ be a finite dimensional $\operatorname{\mathbb{Z}}$-graded nilpotent Lie algebra, ${\mathfrak{g}}_{0}$ a Lie subalgebra of ${\mathpzc{Der}}_{0}({\textswab{m}})$ and ${\mathfrak{P}}_{0}({\textswab{m}},{\mathfrak{g}}_{0})$ the collection of all finite dimensional almost effective prolongations of type ${\mathfrak{g}}_{0}$ of ${\textswab{m}}.$ Then the following are equivalent: - ${\mathfrak{g}}_{0}\,{\oplus}\,{\textswab{m}}$ is of reductive type; - ${\mathfrak{P}}_{0}({\textswab{m}},{\mathfrak{g}}_{0})$ contains a ${\mathfrak{g}}$ of reductive type; - all ${\mathfrak{g}}$ in ${\mathfrak{P}}_{0}({\textswab{m}},{\mathfrak{g}}_{0})$ are of reductive type. Let ${\mathfrak{g}}$ be a finite dimensional $\operatorname{\mathbb{Z}}$-graded Lie algebra. If ${\mathfrak{g}}$ is of reductive type, then ${\mathfrak{g}}_{0}$ is reductive and the subalgebra ${\mathfrak{r}}_{0}$ of the degree $0$ homogeneous elements of its radical ${\mathfrak{r}}$ is contained in the center of ${\mathfrak{g}}_{0}.$ Since $[{\mathfrak{g}}_{0},{\mathfrak{r}}_{0}]$ is contained in ${\mathfrak{n}}_{0},$ the assumption that ${\mathfrak{n}}_{0}\,{=}\,\{0\}$ implies that ${\mathfrak{r}}_{0}$ is contained in the center of ${\mathfrak{g}}_{0}.$ By Theorem \[lmc-dec\], ${\mathfrak{g}}$ has a $\operatorname{\mathbb{Z}}$-graded semisimple Levi factor $\operatorname{\mathfrak{s}}$ and $\operatorname{\mathfrak{s}}_{0}{\coloneqq}\operatorname{\mathfrak{s}}\,{\cap}\,{\mathfrak{g}}_{0}$ is reductive. Then ${\mathfrak{g}}_{0}$ is reductive, being the sum of a reductive Lie subalgebra and of its centraliser. In the following lemma we do not require that ${\mathfrak{g}}$ is $\operatorname{\mathbb{Z}}$-graded. \[ll8.16\] Let ${\mathfrak{g}}_{0}$ be a Lie subalgebra of a finite dimensional Lie algebra ${\mathfrak{g}}$ and ${\textsf{W}}$ an $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}({{\mathfrak{g}}_{0}})$-invariant subspace of ${\mathfrak{g}}.$ If the representation of $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}({{\mathfrak{g}}_{0}})$ on ${\textsf{W}}$ is semisimple, then also the Lie subalgebra ${\mathfrak{w}}$ of ${\mathfrak{g}}$ generated by ${\textsf{W}}$ is a semismimple $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}({{\mathfrak{g}}_{0}})$-module. If ${\mathfrak{n}}$ is the maximal nilpotent ideal of ${\mathfrak{g}},$ then $[{\mathfrak{g}}_{0}\,{\cap}\,{\mathfrak{n}},{\mathfrak{w}}]\,{=}\,\{0\}.$ The tensor product representation of finite dimensional semisimple representations of a Lie algebra ${\mathfrak{g}}_{0}$ is semisimple (see e.g. [@n1998lie Ch.I, §[6]{}, Cor.1]). Therefore each subspace $\operatorname{\mathfrak{f}}_{{\mathpzc{p}}}({\textsf{W}})$ of the free Lie algebra $\operatorname{\mathfrak{f}}({\textsf{W}})$ generated by ${\textsf{W}}$ is a semisimple ${\mathfrak{g}}_{0}$-module. Let $\varpi\,{:}\,\operatorname{\mathfrak{f}}({\textsf{W}})\,{\to}\,{\mathfrak{w}}$ be the natural projection. Since the image by $\varpi$ of a finite dimensional irreducible ${\mathfrak{g}}_{0}$-submodule is either $\{0\}$ or semisimple, it turns out that ${\mathfrak{w}}$ is a sum of irreducible ${\mathfrak{g}}_{0}$-submodules and hence a semisimple ${\mathfrak{g}}_{0}$-module. \[pp8.13\] Let ${\mathfrak{g}}$ be a finite dimensional $\operatorname{\mathbb{Z}}$-graded Lie algebra. If ${\mathfrak{g}}$ is almost effective, then the following are equivalent 1. ${\mathfrak{g}}$ is decomposable and of reductive type; 2. $\rhoup\,{:}\,{\mathfrak{g}}_0\,{\to}\,{\mathfrak{gl}}_{{\mathbb{K}}}({\textswab{m}})$ is semisimple; 3. ${\mathfrak{g}}$ is a semisimple $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}({\mathfrak{g}}_{0})$-module. Clearly ($iii$)$\Rightarrow$($ii$) because ${\textswab{m}}$ is $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}({\mathfrak{g}}_{0})$-invariant and $(i){\Rightarrow}(ii),(iii).$ Let us show that $(ii){\Rightarrow}(i),(iii).$ Let ${\mathfrak{g}}\,{=}\,\operatorname{\mathfrak{s}}\,{\oplus}\,{\mathfrak{r}}$ be a Levi-Malčev decomposition of ${\mathfrak{g}},$ with a $\operatorname{\mathbb{Z}}$-graded semisimple Levi factor $\operatorname{\mathfrak{s}}$. Since the adjoint representation restricts to a semisimple representation of $\operatorname{\mathfrak{s}}_{0}$ on ${\mathfrak{g}},$ it suffices to show that the inner derivations corresponding to the elements of ${\mathfrak{r}}_{0}$ are semisimple. Let indeed $A\,{\in}\,{\mathfrak{r}}_{0}$ and $D_{s},D_{n}\,{\in}\,{\mathpzc{Der}}_{0}({\mathfrak{g}})$ be the semisimple and nilpotent summands of the Jordan-Chevalley decomposition of $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}(A).$ If we assume that $\rhoup$ is semisimple, by the uniqueness of the Jordan-Chevalley decomposition in ${\mathfrak{gl}}_{{\mathbb{K}}}({\textswab{m}})$ we obtain that $D_{s}{-}\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}(A)$ and $D_{n}$ vanish on ${\textswab{m}}.$ By Lemma \[lm8.7\] this yields $D_{n}\,{=}\,0$ and $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}(A)\,{=}\,D_{s}$ is a semisimple derivation on ${\mathfrak{g}},$ proving at the same time that $(iii)$ holds, because $\operatorname{\mathrm{ad}}_{{\mathfrak{g}}}(A)$ is semisimple for $A\,{\in}{\mathfrak{r}}_{0}$ and that $(i)$ holds, because ${\mathfrak{g}}_{0}$ is the sum of a reductive ideal $\operatorname{\mathfrak{s}}_{0}$ and an abelian Lie algebra of semisimple elements. If ${\textswab{m}}$ is fundamental, then ${\mathfrak{g}}$ is decomposable and of the reductive type if and only if $\rhoup_{-1}$ is semisimple. Quasi-effective $\operatorname{\mathbb{Z}}$-graded Lie algebras {#sec-quasi} =============================================================== Under the stronger assumption of quasi-effectiveness we obtain better structure theorems. \[lm8.16\] Let ${\mathfrak{g}}$ be a quasi-effective finite dimensional $ \operatorname{\mathbb{Z}}$-graded Lie algebra. If ${\mathfrak{g}}$ is of reductive type, then the subspaces ${\mathfrak{n}}_{{\mathpzc{p}}}$ of homogeneous elements of positive degree ${\mathpzc{p}}$ of its maximal nilpotent ideal ${\mathfrak{n}}$ are trivial. Let us prove recursively that ${\mathfrak{n}}_{{\mathpzc{p}}}\,{=}\,\{0\}$ for ${\mathpzc{p}}\,{\geq}\,0.$ For ${\mathpzc{p}}\,{=}\,0$ this is true by assumption. Let ${\mathpzc{p}}\,{>}\,0$ and suppose we already know that ${\mathfrak{n}}_{{\mathpzc{q}}}\,{=}\,\{0\}$ for $0{\leq}{\mathpzc{q}}{<}{\mathpzc{p}}.$ Then we have $$[{\mathfrak{n}}_{{\mathpzc{p}}},{\textswab{m}}] \subseteq{\sum}_{{\mathpzc{q}}<{\mathpzc{p}}}{\mathfrak{n}}_{{\mathpzc{q}}} \subseteq{\sum}_{{\mathpzc{q}}<0}{\mathfrak{n}}_{{\mathpzc{q}}}\subseteq{\textswab{m}}.$$ By the quasi-effectiveness assumption this yields ${\mathfrak{n}}_{{\mathpzc{p}}}\,{\subseteq}\,{\mathfrak{n}}_{0}\,{=}\,\{0\},$ proving the statement. We recall that the *depth* of a $\operatorname{\mathbb{Z}}$-graded Lie algebra ${\mathfrak{g}}$ is the largest integer $\muup$ for which ${\mathfrak{g}}_{-\muup}\,{\neq}\{0\}.$ An almost effective ${\mathfrak{g}}\,{\neq}\,\{0\}$ has positive depth. \[pp8.18\] Let ${\mathfrak{g}}$ be a quasi-effective finite dimensional $\operatorname{\mathbb{Z}}$-graded Lie algebra. If ${\mathfrak{g}}$ is of reductive type and depth $\muup,$ then ${\mathfrak{g}}_{{\mathpzc{p}}}\,{=}\,\{0\}$ for ${\mathpzc{p}}\,{>}\,\muup.$ We note that ${\mathfrak{g}}_{{\mathpzc{p}}}$ is orthogonal to ${\mathfrak{g}},$ and hence belongs to ${\mathfrak{r}}_{{\mathpzc{p}}},$ for ${\mathpzc{p}}{>}\muup.$ For positive ${\mathpzc{p}}$ we have ${\mathfrak{r}}_{{\mathpzc{p}}}\,{=}\,{\mathfrak{n}}_{{\mathpzc{p}}}$ and thus the statement is a consequence of Lemma \[lm8.16\]. A finite dimensional $\operatorname{\mathbb{Z}}$-graded Lie algebra ${\mathfrak{g}}$ has a $\operatorname{\mathbb{Z}}$-graded solvable radical ${\mathfrak{r}},$ whose depth $\muup_{r}$ we call its *solvable depth*; besides, all its $\operatorname{\mathbb{Z}}$-graded semisimple Levi factors have the same depth $\muup_{s},$ that we call its *semisimple depth*. In case ${\mathfrak{g}}$ is solvable, we set $\muup_{s}{=}0$ and, likewise, we set $\muup_{r}{=}0$ when it is semisimple. We have the following \[thm8.22\] A finite dimensional $\operatorname{\mathbb{Z}}$-graded quasi-effective Lie algebra of reductive type has a $\operatorname{\mathbb{Z}}$-graded Levi-Malčev decomposition $$\label{q8.17} {\mathfrak{g}}=\operatorname{\mathfrak{s}}\oplus\,{\mathfrak{r}}$$ with a $\operatorname{\mathbb{Z}}$-graded semisimple Levi factor $$\label{q8.18} \operatorname{\mathfrak{s}}={\sum}_{{\mathpzc{p}}=-\muup_{s}}^{\muup_{s}}\operatorname{\mathfrak{s}}_{{\mathpzc{p}}}$$ and a $\operatorname{\mathbb{Z}}$-graded radical $$\label{q8.19} {\mathfrak{r}}={\sum}_{{\mathpzc{p}}=-\muup_{r}}^0{\mathfrak{r}}_{{\mathpzc{p}}}$$ with no homogeneous term of positive degree. Moreover, ${\mathfrak{g}}$ is decomposable if and only if $\rhoup(A)$ is semisimple on ${\textswab{m}}$ for all $A\,{\in}\,{\mathfrak{r}}_0.$ When ${\mathfrak{g}}$ is decomposable, quasi-effective and of the reductive type, the $\operatorname{\mathbb{Z}}$-graded Levi factor $\operatorname{\mathfrak{s}}$ can be chosen in such a way that $$\label{q8.20} [{\mathfrak{r}}_0,\operatorname{\mathfrak{s}}]=\{0\}.$$ The fact that for every $\operatorname{\mathbb{Z}}$-graded Levi-Malčev decomposition the gradings of the Levi factor $\operatorname{\mathfrak{s}}$ and of the solvable radical ${\mathfrak{r}}$ are as in and follows from Lemma \[lm8.16\]. By Proposition \[pp8.13\] (see also [@Bou82 Thm.2, Ch.VII, §[5]{}]) a quasi-effective and therefore an almost effective ${\mathfrak{g}}$ is decomposable if and only if $\rhoup({\mathfrak{r}}_{0})\,{\subseteq}\,{\mathfrak{gl}}_{{\mathbb{K}}}({\textswab{m}})$ is decomposable. If the elements of ${\mathfrak{r}}_{0}$ are $\operatorname{\mathrm{ad}}$-semisimple, then for each $A\,{\in}\,{\mathfrak{r}}_{0}$ we have ${\mathfrak{g}}\,{=}\,[A,{\mathfrak{g}}]{\oplus}\{X\,{\in}\,{\mathfrak{g}}\,{\mid}\,[A,X]\,{=}0\}.$ Hence $\operatorname{{\kappaup}}\,{=}\,\{X\,{\in}\,{\mathfrak{g}}\,{\mid}\,[X,{\mathfrak{r}}_{0}]\,{=}\,\{0\}\}$ is a $\operatorname{\mathbb{Z}}$-graded Lie subalgebra of ${\mathfrak{g}}$ such that ${\mathfrak{g}}\,{=}\,\operatorname{{\kappaup}}{+}{\mathfrak{r}}$ and a $\operatorname{\mathbb{Z}}$-graded semisimple Levi factor of $\operatorname{{\kappaup}}$ is then a $\operatorname{\mathbb{Z}}$-graded semisimple Levi factor of ${\mathfrak{g}}$ satisfying . This completes the proof. Assume that ${\mathfrak{g}}$ is an effective finite dimensional $\operatorname{\mathbb{Z}}$-graded Lie algebra with ${\textswab{m}}$ fundamental. If $\rhoup_{-1}$ is simple, either ${\mathfrak{g}}$ is semisimple or ${\mathfrak{n}}\,{=}\,{\textswab{m}}$ and ${\mathfrak{g}}\,{=}\,{\textswab{m}}\,{\oplus}{\mathfrak{g}}_{0}.$ By Proposition \[pp8.13\] and Lemma \[ll8.16\] we know that ${\mathfrak{g}}$ is decomposable and of reductive type. Consider a $\operatorname{\mathbb{Z}}$-graded Levi-Malčev decomposition for which , , are satisfied. Since we assumed that $\rhoup_{-1}$ is irreducible, either ${\textswab{m}}\,{\subseteq}\,\operatorname{\mathfrak{s}},$ or ${\textswab{m}}\,{\subseteq}\,{\mathfrak{r}}.$ In the first case ${\mathfrak{r}}\,{=}\,{\mathfrak{r}}_{0}\,{=} \{0\}$ by the assumption that ${\mathfrak{g}}$ is effective, because $[{\mathfrak{r}}_{0},{\textswab{m}}]=0$ and thus ${\mathfrak{g}}\,{=}\,\operatorname{\mathfrak{s}}.$ In the second case we have $\operatorname{\mathfrak{s}}\,{\subseteq}\,{\mathfrak{g}}_{0}$ and ${\textswab{m}}\,{=}\,{\textswab{m}}\,{\oplus}\,{\mathfrak{g}}_{0},$ because ${\mathfrak{g}}_{{\mathpzc{p}}}\,{=}\,{\mathfrak{n}}_{{\mathpzc{p}}}\,{=}\,\{0\}$ for ${\mathpzc{p}}\,{>}\,0.$ For a positive integer ${\mathpzc{h}}{>}1,$ we consider on ${\textsf{V}}\,{=}\,{\mathbb{K}}^{3}$ the $\operatorname{\mathbb{Z}}$-gradation described by $E_{{\textsf{V}}}{=}\!\left( \begin{smallmatrix} 0\\ &{\mathpzc{h}}\\ && 1 \end{smallmatrix}\right).$ Then the upper triangular nilpotent $3{\times}3$ matrices are a $\operatorname{\mathbb{Z}}$-graded Lie subalgebra ${\mathfrak{g}}{=}{\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}\,{\mathfrak{g}}_{{\mathpzc{p}}}$ with ${\mathfrak{g}}_{{\mathpzc{p}}}{=}\{0\}$ for ${\mathpzc{p}}\neq{-}{\mathpzc{h}}, {\mathpzc{h}}{-}1, {-}1$ and $${\mathfrak{g}}_{-{\mathpzc{h}}}=\left\{\left( \begin{smallmatrix} 0 & \lambdaup & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{smallmatrix}\right)\right\}, \;\; {\mathfrak{g}}_{-1}=\left\{\left( \begin{smallmatrix} 0 & 0 & \lambdaup\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{smallmatrix}\right)\right\},\;\; {\mathfrak{g}}_{{\mathpzc{h}}-1}=\left\{\left( \begin{smallmatrix} 0 & 0 & 0\\ 0 & 0 &\lambdaup\\ 0 & 0 & 0 \end{smallmatrix}\right)\right\},$$ which is almost, but not quasi-effective. Moreover, ${\mathfrak{g}}$ is trivially of reductive type, because ${\mathfrak{g}}_{0}{=}\{0\}.$ Note that is not valid in this case. Let ${\textsf{V}}\,{=}\,{\mathbb{K}}^{4}$ and, for a pair ${\mathpzc{h}},{\mathpzc{k}}$ of positive integers, with ${\mathpzc{h}}{+}{\mathpzc{k}}{>}2,$ consider on ${\textsf{V}}$ the $\operatorname{\mathbb{Z}}$-gradation provided by $E_{{\textsf{V}}}=\left( \begin{smallmatrix} 0 \\ & {\mathpzc{h}}\\ && {-}{\mathpzc{k}}\\ &&& 1 \end{smallmatrix} \right).$ We consider the $\operatorname{\mathbb{Z}}$-graded subalgebra ${\mathfrak{g}}$ of ${\mathfrak{gl}^{\vee}}_{{\mathbb{K}}}({\textsf{V}})$ consisting of the upper triangular nilpotent $4{\times}4$ matrices. Then ${\mathfrak{g}}{=}{\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}\,{\mathfrak{g}}_{{\mathpzc{p}}}$ with ${\mathfrak{g}}_{{\mathpzc{p}}}\,{=}\,\{0\}$ for ${\mathpzc{p}}\notin\{ {-}{\mathpzc{h}}, -(1{+}k), {-}1, {\mathpzc{k}}, ({\mathpzc{h}}{-}1), ({\mathpzc{h}}{+}{\mathpzc{k}})\}$ and $$\begin{aligned} {\mathfrak{g}}_{-h}=\left\{\left( \begin{smallmatrix} 0 & \lambdaup & 0 & 0\\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{smallmatrix}\right)\right\},\; {\mathfrak{g}}_{-{\mathpzc{k}}-1}=\left\{\left( \begin{smallmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \lambdaup \\ 0 & 0 & 0 & 0 \end{smallmatrix}\right)\right\},\; {\mathfrak{g}}_{-1}=\left\{\left( \begin{smallmatrix} 0 & 0 & 0 & \lambdaup\\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{smallmatrix}\right)\right\},\\ {\mathfrak{g}}_{{\mathpzc{k}}}=\left\{\left( \begin{smallmatrix} 0 & 0 & \lambdaup & 0\\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{smallmatrix}\right)\right\},\; {\mathfrak{g}}_{{\mathpzc{h}}-1}=\left\{\left( \begin{smallmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \lambdaup \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{smallmatrix}\right)\right\},\; {\mathfrak{g}}_{{\mathpzc{h}}+{\mathpzc{k}}}=\left\{\left( \begin{smallmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & \lambdaup & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{smallmatrix}\right)\right\}.\end{aligned}$$ Then ${\mathfrak{g}}$ is almost, but not quasi-effective. Moreover, ${\mathfrak{g}}$ is of reductive type if and only if ${\mathpzc{h}}{>}1.$ Its depth is $\sup\{{\mathpzc{h}},{\mathpzc{k}}{+}1\},$ which, if ${\mathpzc{k}},{\mathpzc{h}}{>}1,$ is strictly smaller than the maximum index ${\mathpzc{p}}$ for which ${\mathfrak{g}}_{{\mathpzc{p}}}{\neq}\{0\},$ which is ${\mathpzc{h}}{+}{\mathpzc{k}}.$ Also in this case is not valid. We will be primarily interested in finite dimensional quasi-effective $Z$-graded ${\mathfrak{g}}$ having a *reductive* [structure algebra]{} ${\mathfrak{g}}_{0}.$ Before getting into this, let us briefly discuss how to construct general finite dimensional nilpotent prolongation . By Engel’s theorem, a faithful representation of ${\mathfrak{g}}$ will lead to realise ${\mathfrak{g}}$ as a Lie algebra of nilpotent upper triangular $n{\times}n$ matrices $$\begin{pmatrix} 0 & x_{1,2}& x_{1,3} & \hdots & x_{1,n-1}&x_{1,n}\\ 0 & 0 & x_{2,3}& \hdots & x_{2,n-1}& x_{2,n}\\ \vdots & \ddots & \ddots & \ddots & \ddots &\vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots& \vdots \\ 0 & 0 & 0 &\hdots & 0 & x_{n-1,n}\\ 0 & 0 & 0 & \hdots & 0 & 0 \end{pmatrix}.$$ A grading on ${\mathfrak{g}}$ can be described by the action of an integral diagonal matrix ${E}{=}{\mathrm{diag}}(k_{1},\hdots,k_{n})$: the entry $x_{i,j}$ is then *homogeneous* of degree $k_{i}{-}k_{j}.$ Let us consider the specific example consisting of the Lie algebra ${\mathfrak{g}}$ of $8{\times}8$ upper triangular matrices with entries in ${\mathbb{K}}$ which are antisymmetric with respect to the second diagonal. These matrices belong to the orthogonal algebra of the bilinear symmetric form ${\textswab{b}}({\mathpzc{v}},{\mathpzc{w}}) \,{=}\,{\mathpzc{v}}^{\intercal}{\textsf{J}}{\mathpzc{w}},$ with $${\textsf{J}}= \left( \begin{smallmatrix} &&&&&&& 1\\ &&&&&& 1 \\ &&&&& 1\\ &&&& 1\\ &&& 1 \\ && 1 \\ & 1\\ 1 \end{smallmatrix}\right)$$ and are indeed the largest nilpotent ideal of a Borel subalgebra of a split form of $\mathbf{D}_{4}.$ By using the gradation with ${E}={\mathrm{diag}}(-1,-2,-1,0,0,1,2,1),$ we obtain on ${\mathfrak{g}}$ a structure of $12$-dimensional nilpotent fundamental graded Lie algebra $$\begin{gathered} {\mathfrak{g}}\,{=}\,{\sum}_{{\mathpzc{p}}=-3}^{1}{\mathfrak{g}}_{{\mathpzc{p}}}, \;\;\;\text{with}\\ \dim_{{\mathbb{K}}}({\mathfrak{g}}_{-3}){=}2, \; \dim_{{\mathbb{K}}}({\mathfrak{g}}_{-2}){=}3,\; \dim_{{\mathbb{K}}}({\mathfrak{g}}_{-1}){=}5,\; \dim_{{\mathbb{K}}}({\mathfrak{g}}_{0}){=}1,\; \dim_{{\mathbb{K}}}({\mathfrak{g}}_{1}){=}1.\end{gathered}$$ The nonzero elements of ${\mathfrak{g}}_{0}$ have rank $2$ on ${\mathfrak{g}}_{-1}$ and therefore the maximal prolongation of type ${\mathfrak{g}}_{0}$ of ${\textswab{m}}{=}{\sum}_{{\mathpzc{p}}=-3}^{-1}{\mathfrak{g}}_{{\mathpzc{p}}}$ is finite dimensional (see e.g. [@MMN2018]). We obtain a solvable prolongation of ${\mathfrak{g}}$ by adding to ${\mathfrak{g}}_{0}$ the $8{\times}8$ diagonal matrices which are antisymmetric with respect to the second diagonal. Semisimple prolongations {#sect9} ======================== By Theorem \[thm8.22\] all positive degree summands of an effective $\operatorname{\mathbb{Z}}$-graded finite dimensional Lie algebra ${\mathfrak{g}}$ of reductive type are contained in its graded semisimple Levi factor. It is therefore of some interest investigating the way $\operatorname{\mathbb{Z}}$-gradations of semisimple Lie algebras relate to (maximal) prolongations of fundamental graded Lie algebras. Let ${\mathfrak{g}}={\sum}_{{\mathpzc{p}}={-}\muup}^\muup{\mathfrak{g}}_{{\mathpzc{p}}}$ be a finite dimensional $\operatorname{\mathbb{Z}}$-graded semisimple Lie algebra over ${\mathbb{K}}.$ We say that its gradation is *not trivial* if $\muup{>}0$ and ${\mathfrak{g}}_{\muup}{\neq}\{0\}.$ Moreover, ${\mathfrak{g}}$ is effective iff none of its nontrivial ideals is contained in ${\mathfrak{g}}_0.$ Its Lie subalgebra ${\mathfrak{g}}_0$ is reductive, since the restriction of the Killing form of ${\mathfrak{g}}$ to ${\mathfrak{g}}_{0}$ is nondegenerate. Set $ {\textswab{m}}={\sum}_{{\mathpzc{p}}={-}\muup}^{{-}1}{\mathfrak{g}}_{{\mathpzc{p}}}$ and $V{=}{\mathfrak{g}}_{{-}1}.$ Assuming that ${\textswab{m}}$ is fundamental, ${\mathfrak{g}}$ is an effective prolongation of ${\textswab{m}}$ if and only if the action of ${\mathfrak{g}}_0$ on $V$ is faithful: in this case ${\mathfrak{g}}_0$ can be identified with a Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}(V).$ The derived algebra $[{\mathfrak{g}}_0,{\mathfrak{g}}_0]$ is semisimple and $V$ decomposes into a direct sum $V=V_1{\oplus}\cdots{\oplus}V_k$ of its irreducible representations. For each $1{\leq}i{\leq}k,$ the set ${\mathbb{A}}_i$ of ${\mathbb{K}}$-endomorphisms of $V_i$ which commute with the action of $[{\mathfrak{g}}_0,{\mathfrak{g}}_0]$ is, by Schur’s lemma, a division ${\mathbb{K}}$-algebra. The elements of ${\mathbb{A}}_i$ uniquely extend to derivations of ${\mathfrak{g}}$ vanishing on $V_{\!{j}}$ for $j{\neq}i.$ In particular, the identity of ${\mathbb{A}}_i$ yields a projection $\etaup_{\,i}:V{\to}V_i.$ Since every derivation of a semisimple finite dimensional Lie algebra is inner, we can consider the ${\mathbb{A}}_i$’s as subalgebras of ${\mathfrak{g}}_0.$ Assume that ${\mathfrak{g}}$ is finite dimensional and semisimple and that the action of ${\mathfrak{g}}_0$ on $V$ is faithful. Then the center of ${\mathbb{A}}_1\oplus\cdots\oplus{\mathbb{A}}_k$ is the center of ${\mathfrak{g}}_0.$ The commutant ${\mathbb{A}}_i$ may contain a simple Lie algebra over ${\mathbb{K}},$ that will contribute as a summand to $[{\mathfrak{g}}_0,{\mathfrak{g}}_0].$ For instance, when ${\mathbb{K}}$ is the field $\operatorname{{\mathbb{R}}}$ of real numbers, the possible ${\mathbb{A}}_i$ are $\operatorname{{\mathbb{R}}}$ itself, or $\operatorname{\mathbb{C}},$ or the non commutative real division algebra ${\mathbb{H}}$ of quaternions, and ${\mathbb{H}}{\simeq}{\mathfrak{o}}(3){\oplus}\operatorname{{\mathbb{R}}}.$ A simple instance of this situation is the simple Lie algebra ${\mathfrak{sl}}_2({\mathbb{H}}),$ with ${\mathfrak{g}}_0{=} {\mathfrak{o}}(3){\oplus}{\mathfrak{o}}(3){\oplus}\operatorname{{\mathbb{R}}}{\simeq}{\mathfrak{o}}(4){\oplus}\operatorname{{\mathbb{R}}}{\simeq} \mathfrak{co}(4),$ with the standard action on $V{=}\operatorname{{\mathbb{R}}}^4{\simeq}{\mathbb{H}},$ where ${\mathfrak{sl}}_2({\mathbb{H}})$ can be viewed as a maximal prolongation of type $\mathfrak{co}(4)$ of $V.$ This presentation corresponds to the *cross marked Satake diagram* (see e.g. [@AMN06]) $$\vspace{-15pt} \xymatrix@R=-.3pc{ \!\!\medbullet\!\!\! \ar@{-}[r]&\!\!\medcirc\!\! \ar@{-}[r]& \!\!\medbullet\\ &{\,\times}}$$ Semisimple and maximal prolongations are related by the following proposition (see e.g. [@MN97; @Tan67]). \[prop-7.3\] Assume that ${\mathfrak{g}}$ is semisimple and that the action of ${\mathfrak{g}}_0$ on $V$ is faithful. Then ${\mathfrak{g}}$ is maximal among the finite dimensional effective prolongations of type ${\mathfrak{gl}}_{{\mathbb{K}}}(V)$ of ${\textswab{m}}.$ Indeed, an effective finite dimensional prolongation ${\mathfrak{G}}{=}{\sum}_{{\mathpzc{p}}{\in}\operatorname{\mathbb{Z}}}{\mathfrak{G}}_{{\mathpzc{p}}}$ of ${\textswab{m}}$ containing ${\mathfrak{g}}$ is a ${\mathfrak{g}}$-module. Since the finite dimensional linear representations of a semisimple ${\mathfrak{g}}$ are completely reducible, ${\mathfrak{g}}$ has in ${\mathfrak{G}}$ a complementary $\operatorname{\mathbb{Z}}$-graded ${\mathfrak{g}}$-module ${\mathfrak{G}}'.$ The conditions that $[{\mathfrak{G}}',{\textswab{m}}]{\subset}{\mathfrak{G}}'$ and that ${\mathfrak{G}}'$ is contained in ${\mathfrak{G}}_{+}{=}{\sum}_{{\mathpzc{p}}{\geq}0}{\mathfrak{G}}_{{\mathpzc{p}}}$ implies by effectiveness that ${\mathfrak{G}}'=\{0\}.$ If ${\textswab{m}}\,{=}\,{\mathfrak{g}}_{-1}\,{=}\,{\mathbb{K}}^{n},$ then ${\mathfrak{sl}}_{n+1}({\mathbb{K}})$ is the unique finite dimensional semisimple prolongation of ${\textswab{m}}.$ However, the maximal prolongation of type ${\mathfrak{gl}}_{n}({\mathbb{K}})$ of ${\textswab{m}}$ is the infinite dimensional graded Lie algebra ${\mathpzc{X}}({\mathbb{K}}^{n})$ of vector fields with polynomial coefficients in ${\mathbb{K}}^{n}$ (see e.g. [@MMN2018 §[3]{}]). In the rest of this section, we will exhibit structures of maximal $\operatorname{\mathbb{Z}}$-graded prolongations on semisimple Lie algebras, that we think could be of some interest in geometry and physics. Let us explain the pattern of our constructions. We start from a semismiple Lie algebra $\operatorname{\mathfrak{L}}_0$ and fix a faithful finite dimensional $\operatorname{\mathfrak{L}}_0$-module $V,$ identifying $\operatorname{\mathfrak{L}}_0$ with a Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}(V).$ The structure algebra ${\mathfrak{g}}_0$ is obtained by adding to $\operatorname{\mathfrak{L}}_0$ its commutant in ${\mathfrak{gl}}_{{\mathbb{K}}}(V).$ Then ${\mathfrak{g}}_0$ is reductive, with $\operatorname{\mathfrak{L}}_0\,{\subseteq}\,[{\mathfrak{g}}_0,{\mathfrak{g}}_0].$ The derived algebra $\operatorname{\mathfrak{L}}{\coloneqq}[{\mathfrak{g}}_0,{\mathfrak{g}}_0]$ is the semisimple ideal of ${\mathfrak{g}}_0.$ The exterior power $\Lambda^2(V)$ is an $\operatorname{\mathfrak{L}}$-module and we can choose ${\mathfrak{g}}_{{-}2}$ equal to any $\operatorname{\mathfrak{L}}$-submodule of $\Lambda^2(V).$ Likewise, all homogeneous summands in the natural gradation of the free Lie algebra $\operatorname{\mathfrak{f}}(V){=}{\sum}_{{\mathpzc{p}}<0}\operatorname{\mathfrak{f}}_{{\mathpzc{p}}}(V)$ of $V$ (cf. [@MMN2018; @Reu93; @Warhurst2007]) are $\operatorname{\mathfrak{L}}$-modules and we can choose ${\mathfrak{g}}_{{-}3}$ as an $\operatorname{\mathfrak{L}}$-submodule of $(V{\otimes}{\mathfrak{g}}_{{-}2})\cap\operatorname{\mathfrak{f}}_{{-}3}(V),$ and next, by recurrence, ${\mathfrak{g}}_{{-}{\mathpzc{p}}{-}1}$ as an $\operatorname{\mathfrak{L}}$-submodule of the intersection $(V{\otimes}{\mathfrak{g}}_{{-}{\mathpzc{p}}})\cap\operatorname{\mathfrak{f}}_{{-}{\mathpzc{p}}{-}1}(V).$ In this way we build up the general fundamental graded Lie algebra ${\textswab{m}}$ with ${\mathfrak{g}}_{{-}1}{=}V$ admitting a maximal prolongation with structure subalgebra $\operatorname{\mathfrak{L}}.$ A randomly constructed ${\textswab{m}}$ may have, in general, no $\operatorname{\mathbb{Z}}$-graded prolongation of positive heigth. For this, we need to make specific choices of ${\textswab{m}}.$ Let ${\mathbb{K}}$ be the field $\operatorname{\mathbb{C}}$ of complex numbers. Fix a Cartan subalgebra ${\mathfrak{h}}$ of $\operatorname{\mathfrak{L}}$ and let ${\mathpzc{R}},$ ${\mathpzc{W}}$ be the corresponding root system and weight lattice, which are contained in a Euclidean space $\operatorname{{\mathbb{R}}}^{\ell}.$ For an $\omegaup\,{\in}\,{\mathpzc{W}},$ we denote by ${\mathpzc{W}}(\omegaup)$ the set of weights of an irreducible $\operatorname{\mathfrak{L}}$-module with extremal weigth $\omegaup.$ As usual, we denote by $(\alphaup|\betaup)$ the standard scalar product of $\alphaup,\betaup\in\operatorname{{\mathbb{R}}}^\ell$ and, for a nonzero element $\alphaup$ of $\operatorname{{\mathbb{R}}}^\ell$ we set $\alphaup^\vee=2\alphaup{/}\|\alphaup\|^2$ and $\langle\alphaup|\betaup\rangle=(\alphaup|\betaup^\vee).$ Then, by choosing a system $\alphaup_1,\hdots,\alphaup_{\ell}$ of simple roots in ${\mathpzc{R}},$ we associate to $\operatorname{\mathfrak{L}}$ the Cartan matrix $A{=} (\langle\alphaup_i\,|\,\alphaup_j\rangle)_{1{\leq}i,j{\leq}\ell}.$ Our next step is to construct, starting from the data of $\operatorname{\mathfrak{L}}$ and $V$ a *generalised Cartan matrix* $\tilde{A}$ extending $A$ (see [@kac_1990; @Moody68]). The matrix $\tilde{A}$ will be associated to an isometric embedding of $\operatorname{{\mathbb{R}}}^{\ell}$ into an orthogonal space $(\operatorname{{\mathbb{R}}}^{\ell{+}k},{\textswab{b}})$ ($k$ is the number of irreducible $\operatorname{\mathfrak{L}}$-modules $V_i$ in $V$). Here ${\textswab{b}}$ is a symmetric bilinear form on $\operatorname{{\mathbb{R}}}^{\ell{+}k},$ which restricts to the Euclidean scalar product on $\operatorname{{\mathbb{R}}}^\ell.$ For each $V_i,$ let $\omegaup_i$ be its *lowest* weight (i.e. the one with $(\omegaup_i|\alphaup_j){\leq}0$ for all $j{=}1,\hdots,\ell$) and define a new *simple root* $\alphaup_{\ell{+}i}=\omegaup_i{+}{\epsilonup}_i$ by adding to $\omegaup_i$ a *marker* ${\epsilonup}_i$ from $\mathrm{E}{=}\{{\mathpzc{w}}{\in}\operatorname{{\mathbb{R}}}^{\ell{+}k} \mid {\textswab{b}}({\mathpzc{v}},{\mathpzc{w}}){=}0,\;\forall{\mathpzc{v}}{\in}\operatorname{{\mathbb{R}}}^\ell\}.$ We choose linearly independent ${\epsilonup}_1,\hdots,{\epsilonup}_k$ in $\mathrm{E}$ such that ${\textswab{b}}(\alphaup_i,\alphaup_i){\neq}0$ for all $1{\leq}i{\leq}\ell{+}k$ and set $$\tilde{A}= (a_{i,j})_{1{\leq}i,j{\leq}\ell{+}k} =\left(\dfrac{2{{\textswab{b}}}(\alphaup_i,\alphaup_j)}{{{\textswab{b}}}(\alphaup_i,\alphaup_i)}\right)_{1{\leq}i,j{\leq}\ell{+}k}.$$ Then $A$ is the submatrix of the first $\ell$ lines and columns of $\tilde{A}.$ The elements $a_{i,j}$ with $i{\leq}\ell$ are determined by $\operatorname{\mathfrak{L}}$ and $V,$ because $a_{i,j}{=}\langle\alphaup_i\,|\,\omegaup_{j{-}\ell}\rangle$ if $\ell{<}j{\leq}\ell{+}k.$ The others depend on the choice of ${\textswab{b}},$ for which we need to keep the constrain that $\tilde{A}$ be a generalised Cartan matrix, having nonpositive integers off the main diagonal. The restriction of ${\textswab{b}}$ to $\mathrm{E}$ must satisfy $$\label{eq7.1} \begin{cases} \|\alphaup_{\ell{+}i}\|^2{=} \|\omegaup_i\|^2{+}{\textswab{b}}({\epsilonup}_i,{\epsilonup}_i)>0,&\text{for $1{\leq}i{\leq}k,$} \\[5pt] \dfrac{2(\omegaup_i\,|\,\alphaup_j)}{\|\omegaup_i\|^2{+}{\textswab{b}}({\epsilonup}_i,{\epsilonup}_i)}\in\operatorname{\mathbb{Z}}, & \text{for $1{\leq}i{\leq}k,$ $1{\leq}j{\leq}\ell,$}\\[14pt] \dfrac{2(\omegaup_i\,|\,\omegaup_j){+}2{\textswab{b}}({\epsilonup}_i,{\epsilonup}_j)}{\|\omegaup_i\|^2{+}{\textswab{b}}({\epsilonup}_i,{\epsilonup}_i)}\in\operatorname{\mathbb{Z}}, &\text{for $1{\leq}i{\neq}j{\leq}k,$}\\[12pt] {(\omegaup_i\,|\,\omegaup_j){+}{\textswab{b}}({\epsilonup}_i,{\epsilonup}_j)}{\leq}0, &\text{for $1{\leq}i{\neq}j{\leq}k.$} \end{cases}$$ Then ${\mathfrak{g}}_0{=}\operatorname{\mathfrak{L}},$ ${\mathfrak{g}}_{{-}1}{=}V,$ for a $\operatorname{\mathbb{Z}}$-gradation of the Kac-Moody algebra ${\mathfrak{g}}$ of $\tilde{A},$ which is finite dimensional iff ${\textswab{b}}$ is positive definite (see e.g. [@kac_1990]). In general, we may consider choices for which ${\textswab{b}}$ has maximal positive/non-negative inertia. Note that ${\textswab{b}}$ is completely determined by the values of the entries of . When ${\mathfrak{g}}$ is infinite dimensional, the maximal effective prolongation of type ${\mathfrak{g}}_0$ of ${\textswab{m}}$ is in general strictly larger than ${\mathfrak{g}}.$ \[rmk7.4\] When $V$ is irreducible, with extremal weight $\omegaup,$ we need to add a unique marker ${\epsilonup}.$ Then conditions reduce to the first two and $\tilde{A}$ is completely determined by the value of ${\textswab{b}}({\epsilonup},{\epsilonup}).$ We require that $$\label{eq7.2} \|\omegaup\|^2{+}{\textswab{b}}({\epsilonup},{\epsilonup}){>}0 \;\;\text{and}\;\; \dfrac{2(\alphaup|\omegaup)}{ \|\omegaup\|^2 {+}{\textswab{b}}({\epsilonup},{\epsilonup})} \in\operatorname{\mathbb{Z}},\;\;\forall\alphaup{\in}{\mathpzc{R}}.$$ In particular, to obtain a finite dimensional semisimple ${\mathfrak{g}},$ we need that $\|\omegaup\|^2<\min\{2|(\omegaup|\alphaup)|>0 \mid\alphaup{\in}{\mathpzc{R}}\}.$ To discuss the case where ${\mathbb{K}}{=}\operatorname{{\mathbb{R}}},$ we observe that the complexification of the effective prolongation of a real fundamental graded Lie algebra ${\textswab{m}}$ is the complex effective prolongation of the complexification of ${\textswab{m}}.$ In this way we reduce to the complex case, by taking into account the way real representations lift to complex ones (see e.g. [@Bou82 Ch.IX,Appendix]). Semisimple effective prolongations ${\mathfrak{g}}$ can be read off their associated diagrams of Dynkin/Satake. In particular, for a Satake diagram $\Sigma$ (see e.g. [@AMN06; @Ara62]) of a real semisimple effective prolongation ${\mathfrak{g}}$ we require that - the eigenspaces corresponding to fundamental roots of $\Sigma$ are homogeneous of degree either $0$ or $1;$ - compact roots have degree $0$ and those joined by an arrow have the same degree; - degree $0$ roots are the nodes of the Satake diagram of $[{\mathfrak{g}}_0,{\mathfrak{g}}_0].$ Crosses can be added under the nodes of a Satake diagram to indicate the roots of positive degree. Complex type representations can be associated to couples of *positive* roots joined by an arrow (see e.g. [@MN98]). Exceptional Lie algebras naturally arise as maximal effective prolongations of fundamental graded Lie algebras with non exceptional structure algebras. These constructions are related to the investigation of their maximal rank reductive subalgebras (see e.g. [@adams1996lectures; @golubitsky1971]). Structure algebras of type $\mathbf{A}$ --------------------------------------- To describe the root system of ${\mathfrak{sl}}_n(\operatorname{\mathbb{C}}),$ it is convenient to use an orthonormal basis ${\mathpzc{e}}_1,\hdots,{\mathpzc{e}}_n$ of $\operatorname{{\mathbb{R}}}^n$ and set $${\mathpzc{R}}=\{{\pm}({\mathpzc{e}}_i{-}{\mathpzc{e}}_j)\mid 1{\leq}i{<}j{\leq}n\}.$$ The Dynkin diagram is $$\xymatrix@R=-.3pc{ \alphaup_1 & \alphaup_2 & &\alphaup_{n{-}2} &\alphaup_{n{-}1} \\ \!\!\medcirc\!\!\! \ar@{-}[r]&\!\!\medcirc\!\! \ar@{-}[r]& \cdots \ar@{-}[r]& \!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc}$$ with simple roots $$\alphaup_i={\mathpzc{e}}_{i}{-}{\mathpzc{e}}_{i{+}1},\;\;\text{for}\;\; 1{\leq}i{\leq}n{-}1.$$ Set ${\mathpzc{e}}_0{=}{\mathpzc{e}}_1{+}\cdots{+}{\mathpzc{e}}_n.$ The simple positive weights in $\langle{\mathpzc{e}}_i{-}{\mathpzc{e}}_j\, {\mid} 1{\leq}i{<}j{\leq}n\rangle {\simeq}\operatorname{{\mathbb{R}}}^{n{-}1}$ are $\omegaup_{k}{=}{\sum}_{j{=}1}^{k}{\mathpzc{e}}_{i}{-}\frac{k}{n}{\mathpzc{e}}_0,$ for $1{\leq}i{\leq}n{-}1,$ with $\omegaup_k$ corresponding to the irreducible representation $\Lambda^k(\operatorname{\mathbb{C}}^n).$ We have $$\begin{aligned} (\omegaup_j|\omegaup_k) =j{\cdot}\left(1-\frac{k}{n}\right)>0,\;\;\forall 1{\leq}j{\leq}k{<}n\end{aligned}$$ and hence, for a dominant weight (${\mathpzc{a}}_j{\geq}0$ for all $j$), $$\begin{aligned} \left\| {\sum}_{j{=}1}^{n{-}1}{\mathpzc{a}}_j\omegaup_j\right\|^2 ={\sum}_{j{=}1}j{\mathpzc{a}}_j\left[{\mathpzc{a}}_j\left(1{-}\frac{j}{n}\right) {+}2{\sum}_{k{=}j{+}1}^{n{-}1}{\mathpzc{a}}_k \left(1{-}\frac{k}{n}\right)\right]\\ > {\sum}_{j{=}1}^{n{-}1}{\mathpzc{a}}_j^2{\cdot} j{\cdot} \left(1-\frac{j}{n}\right).\end{aligned}$$ By Remark \[rmk7.4\], to find a semisimple effective fundamental graded Lie algebra ${\mathfrak{g}}$ with ${\mathfrak{g}}_0{=}{\mathfrak{gl}}_n(\operatorname{\mathbb{C}})$ and ${\mathfrak{g}}_{{-}1}$ equal to an irreducible ${\mathfrak{sl}}_n(\operatorname{\mathbb{C}})$-module $V{=}V_{\omegaup},$ we need that $\omegaup{\sim}\omegaup_i$ with - either $i\,{=}\,1, n{-}1,$ and any $n{\geq}{2},$ - or $i{=}2,n{-}2$ and $n{\geq}{4},$ - or $i{=}3,n{-}3$ and $6{\leq}n{\leq}9.$ A *marker* could be taken of the form ${\epsilonup}{=}{\mathpzc{c}}{\cdot}{\mathpzc{e}}_{\,0}.$ ### ${\mathfrak{g}}_{{-}1}{\simeq}V_{\omegaup_1}:$ construction of $\mathbf{B}_n$ and $\mathbf{G}_2$ {#sub7.1.1} Let $\omegaup{=}{\mathpzc{e}}_n{-}\tfrac{1}{n}{\mathpzc{e}}_0{\sim}\omegaup_1.$ By Remark \[rmk7.4\], the possible choices for ${\epsilonup}$ are $${\epsilonup}= \begin{cases} \tfrac{\sqrt{1{+}n}}{n}{\mathpzc{e}}_0 \Longrightarrow \|\omegaup{+}{\epsilonup}\|^2 {=}2,\\ \tfrac{1}{n}{\mathpzc{e}}_0 \Longrightarrow \|\omegaup{+}{\epsilonup}\|^2{=}1,\\ \tfrac{1}{2\sqrt{3}}({\mathpzc{e}}_1{+}{\mathpzc{e}}_2) \Longrightarrow \|\omegaup {+}{\epsilonup}\|^2{=}\tfrac{2}{3}, & \text{if $n{=}2.$} \end{cases}$$ The first corresponds to an ${\textswab{m}}$ of kind $1,$ the second to an ${\textswab{m}}$ of kind $2,$ the third to an ${\textswab{m}}$ of kind $3.$ The kind one abelian Lie algebra ${\textswab{m}}{=}V_{\omegaup_1}{\simeq}\operatorname{\mathbb{C}}^n$ has the simple effective prolongation ${\mathfrak{sl}}_{n{+}1}(\operatorname{\mathbb{C}}),$ but has the infinite dimensional maximal effective prolongation ${\mathpzc{X}}(V)$ (see e.g. [@MMN2018 §[3]{}]). Since $\Lambda^2(V_{\omegaup_1})$ is an irreducible ${\mathfrak{sl}}_n(\operatorname{\mathbb{C}})$-module, with dominant weight $\omegaup_2,$ the only possible choice for an ${\textswab{m}}$ of kind two is to take ${\textswab{m}}\,{=}\,\operatorname{\mathbb{C}}^n{\oplus}\Lambda^2(\operatorname{\mathbb{C}}^n).$ Indeed, with ${\epsilonup}=\tfrac{1}{n}{\mathpzc{e}}_0,$ we have $$\|\,\omegaup_1{+}{\epsilonup}\,\|^2{=}\|\,{\mathpzc{e}}_1\,\|^2{=}1, \quad \|\,\omegaup_2{+}2{\epsilonup}\,\|^2=\|{\mathpzc{e}}_1{+}{\mathpzc{e}}_2\|^2=2.$$ By [@MMN2018 Cor.4.11] when $n{\leq}2,$ $\dim({\mathfrak{g}}_{{-}2}){\leq}1$ and thus the maximal effective prolongation of this ${\textswab{m}}$ is infinite dimensional. It is finite dimensional by [@MMN2018 Thm.4.8] when $n{\geq}3,$ and in fact is isomorphic to ${\mathfrak{o}}(2n{+}1,\operatorname{\mathbb{C}})$ (see e.g. [@Yam93]). Indeed, setting $$\begin{cases} {\mathpzc{R}}_{\;{-}2}=\{\omegaup+2{\epsilonup}\mid \omegaup{\in}{\mathpzc{W}}(\omegaup_2)\}=\{{\mathpzc{e}}_i{+}{\mathpzc{e}}_j\mid 1{\leq}i{<}j{\leq}n\},\\ {\mathpzc{R}}_{\;{-}1}=\{\omegaup+{\epsilonup}\mid \omegaup{\in}{\mathpzc{W}}(\omegaup_1)\}=\{{\mathpzc{e}}_i\mid 1{\leq}i{\leq}n\},\\ {\mathpzc{R}}_{\;0}= \{{\mathpzc{e}}_i{-}{\mathpzc{e}}_j\mid 1{\leq}i{\neq}j{\leq}n\},\\ {\mathpzc{R}}_{\;{}1}=\{\omegaup-{\epsilonup}\mid \omegaup{\in} {\mathpzc{W}}(\omegaup_{n{-}1})\}=\{{-}{\mathpzc{e}}_i\mid 1{\leq}i{\leq}n\},\\ {\mathpzc{R}}_{\;2}=\{\omegaup-2{\epsilonup}\mid \omegaup {\in}{\mathpzc{W}}(\omegaup_{n{-}2})\}=\{{-}{\mathpzc{e}}_i{-}{\mathpzc{e}}_j \mid 1{\leq}i{<}j{\leq}n\}, \end{cases}$$ the union ${\mathpzc{R}}={\bigcup}_{{\mathpzc{p}}{=}{-}2}^2{\mathpzc{R}}_{\;{\mathpzc{p}}}$ is a root system of type $\mathbf{B}_n$ and the prolongation $${\mathfrak{g}}={\sum}_{{\mathpzc{p}}{=}{-}2}^2{\mathfrak{g}}_{{\mathpzc{p}}}\simeq{\mathfrak{o}}(2n{+}1,\operatorname{\mathbb{C}}),\;\;\text{with}\;\; {\mathfrak{g}}_0 {=}{\mathfrak{h}}_n{\oplus}\langle{\mathpzc{R}}_{\;0}\rangle,\;\; {\mathfrak{g}}_{{\mathpzc{p}}}=\langle{\mathpzc{R}}_{\;{\mathpzc{p}}}\rangle,$$ where ${\mathfrak{h}}_n$ is an $n$-dimensional Cartan subalgebra, is by Proposition \[prop-7.3\], a maximal effective prolongation of ${\textswab{m}},$ because ${\mathfrak{o}}(2n{+}1,\operatorname{\mathbb{C}})$ is simple. We got indeed $${\mathpzc{R}}=\{{\pm}{\mathpzc{e}}_i\mid 1{\leq}i{\leq}n\} \cup\{{\pm}{\mathpzc{e}}_i{\pm}{\mathpzc{e}}_j\mid 1{\leq}i{<}j{\leq}n\}$$ for an orthonormal basis ${\mathpzc{e}}_1,\hdots,{\mathpzc{e}}_n$ of $\operatorname{{\mathbb{R}}}^n.$ The grading of ${\mathfrak{g}}$ could also have been obtained from the cross marked Dynkin diagram (see [@Bou68]) $$\xymatrix@R=-.3pc{ \alphaup_1 & &\alphaup_{n-1} &\alphaup_{n{-}1} & \alphaup_n\\ \!\!\medcirc\!\! \ar@{-}[r]& \cdots \ar@{-}[r]& \!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc\!\! \ar@{=>}[r]&\!\!\medcirc\\ &&&& \times}$$ with $$\alphaup_i={\mathpzc{e}}_{i}-{{\mathpzc{e}}}_{i{-}1},\;\text{for}\, 1{\leq}i{\leq}n{-}1,\;\;\alphaup_n={\mathpzc{e}}_n.$$ by setting $\deg(\alphaup_n){=}1,$ $\deg(\alphaup_i){=}0$ for $1{\leq}i{\leq}n{-}1.$ For $n{=}2,$ the summands $\operatorname{\mathfrak{f}}_{{-}{\mathpzc{p}}}(\operatorname{\mathbb{C}}^2)$ (for ${\mathpzc{p}}{\geq}1$) are all irreducible and isomorphic either to the trivial one-dimensional representation on $\Lambda^2(\operatorname{\mathbb{C}}^2)\simeq\operatorname{\mathbb{C}},$ for ${\mathpzc{p}}$ even, or to the two-dimensional standard representation $\Lambda^1(\operatorname{\mathbb{C}}^2){=}\operatorname{\mathbb{C}}^2$ for ${\mathpzc{p}}$ odd. Therefore we will consider the fundamental graded Lie algebra of the third kind ${\textswab{m}}=\operatorname{\mathbb{C}}^2{\oplus}\Lambda^2(\operatorname{\mathbb{C}}^2){\oplus}\operatorname{\mathfrak{f}}_{3}(\operatorname{\mathbb{C}}^2),$ with $\operatorname{\mathfrak{f}}_{3}(\operatorname{\mathbb{C}}^2){\simeq}\operatorname{\mathbb{C}}^2.$ Let $\omegaup{=}\tfrac{1}{2}({\mathpzc{e}}_2{-}{\mathpzc{e}}_1)$ and take the marker ${\epsilonup}=\tfrac{1}{2\sqrt{3}}({\mathpzc{e}}_1{+}{\mathpzc{e}}_2).$ Then $$\|\,\omegaup{+}{\epsilonup}\,\|^2=\frac{2}{3},\;\; \|\,2{\epsilonup}\,\|^2 =\frac{2}{3},\;\; \|\,\omegaup{+}3{\epsilonup}\,\|^2=2.$$ By setting $${\mathpzc{R}}={\bigcup}_{{\mathpzc{p}}{=}{-}3}^3{\mathpzc{R}}_{\;{\mathpzc{p}}},\;\;\;\text{with}\;\;\; \begin{cases} {\mathpzc{R}}_{\;0}=\{{\pm}2\omegaup,\},\\ {\mathpzc{R}}_{\;{\pm}1}=\{{\mp}{\epsilonup}+\omegaup,{\mp}{\epsilonup}-\omegaup\},\\ {\mathpzc{R}}_{\;{\pm}2}=\{{\mp}2{\epsilonup}\},\\ {\mathpzc{R}}_{\;{\pm}3}=\{{\mp}3{\epsilonup}{+}\omegaup,{\mp}3{\epsilonup}{-}\omegaup\}, \end{cases}$$ we obtain a root system of type $\mathbf{G}_2.$ With a $2$-dimensional Cartan subalgebra ${\mathfrak{h}}_2,$ the graded Lie algebra $${\sum}_{{\mathpzc{p}}{=}{-}3}^3{\mathfrak{g}}_{{\mathpzc{p}}},\;\; \text{with}\;\; {\mathfrak{g}}_0{=}{\mathfrak{h}}_2{\oplus}\langle{\mathpzc{R}}_{\;0}\rangle \;\text{and}\; {\mathfrak{g}}_{\mathpzc{p}}{=}\langle{\mathpzc{R}}_{\;{\mathpzc{p}}}\rangle \;\text{if ${\mathpzc{p}}{\neq}0,$}$$ is an effective prolongation of an fundamental graded Lie algebra of the third kind. It is the maximal one. Indeed, by [@MMN2018 Thm.5.3] the effective prolongations of ${\textswab{m}}{=}{\sum}_{{\mathpzc{p}}{<}0}{\mathfrak{g}}_{{\mathpzc{p}}}$ are finite dimensional, because in this case $W{=}\{0\}.$ Then, since ${\mathfrak{g}}$ is simple, it is maximal by Proposition \[prop-7.3\]. The cross marked Dynkin diagram is $$\xymatrix@R=-.3pc{ \alphaup_1 & \alphaup_2 \\ \!\!\medcirc\!\!\! \ar@3{->}[r]&\!\!\medcirc\!\! \\ &\times}\quad \qquad\begin{matrix} \qquad\quad \\ \alphaup_1{=}2\omegaup,\;\; \alphaup_2{=}\omegaup{-}{\epsilonup}, \end{matrix}$$ with $\deg(\alphaup_1){=}0,$ $\deg(\alphaup_2){=}1$ see e.g. [@Cartan1910; @golubitsky1971; @Wil1971]). This gradation and those introduced for $\mathbf{B}_n$ are compatible only with the split real forms of the complex simple Lie algebras considered above: we obtain structures of effective prolongation of type ${\mathfrak{gl}}_n(\operatorname{{\mathbb{R}}})$ and second kind on ${\mathfrak{o}}(n,n{+}1)$ for $n{\geq}3$ and of the third kind on the split real form of $\mathbf{G}_2$ for $n{=}2.$ ### ${\mathfrak{g}}_{{-}1}{\simeq}V_{2\omegaup_1},V_{3\omegaup_1} :$ construction of $\mathbf{C}_n$ and $\mathbf{G}_2$ Let us consider now the irreducible faithful representation of ${\mathfrak{sl}}_n(\operatorname{\mathbb{C}})$ on the space ${\mathpzc{S}}_2(\operatorname{\mathbb{C}}^n)$ of degree two symmetric tensors. From [@MMN2018 §[3.1]{}] we know that the maximal effective prolongation of type ${\mathfrak{gl}}_n(\operatorname{\mathbb{C}})$ of ${\mathpzc{S}}_2(\operatorname{\mathbb{C}}^n)$ is finite dimensional if $n{\geq}3.$ In fact, it is isomorphic to a Lie algebra of type $\mathbf{C}_n.$ The dominant weight of ${\mathpzc{S}}_2(\operatorname{\mathbb{C}}^n)$ is $2\omegaup_1$ and we have $\|\,2\omegaup_1\,\|^2=4{-}\frac{4}{n}.$ We consider the extremal weight $\omegaup{=}2{\mathpzc{e}}_n{-}\tfrac{2}{n}{\mathpzc{e}}_0$ and take the marker ${\epsilonup}{=}\frac{2}{n}{\mathpzc{e}}_0.$ Set $$\begin{cases} {\mathpzc{R}}_{\;{-}1}=\{{\mathpzc{w}}{+}{\epsilonup}\mid {\mathpzc{w}}{\in} {\mathpzc{W}}(2\omegaup_1)\}=\{({\mathpzc{e}}_i{+}{\mathpzc{e}}_j)\mid 1{\leq}i{\leq}j{\leq}n\},\\ {\mathpzc{R}}_{\;0} =\{{\mathpzc{e}}_i{-}{\mathpzc{e}}_j\mid 1{\leq}i{\neq}j{\leq}n\},\\ {\mathpzc{R}}_{\;1}=\{{\mathpzc{w}}{-}{\epsilonup}\mid {\mathpzc{w}}{\in}{\mathpzc{W}}(2\omegaup_{n{-}1})\}=\{-({\mathpzc{e}}_i{+}{\mathpzc{e}}_j)\mid 1{\leq}i{\leq}j{\leq}n\}. \end{cases}$$ Then ${\mathpzc{R}}={\bigcup}_{{\mathpzc{p}}{=}{-}1}^1{\mathpzc{R}}_{\;{\mathpzc{p}}}$ is a root system of type $\mathbf{C}_n.$ With a Cartan subalgebra ${\mathfrak{h}}_n$ of dimension $n,$ we obtain the maximal EPFGA of type ${\mathfrak{gl}}_n(\operatorname{\mathbb{C}})$ of ${\mathpzc{S}}_2(\operatorname{\mathbb{C}}^n)$ in the form $${\mathfrak{g}}={\sum}_{{\mathpzc{p}}{=}{-}1}^1{\mathfrak{g}}_{{\mathpzc{p}}}\simeq{\mathfrak{sp}}(n,\operatorname{\mathbb{C}}),\;\;\;\text{with}\;\;{\mathfrak{g}}_0{=}{\mathfrak{h}}_n{\oplus}\langle{\mathpzc{R}}_{\;0}\rangle, \;\; {\mathfrak{g}}_{{\mathpzc{p}}}=\langle{\mathpzc{R}}_{\;{\mathpzc{p}}}\rangle,\;\text{for ${\mathpzc{p}}{\neq}0.$}$$ Indeed, $${\mathpzc{R}}=\{{\pm}{\mathpzc{e}}_i{\pm}{\mathpzc{e}}_j\mid 1{\leq}i{<}j\leq{n}\}\cup\{2{\mathpzc{e}}_i\mid 1{\leq}i{\leq}n\}$$ is the set of roots of ${\mathfrak{sp}}(n,\operatorname{\mathbb{C}})$ and we can obtain the gradation above from its cross marked Dynkin diagram $$\xymatrix@R=-.3pc{ \alphaup_1 & \alphaup_2 & &\alphaup_{n-2} &\alphaup_{n-1} & \alphaup_n\\ \!\!\medcirc\!\! \ar@{-}[r]& \!\!\medcirc\!\! \ar@{-}[r]& \cdots \ar@{-}[r]& \!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc\!\! \ar@{<=}[r] & \!\!\medcirc \\ &&&& & \times}$$ with $ \alphaup_i={\mathpzc{e}}_{i}-{{\mathpzc{e}}}_{i+1},\;\text{for}\, 1{\leq}i{\leq}n{-}1,\;\;\alphaup_n=2{\mathpzc{e}}_n, $ by requiring that $\deg(\alphaup_i){=}0$ for $1{\leq}i{\leq}n{-}1$ and $\deg(\alphaup_n){=}1.$ For $n{=}2,$ we consider the irreducible ${\mathfrak{sl}}_2(\operatorname{\mathbb{C}})$-module ${\mathpzc{S}}_3(\operatorname{\mathbb{C}}^2){\simeq} V_{3\omegaup_1}.$ Its exterior square contains the irreducible representation $V_{0} \simeq \Lambda^2(\operatorname{\mathbb{C}}^2) \simeq \operatorname{\mathbb{C}}.$ One can check, by using [@MMN2018 §[3]{},§[4]{},§[6]{}] that the maximal effective prolongation of the fundamental graded Lie algebra ${\textswab{m}}={\mathfrak{g}}_{{-}1}\oplus{\mathfrak{g}}_{{-}2},$ with ${\mathfrak{g}}_{{-}1}={\mathpzc{S}}_3(\operatorname{{\mathbb{R}}}^2)$ and ${\mathfrak{g}}_{{-}2}=\Lambda^2(\operatorname{\mathbb{C}}^2),$ is finite dimensional. It is in fact a simple Lie algebra of type $\mathbf{G}_2.$ With the marker ${\epsilonup}=\frac{\sqrt{3}}{2}{\mathpzc{e}}_0$ we obtain $$\|\pm 3\omegaup_1{+}{\epsilonup}\|^2=6,\;\; \|\pm\omegaup_1{+}\,{\epsilonup}\,\|^2 =2,\;\;\|\,2{\epsilonup}\,\|^2=6.$$ Set $$\begin{cases} {\mathpzc{R}}_{\;{-}2}=\{2{\epsilonup}\},\\ {\mathpzc{R}}_{\;{-}1}=\{{\epsilonup}{\pm}\omegaup_1,{\epsilonup}{\pm}3\omegaup_1\},\\ {\mathpzc{R}}_{\;0}=\{\pm2\omegaup_1\},\\ {\mathpzc{R}}_{\;1}=\{{-}{\epsilonup}{\pm}\omegaup_1,{-}{\epsilonup}{\pm}3\omegaup_1\},\\ {\mathpzc{R}}_{\;2}=\{{-}2{\epsilonup}\}. \end{cases}$$ The union ${\mathpzc{R}}={\bigcup}_{{\mathpzc{p}}{=}{-}2}^2{\mathpzc{R}}_{\;{\mathpzc{p}}}$ is a root system of type $\mathbf{G}_2$ and the prolongation $${\mathfrak{g}}={\sum}_{{\mathpzc{p}}{=}{-}2}^2{\mathfrak{g}}_{{\mathpzc{p}}},\;\;\text{with}\;\; {\mathfrak{g}}_0 {=}{\mathfrak{h}}_2{\oplus}\langle{\mathpzc{R}}_{\;0}\rangle,\;\; {\mathfrak{g}}_{{\mathpzc{p}}}=\langle{\mathpzc{R}}_{\;{\mathpzc{p}}}\rangle,$$ with ${\mathfrak{h}}_2$ a Cartan subalgebra of dimension $2,$ is by Proposition \[prop-7.3\], a maximal effective prolongation of ${\textswab{m}},$ because ${\mathfrak{g}}$ is simple. The corresponding cross-marked Dynkin diagram is $$\xymatrix@R=-.3pc{ \alphaup_1 & \alphaup_2 \\ \!\!\medcirc\!\!\! \ar@3{<-}[r]&\!\!\medcirc\!\! \\ &\times}\quad \qquad\begin{matrix} \qquad\quad \\ \alphaup_1{=}2\omegaup_1,\;\; \alphaup_2{=}{-}(3\omegaup_1{+}{\epsilonup}), \end{matrix}$$ The discussion above applies to the split real form ${\mathfrak{gl}}_{n}(\operatorname{{\mathbb{R}}})$ of ${\mathfrak{gl}}_n(\operatorname{\mathbb{C}}),$ exhibiting ${\mathfrak{sp}}(n,\operatorname{{\mathbb{R}}})$ and the real split form of $\mathbf{G}_2$ as (EPGFLA)’s of a fundamental graded Lie algebra with structure algebra ${\mathfrak{gl}}_n(\operatorname{{\mathbb{R}}}).$ ### ${\mathfrak{g}}_{{-}1}{\simeq}V_{\omegaup_2}:$ construction of $\mathbf{D}_n$ {#sec-7.1.3} The next example refers to the representation of ${\mathfrak{sl}}_n(\operatorname{\mathbb{C}})$ on ${V}_{\omegaup_2}{=}\Lambda^2(\operatorname{\mathbb{C}}^n),$ for $n{\geq}4.$ Since the Dynkin diagram of a simple prolongation would have a ramification node, all its roots would have the same square lenght $2$. Take the extremal weight $\omegaup{=}{\mathpzc{e}}_{n{-}1}{+}{\mathpzc{e}}_n{-}\tfrac{2}{n}{\mathpzc{e}}_0$ and the marker ${\epsilonup}{=}\frac{2}{n}{\mathpzc{e}}_0$; then $\|{\epsilonup}{+}{\mathpzc{w}}\|^2{=}2,$ for all ${\mathpzc{w}}{\in}\Lambda(\omegaup_2).$ We set ${\textswab{m}}{=}{\mathfrak{g}}_{{-}1}{=}\Lambda^2(\operatorname{\mathbb{C}}^n)$ and $$\begin{cases} {\mathpzc{R}}_{\;{-}1}=\{\omegaup{+}{\epsilonup}\mid \omegaup{\in}{\mathpzc{W}}(\omegaup_2)\}=\{{\mathpzc{e}}_i{+}{\mathpzc{e}}_j\mid 1{\leq}i{<}j{\leq}n\},\\ {\mathpzc{R}}_{\;0} =\{{\mathpzc{e}}_i{-}{\mathpzc{e}}_j\mid 1{\leq}i{\neq}j{\leq}n\},\\ {\mathpzc{R}}_{\;1}=\{\omegaup{-}{\epsilonup}\mid \omegaup{\in}{\mathpzc{W}}(\omegaup_{n{-}2})\}=\{-({\mathpzc{e}}_i{+}{\mathpzc{e}}_j)\mid 1{\leq}i{<}j{\leq}n\}. \end{cases}$$ The union ${\mathpzc{R}}={\bigcup}_{{\mathpzc{p}}{=}{-}2}^2{\mathpzc{R}}_{\;{\mathpzc{p}}}$ is a root system of type $\mathbf{D}_n$ and the prolongation $${\mathfrak{g}}={\sum}_{{\mathpzc{p}}{=}{-}2}^2{\mathfrak{g}}_{{\mathpzc{p}}}\simeq{\mathfrak{o}}(2n,\operatorname{\mathbb{C}}), \;\;\text{with}\;\; {\mathfrak{g}}_0 {=}{\mathfrak{h}}_{n}{\oplus}\langle{\mathpzc{R}}_{\;0}\rangle,\;\; {\mathfrak{g}}_{{\mathpzc{p}}}=\langle{\mathpzc{R}}_{\;{\mathpzc{p}}}\rangle,$$ with ${\mathfrak{h}}_n$ a Cartan subalgebra of dimension $n,$ is, by Proposition \[prop-7.3\], a maximal effective prolongation of ${\textswab{m}},$ because ${\mathfrak{g}}$ is simple. Indeed, $${\mathpzc{R}}=\{{\pm}{\mathpzc{e}}_i{\pm}{\mathpzc{e}}_j\mid 1{\leq}i{<}j{\leq}n\},$$ is the root system of ${\mathfrak{o}}(2n,\operatorname{\mathbb{C}}),$ with cross-marked Dynkin diagram $$\xymatrix@R=.1pc{ \alphaup_1\\ \!\! \medcirc\!\!\! \ar@{-}[rd] &\alphaup_{n{-}3} &\alphaup_{n{-}4} &&& \alphaup_{1}\\ \alphaup_n & \!\!\medcirc\!\!\! \ar@{-}[r] & \!\!\medcirc\!\!\! \ar@{-}[r] & \ar@{--}[r] &{} \ar@{-}[r] & \!\!\medcirc\!\!\! \\ \!\! \medcirc\!\!\! \ar@{-}[ru] \\ \times}$$ with $\alphaup_i={\mathpzc{e}}_i{-}{\mathpzc{e}}_{i{+}1}$ for $1{\leq}i{\leq}n{-}1$ and $\alphaup_n{=}({\mathpzc{e}}_{n{-}1}{+}{\mathpzc{e}}_n).$ By setting $\deg({\mathpzc{e}}_i)=\tfrac{1}{2}$ for $1{\leq}i{\leq}n$ we obtain $\deg(\alphaup_i){=}0$ for $1{\leq}i{\leq}n{-}1,$ $\deg(\alphaup_n){=}1$ and hence the gradation above for ${\mathfrak{g}}.$ The gradation above is compatible both with the Satake diagram of the split form ${\mathfrak{gl}}_n(\operatorname{{\mathbb{R}}}),$ yielding ${\mathfrak{o}}(n,n),$ and, for $n{=}2m$ even, also with ${\mathfrak{sl}}_m({\mathbb{H}}),$ which has the Satake diagram $$\xymatrix@R=-.3pc{ \alphaup_{2m{-}1} & \alphaup_{2m{-}2} & \alphaup_{2m{-}3} & &\alphaup_{2} &\alphaup_{1} \\ \!\!\medbullet\!\!\! \ar@{-}[r]&\!\!\medcirc\!\! \ar@{-}[r]& \!\!\medbullet\!\!\ar@{-}[r]& \cdots \ar@{-}[r]& \!\!\medcirc\!\! \ar@{-}[r] &\!\!\medbullet}$$ In this case the corresponding maximal effective prolongation of type ${\mathfrak{gl}}_n(\operatorname{{\mathbb{R}}})$ of the fundamental graded Lie algebra of the first kind $V{\simeq}\Lambda^2(\operatorname{{\mathbb{R}}}^{2m})$ is isomorphic to ${\mathfrak{o}}^*(2m),$ having cross-marked Satake diagram $$\xymatrix@R=.1pc{ \alphaup_{2m{-}1}\\ \!\!\medbullet\!\!\! \ar@{-}[rd] &\alphaup_{2m{-}2} &\alphaup_{2m{-}3} &\alphaup_{2m{-}4} && \alphaup_{1}\\ \alphaup_{2m} & \!\!\medcirc\!\!\! \ar@{-}[r] & \!\!\medbullet\!\!\! \ar@{-}[r] & \!\!\medcirc\!\! \ar@{--}[r] &{} \ar@{-}[r] & \!\!\medbullet\!\!\! \\ \!\!\medcirc\!\!\! \ar@{-}[ru] \\ \times }$$ ### ${\mathfrak{g}}_{{-}1}{\simeq}{V}_{\omegaup_3}:$ construction of $\mathbf{E}_6,$ $\mathbf{E}_7,$ $\mathbf{E}_8$ Let us consider effective prolongations which are constructed on the representation of ${\mathfrak{sl}}_n(\operatorname{\mathbb{C}})$ on $\Lambda^3(\operatorname{\mathbb{C}}^n),$ for $n{\geq}6.$ We know from [@MMN2018 §[3.1]{},§[6]{}] that they are finite dimensional. The dominant weight for $\Lambda^3(\operatorname{\mathbb{C}}^n)$ is $\omegaup_3={\mathpzc{e}}_1{+}{\mathpzc{e}}_2{+}{\mathpzc{e}}_3{-}\frac{3}{n}{\mathpzc{e}}_{\,0}.$ By Remark \[rmk7.4\] we know that the necessary and sufficient condition for the existence of a semisimple effective prolongation ${\mathfrak{g}}$ of an ${\textswab{m}}$ with ${\mathfrak{g}}_{{-}1}{\simeq}\Lambda^3(\operatorname{\mathbb{C}}^n)$ is that $$\|\omegaup_3\|^2=3-\frac{9}{n}<2.$$ Thus the only possible choices are $n{=}6,7,8$ with corresponding markers $${\epsilonup}_6=\frac{1}{2\sqrt{3}}\,{\mathpzc{e}}_0,\;\; {\epsilonup}_7=\frac{\sqrt{2}}{7}{\mathpzc{e}}_0,\;\; {\epsilonup}_8=\frac{1}{8}\,{\mathpzc{e}}_0.$$ Let us set $$\begin{gathered} \begin{cases} {\mathpzc{R}}^{(6)}_{\;{-}2}{=}\{2\epsilonup_6\},\\ {\mathpzc{R}}^{(6)}_{\;{-}1}{=}\{\epsilonup_6{+}\omegaup\mid\omegaup{\in} {\mathpzc{W}}(\omegaup_3)\}, \\ {\mathpzc{R}}^{(6)}_{\;0}{=}\left\{{\mathpzc{e}}_i{-}{\mathpzc{e}}_j\mid 1{\leq}i{\neq}j{\leq}6 \right\},\\ {\mathpzc{R}}^{(6)}_{\;1}{=}\{-\epsilonup_6{+}\omegaup\mid \omegaup{\in}{\mathpzc{W}}(\omegaup_{n{-}3})\}, \\ {\mathpzc{R}}^{(6)}_{\;2}{=}\{-2\epsilonup_6\}, \end{cases} \qquad \begin{cases} {\mathpzc{R}}^{(7)}_{\;{-}2}{=}\{2\epsilonup_7{+}\omegaup\mid\omegaup {\in}{\mathpzc{W}}(\omegaup_6)\}, \\ {\mathpzc{R}}^{(7)}_{\;{-}1}{=}\{\epsilonup_7{+}\omegaup \mid \omegaup{\in}{\mathpzc{W}}(\omegaup_3)\},\\ {\mathpzc{R}}^{(7)}_{\;0}{=}\left\{{\mathpzc{e}}_i{-}{\mathpzc{e}}_j\mid 1{\leq}i{\neq}j{\leq}7 \right\},\\ {\mathpzc{R}}^{(7)}_{\;1}{=}\{{-}\epsilonup_7{+}\omegaup\mid \omegaup{\in}{\mathpzc{W}}(\omegaup_4)\},\\ {\mathpzc{R}}^{(7)}_{\;2}{=}\{{-}2\epsilonup_7{+}\omegaup\mid \omegaup{\in}{\mathpzc{W}}(\omegaup_1)\}, \end{cases} \\ \begin{cases} {\mathpzc{R}}^{(8)}_{\;{-}2}{=}\{2\epsilonup_8{+}\omegaup\mid\omegaup {\in}{\mathpzc{W}}(\omegaup_6)\}, \\ {\mathpzc{R}}^{(8)}_{\;{-}1}{=}\{\epsilonup_8{+}\omegaup \mid \omegaup{\in}{\mathpzc{W}}(\omegaup_3)\},\\ {\mathpzc{R}}^{(8)}_{\;0}{=}\left\{{\mathpzc{e}}_i{-}{\mathpzc{e}}_j\mid 1{\leq}i{\neq}j{\leq}8 \right\},\\ {\mathpzc{R}}^{(8)}_{\;1}{=}\{{-}\epsilonup_8{+}\omegaup\mid \omegaup{\in}{\mathpzc{W}}(\omegaup_5)\},\\ {\mathpzc{R}}^{(8)}_{\;2}{=}\{{-}2\epsilonup_8{+}\omegaup\mid \omegaup{\in}{\mathpzc{W}}(\omegaup_2)\}. \end{cases}\end{gathered}$$ Then one can show that, for each $n{=}6,7,8,$ the sets ${\mathpzc{R}}^{(n)}{=}{\bigcup}_{{\mathpzc{p}}{=}{-}2}^2{\mathpzc{R}}^{(n)}_{{\mathpzc{p}}}$ are root systems of type $\mathbf{E}_n,$ with Dynkin diagrams [$$\xymatrix@C=.9pc@R=-.3pc{ \alphaup_1 & \alphaup_2 &\alphaup_3 &\alphaup_4 &\alphaup_5\\ \!\!\medcirc\!\!\! \ar@{-}[r]&\!\!\medcirc\!\! \ar@{-}[r] &\!\!\medcirc\!\!\ar@{-}[r] \ar@{-}[dddddd] &\!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc& && \alphaup_i{=}{\mathpzc{e}}_i{-}{\mathpzc{e}}_{i{+}1}\\ && \\ && \\ && \\ && \\ && \\ & & \!\medcirc\! &\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\alphaup_6 &&&& \alphaup_6{=}{-}\epsilonup_6{+}\tfrac{1}{2}( {\mathpzc{e}}_4{+}{\mathpzc{e}}_5{+}{\mathpzc{e}}_6 {-}{\mathpzc{e}}_1{-}{\mathpzc{e}}_2{-}{\mathpzc{e}}_3)\\ & & \times}$$ $$\xymatrix@C=.9pc@R=-.3pc{ \alphaup_1 & \alphaup_2 &\alphaup_3 &\alphaup_4 &\alphaup_5 &\alphaup_6\\ \!\!\medcirc\!\!\! \ar@{-}[r]&\!\!\medcirc\!\! \ar@{-}[r] &\!\!\medcirc\!\!\ar@{-}[r] \ar@{-}[dddddd] &\!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc& \alphaup_i{=}{\mathpzc{e}}_i{-}{\mathpzc{e}}_{i{+}1} \\ && \\ && \\ && \\ && \\ && \\ & & \!\medcirc\! &\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\alphaup_7&&& \alphaup_7{=}{-}{\epsilonup}_7{+}\tfrac{1}{7}[3 ({\mathpzc{e}}_1{+}\cdots{+}{\mathpzc{e}}_7){-}7 ({\mathpzc{e}}_1{+}{\mathpzc{e}}_2{+}{\mathpzc{e}}_3)]\\ & & \times}$$ $$\xymatrix@C=.9pc@R=-.3pc{ \alphaup_1 & \alphaup_2 &\alphaup_3 &\alphaup_4 &\alphaup_5 &\alphaup_6&\alphaup_7\\ \!\!\medcirc\!\!\! \ar@{-}[r]&\!\!\medcirc\!\! \ar@{-}[r]&\!\!\medcirc\!\!\ar@{-}[r] \ar@{-}[dddddd] &\!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc &\alphaup_i{=}{\mathpzc{e}}_i{-}{\mathpzc{e}}_{i{+}1}\\ && \\ && \\ && \\ && \\ && \\ & & \!\medcirc\! &\! \!\!\!\!\!\!\!\! \!\!\!\!\! \!\!\!\!\alphaup_8 &&&&\alphaup_8{=}{-}\epsilonup_8{+}\tfrac{1}{8}[ 8({\mathpzc{e}}_i{+}{\mathpzc{e}}_j{+}{\mathpzc{e}}_k){-}3({\mathpzc{e}}_1{+}\cdots{+}{\mathpzc{e}}_8)]\\ & & \times }$$]{} Accordingly, we obtain on the complex Lie algebras of type $\mathbf{E}_6,$ $\mathbf{E}_7,$ $\mathbf{E}_8$ structures of effective prolongations of type ${\mathfrak{gl}}_n(\operatorname{\mathbb{C}}),$ for $6{\leq}n{\leq}8,$ by setting $$\begin{aligned} {\mathfrak{g}}={\sum}_{{\mathpzc{p}}{=}{-}2}^2{\mathfrak{g}}_{{\mathpzc{p}}},\quad\text{with}\quad \begin{cases} {\mathfrak{g}}_{{\mathpzc{p}}}{=}\langle {\mathpzc{R}}^{(n)}_{{\mathpzc{p}}}\rangle, \quad \text{for ${\mathpzc{p}}{=}{\pm}{1},{\pm}{2},$}\\ {\mathfrak{g}}_0{=}{\mathfrak{h}}_n\oplus \langle{\mathpzc{R}}^{(n)}_0\rangle \simeq{\mathfrak{gl}}_n(\operatorname{\mathbb{C}}), \end{cases}\end{aligned}$$ where ${\mathfrak{h}}_n$ is an $n$-dimensional Cartan subalgebra. In all cases we have ${\mathfrak{g}}_{{-}1}{\simeq}\Lambda^3(\operatorname{\mathbb{C}}^n)$ and ${\mathfrak{g}}_{{-}2}{\simeq}\Lambda^6(\operatorname{\mathbb{C}}^n),$ with the Lie brackets defined by the exterior product. This gradation is compatible with the non compact real forms $\mathbf{E}\mathrm{I},\mathrm{I\!{I}}, \mathrm{I\!{I}\!{I}}$ of $\mathbf{E}_6$ (see below). In these cases ${\mathfrak{g}}_{{-}1}{\simeq}\Lambda^3(\operatorname{{\mathbb{R}}}^6),$ while $[{\mathfrak{g}}_0,{\mathfrak{g}}_0]$ is simple and of type ${\mathfrak{sl}}_6(\operatorname{{\mathbb{R}}}),$ ${\mathfrak{su}}(3,3)$ and ${\mathfrak{su}}(1,4),$ respectively. For $n{=}7,8,$ the gradation is only consistent with the real split forms of $\mathbf{E}_7$ and $\mathbf{E}_8$ yielding the real analogue of the examples above. Structure algebras of type $\mathbf{B}$ --------------------------------------- The root system of a complex Lie algebra of type $\mathbf{B}_m$ (isomorphic to ${\mathfrak{o}}(2n{+}1,\operatorname{\mathbb{C}})$) is $${\mathpzc{R}}=\{{\pm}{\mathpzc{e}}_i\mid 1{\leq}i{\leq}m\}\cup\{{\pm}{\mathpzc{e}}_i{\pm}{\mathpzc{e}}_j\mid 1{\leq}i{<}j{\leq}m\}$$ for an orthonormal basis ${\mathpzc{e}}_1,\hdots,{\mathpzc{e}}_m$ of $\operatorname{{\mathbb{R}}}^m$ and its Dynkin diagram $$\xymatrix@R=-.3pc{ \alphaup_1 &\alphaup_{2} & &\alphaup_{m-1}& \alphaup_m \\ \!\!\medcirc\!\!\! \ar@{-}[r]&\!\!\medcirc\!\! \ar@{-}[r]& \cdots \ar@{-}[r]& \!\!\medcirc\!\!\ar@2{->}[r] &\!\!\medcirc}$$ has simple roots that can be chosen to be $\alphaup_i={\mathpzc{e}}_{i}{-}{\mathpzc{e}}_{i{+}1}$ for $1{\leq}i{<}m$ and $\alphaup_m{=}{\mathpzc{e}}_m.$ Its fundamental weights are $$\sigmaup_1{=}{\mathpzc{e}}_1, \; \sigmaup_2{=}{\mathpzc{e}}_1{+}{\mathpzc{e}}_2,\; \hdots, \sigmaup_{m{-}1}{=}{\mathpzc{e}}_1{+}\cdots{+}{\mathpzc{e}}_{m{-}1},\; \sigmaup_m{=} \tfrac{1}{2}({\mathpzc{e}}_1{+} \cdots{+}{\mathpzc{e}}_m).$$ Its irreducible representation with maximal weight $\sigmaup_m$ is called its *complex spin representation* and indicated by $S_{\!2m+1}^{\operatorname{\mathbb{C}}}.$ Its weights $${\mathpzc{W}}(\sigmaup_m) {=}\{\tfrac{1}{2}({\pm}{\mathpzc{e}}_1{\pm}\cdots{\pm}{\mathpzc{e}}_m)\}$$ are all simple, so that $\dim(S_{2m+1}^{\operatorname{\mathbb{C}}}){=}2^m.$ We know (see [@MMN2018 Ex.3.3,3.8]) that all the effective prolongations of type ${\mathfrak{o}}(n,\operatorname{\mathbb{C}}),$ with $n{\geq}2,$ or $\mathfrak{co}(n,\operatorname{\mathbb{C}}), $ with $n{\geq}3,$ of a fundamental graded Lie algebra of the first kind are finite dimensional. Then this holds also for fundamental graded Lie algebras of any finite kind. ### Spin representation for $\mathbf{B}_m$ Let us take an ${\textswab{m}}$ with ${\mathfrak{g}}_{{-}1}$ equal to the spin representation $S{=}S_{2m{+}1}^{\operatorname{\mathbb{C}}}$ of ${\mathfrak{o}}(2m{+}1,\operatorname{\mathbb{C}}).$ They will be complexifications of *real* spin representations of real orthogonal Lie algebras. To obtain a semisimple effective prolongation of type ${\mathfrak{co}}(2m{+}1,\operatorname{\mathbb{C}})$ of a fundamental graded Lie algebra with ${\mathfrak{g}}_{{-}1}{=}S,$ since its dominant weight is attached to a simple root of length $1,$ by it is necessary to produce a *new lenght $1$ root* by adding a *marker* to the dominant weight $\sigmaup_m$ of $S.$ Since $\|\sigma_m\|^2=\frac{m}{4},$ this is possible iff $m{\leq}3.$ For $m{=}1,$ we have ${\mathfrak{o}}(3,\operatorname{\mathbb{C}}){\simeq}{\mathfrak{sl}}_2(\operatorname{\mathbb{C}})$ and hence we refer to §\[sub7.1.1\] (in particular, we may consider the split real form of $\mathbf{G}_2$ as related to the real spin representation of ${\mathfrak{o}}(1,2)$). For $m{=}2$ we obtain a maximal effective prolongation which is isomorphic to ${\mathfrak{sp}}(3,\operatorname{\mathbb{C}})$ and whose real form ${\mathfrak{sp}}_{1,2}$ can be associated to the real spin representation of ${\mathfrak{o}}(2,3).$ When $m{=}3$ we obtain a presentation of the exceptional Lie algebra of type $\mathbf{F}_4,$ as an effective prolongation of type ${\mathfrak{co}}(7,\operatorname{\mathbb{C}})$ of a fundamental graded Lie algebra of depth $2$ with ${\mathfrak{g}}_{{-}1}$ equals to the $8$-dimensional spin representation of ${\mathfrak{o}}(7,\operatorname{\mathbb{C}}).$ Taking ${\epsilonup}$ orthogonal to $\langle{\mathpzc{e}}_1,{\mathpzc{e}}_2,{\mathpzc{e}}_3\rangle$ and of lenght $\frac{1}{2},$ we obtain $$\|\,\sigmaup_3{+}{\epsilonup}\,\|^2=1,\quad \|\,{\mathpzc{e}}_1{+}2{\epsilonup}\,\|^2=2,\;\; \|\,2{\epsilonup}\,\|^2{=}2.$$ We note that the vector representation $V_{{\mathpzc{e}}_1}\simeq\operatorname{\mathbb{C}}^7$ of ${\mathfrak{o}}(7,\operatorname{\mathbb{C}})$ is an irreducible summand of $\Lambda^2(S).$ Then we can take ${\textswab{m}}{=}S{\oplus}V_{{\mathpzc{e}}_1}.$ Set $$\begin{aligned} \begin{cases} {\mathpzc{R}}_{\;{-}2}=\{2{\epsilonup}+\omegaup\mid \omegaup{\in}{\mathpzc{W}}({\mathpzc{e}}_1)\}= \{2{\epsilonup}\}\cup\{2{\epsilonup}{\pm}{\mathpzc{e}}_i\mid 1{\leq}i{\leq}3\},\\ {\mathpzc{R}}_{\;{-}1}=\{{\epsilonup}+\omegaup\mid\omegaup{\in}{\mathpzc{W}}(\sigmaup_3)\} =\{{\epsilonup}{+}\tfrac{1}{2}({\pm}{\mathpzc{e}}_1{\pm}{\mathpzc{e}}_2{\pm}{\mathpzc{e}}_3) \},\\ {\mathpzc{R}}_{\;0}=\{{\pm}{\mathpzc{e}}_i\mid 1{\leq}i{\leq}3\} \cup\{{\pm}{\mathpzc{e}}_i{\pm} {\mathpzc{e}}_j\mid 1{\leq}i{<}j{\leq}3\},\\ {\mathpzc{R}}_{\;1}=\{{-}{\epsilonup}+\omegaup\mid\omegaup{\in}{\mathpzc{W}}(\sigmaup_3)\} =\{-{\epsilonup}{+}\tfrac{1}{2}({\pm}{\mathpzc{e}}_1{\pm}{\mathpzc{e}}_2{\pm}{\mathpzc{e}}_3) \},\\ {\mathpzc{R}}_{\;2}=\{{-}2{\epsilonup}+\omegaup\mid \omegaup{\in}{\mathpzc{W}}({\mathpzc{e}}_1)\}= \{{-}2{\epsilonup}\}\cup\{{-}2{\epsilonup}{\pm}{\mathpzc{e}}_i\mid 1{\leq}i{\leq}3\}. \end{cases} \end{aligned}$$ Then ${\mathpzc{R}}={\bigcup}_{{\mathpzc{p}}{=}{-}2}^2{\mathpzc{R}}_{{\mathpzc{p}}}$ is a root system of type $\mathbf{F}_4.$ With the four dimensional Cartan algebra ${\mathfrak{h}}_4$ of $\mathfrak{co}(7,\operatorname{\mathbb{C}}),$ we obtain an effective prolongation of ${\textswab{m}}$ in the form $${\mathfrak{g}}{=}{\sum}_{{\mathpzc{p}}{=}{-}2}^2{\mathfrak{g}}_{{\mathpzc{p}}}, \;\;\text{with}\;\; {\mathfrak{g}}_{{\mathpzc{p}}} = \begin{cases} \langle{\mathpzc{R}}_{\;{\mathpzc{p}}}\rangle ,\quad\text{for}\; {\mathpzc{p}}{=}\pm{1},{\pm}2,\\ {\mathfrak{h}}_4\oplus \langle{\mathpzc{R}}_{\;0}\rangle\simeq \mathfrak{co}(7,\operatorname{\mathbb{C}}), \;\; \text{for}\; {\mathpzc{p}}{=}0. \end{cases}$$ There are two non compact real forms of $\mathbf{F}_4,$ with Satake diagrams $$\tag{$\mathbf{F}\mathrm{I}$} \xymatrix@R=-.3pc{ \alphaup_1 & \alphaup_2 &\alphaup_3 &\alphaup_4 \\ \!\!\medcirc\!\!\! \ar@{-}[r]&\!\!\medcirc\!\! \ar@2{->}[r] &\!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc}$$ $$\tag{$\mathbf{F}\mathrm{I\!{I}}$} \xymatrix@R=-.3pc{ \alphaup_1 & \alphaup_2 &\alphaup_3 &\alphaup_4 \\ \!\!\medbullet\!\!\! \ar@{-}[r]&\!\!\medbullet\!\! \ar@2{->}[r] &\!\!\medbullet\!\!\ar@{-}[r] &\!\!\medcirc}$$ They correspond to the *real* spin representations for a $7$-dimensional vector representation. This means that we have to take the orthogonal algebras ${\mathfrak{o}}(3,4)$ and ${\mathfrak{o}}(7),$ discarding the other possible two, namely ${\mathfrak{o}}(2,5)$ and ${\mathfrak{o}}(1,6),$ which have quaternionic spin representations (see e.g. [@Deligne99 p.103]). The lifting of the quaternionic spin representations of ${\mathfrak{o}}(2,5)$ and ${\mathfrak{o}}(1,6)$ yields an irreducible representation of ${\mathfrak{o}}(7,\operatorname{\mathbb{C}}){\oplus}{\mathfrak{sl}}_2(\operatorname{\mathbb{C}})$ for which we cannot comply . In particular, there are infinitely many summands with negative indices in the associated graded Kac-Moody algebra. In contrast, for $m{=}2,$ the quaternionic spin representations of ${\mathfrak{o}}(5)$ and ${\mathfrak{o}}(1,4)$ can be lifted to maximal effective prolongations of type ${\mathfrak{co}}(5,\operatorname{\mathbb{C}})\oplus{\mathfrak{gl}}_2(\operatorname{\mathbb{C}})$ isomorphic to ${\mathfrak{sp}}(4,\operatorname{\mathbb{C}}).$ The corresponding real models are ${\mathfrak{sp}}_{1,3}$ and ${\mathfrak{sp}}_{2,2},$ respectively. Structure algebras of type $\mathbf{C}$ --------------------------------------- Let us consider the complex Lie algebra ${\mathfrak{sp}}(m,\operatorname{\mathbb{C}}),$ of type $\mathbf{C}_m.$ Its root system is $${\mathpzc{R}}=\{{\pm}2{\mathpzc{e}}_i\mid 1{\leq}i{\leq}m\} \cup\{{\pm}{\mathpzc{e}}_i{\pm}{\mathpzc{e}}_j\mid 1{\leq}i{<}j{\leq}m\}$$ for an orthonormal basis ${\mathpzc{e}}_1,\hdots,{\mathpzc{e}}_m$ of $\operatorname{{\mathbb{R}}}^m$ and its Dynkin diagram $$\xymatrix@R=-.3pc{ \alphaup_1 &\alphaup_{2} & &\alphaup_{m-1} & \alphaup_m \\ \!\!\medcirc\!\!\! \ar@{-}[r]&\!\!\medcirc\!\! \ar@{-}[r] & \cdots \ar@{-}[r]& \!\!\medcirc\!\!\ar@2{<-}[r] &\!\!\medcirc}$$ has simple roots that can be chosen to be $\alphaup_i{=}{\mathpzc{e}}_{i}{-}{\mathpzc{e}}_{i{+}1}$ for $1{\leq}i{<}m$ and $\alphaup_m{=}2{\mathpzc{e}}_m.$ Its weights lattice is the $\operatorname{\mathbb{Z}}$-module $\operatorname{\mathbb{Z}}^m$ in $\operatorname{{\mathbb{R}}}^m,$ with fundamental weights $\omegaup_j{=}{\sum}_{i{=}1}^j{\mathpzc{e}}_i,$ for $j{=}1,\hdots,m.$ We know that the maximal effective prolongation of type ${\mathfrak{sp}}(m,\operatorname{\mathbb{C}})$ of the fundamental graded Lie algebra of the first kind $\operatorname{\mathbb{C}}^{2m}$ are infinite dimensional (see [@MMN2018 Ex.3.10], or [@Kob p.10]). On the other hand, if we choose another faithful irreducible representation $V$ of ${\mathfrak{sp}}(m,\operatorname{\mathbb{C}}),$ then its maximal effective prolongation of type $\mathfrak{csp}(m,\operatorname{\mathbb{C}})$ is finite dimensional by [@MMN2018 Prop.3.13]. Let us take for instance $V{=}V_{\omegaup_m}{\subset}\Lambda^m(\operatorname{\mathbb{C}}^{2m}).$ Then ${\mathfrak{sp}}(m,\operatorname{\mathbb{C}})$ acts on $V$ as an algebra of transformations that keep invariant the bilinear form on $V$ that can be obtained from the exterior product of elements of $\Lambda^m(\operatorname{\mathbb{C}}^{2m}).$ If $m$ is even, this is a nondegenerate symmetric bilinear form and the finite dimension of the maximal prolongation can be also checked by using [@MMN2018 Ex.3.3,3.8]. When $m$ is odd, we can use the exterior product on $\Lambda^m(\operatorname{\mathbb{C}}^{2m})$ to define a Lie product on $V,$ yielding a fundamental graded Lie algebra ${\textswab{m}}{=}V{\oplus}\Lambda^{2m}(\operatorname{\mathbb{C}}^{2m})$ of the second kind, which has a finite dimensional maximal effective prolongation of type $\mathfrak{csp}(m,\operatorname{\mathbb{C}}).$ The fundamental weight $\omegaup_m$ is attached to the long root $2{\mathpzc{e}}_m.$ In order to be able to find a marker ${\epsilonup}$ to embed $\omegaup_m{+}{\epsilonup}$ into the root system of a simple Lie algebra, we need that $\|\omegaup\|^2{=}m{<}4,$ i.e. that $m{\leq}3.$ Te case $m{=}3$ leads to another presentation of the exceptional Lie algebra $\mathbf{F}_4.$ ### The exceptional Lie algebra of type $\mathbf{F}_4$ We take the marker ${\epsilonup}$ as a unit vector orthogonal to ${\mathpzc{e}}_1,{\mathpzc{e}}_2,{\mathpzc{e}}_3.$ We note that $$\|\,\omegaup_3{+}{\epsilonup}\,\|^2{=}4,\;\; \|\,{\mathpzc{e}}_1{+}{\epsilonup}\,\|^2=2,\;\; \|\,2{\epsilonup}\,\|^2=4$$ and set $$\begin{aligned} \begin{cases} {\mathpzc{R}}_{\;{-}2}=\{2{\epsilonup}\},\\ {\mathpzc{R}}_{\;{-}1}=\{{\epsilonup}{+}\omegaup\mid\omegaup{\in}{\mathpzc{W}}(\omegaup_3)\} = \{{\epsilonup}{\pm}{\mathpzc{e}}_1{\pm}{\mathpzc{e}}_2{\pm}{\mathpzc{e}}_3 \}{\cup}\{{\epsilonup}{\pm}{\mathpzc{e}}_i\mid 1{\leq}i{\leq}3\},\\ {\mathpzc{R}}_{\;\;0}=\{{\pm}2{\mathpzc{e}}_i\mid 1{\leq}i{\leq}3\}\cup\{{\pm}{\mathpzc{e}}_i{\pm} {\mathpzc{e}}_j\mid 1{\leq}i{<}j{\leq}3\},\\ {\mathpzc{R}}_{\;1}=\{{-}{\epsilonup}{+}\omegaup\mid \omegaup{\in}{\mathpzc{W}}(\omegaup_3)\} = \{{-}{\epsilonup}{\pm}{\mathpzc{e}}_1{\pm}{\mathpzc{e}}_2{\pm}{\mathpzc{e}}_3 \}{\cup}\{{\epsilonup}{\pm}{\mathpzc{e}}_i\mid 1{\leq}i{\leq}3\},\\ {\mathpzc{R}}_{\;2}=\{-2{\epsilonup}\}. \end{cases} \end{aligned}$$ Then $${\mathpzc{R}}{=}{\bigcup}_{{\mathpzc{p}}{=}{-}2}^2{\mathpzc{R}}_{\;{\mathpzc{p}}}=\{{\pm}2{{\mathpzc{e}}}_i\mid 1{\leq}i{\leq}4 \}\cup\{{\pm}{\mathpzc{e}}_i{\pm}{\mathpzc{e}}_j\mid 1{\leq}i{<}j{\leq}4 \} \cup\left\{{\pm}{\mathpzc{e}}_1{\pm}{\mathpzc{e}}_2{\pm}{\mathpzc{e}}_3{\pm}{\mathpzc{e}}_{4})\right\},$$ is a root system of type $\mathbf{F}_4.$ We have, with a $4$-dimensional Cartan subalgebra ${\mathfrak{h}},$ $${\mathfrak{g}}{=}{\sum}_{{\mathpzc{p}}{=}{-}2}^2{\mathfrak{g}}_{{\mathpzc{p}}}, \;\;\text{with}\;\; {\mathfrak{g}}_{{\mathpzc{p}}} = \begin{cases} \langle{\mathpzc{R}}_{\;{\mathpzc{p}}}\rangle ,\quad\text{for}\; {\mathpzc{p}}{=}\pm{1},{\pm}2,\\ {\mathfrak{h}}_4\oplus \langle{\mathpzc{R}}_{\;0}\rangle\simeq{\mathfrak{o}}(7,\operatorname{\mathbb{C}}){\oplus}\operatorname{\mathbb{C}}, \;\; \text{for}\; {\mathpzc{p}}{=}0. \end{cases}$$ We have $\dim({\mathfrak{g}}_0){=}22,$ $\dim({\mathfrak{g}}_{{\pm}1}){=}14,$ $\dim({\mathfrak{g}}_{{\pm}2})=1.$ This is the maximal effective prolongation of a fundamental graded Lie algebra of the second kind. The cross marked Dynkin diagram associated to ${\mathfrak{g}}$ is $$\xymatrix@R=-.3pc{ \alphaup_1 & \alphaup_2 &\alphaup_3 &\alphaup_4 \\ \!\!\medcirc\!\!\! \ar@{-}[r]&\!\!\medcirc\!\! \ar@2{<-}[r] &\!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc\\ &&& \times}$$ with simple roots $ \alphaup_1{=}{\mathpzc{e}}_1{-}{\mathpzc{e}}_2,\;\alphaup_2{=} {\mathpzc{e}}_2{-}{\mathpzc{e}}_3,\;\alphaup_3{=}2{\mathpzc{e}}_3,\;\alphaup_4{=} {\mathpzc{e}}_4{-}{\mathpzc{e}}_{1}{-}{\mathpzc{e}}_{2}{-}{\mathpzc{e}}_3. $ The gradation is obtained by setting $\deg({\mathpzc{e}}_i){=}0$ for $1{\leq}i{\leq}3$ and $\deg({\mathpzc{e}}_4){=}1.$ Only the split real form $\mathbf{F}\mathrm{I}$ is compatible with this grading, yielding a real equivalent of the complex case. Structure algebras of type $\mathbf{D}$ --------------------------------------- The diagram $\mathbf{D}_m$ (we assume $m{\geq}4$) corresponds to the orthogonal algebra ${\mathfrak{o}}(2m,\operatorname{\mathbb{C}}).$ Its root system is defined, in an orthonormal basis ${\mathpzc{e}}_1,\hdots,{\mathpzc{e}}_m$ of $\operatorname{{\mathbb{R}}}^m,$ by $${\mathpzc{R}}(\mathbf{D}_m)=\{{\pm}{\mathpzc{e}}_i{\pm}{\mathpzc{e}}_j\mid 1{\leq}i{<}j{\leq}m\},$$ and we consider the corresponding Dynkin diagram $$\xymatrix@R=.1pc{ \alphaup_{m{-}1} \\ \!\!\medcirc\!\!\! \ar@{-}[rd] &\alphaup_{m{-}2} &\alphaup_{m{-}3} &&& \alphaup_{1}\\ & \!\!\medcirc\!\!\! \ar@{-}[r] & \!\!\medcirc\!\!\! \ar@{-}[r] & \ar@{--}[r] &{} \ar@{-}[r] & \!\!\medcirc\!\!\! \\ \!\!\medcirc\!\!\! \ar@{-}[ru] \\ \alphaup_m\\ }$$ with $\alphaup_i={\mathpzc{e}}_i{-}{\mathpzc{e}}_{i{+}1}$ for $1{\leq}i{\leq}m{-}1$ and $\alphaup_m{=}{\mathpzc{e}}_{m{-}1}{+}{\mathpzc{e}}_m.$ The maximal root is ${\mathpzc{e}}_1{+}{\mathpzc{e}}_m$ and the fundamental weights are $$\begin{aligned} \omegaup_j{=}{\sum}_{i{=}1}^j{\mathpzc{e}}_i,\;\;\text{for $1{\leq}j{\leq}m{-}2,$},\;\; \omegaup_{m{-}1}{=}\tfrac{1}{2}({\mathpzc{e}}_1{+} \cdots{+}{\mathpzc{e}}_{m{-}1}{-}{\mathpzc{e}}_m),\qquad\\ \omegaup_{m}{=}\tfrac{1}{2}({\mathpzc{e}}_1{+}\cdots{+}{\mathpzc{e}}_m).\end{aligned}$$ The last two are the dominant weights of two complex spin representations $S^{\operatorname{\mathbb{C}}}_{m,\pm},$ with opposite chiralities and simple weights $$\begin{aligned} {\mathpzc{W}}(\omegaup_{m{-}1}){=}{\mathpzc{W}}_-(m) &{=}\left.\left\{\tfrac{1}{2}{\sum}_{i{=}1}^m{\mathpzc{a}}_i{\mathpzc{e}}_i\right| {\mathpzc{a}}_i{=}\pm{1},\; {\mathpzc{a}}_1{\cdots}{\mathpzc{a}}_m{=}{-}1\right\},\\ {\mathpzc{W}}(\omegaup_m){=} {\mathpzc{W}}_+(m)&{=} \left.\left\{\tfrac{1}{2}{\sum}_{i{=}1}^m{\mathpzc{a}}_i{\mathpzc{e}}_i\right| {\mathpzc{a}}_i{=}\pm{1},\; {\mathpzc{a}}_1{\cdots}{\mathpzc{a}}_m{=}1\right\}.\end{aligned}$$We call $V^{\operatorname{\mathbb{C}}}_m{=}V_{\omegaup_1}{\simeq}\operatorname{\mathbb{C}}^{2m}$ the complex vector representation. ### Real Spin representations of real Lie algebras of type $\mathbf{D}$ Let us first consider complex effective prolongations with structure algebra ${\mathfrak{o}}(2m,\operatorname{\mathbb{C}})$ in which ${\mathfrak{g}}_{{-}1}$ is a spin representation. A necessary condition for finding a marker ${\epsilonup}$ for which $\omegaup_{m{-}1}{+}{\epsilonup}$ or $\omegaup_{m}{+}{\epsilonup}$ could be embedded into the root system of a simple Lie algebra is that $\|\omegaup_{m{-}1}\|^2=\|\omegaup_m\|^2=\frac{m}{4}{<}2,$ i.e. that $m{=}4,5,6,7.$ When $m{=}4,$ the spin and the vector representations are isomorphic and the maximal effective prolongation of type ${\mathfrak{o}}(8,\operatorname{\mathbb{C}})$ of the abelian Lie algebra $V^{\operatorname{\mathbb{C}}}_4{\simeq}S^{\operatorname{\mathbb{C}}}_{4,\pm}$ is just the orthogonal algebra ${\mathfrak{o}}(10,\operatorname{\mathbb{C}})$ (see [@MMN2018 Ex.3.8]). For $m{=}5,6,7$ we obtain the three exceptional Lie algebras of type $\mathbf{E}.$ Denote by ${\epsilonup}_{m{+}1}$ a vector of $\operatorname{{\mathbb{R}}}^{m{+}1},$ orthogonal to ${\mathpzc{e}}_1,\hdots,{\mathpzc{e}}_{m}$ and with $\|\,{\epsilonup}_{m{+}1}\,\|^2=\frac{8{-}m}{4}.$ We define the sets $$\begin{gathered} \begin{cases} {\mathpzc{R}}^{(6)}_{\;-1}=\{{\epsilonup}_6+{\mathpzc{w}}\mid {\mathpzc{w}}{\in}{\mathpzc{W}}_{-}(5)\},\\ {\mathpzc{R}}^{(6)}_{\;0}=\{{\pm}{\mathpzc{e}}_i{\pm}{\mathpzc{e}}_j\mid 1{\leq}i{<}j\leq{5}\},\\ {\mathpzc{R}}^{(6)}_{\;1}=\{{-}{\epsilonup}_6+{\mathpzc{w}}\mid {\mathpzc{w}}{\in}{\mathpzc{W}}_{+}(5)\} \end{cases} \\ \begin{cases} {\mathpzc{R}}^{(7)}_{\;{-}2}=\{2{\epsilonup}_7\},\\ {\mathpzc{R}}^{(7)}_{\;-1}=\{{\epsilonup}_7{+}{\mathpzc{w}}\mid {\mathpzc{w}}{\in}{\mathpzc{W}}_{+}(6)\},\\ {\mathpzc{R}}^{(7)}_{\;0}=\{{\pm}{\mathpzc{e}}_i{\pm}{\mathpzc{e}}_j\mid 1{\leq}i{<}j\leq{6}\},\\ {\mathpzc{R}}^{(7)}_{\;1}=\{{-}{\epsilonup}_7+{\mathpzc{w}}\mid {\mathpzc{w}}{\in}{\mathpzc{W}}_{+}(6)\},\\ {\mathpzc{R}}^{(7)}_{\;2}=\{{-}2{\epsilonup}_7\}, \end{cases} \qquad \begin{cases} {\mathpzc{R}}^{(8)}_{\;{-}2}=\{2{\epsilonup}_8{\pm}{\mathpzc{e}}_i\mid 1{\leq}i{\leq}7\},\\ {\mathpzc{R}}^{(8)}_{\;-1}=\{{\epsilonup}_8+{\mathpzc{w}}\mid {\mathpzc{w}}{\in}{\mathpzc{W}}_{-}(7)\},\\ {\mathpzc{R}}^{(8)}_{\;0}=\{{\pm}{\mathpzc{e}}_i{\pm}{\mathpzc{e}}_j\mid 1{\leq}i{<}j\leq{7}\},\\ {\mathpzc{R}}^{(8)}_{\;1}=\{{-}{\epsilonup}_8+{\mathpzc{w}}\mid {\mathpzc{w}}{\in}{\mathpzc{W}}_{+}(7)\},\\ {\mathpzc{R}}^{(8)}_{\;2}=\{{-}2{\epsilonup}_8{\pm}{\mathpzc{e}}_i\mid 1{\leq}i{\leq}7\}, \end{cases}\end{gathered}$$ Then ${\mathpzc{R}}^{(6)}={\bigcup}_{{\mathpzc{p}}{=}{-}1}^1{\mathpzc{R}}^{(6)}_{\;{\mathpzc{p}}},$ ${\mathpzc{R}}^{(7)}={\bigcup}_{{\mathpzc{p}}{=}{-}2}^2{\mathpzc{R}}^{(7)}_{\;{\mathpzc{p}}},$ ${\mathpzc{R}}^{(8)}={\bigcup}_{{\mathpzc{p}}{=}{-}2}^2{\mathpzc{R}}^{(8)}_{\;{\mathpzc{p}}},$ are root systems of type $\mathbf{E}_6,$ $\mathbf{E}_7,$ $\mathbf{E}_8,$ respectively and we obtain on the complex Lie algebras of type $\mathbf{E}_n$ structures of effective prolongation for structure algebras of type $\mathbf{D}_{n{-}1}$ and their spin representations, of the form $${\mathfrak{g}}={\sum}{\mathfrak{g}}_{{\mathpzc{p}}},\;\;\text{with}\;\; {\mathfrak{g}}_0 {=}\mathfrak{co}(2n{-}2,\operatorname{\mathbb{C}}),\;\; {\mathfrak{g}}_{{\mathpzc{p}}}=\langle{\mathpzc{R}}^{(n)}_{\;{\mathpzc{p}}}\rangle \;\text{for}\; {\mathpzc{p}}{\neq}0.$$ We note that ${\mathfrak{g}}_{{\pm}1}$ are for $n{=}6,8$ spin representations with opposite chirality, while for $\mathbf{E}_7$ the representations ${\mathfrak{g}}_{{\pm}1}$ have equal chiralities. For $\mathbf{E}_8$ the ${\mathfrak{g}}_{{\pm}2}$ representations are vectorial. Since the complexification of the spin representation that we found in these cases are by construction irreducible over $\operatorname{\mathbb{C}},$ these complex effective prolongation must be complexifications of *real* spin representations. We recall that the spin representations of ${\mathfrak{o}}({\mathpzc{p}},{\mathpzc{q}})$ are (see e.g. [@Deligne99 p.103]) $$\begin{cases} \text{real} & \text{if}\;\; {\mathpzc{q}}{-}{\mathpzc{p}}\equiv 0,1,7\mod 8,\\ \text{complex} & \text{if}\;\; {\mathpzc{q}}{-}{\mathpzc{p}}\equiv 2,6\;\; \;\mod 8,\\ \text{quaternionic} & \text{if}\;\; {\mathpzc{q}}{-}{\mathpzc{p}}\equiv 3,4,5\mod 8.\\ \end{cases}$$ Thus, for the real forms of ${\mathfrak{o}}(10,\operatorname{\mathbb{C}}),$ the semisimple ideal of ${\mathfrak{g}}_0$ can only be ${\mathfrak{o}}(5,5)$ ${\mathfrak{o}}(1,9).$ Corresponding, we obtain the real forms of $\mathbf{E}_6$ having cross-marked Satake diagrams $$\tag{$\mathbf{E}{\mathrm{I}}$} \xymatrix@C=.9pc@R=-.3pc{ \alphaup_1 & \alphaup_3 &\alphaup_4 &\alphaup_5 &\alphaup_6\\ \!\!\medcirc\!\!\! \ar@{-}[r]&\!\!\medcirc\!\! \ar@{-}[r]&\!\!\medcirc\!\!\ar@{-}[r] \ar@{-}[dddddd] &\!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc\\ \times && \\ && \\ && \\ && \\ && \\ & & \!\medcirc\! &\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\alphaup_2}$$ $$\tag{$\mathbf{E}{\mathrm{I\!{V}}}$} \xymatrix@C=.9pc@R=-.3pc{ \alphaup_1 & \alphaup_3 &\alphaup_4 &\alphaup_5 &\alphaup_6\\ \!\!\medcirc\!\!\! \ar@{-}[r]&\!\!\medbullet\!\! \ar@{-}[r]&\!\!\medbullet\!\!\ar@{-}[r] \ar@{-}[dddddd] &\!\!\medbullet\!\!\ar@{-}[r] &\!\!\medcirc\\ \times&& \\ && \\ && \\ && \\ && \\ & & \!\medbullet\! &\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\alphaup_2}$$ For $\mathbf{E}_7$ we obtain a graded algebra of the second kind. The real forms of ${\mathfrak{o}}(12,\operatorname{\mathbb{C}})$ having real spin representations are ${\mathfrak{o}}(6,6),$ ${\mathfrak{o}}(2,10)$ and ${\mathfrak{o}}^*(12)$ (see e.g. [@barut]). Thus we obtain all non compact real forms of $\mathbf{E}_7$ as (EPGFLA)’s of the real spin representations for vector dimension $12,$ which correspond to the cross-marked Satake diagrams $$\tag{$\mathbf{E}\mathrm{V}$} \xymatrix@C=.9pc@R=-.3pc{ \alphaup_1 & \alphaup_3 &\alphaup_4 &\alphaup_5 &\alphaup_6 &\alphaup_7\\ \!\!\medcirc\!\!\! \ar@{-}[r]&\!\!\medcirc\!\! \ar@{-}[r]&\!\!\medcirc\!\!\ar@{-}[r] \ar@{-}[dddddd] &\!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc\\ \times && \\ && \\ && \\ && \\ && \\ & & \!\medcirc\! &\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\alphaup_2}$$ $$\tag{$\mathbf{E}\mathrm{V\!{I}\!{I}}$} \xymatrix@C=.9pc@R=-.3pc{ \alphaup_1 & \alphaup_3 &\alphaup_4 &\alphaup_5 &\alphaup_6 &\alphaup_7\\ \!\!\medcirc\!\!\! \ar@{-}[r]&\!\!\medbullet\!\! \ar@{-}[r]&\!\!\medbullet\!\!\ar@{-}[r] \ar@{-}[dddddd] &\!\!\medbullet\!\!\ar@{-}[r] &\!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc\\ \times&& \\ && \\ && \\ && \\ && \\ & & \!\medbullet\! &\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\alphaup_2}$$ $$\tag{$\mathbf{E}\mathrm{V\!{I}}$} \xymatrix@C=.9pc@R=-.3pc{ \alphaup_1 & \alphaup_3 &\alphaup_4 &\alphaup_5 &\alphaup_6 &\alphaup_7\\ \!\!\medcirc\!\!\! \ar@{-}[r]&\!\!\medcirc\!\! \ar@{-}[r]&\!\!\medcirc\!\!\ar@{-}[r] \ar@{-}[dddddd] &\!\!\medbullet\!\!\ar@{-}[r] &\!\!\medcirc\!\!\ar@{-}[r] &\!\!\medbullet\\ \times && \\ && \\ && \\ && \\ && \\ & & \!\medbullet\! &\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\alphaup_2}$$ For $\mathbf{E}\mathrm{V}$ the semisimple part of the degree zero subalgebra is ${\mathfrak{o}}(6,6)$ and for $\mathbf{E}\mathrm{V\!{I}\!{I}}$ it is ${\mathfrak{o}}(2,10).$ In both cases the spin representations involved are real. We recall that the real form ${\mathfrak{o}}^*(12)$ is obtained from ${\mathfrak{o}}(12,\operatorname{\mathbb{C}})$ by the conjugation which is described on the roots by $$\bar{{\mathpzc{e}}}_{2h-1}={\mathpzc{e}}_{2h},\;\text{for}\; 1{\leq}h{\leq}3.$$ Accordingly, $\mathfrak{so}^*(12)$ has both a real and a quaternionic spin representation. The real forms of ${\mathfrak{o}}(14,\operatorname{\mathbb{C}})$ admitting real spin representations are ${\mathfrak{o}}(7,7)$ and ${\mathfrak{o}}(3,11)$ ($\mathfrak{so}^*(14)$ has a complex spin representation). In this case, ${\mathfrak{g}}_{{\pm}1}$ are spin representations with opposite chiralities and ${\mathfrak{g}}_{{\pm}2}$ dual copies of the vector representation of ${\mathfrak{o}}(14,\operatorname{\mathbb{C}}),$ which correspond to real vector representations $V^{\operatorname{{\mathbb{R}}}}_7{=}\operatorname{{\mathbb{R}}}^{14}.$ Thus we obtain all non compact real Lie algebras of type $\mathbf{E}_8$ as effective prolongations of type ${\mathfrak{o}}({\mathpzc{p}},{\mathpzc{q}})$ for FGLA’s ${\textswab{m}}$ whose ${\mathfrak{g}}_{{-}1}$ component is a *real* spin representation. Their cross-marked Satake diagrams are $$\tag{$\mathbf{E}\mathrm{V\!{I}\!{I}\!{I}}$} \xymatrix@C=.9pc@R=-.3pc{ \alphaup_1 & \alphaup_3 &\alphaup_4 &\alphaup_5 &\alphaup_6 &\alphaup_7&\alphaup_8\\ \!\!\medcirc\!\!\! \ar@{-}[r]&\!\!\medcirc\!\! \ar@{-}[r]&\!\!\medcirc\!\!\ar@{-}[r] \ar@{-}[dddddd] &\!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc\\ \times && \\ && \\ && \\ && \\ && \\ & & \!\medcirc\! &\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\alphaup_2}$$ $$\tag{$\mathbf{E}\mathrm{I\!{X}}$} \xymatrix@C=.9pc@R=-.3pc{ \alphaup_1 & \alphaup_3 &\alphaup_4 &\alphaup_5 &\alphaup_6 &\alphaup_7&\alphaup_8\\ \!\!\medcirc\!\!\! \ar@{-}[r]&\!\!\medbullet\!\! \ar@{-}[r]&\!\!\medbullet\!\!\ar@{-}[r] \ar@{-}[dddddd] &\!\!\medbullet\!\!\ar@{-}[r] &\!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc\!\!\ar@{-}[r] &\!\!\medcirc\\ \times && \\ && \\ && \\ && \\ && \\ & & \!\medbullet\! &\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\alphaup_2}$$ The corresponding simple ideals in the degree $0$ subalgebras are ${\mathfrak{o}}(7,7)$ and ${\mathfrak{o}}(3,11),$ respectively. ### Complex spin representations of real Lie algebras of type $\mathbf{D}$ The complexification of an irreducible representation of the complex type of a real Lie algebra is the direct sum of two non isomorphic representations. We should therefore start by considering the complex effective prolongations of type ${\mathfrak{g}}_0$ for a reductive ${\mathfrak{g}}_0$ with $[{\mathfrak{g}}_0,{\mathfrak{g}}_0]{\simeq}{\mathfrak{o}}(2m,\operatorname{\mathbb{C}})$ and a two-dimensional center, having a ${\mathfrak{g}}_{{-}1}{\simeq}S^{\operatorname{\mathbb{C}}}_{-}(m){\oplus}S^{\operatorname{\mathbb{C}}}_{+}(m).$ We need to add a marker ${\epsilonup}_{\pm}$ for each irreducible component $S^{\operatorname{\mathbb{C}}}_{\pm}(m)$ of ${\mathfrak{g}}_{{-}1}.$ To find a semisimple effective prolongation, we need that $\omegaup_{m{-}1}{+}{\epsilonup}_-$ and $\omegaup_{m}{+}{\epsilonup}_+$ have length $\sqrt{2}$ and are orthogonal to each other. Then $m{\in}\{4,5,6,7\}$ and $$\|{\epsilonup}_{\pm}\|^2=\frac{8{-}m}{4},\qquad ({\epsilonup}_{+}|{\epsilonup}_{-}){=}-(\omegaup_{+}|\omegaup_{-})= \frac{m-2}{4}.$$ By Cauchy’s ineguality we need that $$\frac{m{-}2}{4}<\frac{(8-m)^2}{16}.$$ This is possible only for $m{=}4.$ Thus we assume $m{=}4$ and take markers ${\epsilonup}_{\pm}$ with $$\|{\epsilonup}_{\pm}\|^2=1,\;\; \; ({\epsilonup}_{+}|{\epsilonup}_{-})={-}\frac{1}{2}.$$ Then we define $$\begin{aligned} & \begin{cases} {\mathpzc{R}}_{\;{-}2}^{(6)} =\{{\epsilonup}_-{+}{\epsilonup}_+{\pm}{\mathpzc{e}}_i\mid 1\leq{i}\leq{4}\},\\ {\mathpzc{R}}_{\;{-}1}^{(6)}=\{{\epsilonup}_-{+}{\mathpzc{w}}\mid{\mathpzc{w}}{\in}{\mathpzc{W}}_{-}(4)\} \cup \{{\epsilonup}_+{+}{\mathpzc{w}}\mid{\mathpzc{w}}{\in}{\mathpzc{W}}_{+}(4)\},\\ {\mathpzc{R}}_{\;0}^{(6)}=\{{\pm}{\mathpzc{e}}_i{\pm}{\mathpzc{e}}_j\mid 1{\leq}i{<}j{\leq}4\},\\ {\mathpzc{R}}_{\;1}^{(6)}=\{{-}{\epsilonup}_{-}{+}{\mathpzc{w}}\mid{\mathpzc{w}}{\in}{\mathpzc{W}}_{-}(4)\} \cup \{{-}{\epsilonup}_+{+}{\mathpzc{w}}\mid{\mathpzc{w}}{\in}{\mathpzc{W}}_{+}(4)\},\\ {\mathpzc{R}}_{\;2}^{(6)}=\{{-}{\epsilonup}_{-} {-}{\epsilonup}_{+}{\pm}{\mathpzc{e}}_i\mid 1\leq{i}\leq{4}\}, \end{cases} $$ One can check that ${\mathpzc{R}}^{(6)}={\bigcup}{\mathpzc{R}}^{(6)}_{\;{\mathpzc{p}}}$ is a root system of type $\mathbf{E}_6,$ and $${\mathfrak{g}}{=}{\sum}_{{\mathpzc{p}}{=}{-}2}^{2}{\mathfrak{g}}_{{\mathpzc{p}}},\;\;\text{with}\;\; \begin{cases} {\mathfrak{g}}_0{=}{\mathfrak{h}}_n{\oplus}\langle{\mathpzc{R}}_{\;0}^{(6)}\rangle,\;\;\dim_{\operatorname{\mathbb{C}}}{\mathfrak{h}}_6{=}6,\\ {\mathfrak{g}}_{{\mathpzc{p}}}=\langle{\mathpzc{R}}_{\;{\mathpzc{p}}}^{(6)}\rangle,\;\;\;{\mathpzc{p}}{\neq}0, \end{cases}$$ is the maximal effective prolongation of type ${\mathfrak{g}}_0,$ with ${\mathfrak{g}}_0$ reductive and $[{\mathfrak{g}}_0,{\mathfrak{g}}_0]{\simeq}{\mathfrak{o}}(8,\operatorname{\mathbb{C}}),$ of a FGLA with ${\mathfrak{g}}_{{-}1}{\simeq}S^{\operatorname{\mathbb{C}}}_-(4){\oplus}S^{\operatorname{\mathbb{C}}}_+(4).$ The real forms of ${\mathfrak{o}}(8,\operatorname{\mathbb{C}})$ having a complex spin representation are ${\mathfrak{o}}(3,5)$ and ${\mathfrak{o}}(1,7),$ corresponding to the two real non compact forms of $\mathbf{E}_6$ having cross-marked Satake diagrams $$\tag{$\mathbf{E}\mathrm{I\!{I}}$} \xymatrix@R=-.3pc{\!\!\medcirc\!\!\ar@{-}[r]\ar@{<->}@/^2pc/[rrrr] &\!\!\medcirc\!\! \ar@{-}[r]\ar@{<->}@/^1pc/[rr]&\!\!\medcirc\!\! \ar@{-}[r] \ar@{-}[dddddd]&\!\!\medcirc\!\! \ar@{-}[r]&\!\!\medcirc\!\!\\ \times &&& &\times \\ && \\ && \\ && \\ &&\\ &&\medcirc }$$ $$\tag{$\mathbf{E}{\mathrm{I\!{I}\!{I}}}$} \xymatrix@R=-.3pc{\!\!\medcirc\!\!\ar@{-}[r]\ar@{<->}@/^1pc/[rrrr] &\!\!\medbullet\!\! \ar@{-}[r]&\!\!\medbullet\!\! \ar@{-}[r] \ar@{-}[dddddd]&\!\!\medbullet\!\! \ar@{-}[r]&\!\!\medcirc\!\!\\ \times &&&&\times \\ && \\ && \\ && \\ &&\\ &&\medcirc }$$ (the irriducible complex representation is pictured in the diagram by two crossed white nodes joined by an arrow). Note that the summands on degree ${\pm}2$ in ${\mathfrak{g}}$ are the vector representations $V^{\operatorname{\mathbb{C}}}_4{\simeq}\operatorname{\mathbb{C}}^8$ in the complex and $V^{\operatorname{{\mathbb{R}}}}_4{\simeq}\operatorname{{\mathbb{R}}}^8$ in the real cases. ### Quaternionic spin representations of real Lie algebras of type $\mathbf{D}$ Let $\etaup_1,\etaup_2$ be an orthonormal basis of $\operatorname{{\mathbb{R}}}^2.$ We denote by $\{{\pm}(\etaup_1{-}\etaup_2)\}$ the root system of ${\mathfrak{sl}}_2(\operatorname{\mathbb{C}}),$ and by $\omegaup{=}\tfrac{1}{2}(\etaup_1{-}\etaup_2)$ its fundamental weight. Then $\{{\pm}(\etaup_1{-}\etaup_2)\} {\cup}\{{\pm}{\mathpzc{e}}_i{\pm}{\mathpzc{e}}_j{\mid}1{\leq}i{<}j{\leq}m\}$ is the root system of ${\mathfrak{sl}}_2(\operatorname{\mathbb{C}}){\oplus}{\mathfrak{o}}(2m,\operatorname{\mathbb{C}})$ and, taking the usual lexicographic orders, its fundamental weights are $\omegaup,\omegaup_1,\hdots,\omegaup_m.$ The complexification of an irreducible spin representation of quaternionic type of a real form of ${\mathfrak{o}}(2m,\operatorname{\mathbb{C}})$ lifts to an irreducible complex representation of ${\mathfrak{sl}}_2(\operatorname{\mathbb{C}}){\oplus}{\mathfrak{o}}(2m,\operatorname{\mathbb{C}})$ whose dominant weight is either $\omegaup{+}\omegaup_{m{-}1},$ or $\omegaup{+}\omegaup_{m}.$ Let us assume it is $\omegaup{+}\omegaup_{m}.$ A necessary condition to find a semisimple complex Lie algebra which is an effective prolongation of type ${\mathfrak{g}}_0,$ with $$[{\mathfrak{g}}_0,{\mathfrak{g}}_0]{=}{\mathfrak{sl}}_2(\operatorname{\mathbb{C}}){\oplus}{\mathfrak{o}}(2m,\operatorname{\mathbb{C}})$$ of an ${\textswab{m}}$ with ${\mathfrak{g}}_{-1}$ equal to the irreducible ${\mathfrak{sl}}_2(\operatorname{\mathbb{C}}){\oplus}{\mathfrak{o}}(2m,\operatorname{\mathbb{C}})$-module with dominant weight $\omegaup{+}\omegaup_m$ is that $$\vspace{-3pt} \|\omegaup{+}\omegaup_{m}\|^2=\frac{1}{2}{+}\frac{m}{4}<2,$$ i.e. that $m{=}4,5.$ With ${\epsilonup}_4{=}\tfrac{1}{2}(\etaup_1{+}\etaup_2),$ ${\epsilonup}_5{=}\tfrac{1}{\sqrt{8}}(\etaup_1{+} \etaup_2),$ set $$\! \begin{cases} {\mathpzc{R}}^{(4)}_{\;{-}2}=\{2{\epsilonup}_4\},\\ {\mathpzc{R}}^{(4)}_{\;{-}1}=\{{\epsilonup}_4{\pm}\etaup {+}{\mathpzc{w}}\mid {\mathpzc{w}}{\in}{\mathpzc{W}}_{+}(4)\},\\ {\mathpzc{R}}^{(4)}_{\;0}=\{\pm(\etaup_1{-}\etaup_2)\} {\cup}\{ {\pm}{\mathpzc{e}}_i {\pm}{\mathpzc{e}}_j {\mid} \begin{smallmatrix} 1{\leq}i{<}j{\leq}4 \end{smallmatrix}\!\},\\ {\mathpzc{R}}^{(4)}_{\;1}=\{{-}{\epsilonup}_4{\pm}\etaup{+}{\mathpzc{w}}\mid {\mathpzc{w}}{\in}{\mathpzc{W}}_{+}(4)\},\\ {\mathpzc{R}}^{(4)}_{\;2}=\{-2{\epsilonup}_4\}, \end{cases} \!\!\!\! \begin{cases} {\mathpzc{R}}^{(5)}_{\;{-}2}=\{2{\epsilonup}_5{\pm}{\mathpzc{e}}_1\mid 1{\leq}i{\leq}\},\\ {\mathpzc{R}}^{(5)}_{\;{-}1}=\{{\epsilonup}_5{\pm}\etaup {+}{\mathpzc{w}}\mid {\mathpzc{w}}{\in}{\mathpzc{W}}_{+}(5)\},\\ {\mathpzc{R}}^{(5)}_{\;0}=\{\pm(\etaup_1{-}\etaup_2)\} {\cup}\{ {\pm}{\mathpzc{e}}_i {\pm}{\mathpzc{e}}_j{\mid} \begin{smallmatrix} 1{\leq}i{<}j{\leq}5 \end{smallmatrix}\! \},\\ {\mathpzc{R}}^{(5)}_{\;1}=\{{-}{\epsilonup}_5{\pm}\etaup {+}{\mathpzc{w}}\mid {\mathpzc{w}}{\in}{\mathpzc{W}}_{+}(5)\},\\ {\mathpzc{R}}^{(5)}_{\;2}=\{-2{\epsilonup}_5{\pm}{\mathpzc{e}}_1 \mid 1{\leq}i{\leq}5\}. \end{cases}$$ Then ${\mathpzc{R}}^{(4)}{=}{\bigcup}_{{\mathpzc{p}}{=}{-}2}^{2}{\mathpzc{R}}^{(4)}_{\;{\mathpzc{p}}}$ is a root system of type $\mathbf{D}_6$ and ${\mathpzc{R}}^{(5)}{=}{\bigcup}_{{\mathpzc{p}}{=}{-}2}^{2}{\mathpzc{R}}^{(5)}_{\;{\mathpzc{p}}}$ a root system of type $\mathbf{E}_7.$ Set $${\mathfrak{g}}^{(m)}{=}{\sum}_{{\mathpzc{p}}{=}{-}2}^{2}{\mathfrak{g}}_{{\mathpzc{p}}}^{(m)},\;\; \text{with}\;\; \begin{cases} {\mathfrak{g}}^{(m)}_0{=}{\mathfrak{h}}_m{\oplus}\langle{\mathpzc{R}}^{(m)}_{\;0} \rangle,\;\;\dim_{\operatorname{\mathbb{C}}}{\mathfrak{h}}_m{=}m{+}2,\\ {\mathfrak{g}}_{{\mathpzc{p}}}^{(m)}=\langle{\mathpzc{R}}^{(m)}_{\;{\mathpzc{p}}}\rangle,\;\;\;{\mathpzc{p}}{\neq}0, \end{cases}$$ for $m{=}4,5.$ Then ${\mathfrak{g}}^{m}_0{=}{\mathfrak{sl}}_2(\operatorname{\mathbb{C}}){\oplus}{\mathfrak{o}}(2m) {\oplus}\langle\varpi\rangle,$ where $\varpi$ satisfies $[\varpi,X]{=}{\mathpzc{p}}{\cdot}{X}$ for $X{\in}{\mathfrak{g}}_{{\mathpzc{p}}}$ and the ${\mathfrak{g}}^{(m)}$ are maximal effective prolongation of type ${\mathfrak{g}}_0^{(m)}.$ For $m{=}4$ each of the summands ${\mathfrak{g}}^{(4)}_{{\pm}{1}}$ consists of two copies of $S^{\operatorname{\mathbb{C}}}_+(4)$ and each of ${\mathfrak{g}}^{(4)}_{{\pm}{2}}$ is the scalar representation. For $m{=}5$ each of the summands ${\mathfrak{g}}^{(4)}_{{\pm}{1}}$ consists of two copies of $S^{\operatorname{\mathbb{C}}}_+(5)$ and each of ${\mathfrak{g}}^{(4)}_{{\pm}{2}}$ is the complex vector representation $V^{\operatorname{\mathbb{C}}}_m{\simeq} \operatorname{\mathbb{C}}^{2m}.$ The only real form of ${\mathfrak{o}}(8,\operatorname{\mathbb{C}})$ having a quaternionic spin representation is ${\mathfrak{o}}(2,6).$ Its prolongation is the real form of ${\mathfrak{g}}^{(4)},$ isomorphic to ${\mathfrak{o}}(4,8),$ whose graded structure is represented by the cross-marked Satake diagram $$\xymatrix@R=.1pc{ \alphaup_{6} \\ \!\!\medbullet\!\!\! \ar@{-}[rd] &\alphaup_{4} &\alphaup_{3} & \alphaup_2 & \alphaup_{1}\\ & \!\!\!\medcirc\!\!\! \!\ar@{-}[r] & \!\!\!\medbullet\!\!\! \ar@{-}[r] &\!\!\medcirc\!\!\ar@{-}[r] & \!\!\medbullet\!\!\! \\ \!\!\medbullet\!\!\! \ar@{-}[ru] & & & \times \\ \alphaup_5 & & \\ }$$ The only real form of ${\mathfrak{o}}(10,\operatorname{\mathbb{C}})$ having a quaternionic spin representation is ${\mathfrak{o}}(3,7)$ and the corresponding real form of ${\mathfrak{g}}^{(5)}$ is of type $\mathbf{E}\mathrm{V\!{I}},$ with cross-marked Satake diagram $$\xymatrix@C=.9pc@R=-.3pc{ \alphaup_1 & \alphaup_3 &\alphaup_4 &\alphaup_5 &\alphaup_6 &\alphaup_7\\ \!\!\medcirc\!\!\! \ar@{-}[r]&\!\!\medcirc\!\! \ar@{-}[r]&\!\! \medcirc\!\!\ar@{-}[r] \ar@{-}[dddddd] &\!\!\medbullet\!\!\ar@{-}[r] &\!\! \medcirc\!\!\ar@{-}[r] &\!\!\medbullet\\ && && \times \\ && \\ && \\ && \\ && \\ & & \!\medbullet\! &\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\alphaup_2}$$

--- abstract: 'The effects of Lorentz boosts on the quantum entanglement encoded by a pair of massive spin one-half particles are described according to the Lorentz covariant structure described by Dirac bispinors. The quantum system considered incorporates four degrees of freedom – two of them related to the bispinor intrinsic parity and other two related to the bispinor spin projection, i.e. the Dirac particle helicity. Because of the natural multipartite structure involved, the Meyer-Wallach global measure of entanglement is preliminarily used for computing global quantum correlations, while the entanglement separately encoded by spin degrees of freedom is measured through the negativity of the reduced two-particle spin-spin state. A general framework to compute the changes on quantum entanglement induced by a boost is developed, and then specialized to describe three particular anti-symmetric two-particle states. According to the obtained results, two-particle spin-spin entanglement cannot be created by the action of a Lorentz boost in a spin-spin separable anti-symmetric state. On the other hand, the maximal spin-spin entanglement encoded by anti-symmetric superpositions is degraded by Lorentz boosts driven by high-speed frame transformations. Finally, the effects of boosts on chiral states are shown to exhibit interesting invariance properties, which can only be obtained through such a Lorentz covariant formulation of the problem.' author: - 'Victor A. S. V. Bittencourt' - 'Alex E. Bernardini' - Massimo Blasone date: - - title: Global Dirac bispinor entanglement under Lorentz boosts --- Introduction ============ Relativistic quantum information is a fast developing field that merges quantum information with relativistic quantum mechanics so to devise communication protocols in relativistic frameworks involving, for instance, clock synchronization [@clock], position verification [@position] and teleportation protocols [@teleport]. The effects of a relativistic frame transformation on quantum correlations have been recently investigated [@relat01; @relat02; @relat03; @relat04; @relat05; @relat06; @relat07; @relat08; @relatvedral] and, considering spin as the natural tool for quantum information engineering, the effects of frame transformations (Lorentz boosts) on the quantum entanglement encoded by a pair of spin one-half particles have been investigated. From the kinematic point of view, the action of the linear transformation given by a Lorentz boost, $\Lambda$, describes the change of space-time coordinates from an inertial frame, $\mathcal{S}$, to another one, $\mathcal{S}^{\prime}$, which moves with respect to $\mathcal{S}$, as to set, for instance, the quadrimomenta transformation relation $p^{\nu\prime} = \Lambda^{\nu}_{\mu} p^{\mu}$, summarized by $p^{\prime} = \Lambda p$ in the matricial representative notation, where $p^{\prime}$ and $p$ are described by coordinates at $\mathcal{S}^{\prime}$ and $\mathcal{S}$, respectively. As seminally stated by Wigner [@wigner], under such a transformation between inertial frames, an observable spin (projector operator) described by the $SU(2)$ adjoint representation realized by ${\boldmath{\mbox{$\sigma$}}} = (\sigma_x,\sigma_y,\sigma_z)$, where $\sigma_{x,y,z}$ are the Pauli matrices, has its spin projection onto the particle momentum direction, $\hat{\bm{e}}_p\cdot\hat{\boldmath{\mbox{$\sigma$}}}$, with ${\bm{e}}_p={\bm{p}}/\vert{\bm{p}}\vert$, changed as to return $\hat{\bm{e}}_{p^{\prime}}\cdot\hat{\boldmath{\mbox{$\sigma$}}} \neq \hat{\bm{e}}_{p}\cdot\hat{\boldmath{\mbox{$\sigma$}}}$, where boldfaced variables, $\bm{v} = (v_x,v_y,v_z)$, denote spatial vectors with modulus $v = \sqrt{\bm{v} \cdot \bm{v}}$, and hats “$~\hat{}~$” denote quantum operators. The rigorous treatment of the above kinematic properties, and of their imprints on quantum states of spin one-half particles, involves a description of their observable related properties in terms of the irreducible representations ([*irreps*]{}) of the Poincaré group [@fonda; @weinberg]. For instance, for a particle with momentum, $\bm{p}$, in an inertial frame $\mathcal{S}$, described by the quantum state $\vert\phi_s \bb{\bm{p}}\rangle$, where $s=1,2$ denote accessible spin states, the action of a Lorentz boost, $\Lambda$, that describes the change from $\mathcal{S}$ to $\mathcal{S}^{\prime}$, is given by the unitary transformation [@fonda; @weinberg; @wigner] $$\label{spinstates} \vert \phi_s \bb{\bm{p}}\rangle \, \, \rightarrow \, \, \hat{D} [\Lambda] \vert\phi_s \bb{\bm{p}}\rangle = \sum_{r} c_{s r}(\Lambda, \bm{p}) \vert\phi_r (\bm{p}^{\prime})\rangle,$$ where the unitary operator, $\hat{D} [\Lambda]$, and its explicit component dependence, $c_{s r}$, according to the Poincaré group representations [@fonda; @weinberg] (cf. Sec. II), are given in agreement with the [*irrep*]{} of the quantum state, $\vert\phi_s \bb{\bm{p}}\rangle$, which can describe, for instance, a spinor (in a doublet representation, like electrons and positrons described either as Weyl or as Majorana fermions), a vector (in a triplet representation, like $^3S_1$ positronium, or even photons), bispinors (in a double doublet representation, like electrons and positrons described by Dirac fermions), or even scalar (in a singlet representation, like $^1S_0$ positronium) and higher order (maybe non-physical) tensor states. The point in this paper is that when quantum states depend on the momentum, namely those described by Dirac equation solutions for spin one-half states, different inertial observers will see different superpositions, and if somehow the momentum degrees of freedom are traced out, the quantum entanglement between spin states might change [@relat01; @relat02; @relat03; @relat04; @relat05; @relat06; @relat07; @relat08; @relatvedral]. Of course, for two-particle states, the question related to the influence of the reference frame in the computation of quantum correlations is much more engendered in the framework of relativistic quantum mechanics supported by the Dirac formalism. Despite the effectiveness of the [*irreps*]{} of the Poincaré group, in the Lorentz covariant Hamiltonian formulation of quantum mechanics, one has to pay attention to inclusion of mass in the relativistic formalism described by the Dirac equation. As one shall see in Sec. II, it requires the inclusion of the parity symmetry and the equalization of its role with the helicity (spin one-half projection, $\hat{\bm{e}}_p\cdot\hat{\boldmath{\mbox{$\sigma$}}} \sim \hat{\sigma}_z$) symmetry, as an accomplished $SU(2)$ symmetry. It supports, for instance, the description of electrons as Dirac Hamiltonian eigenstates in the double doublet [*irrep*]{} of the $SU(2) \otimes SU(2)$. Spatial parity couples positive and negative parity states with positive and negative helicity states as they were described by [*irreps*]{} of the Poincaré group [@fonda] and, in order to have complete invariance under parity, one needs to consider the *extended Poincaré group* [@weinberg; @WuTung]. In this case, spin one-half is carried by Dirac four component spinors, the bispinors satisfying the Dirac equation, in a representation supported by a subgroup of $SL(2,\mathbb{C}) \otimes SL(2,\mathbb{C})$, the $SU(2) \otimes SU(2)$. In fact, the description of massive charged fermions (such as electrons, muons, quarks, etc...) requires the [*irreps*]{} of the [*complete Lorentz group*]{}[^1], namely the Dirac (bi)spinors [@WuTung]. The intrinsic spin-parity (or helicity-parity) entanglement exhibited by a single Dirac bispinor has already been investigated in the context of quantum correlations driven by interactions with external fields [@extfields], which has been used for simulating Dirac-like systems as, for instance, four level ion traps [@diraclike01] and lattice/layer schemes in bilayer graphene [@diraclike02]. Generically, such an intrinsic entanglement encoded by Dirac-like $SU(2) \otimes SU(2)$ structures can also be generated, for example, by quantum electrodynamics (QED) scattering processes [@solano]. Not in the same scope, but also emphasizing the bispinor structure of fermionic quantum states, states constructed with the solutions of Dirac equation have been considered in the scrutinization of Bell inequalities [@bellDirac] and to obtain proper covariant spin density matrices and definitions of the position operator in the context of relativistic quantum mechanics [@spins; @celeri]. Likewise, the effects of Lorentz boosts in quantum entanglement encoded in bispinors were described in connection with Wigner rotations for a specific class of states [@bi-spinorarxiv02], and in the context of Fouldy-Wouthuysen (FW) spin operator [@greiner], with a focus on properties of transformation of spin-spin entanglement encoded in FW eigenstates [@bi-spinorFW]. However, considering the focus on the most phenomenologically appealing measurement of two-particle spin-spin entanglement, the intrinsic $SU(2) \otimes SU(2)$ covariant structure of Dirac bispinors, which is associated with intrinsic parity and spin [@SU2] for each particle, has not yet been completely incorporated into such an overall relativistic framework. The aim of this work is therefore to estimate the influence of Lorentz boosts on the quantum entanglement encoded in the intrinsic $SU(2) \otimes SU(2)$ structure of two spin one-half Dirac particles which are also spin-spin entangled. As each particle described by Dirac bispinors carries two qubits, the whole state is a four-qubit one, and since multipartite entanglement is generally present in such states, the Meyer-Wallach global measure of entanglement [@globalMW] shall be considered as a measure of the entanglement encoded in the four qubits of the system. Alternatively, the net result for the spin-spin entanglement, encoded only in a two-qubit mixed state, shall be computed through the negativity [@negativity01; @negativity02]. The effects of a Lorentz boost on the entanglement content of generic two-particle Dirac bispinor states shall be obtained for the case where superpositions of helicity plane waves are considered. The obtained results shall be specialized to anti-symmetric states showing, for example, that a Lorentz boost cannot create spin-spin entanglement in an initial separable anti-symmetric state. The paper is structured as follows. In Sec. II, the complete Lorentz covariant structure of the Dirac equation solutions, namely associated to the properties of $SU(2)$ spinor doublet representations, and to the composition of higher order multiplet representations, is reported about, and the foundations for establishing and discussing the spin-parity intrinsic entanglement are introduced. In Sec. III, by using the intrinsic $SU(2) \otimes SU(2)$ structure of the Dirac equation, the entanglement profile of a generic superposition of Dirac bispinors is described. In Sec. IV, by using the transformation laws of bispinors under Lorentz boosts, the effects of such transformations on the quantum correlations encoded by two-particle states are computed, with particular emphasis for anti-symmetric states. In addition, the investigation of the effects of boosts onto the superposition of chiral bispinors shows that some subtle invariance properties can be obtained. Our conclusions are given in Sec. V, where lessons concerning the importance of accounting for the Lorentz covariant structures in the computation of quantum correlations are drawn. Lorentz covariant structure of the Dirac equation and spin-parity intrinsic entanglement ======================================================================================== In quantum mechanics, the free particle Dirac Hamiltonian in the coordinate space reads $$\label{diracequation} \hat{H} \, \psi\bb{x} = i \frac{\partial \, \psi\bb{x}}{\partial t} = (-i\bm{\nabla} \cdot \hat{\bm{\alpha}} + m \hat{\beta}) \,\psi\bb{x} = (-i \hat{\alpha}_i\partial^i + m \hat{\beta}) \,\psi\bb{x} = \pm E_p \,\psi\bb{x} ,$$ where $E_p = \sqrt {p^2 + m^2}$, the space-time dependence has been resumed by $x\sim(t,\bm{x})$, and the Dirac matrix operators $ \hat{\bm{\alpha}} =(\hat{\alpha}_x,\,\hat{\alpha}_y,\,\hat{\alpha}_z)$ and $\hat{\beta}$ satisfy the anticommuting relations, $ \hat{\alpha}_i \hat{\alpha}_j + \hat{\alpha}_j \hat{\alpha}_i = 2 \delta_{ij} \hat{I}_4$, and $\hat{\alpha}_i \hat{\beta} + \hat{\beta} \hat{\alpha}_i =0$, for $i,j = x,y,z$, with $ \hat{\beta}^2 = \hat{I}_4$, $I_N$ the $N$-dim identity matrix, and $\hat{H}$ expressed in natural units, i.e. with $c = \hbar = 1$. The above Dirac Hamiltonian dynamics exhibits some symmetries that are supported by a group representation described by a direct product between two algebras which compose a subset of the group $SL(2,\mathbb{C})\otimes SL(2,\mathbb{C})$, the group $SU(2)\otimes SU(2)$. To clear up this point, before discussing the above statement in the enhanced language of Lie algebra and Lie groups, one simply notices that [*left-handed*]{} [*spinors*]{} are described by a doublet ($2$-dim) representation of the $SU(2)$ (left) and a singlet ($1$-dim) representation of the $SU(2)$ (right), $(\bm{2},\bm{1}) \equiv \psi^{\dagger} _{\L}\bb{x} =\left( \psi _{\L\1}\bb{x},\, \psi _{\L\2}\bb{x}\right)$ and, analogously, [*right-handed*]{} [*spinors*]{} are described by a doublet ($2$-dim) representation of the $SU(2)$ (right) and a singlet ($1$-dim) representation of the $SU(2)$ (left), $(\bm{1},\bm{2}) \equiv \psi^{\dagger} _{\R}\bb{x} =\left( \psi _{\R\1}\bb{x},\, \psi _{\R\2}\bb{x}\right)$, in order to support the following decomposition for the Dirac state vectors, $ \psi^{\dagger} \bb{x} =\left( \psi _{\L\1}\bb{x},\, \psi _{\L\2}\bb{x},\, \psi _{\R\1}\bb{x},\, \psi _{\R\2}\bb{x}\right) \equiv (\bm{2},\bm{1})\oplus(\bm{1},\bm{2})$, in a not unique double doublet representation of the $SU(2)\otimes SU(2)$ group. Therefore, the free particle Dirac equation is thus mapped into coupled differential equations for [*left-*]{} and [*right-handed*]{} components, respectively, $$\begin{aligned} i{\overline{\sigma}}^{\mu }\partial _{\mu }\psi _{\L}\bb{x} -m\psi _{\R}\bb{x} &=&0, \\ i{\sigma }^{\mu }\partial _{\mu }\psi _{\R}\bb{x} -m\psi _{\L}\bb{x} &=&0,\end{aligned}$$ in the so-called [*chiral representation*]{}, $(\hat{I}_{\2} , \hat{\bm{\boldsymbol{\sigma }}}) \equiv \sigma^{\mu}$ and $(\hat{I}_{\2}, -\hat{\bm{\boldsymbol{\sigma }}}) \equiv {\overline{\sigma}}^{\mu}$, for which the Lagrangian density reads $$\mathcal{L}=i\psi _{\L}^{\dagger }{\overline{\sigma}}^{\mu }\partial _{\mu }\psi _{\L}+i\psi _{\R}^{\dagger }\mathbf{\sigma }^{\mu }\partial _{\mu }\psi _{\R}-m\left( \psi _{\L}^{\dagger }\psi _{\R}+\psi _{\R}^{\dagger }\psi _{\L}\right),$$ from which a correspondence with the [*spinor*]{} chirality is identified. As it has been mentioned, the above choice is not unique. Another particular representation of the Dirac matrices is the Pauli-Dirac representation in which the Dirac matrices are decomposed into tensor products of Pauli matrices [@SU2], as $\hat{\alpha}_i = \hat{\sigma}_x^{(P)} \otimes \hat{\sigma}_i^{(S)}$, for $i = x,y,z$ and $\hat{\beta} = \hat{\sigma}_z^{(P)} \otimes \hat{I}^{(S)}$, which has another subjacent $su(2) \oplus su(2)$ algebra from the $SU(2)\otimes SU(2)$ group which, in this case, does not correspond to [*left-*]{} and [*right-handed*]{} chiral projection representations, instead, are associated to intrinsic parity, $P$, and spin (or helicity), $S$. In this case, the Dirac Hamiltonian is thus re-written in terms of Kronecker products between Pauli matrices as $$\label{twoqubithamiltonian} \hat{H} = \bm{p}\cdot (\hat{\sigma}_x^{(P)} \otimes \hat{\bm{\sigma}}^{(S)}) + m ( \hat{\sigma}_z ^{(P)} \otimes \hat{I}^{(S)}),$$ from which, according to the interpretation of quantum mechanics as an information theory for particles, where the superscripts $P$ and $S$ refer to the [*qubits*]{} of parity and spin, one can identify the Dirac equation solutions as they were described by two [*qubit*]{} states encoded in a massive particle whose dynamics is constrained by continuous variables. Within this framework, from the Hamiltonian Eq. (\[twoqubithamiltonian\]), the normalized stationary eigenstates in the momentum coordinate are written in terms of a sum of direct products describing *spin-parity* entangled states, $$\begin{aligned} \label{twoqubitspinor} \vert \, u_s\bb{\bm{p}} \, \rangle &=& \frac{1}{\sqrt{2 E_{p} (E_{p} + m)}} \left[ (E_{p} + m)\,\, \vert + \rangle \otimes \vert \chi_s \bb{\bm{p}} \rangle \,\,+\,\, \vert - \rangle \otimes \,(\, \bm{p} \cdot \bm{\sigma} \, \vert \chi_s \bb{\bm{p}} \rangle )\right],\\ \vert \, v_s\bb{\bm{p}} \, \rangle &=& \frac{1}{\sqrt{2 E_{p} (E_{p} + m)}} \left[ (E_{p} + m)\,\, \vert - \rangle \otimes \vert \chi_s \bb{\bm{p}} \rangle\,\, + \,\,\vert + \rangle \otimes \,(\, \bm{p} \cdot \bm{\sigma} \, \vert \chi_s \bb{\bm{p}} \rangle )\right],\end{aligned}$$ for positive and negative eigenvalues (associated frequencies), $\pm E_p = \pm\sqrt{p^2+m^2}$, respectively[^2], where $\vert \chi_s\bb{\bm{p}}\rangle \in \mathcal{H}_S$, with $s=1,\,2$, are the spinors related to the spatial motion of the particle, i.e. the particle’s helicity, which describes the dynamics of a fermion (in momentum representation) coupled to its spin, and $\vert \pm \rangle \in \mathcal{H}_P$ are intrinsic parity states. States as described by Eqs. (\[bi-spinors\]) are composite quantum systems in a total Hilbert space $\mathcal{H} = \mathcal{H}_P \otimes \mathcal{H}_S$ and, in the general form of Eq. (\[twoqubitspinor\]), they are spin-parity entangled [@SU2]. Of course, they are superposition of orthonormal parity eigenstates, $\vert \pm \rangle $, and therefore, they do not have a defined intrinsic parity quantum number[^3]. To summarize, the spin degree of freedom (DoF) identified by the index “$s$” is associated to [*irreps*]{} of the proper Poincaré group, and the positive/negative associated energy eigenstates of the spin one-half particles can be re-indexed through the notation $$\begin{aligned} \label{twoqubitspinor22} \vert \, u_{\pm,s}\bb{\bm{p}} \, \rangle &=& \frac{1}{\sqrt{2 E_{p} (E_{p} + m)}} \left[ (E_{p} + m)\,\, \vert \pm \rangle \otimes \vert \chi_s \bb{\bm{p}} \rangle \,\,+\,\, \vert \mp \rangle \otimes \,(\, \bm{p} \cdot \bm{\sigma} \, \vert \chi_s \bb{\bm{p}} \rangle )\right],\end{aligned}$$ for vectors belonging to the [*irrep*]{} labeled by $(\pm, \frac{1}{2})$, associated to the $SU(2)\otimes SU(2)$ group [@fonda; @weinberg; @wigner]. Therefore, the invariance under spatial parity symmetry requires an analysis with the [*complete Lorentz group*]{} in order to include [*irreps*]{} of $SU(2) \otimes SU(2)$ which merge spin with the additional DoF of intrinsic parity [@fonda; @WuTung]. In the context of a group theory, the above assertion can be better understand when the representations of $sl(2,\mathbb{C})\oplus sl(2,\mathbb{C})$, which corresponds to the Lie algebra of the $SL(2,\mathbb{C})\otimes SL(2,\mathbb{C})$ Lie group, are irreducible, i.e. they correspond to tensor products between linear complex representations of $sl(2,\mathbb{C})$, as it has been observed by considering the subgroup $SU(2)\otimes SU(2) \subset SL(2,\mathbb{C})\otimes SL(2,\mathbb{C})$. Unitary [*irreps*]{} of the $SU(2)\otimes SU(2)$ are built through tensor products between unitary representations of $SU(2)$, in a one-to-one correspondence with the group $SL(2,\mathbb{C})\otimes SL(2,\mathbb{C})$. Since it is a simply connected group, one also has a unique correspondence with the $sl(2,\mathbb{C})\oplus sl(2,\mathbb{C})$ algebra. As it has been above identified for the chiral basis and for the spin-parity basis, the existence of [*inequivalent representations*]{} of $SU(2) \otimes SU(2)$ follows from the above mentioned one-to-one correspondences. Inequivalent representations may not correspond to all the complete set of representations of $SL(2,\mathbb{C})\otimes SL(2,\mathbb{C})$, and therefore, of the proper Lorentz transformations that compose the $SO(1,3)$ group, i.e. the Lorentz group[^4]. Turning back to our point, as the transformations of $SU(2)\otimes SU(2)$ can be described by a subset of $SL(2,\mathbb{C})\otimes SL(2,\mathbb{C})$, one may choose at least two [*inequivalent*]{} subsets of $SU(2)$ generators, such that $SU(2)\otimes SU(2) \subset SL(2,\mathbb{C})\otimes SL(2,\mathbb{C})$, with each group transformation generator having its own [*irrep*]{}. Therefore, a fundamental object of the $SU_{\xi}(2)$, a spinor-like object $\xi$ described by $(\pm,\,0)$, transforms as a [*doublet*]{} – the fundamental representation – of $SU_{\xi}(2)$, and as a singlet – a transparent object under any $SU_{\chi}(2)$ transformations. Reciprocally, the fundamental object of the $SU_{\chi}(2)$, a typical spinor $\chi$ described by $(0,\frac{1}{2})$, transforms as a [*doublet*]{} of the $SU_{\chi}(2)$, and as a singlet of the $SU_{\xi}$. Under an improved notation generalized to higher dimension representations, $(\bm{dim}(SU_{\xi}(2)),\bm{dim}(SU_{\chi}(2)))$, the [*spinor*]{} $\xi$ is an object given by (,). Following the generalized idea for an arbitrary $SU_(2)\otimes SU_(2)$ composition, one has the representations as given by $(\bm{1},\bm{1})$ – for [*scalar*]{} or [*singlet*]{}, with angular momentum projection $j = 0$; $(\bm{2},\bm{1})$ – for [*spinor*]{} $(\frac{1}{2},\,0)$, with angular momentum projection $j = 1/2$, which corresponds to $(\pm,\,0)$ in case of $SU_{\xi}(2)\otimes SU_{\chi}(2)$ and also applies for designating [*left-handed*]{} spinors in case of an inequivalent representation of the $SU_{\tiny\mbox{Left}}(2)\otimes SU_{\tiny\mbox{Right}}(2)$ group; $(\bm{1},\bm{2})$ – for [*spinor*]{} $(0,\,\frac{1}{2})$, with angular momentum projection $j = 1/2$, which also applies for designating [*right-handed*]{} spinors in case of an inequivalent representation of the $SU_{\tiny\mbox{Left}}(2)\otimes SU_{\tiny\mbox{Right}}(2)$ group; $(\bm{2},\bm{2})$ – for [*vector*]{}, with angular momentum projection $j = 0$ and $j = 1$; etc. With respect to the fundamental representations of $SL(2,\mathbb{C})$, one may construct more complex objects like $ (\bm{1},\bm{2}) \otimes (\bm{1},\bm{2}) \equiv (\bm{1},\bm{1}) \oplus (\bm{1},\bm{3})$, a representation that composes Lorentz tensors like $$C_{\alpha\beta}\bb{x} = \epsilon_{\alpha\beta} D\bb{x} + G_{\alpha\beta}\bb{x},$$ where $D\bb{x}$ is a scalar, and $G_{\alpha\beta} = G_{\beta\alpha}$ is totally symmetric, or even $ (\bm{2},\bm{1}) \otimes (\bm{1},\bm{2}) \equiv (\bm{2},\bm{2})$, such that $ (\bm{2},\bm{2}) \otimes (\bm{2},\bm{2}) \equiv (\bm{1},\bm{1}) \oplus (\bm{1},\bm{3}) \oplus (\bm{3},\bm{1}) \oplus (\bm{3},\bm{3}) $, that composes Lorentz tensors like $$\varphi^{\mu\nu}\bb{x} = A^{\mu\nu}\bb{x} + S^{\mu\nu}\bb{x} + \frac{1}{4}g^{\mu\nu} \Omega\bb{x},$$ which correspond to a decomposition into smaller [*irreps*]{} related to the Poincaré classes quoted at [@extfields], with $A^{\mu\nu} \equiv (\bm{1},\bm{3}) \oplus (\bm{3},\bm{1})$ totally anti-symmetric under $\mu\leftrightarrow \nu$, $S^{\mu\nu}\equiv (\bm{3},\bm{3})$ totally symmetric under $\mu\leftrightarrow \nu$, and $\Omega \equiv (\bm{1},\bm{1})$ transforming as a Lorentz scalar, multiplied by the metric tensor, $g^{\mu\nu}$. As a matter of completeness, the above properties, as discussed in Ref. [@extfields], support the inclusion of interacting fields which also transform according to Poincaré symmetries described by the extended Poincaré group [@WuTung], as they appear in a full Dirac Hamiltonian like [@diraclike01; @diraclike02] $$\begin{aligned} \label{E04T} \hat{H} &=& A^0\bb{\bm{x}}\,\hat{I}_4+ \hat{\beta}( m + \phi_S \bb{\bm{x}} ) + \hat{\bm{\alpha}} \cdot (\hat{\bm{p}} - \bm{A}\bb{\bm{x}}) + i \hat{\beta} \hat{\gamma}_5 \mu\bb{\bm{x}} - \hat{\gamma}_5 q\bb{\bm{x}} + \hat{\gamma}_5 \hat{\bm{\alpha}}\cdot\bm{W}\bb{\bm{x}} \nonumber \\ &+& i \hat{\bm{\gamma}} \cdot [ \zeta_a \bm{B}\bb{\bm{x}} + \kappa_a\, \bm{E}\bb{\bm{x}} \,] + \hat{\gamma}_5 \hat{\bm{\gamma}}\cdot[\kappa_a\, \bm{B}\bb{\bm{x}} - \zeta_a \bm{E}\bb{\bm{x}} \,],\end{aligned}$$ where a fermion with mass $m$ and momentum $\bm{p}$ interacts with an external vector field with time component $A^0\bb{\bm{x}}$ and spatial components $\bm{A} \bb{\bm{x}}$, and is non-minimally coupled to magnetic and electric fields $\bm{B}\bb{\bm{x}}$ and $\bm{E}\bb{\bm{x}}$ (via $\kappa_a$ and $\zeta_a$). The above Hamiltonian also admits the inclusion of pseudovector field interactions with time component $q\bb{\bm{x}}$, and spatial components $\bm{W}\bb{\bm{x}}$, besides both scalar and pseudoscalar field interactions through $\phi_S\bb{\bm{x}}$ and $\mu\bb{\bm{x}}$, respectively. Algebraic strategies [@extfields] for obtaining the analytical expression for the matrix density of the associated eigenstates of the above Hamiltonian problem have been developed, however, they are out of the central scope of this paper. Bispinor entanglement under Lorentz boosts ========================================== With the normalized bispinors from Eq. (\[twoqubitspinor22\]), one can construct a general quantum state of two-particles, $A$ and $B$, respectively with momentum (energy) $\bm{p}$ ($E_p$) and $\bm{q}$ ($E_q$), as a generic $M$-term normalized superposition, $$\begin{aligned} \label{generalstate} \vert \, \Psi^{AB}\bb{\bm{p}, \bm{q}} \, \rangle &=& \frac{1}{\sqrt{N}} \displaystyle \sum_{i=1}^M c_i \, \vert \, u_{s_i}\bb{\bm{p}} \rangle^A \otimes \vert \, u_{r_i}\bb{\bm{q}} \, \rangle^B,\end{aligned}$$ with the normalization given by $\sum_{i=1}^M \vert c_i\vert^2 = N$, and where the subindex “$_{\pm}$” has been suppressed from the notation. Such two-particle states can be generated, for instance, in a QED elastic scattering process [@solano] [^5] As a matter of convenience, $u_{s_i}\bb{\bm{p}}$ (as well as $u_{r_i}\bb{\bm{q}}$) can be described by helicity eigenstates such that $\bm{e}_{p} \cdot \hat{\bm{\sigma}}^{(S)} \vert \chi_{s_i} \bb{\bm{p}} \rangle = (-1)^{s_i} \vert \chi_{s_i} \bb{\bm{p}}\rangle$ (where $\bm{e}_p = \bm{p}/\vert \bm{p} \vert$) can be factorized from Eq. (\[twoqubitspinor22\]) to set $u_{s_i}\bb{\bm{p}}$ a spin-parity separable state. In terms of projected states $\vert z_\pm \rangle$, eigenstates of $\hat{\sigma}_z^{(S)}$, one can write $$\vert \chi_{s_i}\bb{\bm{p}} \rangle = \frac{\hat{I}^{(S)}_2 + (-1)^{s_i} \bm{e}_{p} \cdot \hat{\bm{\sigma}}^{(S)}}{\sqrt{1 + \vert \bm{e}_{p} \cdot \bm{e}_z} \vert}\vert z_\pm \rangle,$$ and, if $\bm{e}_{p}$ is in the $z$-direction, $\bm{e}_{p} \equiv \bm{e}_z$, one has $\vert \chi_1\bb{\bm{p}} \rangle = \vert z_+ \rangle$ and $\vert \chi_2\bb{\bm{p}} \rangle = \vert z_- \rangle$, such that, from now on, the labels $s_i$ (and also $r_i$), when they are set equal to $1$ and $2$, denote positive and negative helicity, respectively. Under the above assumptions, the density matrix of the generic superposition from Eq. (\[generalstate\]) is written as $$\label{2partDM} \rho\bb{\bm{p}, \bm{q}} = \frac{1}{N} \displaystyle \sum_{i, j}^M c_i c_j^*\, \rho_{s_i s_j}^{A}\bb{\bm{p}} \otimes \rho_{r_i r_j}^{B}\bb{\bm{q}},$$ where $$\begin{aligned} \rho_{s_i s_j}^{A}\bb{\bm{p}} &=&\left( \vert u_{s_i}\bb{\bm{p}} \rangle \langle u_{s_j}\bb{\bm{p}}\vert\right)^A \nonumber \\ &=&\frac{1}{2 E_{p}} \Bigg[ \left( E_p \delta_{s_i s_j} + m \delta_{s_i s_j+1} \right) \hat{I}^{(P)A}_2 + \left(E_p \delta_{s_i s_j+1} + m \delta_{s_i s_j} \right) \hat{\sigma}_z^{(P)A} + \nonumber \\ &&\qquad \qquad + \sqrt{E_{p}^2 - m^2}\left( (-1)^{s_j} \, \hat{\sigma}_+^{(P)A} + (-1)^{s_i} \, \hat{\sigma}_-^{(P)A} \right)\Bigg] \otimes \Xi_{s_i s_j}^{(S)A}\bb{\bm{p}}, \end{aligned}$$ where $\hat{\sigma}_{\pm} = \hat{\sigma}_x \pm i\hat{\sigma}_y$ and the factorized dependence on the momentum direction is implicitly given by $$\Xi_{s_i s_j}^{(S)A}\bb{\bm{p}} = \left( \vert \chi_{s_i}\bb{\bm{p}} \rangle \langle \chi_{s_j}\bb{\bm{p}} \vert\right)^A,$$ with a similar expression for $\rho_{r_i r_j}^{B}\bb{\bm{q}}$ by replacing $\{\bm{p}; s_{i(j)} \}$ by $\{\bm{q}; r_{i(j)}\}$ and $A$ by $B$. As each of the components of the state (\[2partDM\]) is a two-qubit state, the joint state $\rho\bb{\bm{p}, \bm{q}}$ is a four-qubit state. Differently from the case where a system composed by two subsystems has the quantum entanglement supported by the Schmidt decomposition theorem, the classification and quantification of entanglement in the above constructed multipartite states is an open problem. Subsystems in a multipartite state can share entanglement in different non-equivalent ways, and the corresponding multipartite entanglement can be approached by different points of view. As the joint state (\[2partDM\]) is a pure state, its corresponding multipartite entanglement can be computed through the Meyer-Wallach global measure of entanglement, $E_{G}[\rho]$, expressed in terms of the linear entropy, $E_{L}[\rho]$, as [@globalMW] $$\label{globalent} E_{G}[\rho] = \bar{E}[\rho^{\{\alpha_k\}}] = \frac{1}{4}\big[\, E_L[\rho^{(S)A}] + E_L[\rho^{(P)A}] + E_L[\rho^{(S)B}] + E_L[\rho^{(P)B}] \, \big],$$ with $$E_L[\rho] = \frac{d}{d-1}(1 - \mbox{Tr}[\rho^2]),$$ where $d$ is the dimension of the underlying Hilbert space in which $\rho$ acts, and the reduced density matrix of a given subsystem $\alpha_k$ is obtained by tracing out all the other subsystems $\rho^{\alpha_j} = \mbox{Tr}_{\{\alpha_k \} \neq \alpha_j} [\rho]$. In the above problem, the subsystems considered correspond to spin and parity, $S$ and $P$, for particles $A$ and $B$, i.e. $\{\alpha_k\} \equiv \{(S)A,\,(S)B,\,(P)A,\,(P)B\}$. In particular, the more the subsystems of a given state are mixed, the more entanglement is encoded among them: the global measure, $E_{G}[\rho]$, captures a picture of the quantum correlations distributed among the four DoF’s here involved.. The linear entropy of a reduced subsystem $\rho^{\alpha_k}$ of (\[generalstate\]), which is a two-qubit state, is evaluated in terms of the components of its Bloch vector $a^{\alpha_k}_n = \mbox{Tr}[\hat{\sigma}_n^{\alpha_k} \rho^{\alpha_k}]$ as $$\label{globalexpr0} E_L[\rho^{\alpha_k}]= 1 - \sum_{n = \{x, \, y, \, z\}} (a^{\alpha_k}_n)^2,$$ and the global measure from Eq. (\[globalent\]) can be simplified into $$\label{globalexpr} E_{G}[\rho] =1 - \frac{1}{4} \, \displaystyle \sum_{\alpha = \{\alpha_k\}} \sum_{n = \{x, \, y, \, z\}} (a^{\alpha}_n)^2,$$ with $ \{\alpha_k\}\equiv \{(S)A,\,(S)B,\,(P)A,\,(P)B\}$. The Bloch vectors of the subsystems of $A$ are explicitly given by $$\begin{aligned} \label{blochvecs} a^{(S)A}_n &=& \frac{1}{N}\displaystyle \sum_{i, j}^M\, c_i c_j^* \, \mathcal{M}_{r_i r_j}\bb{\bm{q}} \, \frac{1}{ E_p} ( \, E_p \delta_{s_i s_j} + m \delta_{s_i s_j+1} \,) \mbox{Tr}[\hat{\sigma}_n^{(S)A} \Xi_{s_i s_j}^{(S)A}\bb{\bm{p}}],\end{aligned}$$ for the spin subsystem, $$\begin{aligned} a^{(P)A}_x &=& \frac{1}{N} \displaystyle \sum_{i}^M (-1)^{s_i} \, \vert \, c_i \, \vert^2 \, \frac{\sqrt{E_p^2 - m^2}}{E_p}, \nonumber \\ a^{(P)A}_y &=& 0, \nonumber \\ a^{(P)A}_z &=& \frac{1}{N}\displaystyle \sum_{i, j}^M c_i c_j^* \, \mathcal{M}_{r_i r_j}\bb{\bm{q}} \, \mbox{Tr}[\Xi_{s_i s_j}^{(S)A}\bb{\bm{p}} ] \, \frac{1}{E_p} (\, E_p \delta_{s_i s_j+1} + m \delta_{s_i s_j} \,),\end{aligned}$$ for the parity subsystem, where $$\begin{aligned} \label{coeff} \mathcal{M}_{r_i r_j}\bb{\bm{q}} = \mbox{Tr}[\rho^{B}_{r_i r_j}\bb{\bm{q}}] = \frac{1}{E_q} ( \, E_q \delta_{r_i r_j} + m \delta_{r_i r_j+1} \,) \mbox{Tr}[\Xi_{r_i r_j}^{(S)B}\bb{\bm{q}}].\end{aligned}$$ Analogous expressions for the Bloch vectors of the subsystems of $B$ are given by (\[blochvecs\]) and (\[coeff\]) with the replacement $\{\bm{p}; s_{i(j)} \}\leftrightarrow\{\bm{q}; r_{i(j)}\}$ and $A\leftrightarrow B$. To evaluate the quantum entanglement encoded only by the spin DoF’s in (\[2partDM\]), one considers the spin-spin reduced density matrix $$\begin{aligned} \label{unboostspin} \rho^{(S)A, (S)B}\bb{\bm{p}, \bm{q}} &=& \mbox{Tr}_{(P)A, (P)B} \left[\rho\bb{\bm{p}, \bm{q}}\right] \\ &=& \frac{1}{N} \displaystyle \sum_{i,j}^M c_i c_j^* \, \frac{ \left( E_p \delta_{s_i s_j} + m \delta_{s_i s_j+1} \right) \, \left( E_q \delta_{r_i r_j} + m \delta_{r_i r_j+1} \right)}{E_p\,E_q}\, \Xi_{s_i s_j}^{(S)A}\bb{\bm{p}}\otimes \Xi_{r_i r_j}^{(S)B}\bb{\bm{q}},\quad\nonumber\end{aligned}$$which is, in general, a mixed state. Entanglement in mixed states cannot be evaluate in terms of the linear entropy, as a mixed subsystem does not imply into a joint entangled state for mixtures. Instead, the characterization of quantum entanglement, in this case, is given by the Peres separability criterion [@negativity01] which asserts that a bipartite state $\rho \in \mathcal{H}_{A} \otimes \mathcal{H}_B$ is separable iff the partial transpose density matrix with respect to the any of its subsystem, $\rho^{T_A}$, has only positive eigenvalues. With respect to a fixed basis on the composite Hilbert space $\{\vert \lambda_i \rangle \otimes \vert \nu_j \rangle \}$ (with $\vert \lambda_i \rangle \in \mathcal{H}_{A}$ and $\vert \nu_i \rangle \in \mathcal{H}_{B}$), the matrix elements of the partial transpose with respect to the $A$ subsystem $\rho^{T_{A}}$ are given by $$\langle \lambda_i \vert \otimes \langle \nu_j \vert (\,\rho \,)\,^{T_{A}} \vert \lambda_k \rangle \otimes \vert \nu_l \rangle = \langle \lambda_k \vert \otimes \langle \nu_j \vert \, \rho \, \vert \lambda_i \rangle \otimes \vert \nu_l \rangle,$$ and in the light of the separability criterion, the negativity $\mathcal{N}[\rho]$ is defined as [@negativity02] $$\label{negativity} \mathcal{N}[\rho] = \displaystyle{\sum}_{i} \vert \lambda_i \vert - 1 ,$$ where $\lambda_i$ are the eigenvalues of $\rho^{T_{A}}$. The spin-spin negativity of (\[2partDM\]) $\mathcal{N} \big[ \rho^{(S)A, (S)B} \big]$ is then evaluated with the eigenvalues of the partial transpose of (\[unboostspin\]) with respect to $(S)A$ as to return $$\begin{aligned} \label{unboostparttranspose} \big ( \, \rho^{(S)A,(S)B)} \, \big )^{T_{A}}\bb{\bm{p}, \bm{q}} &=& \big( \,\mbox{Tr}_{(P)A, (P)B} \left[\rho\bb{\bm{p}, \bm{q}}\right] \, \big)^{T_{A}} \\ &=& \frac{1}{N} \displaystyle \sum_{i,j}^M c_i c_j^* \, \frac{ \left( E_p \delta_{s_i s_j} + m \delta_{s_i s_j+1} \right) \, \left( E_q \delta_{r_i r_j} + m \delta_{r_i r_j+1} \right)}{E_p\,E_q}\, \Xi_{s_j s_i}^{(S)A}\bb{\bm{p}}\otimes \Xi_{r_i r_j}^{(S)B}\bb{\bm{q}},\quad\nonumber\end{aligned}$$where the subtle change with respect to (\[unboostspin\]) is in the subindex of $\Xi^{(S)A}$. Covariance of the Dirac equation and the effects of Lorentz boosts ================================================================== Once the global and the spin-spin entanglement of the general superposition (\[generalstate\]) are characterized by Eqs. (\[globalexpr\]) and (\[blochvecs\]), and the spin-spin negativity is evaluated through the eigenvalues of Eq. (\[unboostparttranspose\]), one can describe how the Lorentz boosts do affect such quantum correlations. First, one notices that the covariant form of the Dirac equation $$(\hat{\gamma}_\mu p^\mu -m \hat{I}_4) \psi\bb{x} = 0,$$ where $\hat{\gamma}_0 = \hat{\beta}$ and $\gamma_{\mu} = (\gamma_{0}, \hat{\bm{\gamma}})$ with $\hat{\bm{\gamma}} = \hat{\beta} \hat{\bm{\alpha}}$, transforms under a Lorentz boost, $x^\mu \rightarrow x^{\mu\prime} = \Lambda^\mu_{\, \, \, \nu} x^\nu$, as $$\begin{aligned} (\hat{\gamma}^\mu p_\mu -m \hat{I}_4) \psi\bb{x} = 0 \, \, \rightarrow \, \, ((\hat{\gamma}^\prime)^\mu p^\prime_\mu -m \hat{I}_4) \psi^\prime(x^\prime) = 0,\end{aligned}$$ and its solution, $\psi \bb{x}$, transforms as $$\label{boooo} \psi^\prime(x^\prime) = \hat{S}[\, \Lambda \,] \psi(\Lambda^{-1} x^\prime),$$ where $\hat{S}[\, \Lambda \,]$ corresponds to the transformation in the bispinor space representation (cf. $\hat{D} [\Lambda]$ from (\[spinstates\])). Lorentz boosts, $\Lambda (\omega)$, can be parameterized in terms of components in the vector representation of the $SO(1,3)$ as $[\Lambda (\omega)]_{ij} = \delta_{ij} + (\cosh (\omega) - 1)\,n_i \, n_j$, $[\Lambda (\omega)]_{i0} = [\Lambda(\omega)]_{0i} = \sinh{(\omega)} \,n_i$, and $[\Lambda (\omega)]_{00} = \cosh{(\omega)}$, where $\omega=\mbox{arccosh}(\sqrt{1-v^2})$ is the (dimensionless) boost rapidity, $v$ is the reference frame velocity (between $\mathcal{S}$ and $\mathcal{S}^{\prime}$) and $n_i$ are the space components of the boost direction, $\bm{n}$, with $\bm{n}\cdot\bm{n}=1$. In the bispinor space representation, $\hat{S}[\Lambda(\omega)]$, reads $$\label{boostrep} \hat{S}[\Lambda(\omega)]= \cosh{\left( \frac{\omega}{2} \right)} \hat{I}_4 - \sinh{\left( \frac{\omega}{2} \right)} \bm{n} \cdot \hat{\bm{\alpha}},$$ which is a non-unitary operator. By following the above introduced two-qubit prescription, the boost operator (\[boostrep\]) can be expressed in the form of $$\label{twoqubitboost} \hat{S}[\Lambda(\omega)]=\cosh{\left( \frac{\omega}{2} \right)} \hat{I}_2^{(P)}\otimes \hat{I}_2^{(S)} - \sinh{\left( \frac{\omega}{2} \right)} \bm{n} \cdot( \, \hat{\sigma}_x^{(P)} \otimes \hat{\bm{\sigma}}^{(S)} \,),$$ from which one can evaluate the effects of boosts in parity and spin subsystems. For instance, keeping the covariant notation for the quadrimomentum, $p$, the density matrix of a single helicity bispinor with quantum number $s$ transforms under boosts as $$\rho_s\bb{p} \rightarrow \rho_s^\prime\bb{p^\prime} = \frac{1}{\cosh(\omega)}\hat{S}[\Lambda(\omega)] \, \rho_s(\Lambda^{-1} p^\prime) \, \hat{S}^\dagger[\Lambda(\omega)],$$ where the term $(\cosh{(\omega)})^{-1}$ was included as to keep the normalization of the spinor, and (\[twoqubitboost\]) can be used to describe the transformation law of the subsystem described by the spin density, $\rho^{(S)}_s\bb{p} = \mbox{Tr}_{(P)}[ \rho_s\bb{p}]$, as $$\begin{aligned} \rho^{(S)}_s\bb{p} \rightarrow \rho^{\prime(S)}_s\bb{p^\prime} = \frac{1}{\cosh{(\omega)}} \Big[ \cosh^2{\left(\frac{\omega}{2} \right)} \rho^{(S)}_s\bb{p} + \sinh^2{\left(\frac{\omega}{2} \right)} (\bm{n} \cdot \hat{\bm{\sigma}}) \rho^{(S)}_s\bb{p} (\bm{n} \cdot \hat{\bm{\sigma}}) \nonumber \\ \qquad \qquad\qquad\qquad\qquad\qquad\qquad - (-1)^s \sinh{(\omega)} \frac{E_p - m}{E_p} \{ \bm{n} \cdot \hat{\bm{\sigma}}, \rho^{(S)}_s\bb{p} \}\Big],\end{aligned}$$ where $\{\,\,,\,\,\}$ denotes anti-commutators, and which, in the limit $E_p - m \simeq E_p$, can be subtly simplified as to give a transformation law in the form of $\rho^{\prime(S)}_s\bb{p^\prime} =\hat{O} \, \rho^{(S)}_s\bb{p} \, \hat{O}^\dagger$, where $\hat{O}$ is the unitary operator $$\hat{O} = \frac{1}{\sqrt{\cosh{(\omega)}}} \left[ \cosh{\left(\frac{\omega}{2} \right)} \hat{I}_2 - \sinh{\left(\frac{\omega}{2} \right)} (\bm{n} \cdot \hat{\bm{\sigma}}) \left({\bm{e}}_{p} \cdot \hat{\bm{\sigma}} \right) \right].$$ In fact, such transformation under a Lorentz boost is the same as that one obtained for states belonging to the [*irrep*]{} $(+, \frac{1}{2})$ of the Poincaré group, which can be recast in terms of a momentum dependent rotation and which is the basis of several results in relativistic quantum information[^6]. Considering the generic two-particle state (\[2partDM\]) in a reference frame $\mathcal{S}$, the transformed density matrix describing the state in an inertial frame $\mathcal{S}^{\prime}$, related to $\mathcal{S}$ by a Lorentz boost, $\Lambda$, is given by $$\begin{aligned} \rho\bb{\bm{p}, \bm{q}} \rightarrow \rho^\prime\bb{\bm{p}^\prime, \bm{q}^\prime} &=& \frac{1}{\nu} \big( \, \hat{S}^{A}[\Lambda] \otimes \hat{S}^{B}[\Lambda] \,\big) \,\rho\bb{\bm{p}, \bm{q}} \, \big( \, (\hat{S}^{A}[\Lambda])^\dagger \otimes (\hat{S}^{B}[\Lambda])^\dagger \,\big) \nonumber \\ &=&\frac{1}{\nu} \sum_{i,j}^M c_i c_j^* \, \varrho_{s_i s_j}^{ A} \bb{\bm{p}}\otimes \varrho_{r_i r_j}^{B}\bb{\bm{q}},\end{aligned}$$ where $\nu = Tr\left[\left( \, \hat{S}^{A}[\Lambda] \otimes \hat{S}^{B}[\Lambda] \right)^2\,\rho\bb{\bm{p}, \bm{q}}\right]$ and the transformed term $ \varrho_{s_i s_j}^{A}\bb{\bm{p}}$ reads $$\begin{aligned} \varrho_{s_i s_j}^{ A}\bb{\bm{p}} &=& \cosh^2{\left( \frac{\omega}{2} \right)} \rho_{s_i \, s_j}^{A}\bb{\bm{p}} - \frac{\sinh(\omega)}{2} \{ \rho_{s_i s_j}^{A}\bb{\bm{p}}, \, (\hat{\sigma}_x^{(P)A} \otimes \bm{n} \cdot \hat{\bm{\sigma}}^{(S)A}) \, \} \nonumber \\ &&\qquad\qquad\qquad+ \sinh^2{\left( \frac{\omega}{2} \right)} (\hat{\sigma}_x^{(P)A} \otimes \bm{n} \cdot \hat{\bm{\sigma}}^{(S)A}) \, \rho_{s_i s_j}^{A}\bb{\bm{p}} \, (\hat{\sigma}_x^{(P)A} \otimes \bm{n} \cdot \hat{\bm{\sigma}}^{(S)A}), \end{aligned}$$ with an analogous expression for $\varrho_{r_i r_j}^{B}\bb{\bm{q}}$. The difference between the global entanglement in $\mathcal{S}^{\prime}$ and $\mathcal{S}$, $$\Delta E_G = E_G[\rho^\prime\bb{\bm{p}^\prime, \bm{q}^\prime}] - E_G[\rho\bb{\bm{p}, \bm{q}}],$$ is evaluated through Eqs. (\[globalexpr\]) and (\[blochvecs\]) replaced by transformed Bloch vectors, now renamed by $a\to \mathcal{A}$, which are given by $$\begin{aligned} \label{transfblochspin} {\mathcal{A}}^{(S)A}_{k} &=& \frac{1}{\nu}\displaystyle \sum_{i,j}^M c_i c_j^* \, \mu_{r_i r_j} \, \frac{1}{E_p}\Bigg[ \, \mbox{Tr}[\hat{\sigma}_k^{(S)A} \Xi_{s_i s_j}^{(S)A}] \cosh^2{\left( \frac{\omega}{2} \right)} ( \, E_p \delta_{s_i s_j} + m \delta_{s_i s_j +1} \,) \nonumber \\ &&\qquad-2 \, (-1)^{s_i} n_k \, \mbox{Tr}[\Xi_{s_i s_j}^{(S)A}] \, \sinh(\omega) \sqrt{E_p^2 - m^2} \, \delta_{s_i s_j} \\ &&\qquad\qquad\qquad+ \mbox{Tr}[\hat{\sigma}_k^{(S)A}\, (\bm{n} \cdot \hat{\bm{\sigma}}^{(S)A})\, \Xi_{s_i s_j}^{(S)A} \, (\bm{n} \cdot \hat{\bm{\sigma}}^{(S)A})] \sinh^2{\left( \frac{\omega}{2} \right)} ( \, E_p \delta_{s_i s_j} + m \delta_{s_i s_j +1} \,) \Bigg]\nonumber,\end{aligned}$$for the spin reduced subsystem of $A$, and $$\begin{aligned} \label{transfblochpar} {\mathcal{A}_x}^{(P)A} &=& \frac{1}{\nu}\displaystyle \sum_{i,j}^M c_i c_j^* \, \mu_{r_i r_j} \, \frac{1}{E_p} \, \Bigg[(-1)^{s_i} \mbox{Tr}\big[\Xi_{s_i s_j}^{(S)A} \big] \cosh(\omega) \sqrt{E_p^2 - m^2} \delta_{s_i s_j} \nonumber \\ &&\qquad -\sinh(\omega) \, \mbox{Tr}\big[\,(\bm{n} \cdot \hat{\bm{\sigma}}^{(S)A} )\Xi_{s_i s_j}^{(S)A} \, \big] \left( E_p \delta_{s_i s_j} + m \delta_{s_i s_j +1} \right) \Bigg], \nonumber \\ {\mathcal{A}_x}^{(P)A} &=& \frac{1}{\nu}\displaystyle \sum_{i,j}^M c_i c_j^* \, \mu_{r_i r_j} \, \frac{E_p \delta_{s_i s_j+1} + m \delta_{s_i s_j}}{E_p} \mbox{Tr}\big[\Xi_{s_i s_j}^{(S)A} \big],\end{aligned}$$ for the parity reduced subsystem, where $$\begin{aligned} \label{transfcoeff} \mu_{r_i r_j} &=& \mbox{Tr}[\varrho_{r_i r_j}^{ B}] \nonumber \\ &=& \frac{1}{E_p}\big[ \, \cosh{(\omega)}(\, E_q \delta_{r_i r_j} + m \delta_{r_i r_j+1}\,) \mbox{Tr}[\Xi_{r_i r_j}^{(S)B}] \nonumber \\ &&\qquad\qquad - (-1)^{r_{i}} \, \sinh{(\omega)} \delta_{r_i r_j} \sqrt{E_q^2 - m^2} \, \mbox{Tr}[\bm{n} \cdot \hat{\bm{\sigma}}^{(S)B} \, \Xi_{r_i r_j}^{(S)B}]\, \big],\end{aligned}$$ and, in all the above expressions, the explicit dependence on $\bm{p}$ and $\bm{q}$ has been suppressed from the notation. Through the above expressions, again, the Bloch vector for the subsystems of $B$ can be obtained with the replacement $\{\bm{p}; s_{i(j)} \}\leftrightarrow\{\bm{q}; r_{i(j)}\}$ and $A\leftrightarrow B$ into Eqs. (\[transfblochspin\])-(\[transfcoeff\]). For any boost one also has ${\mathcal{A}_y}^{(P)A} = {\mathcal{A}_y}^{(P)B}=0$. The effects of the boost on the spin-spin entanglement, on the other hand, are described by the change on the negativity $$\Delta \mathcal{N}^{(S)A,(S)B} = \mathcal{N}[\varrho^{(S)A,(S)B}] -\mathcal{N}[\rho^{(S)A,(S)B}] ,$$ with the transformed spin-spin density matrix given by $$\begin{aligned} \label{spintransformed} \varrho^{(S)A,(S)B} =\frac{1}{\nu} \displaystyle \sum_{i,j}^M c_i c_j^* \, \varrho_{s_i s_j}^{ (S)A} \otimes \varrho_{r_i r_j}^{(S)B},\end{aligned}$$ where $$\begin{aligned} \varrho_{s_i s_j}^{ (S)A} &=&\cosh^2{\left( \frac{\omega}{2} \right)}\, \frac{E_p \delta_{s_i s_j} + m \delta_{s_i s_j +1}}{E_p} \, \Xi_{s_i s_j}^{(S)A} \nonumber \\ &&\qquad - (-1)^{s_i} \frac{ \sinh(\omega) }{2} \frac{\sqrt{E_p^2 - m^2}}{E_p} \delta_{s_i s_j} \{ \Xi_{s_i s_j}^{(S)A}, \bm{n} \cdot \hat{\bm{\sigma}}^{(S)A} \} \nonumber \\ &&\qquad\qquad +\sinh^2{\left( \frac{\omega}{2} \right)} \,\frac{E_p \delta_{s_i s_j} + m \delta_{s_i s_j +1}}{E_p} (\bm{n} \cdot \hat{\bm{\sigma}}^{(S)A}) \, \Xi_{s_i s_j}^{(S)A}\,( \bm{n} \cdot \hat{\bm{\sigma}}^{(S)A}),\end{aligned}$$ with a corresponding expression for $\varrho_{r_i r_j}^{(S)B}$. From the above expression one concludes that if the boost is performed in a direction $\bm{n}$ such that $ \{ \Xi_{s_i s_j}^{(S)A}, \bm{n} \cdot \hat{\bm{\sigma}}^{(S)A} \}= \{ \Xi_{r_i r_j}^{(S)B}, \bm{n} \cdot \hat{\bm{\sigma}}^{(S)B} \} = 0$, then the spin reduced density matrix (\[spintransformed\]) is invariant. Entanglement for an overall class of anti-symmetric states ---------------------------------------------------------- The above framework describes quantitatively the changes on multipartite quantum correlations, as quantified by $E_G$, and on spin-spin entanglement induced by Lorentz boosts acting on a generic superposition of two-particle helicity bispinors, as quantified by $\mathcal{N}$. As the nature of fermionic particles requires anti-symmetric wave functions, states that are given by the anti-symmetric superpositions have to be considered in the form of $$\label{antisymmetric} \vert \Psi^{odd}_{sr}\bb{\bm{p}, \bm{q}} \rangle=\frac{\vert u_s\bb{\bm{p}} \rangle^{A} \otimes \vert u_r\bb{\bm{q}} \rangle^{B} - \vert u_r\bb{\bm{q}} \rangle^{A} \otimes \vert u_s\bb{\bm{p}} \rangle^{B}}{\sqrt{2}}.$$ Talking about Dirac particles like electrons, quarks, neutrinos, etc, some of the above configurations are very difficult to be produced phenomenologically. Thus, only some examples shall be considered in the following, from the less to more relevant ones. At the reference frame $\mathcal{S}$ with $\bm{p} = - \bm{q}$, the center of momentum frame, positive and negative helicity eigenstates are given by $$\begin{aligned} \label{hel} \vert \chi_1 \bb{\bm{p}} \rangle = \vert \chi_2 \bb{\bm{q}} \rangle = \vert z_+ \rangle, \nonumber \\ \vert \chi_2 \bb{\bm{p}} \rangle = \vert \chi_1 \bb{\bm{q}} \rangle = \vert z_- \rangle,\end{aligned}$$ and, in the unboosted frame $\mathcal{S}$, the states are also eigenstates of the Pauli spin operator, $\sigma_z$. It is sufficient to consider the boost with direction $\bm{n}$ in a plane defined by the unitary vectors, $\bm{e}_z$ and $\bm{e}_x$, with $\bm{n} = \sin{(\theta)} \bm{e}_x + \cos{(\theta)} \bm{e}_z$ as pictorially depicted in Fig. \[Scheme\]. By adapting the notation to the simplifications from Eq. (\[hel\]), one has the anti-symmetric state given by$$\label{state1} \vert \psi_1 \rangle = \frac{\vert u_1\bb{\bm{p}} \rangle^{A} \otimes \vert u_2\bb{\bm{q}} \rangle^{B} - \vert u_2\bb{\bm{q}} \rangle^{A} \otimes \vert u_1\bb{\bm{p}} \rangle^{B}}{\sqrt{2}},$$ in a superposition of helicities which, however, is spin-spin separable. Since $\Xi_{s \, r}^{(S)A}= \Xi_{s \, r}^{(S)B}=\vert z_+ \rangle \langle z_+ \vert$ for all $s$ and $r$, the transformed spin-spin density matrix Eq. (\[spintransformed\]) is invariant under partial transposition with respect to any of its subsystems, and thus a Lorentz boost does not create spin-spin entanglement. Nevertheless, the global entanglement $E_G$ is not invariant, as depicted in Fig. \[Graph01\] which shows $\Delta E_G$ as function of the boost rapidity $\omega$ and of the boost angle $\theta$. Of course, this is because $\vert \psi_1 \rangle$ mixes different momentum eigenstates, in a kind of artificial and unrealistic physical composition of particles $A$ and $B$. Boosts parallel to the momenta in $\mathcal{S}$ does not increase the amount of global entanglement in the state, although for any non-parallel boosts the global entanglement increases due to an increasing in both parity and spin reduced entropies, which are essentially constrained by the dependence on the momentum components. It tends to the maximum value ($\sim 1$) for high-speed boosts. Otherwise, a maximally entangled spin-spin state in $\mathcal{S}$ can be constructed through an anti-symmetric superposition between positive helicities $$\label{state2B} \vert \psi_2 \rangle = \frac{\vert u_1\bb{\bm{p}} \rangle^{A} \otimes \vert u_1\bb{\bm{q}} \rangle^{B} - \vert u_1\bb{\bm{q}} \rangle^{A} \otimes \vert u_1\bb{\bm{p}} \rangle^{B}}{\sqrt{2}},$$ which, according to the correspondence from (\[hel\]), indeed can be recast as $$\label{state2} \vert \psi_2 \rangle = \frac{\vert u_1\bb{\bm{p}} \rangle^{A} \otimes \vert u_1\bb{\bm{q}} \rangle^{B} - \vert u_2\bb{\bm{p}} \rangle^{A} \otimes \vert u_2\bb{\bm{q}} \rangle^{B}}{\sqrt{2}},$$ which corresponds to a much more realistic configuration, for which particles in the subspace $A$ and $B$ have well defined momenta, $\bm{p}$ and $\bm{q}$, respectively, in agreement with the construction from the previous section. Fig. \[Graph02\] depicts the variation of the global and the spin-spin entanglement of $\vert \psi_2 \rangle$ as function of the boost rapidity $\omega$. In this case, the variation of entanglement is independent of the boost angle and, as for the state from Eq. (\[state1\]), the global entanglement increases under Lorentz boosts. On the other hand, spin-spin entanglement is degraded by the boost transformation and for high speed boosts the spin-spin state is completely separable. A third anti-symmetric configuration is given by $$\label{state3} \vert \psi_3 \rangle = \frac{\vert u_1\bb{\bm{p}} \rangle^{A} \otimes \vert u_2\bb{\bm{p}} \rangle^{B} -\vert u_2\bb{\bm{p}} \rangle^{A} \otimes \vert u_1\bb{\bm{p}} \rangle^{B}}{\sqrt{2}},$$ which describes a two-particle helicity superposition moving in the $\bm{e}_z$ direction where both particles have the same momenta. This case is phenomenologically interesting because $\Delta v = 0$ is a kinematical Lorentz invariant. Two electrons in a common rest frame will have $\Delta v = 0$ for any relativistic boost. In this case, the spin-spin entanglement depends on the momentum $p$ even in the unboosted frame. Differently from the preliminary examples, both global entanglement, depicted in Fig. \[Graph03\], and spin-spin entanglement, depict in Fig. \[Graph04\], exhibit a non-monotonous behavior under Lorentz boosts. In particular, for a boost parallel to the momentum $\bm{p}$ with rapidity equals to $\mbox{arccosh}(\, E_p/m \,)$, the global entanglement is minimum, as this frame corresponds to the common rest frame of the particles where there is only spin-spin entanglement. For a high speed boost, the entanglement shared between the DoF’s of the state is enhanced, although the spin-spin entanglement, as in the case of state (\[state2\]), is completely degraded. It is worth to mention that, although the global measure from Eq. (\[globalent\]) was considered, four-qubit state entanglement can be computed through another global measure of entanglement defined in a similar fashion of (\[globalent\]), but with linear entropies of the reduced subsystems of two qubits. This quantity is calculated with terms of the form $\mbox{Tr}[\hat{\sigma}^{\alpha_k}_i \hat{\sigma}^{\beta_l}_j \rho^{\{\alpha_k; \beta_l\}}]$ and contains, in addition to the information encoded in $E_G$ (\[globalent\]), also correlations between pairs of the subsystems [@rigolin]. Nevertheless, in the case of the anti-symmetric states considered here, the behavior of this quantity is qualitatively similar to the behavior depicted in Figs. \[Graph02\] - \[Graph03\] and add no information about the variation of quantum entanglement encoded by bispinors under Lorentz boosts. Other point of view of multipartite entanglement is provided by considering the geometry of the composite Hilbert space, and by studying distances between a given multipartite state and the set of the so-called $K$-separable states [@MassimoEnt]. In this case, the quantification of multipartite entanglement can capture more information about different multipartite components that contribute to the total amount of quantum correlations in a given state, requiring an extremization process. This more complete picture of multipartite entanglement for the two spinors states considered here is postponed to future investigations. Transformation of entanglement in chiral states ----------------------------------------------- Superpositions of eigenstates of the chiral operator $\hat{\gamma}_5 = \hat{\sigma}_x^{(P)} \otimes, \hat{I}^{(S)}$ defined in terms of the free bispinors $u_s\bb{\bm{p}}$ as $$u^{f}_s\bb{\bm{p}} = \frac{\hat{I} + (-1)^f \hat{\gamma}_5}{2} u_s\bb{\bm{p}},$$ with $f = 0, 1$, can also be investigated in the above context. Differently from the helicity, the chirality is a Lorentz invariant given that the chiral and the boost operator commute, i.e. $[\hat{\gamma}_5, \hat{S}[\Lambda(\omega)]\,] = 0$. However, for massive particles, it is not a dynamical conserved quantity as $[\hat{\gamma}_5, \hat{H}] \neq 0$ [@Alex001; @Alex002]. This invariance property has implications for the transformation laws of quantum entanglement encoded by superpositions of chiral states $$\label{chiral01} \psi^{Chiral}\bb{\bm{p},\bm{q}} = \frac{1}{N}\displaystyle \sum_{i}^M c_i \, \vert u^{f_i} _{s_i}\bb{\bm{p}}\rangle^A \otimes \vert u^{g_i}_{r_i}\bb{\bm{q}}\rangle^B,$$ where $f_i$ is the chirality of the bispinor $\vert u^{f_i} _{s_i}\bb{\bm{p}}\rangle^A$ and $g_i$ is the chirality of $\vert u^{g_i}_{r_i}\bb{\bm{q}}\rangle^A$. Chiral states constructed through projection of helicity states can be written in the simplified form of $$\vert u^{f}_s\bb{\bm{p}} \rangle = \vert f \rangle \otimes \vert \chi_s \bb{\bm{p}} \rangle$$ where $\vert f \rangle = (\vert z_+ \rangle + (-1)^f \vert z_- \rangle)/2$ are the eigenstates of $\hat{\sigma}_x$ operator, and thus the density matrix of (\[chiral01\]) reads $$\label{chiral02} \rho_{Chiral} = \frac{1}{N}\displaystyle \sum_{i,j}^M c_i c_j^* \,( \vert f_i \rangle \langle f_j \vert )^{A}\otimes \Xi^{(S)A}_{s_i s_j} \otimes ( \vert g_i \rangle \langle g_j \vert )^{B} \otimes \Xi^{(S)B}_{r_i r_j},$$ where, again, the explicit dependence on momenta has been suppressed. Since the chiral eigenstates are invariant under boosts, the density matrix (\[chiral02\]) transforms as $$\rho_{Chiral}^\prime = \frac{1}{N}\displaystyle \sum_{i,j}^M c_i c_j^* \,( \vert f_i \rangle \langle f_j \vert )^{A}\otimes \Xi^{\prime \, (S)A}_{s_i s_j} \otimes ( \vert g_i \rangle \langle g_j \vert )^{B} \otimes \Xi^{\prime \, (S)B}_{r_i r_j},$$ where $\Xi^{\prime \,(S)A}_{i\, j} = \hat{\mathcal{O}}_{f_i} \, \Xi^{(S)A}_{s_i s_j} \, \hat{\mathcal{O}}_{f_j}$, with $$\begin{aligned} \hat{\mathcal{O}}_{f_i} &=& \cosh{\left( \frac{\omega}{2}\right)} \hat{I} - (-1)^{(f_i)} \sinh{\left( \frac{\omega}{2}\right)} \bm{n} \cdot \hat{\bm{\sigma}},\end{aligned}$$ and changes on the global entanglement are exclusively due to changes on the spin terms $\Xi_{s_i \, s_j}^{(S)A}$. A particular situation is for $f_i = f$ and $g_i = g$ for which $\Xi^{\prime \, (S)A}_{i\, j} = \hat{\mathcal{O}}_{f} \, \Xi^{(S)A}_{s_i s_j} \, \hat{\mathcal{O}}_{f} $, and $$\begin{aligned} \rho_{Chiral}^\prime &=& \frac{1}{N}\displaystyle \sum_{i,j}^M c_i c_j^* \,( \vert f \rangle \langle f \vert )^{A}\otimes \Xi^{\prime \, (S)A}_{i\, j} \otimes ( \vert g \rangle \langle g \vert )^{B} \otimes \Xi^{\prime \, (S)B}_{i\, j}, \nonumber\end{aligned}$$ which exhibits an invariant quantum correlation when anti-symmetric states as from Eqs. (\[state2\])-(\[state3\]) are considered. In fact, the chiral states $$\vert \, \psi_{2(3)}^{Chiral} \, \rangle = \left( \frac{\hat{I} + (-1)^f \hat{\gamma}_5}{2} \right)^{A} \otimes \, \left( \frac{\hat{I} + (-1)^g \hat{\gamma}_5}{2} \right)^{B}\vert \psi_{2 (3)} \rangle,$$ with $f, g = 0,1$, are such that, for a boost direction given by $\bm{n} = (\sin(\theta), \, 0, \, \cos(\theta) )$, one has $\rho_{2 (3)}^{Chiral} \rightarrow \rho_{2 (3)}^{\prime \, Chiral} = \rho_{2 (3)}^{Chiral}, $ and the states are completely Lorentz invariant. Conclusions =========== The relativistic transformation properties of quantum entanglement have been on the focus of many recent investigations, with a special interest in describing how the spin-spin entanglement does change under Lorentz boosts. Although the setup usually adopted to describe transformation properties of quantum entanglement has given some interesting insights into the physics of relativistic quantum information, when massive charged fermions are considered as the physical carriers of spin one-half, a more complete description of the problem is required. The physical particles, such as electrons, muons, etc., are described by QED including, apart from the usual Poincaré symmetry, also invariance under parity transformation. This last symmetry operation exchange two [*irreps*]{} of the Poincaré group, and a proper formulation is given in terms of [*irreps*]{} of the so called complete Lorentz group. The states of the particles are then described by four component objects, the Dirac bispinors, which satisfy the Dirac equation. In this paper we have described how Lorentz boosts do affect quantum entanglement shared among the DoF’s of a pair of bispinorial particles in a generic framework. As each of the bispinors is supported by a $SU(2) \otimes SU(2)$ structure associated with the spin and intrinsic parity, the corresponding multipartite entanglement was quantified by means of the Meyer-Wallach global measure of entanglement, given in terms of the linear entropies of each subsystem. Additionally, since the reduced spin state is mixed, the spin-spin entanglement was quantified through the appropriate negativity. By means of the $SU(2) \otimes SU(2)$ decomposition of the boost operator, $\hat{S}[\Lambda]$, the transformation laws for the Bloch vectors (and for the reduced spin density matrix) of each subsystem were recovered for a generic state, setting a framework to describe changes on both global and spin-spin entanglements. In order to specialize our results we have considered the action of Lorentz boosts in three different anti-symmetric states. First we considered a spin-spin separable state in which the particles are moving in opposite directions in the unboosted frame. In such scenario, Lorentz boosts cannot create spin-spin entanglement and the global entanglement monotonously increase as a function of the boost rapidity. The second anti-symmetric state considered here describes particles with opposite momenta and maximal spin-spin entanglement. As in the first case, the global entanglement increases as consequence of the boost, although a degradation of spin-spin entanglement is induced by the frame transformation. The last specific case consists of a pair of particles with same momentum and spin-spin entanglement, exhibiting a non-monotonous behavior of both global and spin-spin entanglement under Lorentz boost. Finally, we addressed the effects of Lorentz boosts on chiral states, which exhibit some subtle invariance properties. In particular, the density matrices obtained through projections of the anti-symmetric states on definite chiral states are completely invariant under boosts. The general formalism developed through this paper sets the framework for some future developments including the computation of quantum entanglement among particles involved in scattering processes [@scattering]. It may also be useful in the aim of a field theoretical description of relativistic entanglement. Finally, given that some low energy systems, such as trapped ions and graphene, emulate the Dirac equation dynamics [@diraclike02], interactions in such systems can be engendered as to reproduce the effects of Lorentz transformations in feasible manipulable platforms which can work as simulating platforms for high energy physics measurements. [*Acknowledgments - The work of AEB is supported by the Brazilian Agencies FAPESP (grant 2017/02294-2) and CNPq (grant 300831/2016-1). 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[^2]: In the bispinorial form, one has $$\begin{aligned} \label{bi-spinors} u_s\bb{\bm{p}} = \frac{1}{\sqrt{2 E_{p} (E_{p} + m)}}\left[ \begin{array}{rl} (E_{p} + m) &\chi_s \bb{\bm{p}}\\ \bm{p} \cdot \bm{\sigma} &\chi_s \bb{\bm{p}} \end{array}\right] \,\, \mbox{and} \,\, v_s\bb{\bm{p}} =\frac{1}{\sqrt{2 E_{p} (E_{p} + m)}} \left[ \begin{array}{rl} \bm{p} \cdot \bm{\sigma} &\chi_s \bb{\bm{p}} \\ (E_{p} + m)& \chi_s \bb{\bm{p}} \end{array}\right],\end{aligned}$$ with the orthogonality relations identified by $u_s^\dagger\bb{\bm{p}} u_r\bb{\bm{p}} = v_s^\dagger\bb{\bm{p}} v_r\bb{\bm{p}} = \delta_{sr}$ and $u_s^\dagger\bb{\bm{p}} v_r\bb{-\bm{p}} = v_s^\dagger\bb{\bm{p}} u_r\bb{-\bm{p}} =0$, and the completeness relation given by $$\displaystyle \sum_{s=1}^2\Big[u_s\bb{\bm{p}} u_s^\dagger\bb{\bm{p}} + v_s\bb{\bm{p}}v_s^\dagger\bb{\bm{p}} \Big] = \hat{I}_4.$$ [^3]: A defined total parity operator $\hat{P}$ acts on the direct product $\left\vert \pm \right\rangle \otimes \left\vert \chi_s\bb{\bm{p}}\right\rangle$ in the form of $$\hat{P}\left( \left\vert \pm \right\rangle \otimes \left\vert \chi_s\bb{\bm{p}}\right\rangle\right) =\pm \left( \left\vert \pm \right\rangle\otimes \left\vert \chi_s(-\bm{p})\right\rangle\right),$$ and, for instance, it corresponds to the Kronecker product of two operators, $\hat{P}^{(P)}\otimes \hat{P}^{(S)}$, where $\hat{P}^{(P)}$ is the intrinsic parity (with two eigenvalues, $\hat{P}^{(P)}\left\vert \pm \right\rangle =\pm \left\vert\pm \right\rangle $) and $\hat{P}^{(S)}$ is the spatial parity (with $\hat{P}^{(S)}\chi_s \left( \bm{p}\right) =\chi_s \left( -\bm{p}\right) $). [^4]: Instead, they describe a subset of transformations of the $SO(4) \equiv SO(3)\otimes SO(3)$ group, as for instance, those which include the double covering rotations. [^5]: The choice of different momenta, $\{\bm{p}_i\}\neq \bm{p}$ for each particle state of the same vector subspace (either $A$ or $B$) introduces additional quantum correlations between spin and momemtum variables, turning the problem into a more complex and non-realistic one. [^6]: The non-unitarity of $\hat{S}[\Lambda(\omega)]$ has also additional implications for the definition of spin operators in the context of relativistic quantum mechanics [@spins]. Apart from the usual Pauli spin operator $\propto\hat{\bm{\Sigma}} = \hat{I}^{(P)}_2 \otimes \hat{\bm{\sigma}}^{(S)}$, other spin operators were also proposed in the literature. For example, the Fouldy-Wouthuysen (FW) spin operator [@FW] was used in the context of transformation properties of Dirac bispinors as to define a covariant spin reduced density matrix [@celeri; @bi-spinorFW], and states constructed with FW eigenstates were then used in describing transformation properties of spin entropy as well as spin-spin Bell’s inequality under Lorentz boosts.

--- abstract: 'Interpreting measurements requires a physical theory, but the theory’s accuracy may vary across the experimental domain. To optimize experimental design, and so to ensure that the substantial resources necessary for modern experiments are focused on acquiring the most valuable data, both the theory uncertainty and the expected pattern of experimental errors must be considered. We develop a Bayesian approach to this problem, and apply it to the example of proton Compton scattering. Chiral Effective Field Theory ($\chi$EFT) predicts the functional form of the scattering amplitude for this reaction, so that the electromagnetic polarizabilities of the nucleon can be inferred from data. With increasing photon energy, both experimental rates and sensitivities to polarizabilities increase, but the accuracy of $\chi$EFT decreases. Our physics-based model of $\chi$EFT truncation errors is combined with present knowledge of the polarizabilities and reasonable assumptions about experimental capabilities at HI$\gamma$S and MAMI to assess the information gain from measuring specific observables at specific kinematics, *i.e.*, to determine the relative amount by which new data is apt to shrink uncertainties. The strongest gains would likely come from new data on the spin observables $\Sigma_{2x}$ and $\Sigma_{2x^\prime}$ at $\omega\simeq140$ to $200$MeV and $40^\circ$ to $120^\circ$. These would tightly constrain $\gamma_{E1E1}-\gamma_{E1M2}$. New data on the differential cross section between $100$ and $200$MeV and over a wide angle range will substantially improve constraints on $\alpha_{E1}-\beta_{M1}$, $\gamma_\pi$ and $\gamma_{M1M1}-\gamma_{M1E2}$. Good signals also exist around $160$MeV for $\Sigma_3$ and $\Sigma_{2z^\prime}$. Such data will be pivotal in the continuing quest to pin down the scalar polarizabilities and refine understanding of the spin polarizabilities.' author: - 'J. A. Melendez' - 'R. J. Furnstahl' - 'H. W. Grießhammer' - 'J. A. McGovern' - 'D. R. Phillips' - 'M. T. Pratola' bibliography: - 'bayesian\_refs.bib' - 'more-bayesian-refs.bib' title: ' Designing Optimal Experiments: An Application to Proton Compton Scattering ' --- Introduction ============ Six low-energy parameters known as polarizabilities characterize the response of the nucleon to low-frequency light: the electric and magnetic dipole polarizabilities $\alpha_{E1}$ and $\beta_{M1}$, and four spin polarizabilities $\gamma_i$; see, [*e.g.*]{}, Refs. [@Griesshammer:2012we; @Holstein:2013kia; @Hagelstein:2015egb] for recent reviews. Despite their fundamental importance to understanding the proton and neutron, the value of only one combination is known with better than 2% accuracy, with current uncertainties for the rest varying from 10% to more than $100$%. Most recent values and uncertainties are collected in Ref. [@Griesshammer:2015ahu] and references therein, and summarized in Table \[tab:polarizability\_info\] below. Recent advances in Chiral Effective Field Theory () have enabled precise quantitative predictions of Compton scattering that take the polarizabilities as inputs [@Griesshammer:2012we; @McGovern:2012ew; @Griesshammer:2017txw; @Margaryan:2018opu; @Griesshammer:2013vga].[^1] This new ability to precisely trace the impact of these fundamental nucleon-structure constants on experimental observables is opportune. It comes at a time when photon facilities of unprecedented luminosity and sensitivity are now available. These concurrent developments have inspired new experimental campaigns to refine knowledge of nucleon polarizabilities; see, for example, recent overviews in Refs. [@Martel:2019tgp; @Martel:2017pln; @Huber:2015uza; @Ahmed:2020hux; @Weller:2009zza]. But not all measurements are created equal, and beam time is not cheap. In Nuclear Physics, as in many other advanced disciplines, the costs of running an experiment include not only the workforce, time and money invested, but also the opportunity cost of measurements that could have been carried out with those same resources but were not. Thus, when planning an experiment, it is important to consider which data are most likely to provide the largest information gain. How, then, does one assess the impact of measurements that have yet to be made? One possibility is to simulate various possible experimental scenarios and compute the extent to which each improves constraints on theory parameters. For example, Ref. [@Catacora-Rios:2019goa] recently investigated the ability of data on proton- and neutron elastic scattering from nuclei to constrain optical-model parameters. In the context of Compton scattering from the proton and neutron, Ref. [@Griesshammer:2017txw] assessed the sensitivity of many different observables to the polarizabilities, but provided only an initial theory perspective on those sensitivities. Neither of these papers used a single quantitative measure to choose between competing experimental designs. Such a measure should assess competing experimental designs in light of existing data and the feasibility of new data, and include a rigorous assessment of theory uncertainties. Here, we argue for the framework of Bayesian statistics—Bayesian experimental design, in particular—as an enlightened way forward. Experimental design, in the context of statistics, provides insight into the allocation of scarce resources that have alternative uses. This is a highly practical field of study, with applications including engineering, biology, and psychology [@chaloner1995bayesian; @Liepe2013a; @Ryan:2015aaa; @Myung2009BayesianAO]. One begins by encoding as a utility function the goals of the experiment and the constraints inherent in carrying it out. Then, one considers the range of possible future experimental measurements, and computes the expected utility for each. The optimal design is then the one that maximizes the expected utility function. In this context, a *design* refers to the choice of observable, the kinematic points at which to measure it, and the relative allocation of beam time. With Bayesian experimental design, we incorporate *a priori* knowledge of the parameters we wish to constrain—in this case the nucleon polarizabilities—via Bayesian priors. The utility function must also account for the accuracy future experiments will reasonably achieve, and factor in the kinematic regimes where collecting data will likely be excessively difficult. As we emphasize in this work, accounting for theory uncertainties in such an analysis is essential for a proper assessment of the optimal design. All models are wrong, but some are useful [@pc; @Box]. Still fewer are wrong in a way that is useful. Effective Field Theory calculations are carried out up to a particular order in a systematically improvable expansion. This predictable character of an EFT’s uncertainty permits quantitative answers regarding this trade-off between increased experimental sensitivity and decreased theoretical accuracy. We address this trade-off using the physics-based Bayesian machine learning model proposed in Ref. [@Melendez:2019izc]. The known order-by-order EFT predictions allow the algorithm to learn the convergence pattern and how it is correlated in kinematic space, and then to use that pattern to formulate a statistical model of the truncation error, thus ensuring that high-energy data are not over-weighted.  incorporates all the physics of Compton scattering at photon energies between $0$ and about $300\MeV$. According to Ref. [@Pascalutsa:2002pi], the  Compton amplitude at a given order in the EFT expansion has a theory truncation error approximately proportional to resolved $(m_\pi/\Lambda_b)^{\nu/2}$. Here, $\Lambda_b$ is the breakdown scale of the theory, and $\nu$ depends on, but is not necessarily equal to, the order of the calculation (cf. Sec. \[sec:bayesian\_methodology\_compton\] below). A naïve application of experimental design that ignored the theory uncertainty might suggest running experiments at energies which are so high that  is unreliable. That would lead to experiments with high precision, whose information content is, however, very small, wasting scarce human and financial resources. Although we devote more space to the explanation of theoretical errors, it is of course mandatory to also account for experimental realities. Our recommended experimental design should not involve kinematics at which measurements are notoriously difficult. Difficulties can arise in two different directions. The first is that beam time is limited. We account for that experimental constraint by considering three levels of possible precision of Compton data. These are intended to cover a range of plausible future experiments. Since these results correspond to different numbers of photons on target, they give us a sense of how knowledge of the different polarizabilities will scale with beam intensity and experimental run time. The second issue is that physical limitations can make it difficult to place detectors at particular locations, or to run a certain machine at specific energies. While these constraints are probably best assessed using a facility-specific factor in the utility function, in this first study we account for them crudely by precluding designs involving photon scattering angles where physical limitations will make it difficult to place detectors. In the interest of accessibility to all readers, we try to ensure the sections listed below are self-contained. For example, an understanding of the EFT truncation model should not be essential to understand the experimental design, or the results of the analysis. We begin in Sec. \[sec:basic-compton-facts\] by recounting the relevant facts of nucleon Compton scattering. Next, Sec. \[sec:bayesian\_methodology\_compton\] describes the important results of the Bayesian methodology employed in this work, including the model of EFT truncation errors and experimental design, adapted to  for Compton scattering. These methods are applied to Compton scattering on a proton for various choices of experimental goals in Sec. \[sec:results\_experimental\_design\]. Finally we conclude in Sec. \[sec:summary\_design\]. We provide details of our derivations in Appendix \[sec:experimental\_design\_details\] and details of the truncation error model in Appendix \[sec:truncation\_model\_details\_compton\]. Results for different levels of experimental precision and our investigation of neutron Compton scattering is reserved for the Supplemental Material: [Appendix \[sec:extra\_compton\_figures\].]{} We provide all data and codes needed to reproduce our results [@BUQEYEgithub]. Basic Facts of Nucleon Compton Scattering {#sec:basic-compton-facts} ========================================= We start with an enumeration of those aspects of Compton scattering on the nucleon relevant for this presentation, to remind experts and introduce the minimal necessary vocabulary for non-experts. Motivations, context and details can be found in Refs. [@Griesshammer:2017txw; @Griesshammer:2015ahu; @McGovern:2012ew; @Griesshammer:2012we] and elsewhere. Let us first consider photon energies up to around the pion mass, $\omega\lesssim{\ensuremath{m_\pi}}\approx140\MeV$. At low energies, the process is dominated by the Born terms: a point-nucleon with anomalous magnetic moment, plus the $\pi^0$ $t$-channel coupling. We define this as LO, or $\calO(Q^0)$ in the power-counting employed here[^2]. There is no NLO \[$\calO(Q^1)$\] correction. The first corrections come at \[$\calO(Q^2)$\] from the pion cloud around the nucleon. It is at this order that the polarizabilities enter first. At \[$\calO(Q^3)$\], effects from the lowest-lying nucleonic resonance, the $\Delta(1232)$, and its pion cloud are added. At \[$\calO(Q^4)$\], corrections to pion-cloud effects are accounted for. Effects of higher order are not included. Secondly, we discuss the regime where the photon energy approaches the excitation energy of the Delta resonance, $\omega\simeq\Delta\approx300\MeV$, the contributions are re-ordered: LO is now the resonance contribution and counts as $\calO(Q^{-1})$; Born effects and corrections to resonance parameters enter at NLO \[now $\calO(Q^0)$\]; and all other terms, including contributions from the polarizabilities, are suppressed further. This region can roughly be estimated from the Delta resonance width as $\Delta\pm\Gamma/2\approx 300\pm70\MeV$. Thus, the power-counting changes in a transition region between about $180\MeV$ and $230\MeV$. Finally, the EFT expansion breaks down entirely as $\omega\to\Lambda_b\approx650\MeV$. These changes in importance as $\omega$ increases are reflected in the EFT power-counting we employ; see Eq.  and discussion in Sec. \[sec:truncation\_error\_compton\], especially Sec. \[sec:compton\_pc\_rearrangement\]. At all these energies, the polarizability contributions are well-described by six dipole polarizabilities which are labeled by the multipolarities of the incoming and outgoing electromagnetic field. In Ref. [@Griesshammer:2017txw], the following linear combinations were identified as most convenient for exploring sensitivities while exploiting the best available prior knowledge: the scalar (dipole) polarizabilities in the combinations $$\begin{aligned} \alpha_{E1}\pm\beta_{M1} \, ,\end{aligned}$$ and the mutually orthogonal spin-polarizability combinations $$\begin{aligned} \gamma_{0,\pi} & \equiv -(\gamma_{E1E1} \pm \gamma_{M1M1} + \gamma_{E1M2} \pm \gamma_{M1E2}) \\ \gamma_{E-} & \equiv \gamma_{E1E1} - \gamma_{E1M2} \\ \gamma_{M-} & \equiv \gamma_{M1M1} - \gamma_{M1E2} \, .\end{aligned}$$ These combinations map onto tight constraints on $\alpha_{E1}+\beta_{M1}$ and $\gamma_0$ from sum rules. For the proton, these have error bars which are better than those from direct Compton experiments. Polarizabilities are fundamental hadron properties. The scalar polarizabilities are also important ingredients in, for example, the proton-neutron mass splitting [@WalkerLoud:2012bg; @Walker-Loud:2019qhh; @Gasser:2015dwa; @Gasser:2020mzy], and the spin polarizabilities parametrize the response of the nucleon spin to electromagnetic fields (such as the nucleonic Faraday effect). Thirteen independent observables per nucleon parametrize the process when at most two of the photon beam, nucleon target or recoil nucleon are polarized. The (unpolarized) differential cross section $\diffcs$ (in $\mathrm{nb}/\mathrm{sr}$) is larger than zero but otherwise unbounded. The beam-target asymmetries $\Sigma_3$, $\Sigma_y$, $\Sigma_{1x}$, $\Sigma_{1z}$, $\Sigma_{2x}$, $\Sigma_{2z}$, $\Sigma_{3y}$ and the polarization-transfer observables from a polarized beam to the recoil nucleon $\Sigma_{1x^\prime}$, $\Sigma_{1z^\prime}$, $\Sigma_{2x^\prime}$, $\Sigma_{2z^\prime}$, $\Sigma_{3y^\prime}$ are ratios of differences over sums of rates and take values between $-1$ and $1$. Below the pion-production threshold $\omega_\pi(\mathrm{lab})\approx150\MeV$, only six observables are non-zero: $\diffcs$, $\Sigma_3$, $\Sigma_{2x}$, $\Sigma_{2z}$, $\Sigma_{2x^\prime}$, $\Sigma_{2z^\prime}$. The following data on these proton observables is available: about $420$ points of widely varying quality for the cross section (see extensive discussions in Refs. [@Griesshammer:2012we; @McGovern:2012ew]), about $120$ points for $\Sigma_3$ [@Blanpied:2001ae; @Sokhoyan:2016yrc; @Martel:2017pln; @CollicottPhD], $9$ for $\Sigma_{2x}$ [@Martel:2014pba; @MartelPhD], and $10$ for $\Sigma_{2z}$ [@Martel:2017pln; @Paudyal:2019mee]; no direct neutron data exists. In Ref. [@Griesshammer:2017txw], all observables are calculated from amplitudes which are complete up to and including \[$\calO(Q^4)$\] for $\omega\lesssim{\ensuremath{m_\pi}}$, and NLO \[$\calO(Q^0)$\] for $\omega\simeq\Delta$. Bayesian Methodology in EFT {#sec:bayesian_methodology_compton} =========================== Problems and Solutions of Design Strategy {#sec:problems+solutions} ----------------------------------------- To estimate the best design strategy, our approach must incorporate two distinct sources of uncertainty: (1) the EFT truncation error and (2) the unknown measurements from future experiments, including their likely measurement uncertainties. Our Bayesian approach can handle both of these problems in one coherent framework while being candid about our uncertainties. The two problems and the solutions we propose are summarized as: 1. *Problem:*  must be truncated at a finite order, leading to a truncation error that is correlated in kinematic space. That is, we trust our theory more in some kinematic regimes than we do in others, and the discrepancy itself is a smooth function. This should be reflected when assessing how well the experimental data from these regimes constrain polarizabilities. *Solution:* An estimate of the truncation error is found by summing over all plausible values for the higher-order terms in the EFT. This results in a covariance matrix for the theory error that weights experimental data from trusted regimes more heavily than data from less trusted regimes [@Wesolowski:2018lzj]. 2. *Problem:* Given a choice of design, we still do not know the results of the yet-to-be-performed experiment. But such results are needed in order to estimate how well they would constrain the polarizabilities. *Solution:* Bayesian experimental design considers all data that could plausibly be measured. For each of these we compute corresponding polarizability posteriors. The *expected* utility, or worth, of such an experiment can then be judged by sampling a utility function over all the data possibilities that have been evaluated. Sampling is often computationally quite expensive, however, in our case, a controlled approximation allows it to be done analytically, leading to a simple and intuitive formula for the expected utility of an experiment. In the following subsections we describe in detail our approach to the problem of truncation errors and experimental design. EFT Truncation Errors {#sec:truncation_error_compton} --------------------- A Bayesian model of EFT truncation errors has been proposed and discussed thoroughly in Refs. [@Furnstahl:2015rha; @Melendez:2017phj; @Melendez:2019izc]. Here we recapitulate the main results of the convergence model that are relevant to Compton scattering, and discuss how it must be modified to account for the rearrangement of the power counting in the regime of the Delta resonance. For a much more thorough introduction to this model of EFT truncation errors, we refer to Ref. [@Melendez:2019izc]. Suppose we are interested in the prediction of an observable $\genobs(\kinparvec)$ at some kinematic point $\kinparvec$. Here, $x \equiv \{\omega, \theta\}$ is the incident-photon energy $\omega$ and scattering angle $\theta$ in the lab frame. EFTs provide a hierarchy of predictions $\{\genobs_n(\kinparvec;\lecs)\}$, with each order $n$ more precise than the last. These predictions depend on low-energy constants—the polarizabilities[^3]—which we denote collectively as a vector $\lecs$. Let $k$ be the highest order at which the complete EFT process has been calculated to date. Then there is a theory truncation error $\delta\genobs_k$ associated with all higher order terms left out of the state-of-the-art EFT prediction. Furthermore, if we are to compare our predictions to experimental measurements $\genobsexp$, there is the problem of experimental noise, $\delta\genobsexp$, to contend with. In Ref. [@Melendez:2019izc], the authors assume the following relationship, where theory and experimental uncertainties are independent: $$\begin{aligned} \label{eq:errormodel_compton} \genobsexp(\kinparvec) = \genobs_k(\kinparvec;\lecs) + \delta\genobs_k(\kinparvec) + \delta\genobsexp(\kinparvec) \, .\end{aligned}$$ Because $\delta\genobs_k(\kinparvec)$ and $\delta\genobsexp(\kinparvec)$ are unknown, they are treated as random variables. Given statistical models for $\delta\genobs_k$ and $\delta\genobsexp$, we can use Eq.  to tell us the kinematics $x$ that will result in the most stringent constraints on the polarizabilities $\lecs$. We extend Refs. [@Furnstahl:2015rha; @Melendez:2017phj; @Melendez:2019izc] by writing the observable expansion as $$\begin{aligned} \label{eq:observable_expansion_compton} \genobs_k(\kinparvec) = \genobsref(\kinparvec) \sum_{n=0}^{k} c_n(\kinparvec) Q^{\nu_n(\omega)}(\kinparvec) \, ,\end{aligned}$$ where $Q$ is the dimensionless expansion parameter of the EFT and $\genobsref$ is a reference scale for the observable $\genobs_k$. For the EFT power counting to hold, the (dimensionless) observable coefficients $c_n$ should be approximately of order unity. Equation (\[eq:observable\_expansion\_compton\]) differs from Refs. [@Furnstahl:2015rha; @Melendez:2017phj; @Melendez:2019izc] by the inclusion of $\nu_n(\omega)$ rather than a simple $n$ as the exponent of the expansion parameter. The use of $\nu_n(\omega)$ reflects the fact that the power counting changes as one moves from $\omega\simeq m_\pi$ to $\omega \simeq \Delta$, resulting in a re-ordering of contributions to the amplitude [@Pascalutsa:2002pi]. Because the EFT amplitudes are squared to compute observables (and in the case of spin observables, further divided by the differential cross section), the $c_n$ are not cleanly related to the polarizabilities; instead they appear naturally sized and randomly distributed. We will exploit this fact in the convergence model. For given choices of $\genobsref$, $Q$ and $\nu_n(\omega)$, the coefficients $c_n$ are in 1-to-1 correspondence with the results $\genobs_n$ for orders $n \leq k$. We will begin by discussing these choices and how they lead to a physically motivated distribution for $\delta\genobs_k$. The choice of $\nu_n(\omega)$ is more technical, and is described in Sec. \[sec:compton\_pc\_rearrangement\]. ### The Distribution of dyk {#sec:truncation_distribution_compton} The reference scale $\genobsref$ should capture the overall size of $\genobs_k$ in the appropriate units. The spin observables $\Sigma_i$ are dimensionless and bounded in $[-1, 1]$, hence the natural choice is $\genobsref = 1$. The cross section varies over orders of magnitude, and can contain cusp-like behavior near the pion-production threshold. We capture its overall trend, without cusps, by using a $\genobsref$ comprised of the basic Born, pion pole, and Delta-pole parts of the proton (and neutron) cross section. With this choice, the proton cross section still shows some growth of the $c_n$ near $\omega_\pi$ and at forward angles, so we multiply this reference by a shifted 2-dimensional Lorentzian $$\begin{aligned} \left[\left(\frac{\omegalab-\omega_\pi}{50\MeV}\right)^2 + \left(\frac{\thetalab}{150^\circ}\right)^2 + \frac{1}{3}\right]^{-1} + 1 \, .\end{aligned}$$ There is no particular physics in this function. It serves only to produce $c_n$ that look similar across kinematic space. EFTs exploit a separation of scales, from which one can construct a small expansion parameter $Q$. Here we choose the expansion parameter $$\begin{aligned} \label{eq:expansion_parameter_compton} Q(\kinparvec) = \sqrt{\frac{\omega_{\text{cm}} + m_\pi}{2\Lambda_b}} \, ,\end{aligned}$$ where the low-momentum scale is the average of $m_\pi$ and $\omega_{\text{cm}}$, the photon momentum in the center-of-momentum frame. The high-momentum scale, $\Lambda_b = 650\MeV$, is the approximate breakdown scale of . This is an extension of the expansion parameter $Q = \sqrt{m_\pi/\Lambda_b}$, proposed in Ref. [@Pascalutsa:2002pi] for $\omega \lesssim m_\pi$. Equation  explicitly builds in our expectation that  degrades with large $\omega$. We also ran the analysis using the expansion parameter $Q\to\sqrt{{\ensuremath{m_\pi}}/\Lambda_b}$ and found the results were essentially unchanged. The crux of the EFT truncation error model is induction on the $c_n$: the coefficients for $n > k$, which we have not yet seen, are assumed to have approximately the same size and dependence on $(\omega,\theta)$ as the lower-order $c_n$ that we already have from our EFT calculation. To formalize this inductive step, we model the $c_n$ as independent and identically distributed () curves and assign them a Gaussian process (GP) prior[^4] $$\begin{aligned} \label{eq:coefficient_prior_compton} c_n(\kinparvec) \given \sdth^2, \ell_\omega, \ell_\theta \overset{\text{\tiny \iid}}{\sim} \GP[0, \sdth^2 r(\kinparvec, \kinparvec';\ell_\omega, \ell_\theta)] \, .\end{aligned}$$ GPs are popular machine-learning algorithms that have been employed in a wide variety of disciplines to perform nonparametric regression and classification [@sacks1989design; @cressie1992statistics; @rasmussen2006gaussian]. The samples from a GP are *functions*, as opposed to numbers or vectors. A brief introduction to GPs in this context is given in Ref. [@Melendez:2019izc]; see also Refs. [@rasmussen2006gaussian; @Mackay:1998introduction; @Mackay:2003information] for more in-depth discussions. We adopt a mean function of 0 since, *a priori*, the  corrections $c_n$ are just as likely to be positive as they are to be negative. The values of the GP hyperparameters $\sdth^2$ and $\ell_i$, whose meaning are discussed below, are tuned to the known $c_n$ with $n \leq k$ (at the best known values of the polarizabilities $\lecs$ for each EFT order). An example of how these hyperparameters, combined with symmetry constraints on the observables, lead to a distribution for higher-order $c_n$ is shown in Fig. \[fig:eft\_coefficients\]. Note the cusps in the observables around the pion-production threshold, $\omega\approx\omega_\pi$, which can grow rather large. These are expected, and are not a problem in and of themselves (as discussed when tuning $\ell_\omega$ below). But, for spin observables $\Sigma_i$, their large size is uncharacteristic compared to the $c_n$ away from $\omega_\pi$; hence we choose to exclude $125< \omegalab < 200\MeV$ when training $\sdth^2$ and $\ell_i$.[^5] Despite providing the most rigorous accounting of uncertainties to date, we are thus less confident in the estimate of the  uncertainty $\delta\genobs_k$ for $\Sigma_i$ very close to $\omega_\pi$. \[The convergence pattern of $\diffcs$, in contrast, remains regular at $\omega_\pi$, giving us confidence in our design results there.\] The details of the fitting procedure, the symmetry constraints on the observables, and figures for the remaining observables, are reserved for Appendix \[sec:truncation\_model\_details\_compton\]. The results of the fits are shown in Table \[tab:truncation\_details\_observables\]. By the inductive step \[Eq. \], these hyperparameters tell us about the unknown higher-order $c_n$. ![ Observable coefficients for $\Sigma_3$. The gray $2\sigma$ bands indicate the expected 95% credible interval for all higher order coefficients. Note that, although $c_0 \neq 0$ at $\omega = 0$, all of the corrections $c_n$, along with their derivatives, do vanish. A similar situation occurs at forwards and backwards angles. These constraints, and the corresponding ones for other observables, are built into the EFT truncation error model. []{data-label="fig:eft_coefficients"}](coeffs_obs-3) [Sl d[2.2]{}\*[2]{}[d[2.0]{}]{} d[2.2]{}\*[2]{}[d[2.0]{}]{}]{} & &\ & & & & & &\ $\diffcs$ & 0.59 & 54 & 56 & 2.8 & 54 & 79\ $\Sigma_{1x}$ & 0.61 & 49 & 48 & 0.47 & 91 & 46\ $\Sigma_{1z}$ & 0.37 & 51 & 53 & 0.32 & 92 & 43\ $\Sigma_{2x}$ & 0.42 & 43 & 38 & 0.57 & 56 & 39\ $\Sigma_{2z}$ & 1 & 50 & 44 & 1.6 & 58 & 52\ $\Sigma_{3}$ & 0.6 & 54 & 35 & 0.44 & 71 & 43\ $\Sigma_{y}$ & 0.47 & 61 & 52 & 0.42 & 95 & 43\ $\Sigma_{3y}$ & 0.64 & 50 & 48 & 0.46 & 90 & 44\ $\Sigma_{3y'}$ & 0.49 & 64 & 46 & 0.47 & 79 & 46\ $\Sigma_{1x'}$ & 0.51 & 56 & 48 & 0.31 & 76 & 41\ $\Sigma_{1z'}$ & 0.24 & 51 & 45 & 0.25 & 82 & 44\ $\Sigma_{2x'}$ & 0.91 & 33 & 56 & 0.95 & 44 & 53\ $\Sigma_{2z'}$ & 0.52 & 38 & 47 & 1.4 & 72 & 55\ The marginal variance $\sdth^2$ in Eq.  controls the size of the $c_n$. If the $c_n$ are naturally sized, then $\sdth$ should be of order unity. We place an inverse chi-squared prior on $\sdth^2$: $$\begin{aligned} \sdth^2 \sim \chi^{-2}(\nu_0, \tau_0^2) \, ,\end{aligned}$$ where $\nu_0$ and $\tau_0$ are the prior degrees of freedom and scale parameters, respectively. This is a conjugate prior and allows the posterior for $\sdth^2$ to be found analytically; see Ref. [@Melendez:2019izc] for details. We choose $\nu_0 = {1}$ and $\tau_0 = {1}$, which is weakly informative. We take the posterior mean as an estimate for $\sdth^2$ in Eq. . The smoothness of the $c_n$ is dictated by the correlation function $r$. We take the correlation of $c_n$ between two kinematic points $x=(\omega,\theta)$ and $x^\prime=(\omega^\prime,\theta^\prime)$ to be given by a radial basis function (RBF) $$\begin{aligned} \label{eq:rbf_kernel_compton} r(x, x'; \ell_\omega, \ell_\theta) = \exp{-\frac{(\omega - \omega')^2}{2\ell_\omega^2} - \frac{(\theta - \theta')^2}{2\ell_\theta^2}} \, ,\end{aligned}$$ where the correlation lengths $\ell_i$ control how quickly the $c_n$ vary as a function of $\omega$ and $\theta$. There is no conjugate prior for $\ell_i$; rather, we use a uniform prior and find the best fits by optimizing the log likelihood. Choosing the RBF as a correlation function for the $c_n$ implies that they are quite smooth. This assumption is validated empirically, except, as already noted, at the pion-production threshold, which occurs at photon energy $\omega_\pi\approx150\MeV$ in the lab frame. Tuning $\ell_\omega$ to data with cusps will bias it towards very small values. To fix this bias, we set the correlations between $c_n(x)$ and $c_n(x')$ to zero if they are on opposite sides of the pion-production threshold, but still use the same correlation lengths $\ell_\omega$ and $\ell_\theta$ below and above the cusps. Assuming we have estimates of $\sdth^2$ and $\ell_i$, we can construct the distribution for $\delta\genobs_k$. It follows from extending Eq.  that $$\begin{aligned} \label{eq:discrepancy_sum_compton} \delta\genobs_k(\kinparvec) = \genobsref(\kinparvec) \sum_{n=0}^\infty c_{n+k+1}(\kinparvec) Q^{\nu_{\delta k}(\omega) + n}(\kinparvec) \, ,\end{aligned}$$ where $\nu_{\delta k}(\omega)$ captures the first incomplete order of the EFT, and we assert for simplicity that powers of $Q$ increment in integer steps afterwards. \[For an EFT with a single power counting, one might expect $\nu_{\delta k}(\omega) = k+1$.\] Equation  is a geometric sum of Gaussian random variables, from which it follows that $$\begin{aligned} \label{eq:discrepancy_gp_compton} \delta\genobs_k(\kinparvec) \given \sdth^2, \ell_\omega, \ell_\theta \sim \GP[0, \sdth^2 \discrcorr{k}(\kinparvec, \kinparvec';\ell_\omega, \ell_\theta)] \, ,\end{aligned}$$ where $$\begin{aligned} \discrcorr{k}(\kinparvec, \kinparvec';\ell_\omega, \ell_\theta) & \equiv \genobsref(\kinparvec)\genobsref(\kinparvec')\frac{Q^{\nu_{\delta k}(\omega)}(\kinparvec)Q^{\nu_{\delta k}(\omega')}(\kinparvec')}{1 - Q(\kinparvec)Q(\kinparvec')} \notag \\ & \times r(\kinparvec, \kinparvec'; \ell_\omega, \ell_\theta) \, . \label{eq:compton_truncation_corrfunc}\end{aligned}$$ Given choices of $\genobsref$, $Q$, $\nu_{\delta k}$, and $r(\kinparvec, \kinparvec')$, along with estimates of $\sdth^2$ and $\ell_i$, the above equations completely define a physics-based uncertainty due to truncation. ### The Power-Counting Rearrangement {#sec:compton_pc_rearrangement} An EFT begins with an infinite set of operators that one orders via a power counting. There is then a finite number of parameters that contributes to the process of interest at any given order in the EFT expansion. Our implementation of  for Compton scattering from the nucleon contains all contributions up to and including for photon energies $\omega \simeq m_\pi$ as well as some terms that are in that regime. They can be represented by a finite set of Feynman diagrams. Their details and amplitudes are discussed in Refs. [@McGovern:2012ew; @Griesshammer:2015ahu; @Griesshammer:2017txw]; see also references therein. Therefore, the theory error $\delta\genobs_k$ in this regime follows Eq.  with $\nu_{\delta k}=k+1$, and proceeds indeed in integer steps. As briefly described in Sec. \[sec:basic-compton-facts\], diagrams are reordered in the vicinity of the Delta resonance, $\omega \simeq \Delta\approx300\MeV$, because of a different hierarchy of physical mechanisms. The power counting of EFT contributions changes to reflect this, and so the first incomplete order is different. We define $\nu_n(\Delta)$ as the lowest order at which those diagrams which are of order $n$ at $\omega\lesssim m_\pi$ contribute when $\omega \simeq \Delta$. The most dramatic reordering involves Delta-pole diagrams, which transition from $\mathcal{O}(Q^3)$ () for $\omega \simeq m_\pi$ to $\calO(Q^{-1})$ (LO) for $\omega \simeq \Delta$. The contributions from pion loops around the Delta also enter at $\calO(Q^3)$ () for $\omega \simeq m_\pi$, and these are relocated to $\calO(Q^{1})$ () for $\omega \simeq \Delta$. Other diagrams contribute at $\calO(Q^{1})$ () for photon energies near the Delta peak, including those from pion loops around the nucleon which are $\calO(Q^2)$ or for $\omega \simeq m_\pi$. The most relevant reorderings therefore turn out to follow the rule that, for diagrams of order $n$ at $\omega\simeq{\ensuremath{m_\pi}}$, $\nu_n(\Delta)=n/2$ for even orders but $\nu_n(\Delta)=(n-5)/2$ for odd orders. Then the first omitted order, $\nu_{\delta k}(\Delta)$, is given by the smaller of $\nu_{k+1}(\Delta)$ or $\nu_{k+2}(\Delta)$. Now we wish to be able to handle data in the transition region between $\omega\simeq{\ensuremath{m_\pi}}$ and $\omega\simeq\Delta$, by defining a $\nu_{n}(\omega)$ that is a function of $\omega$. If we define a suitable monotonic function $f(\omega)$ satisfying $$\label{eq:fofomega} f(\omega\approx{\ensuremath{m_\pi}})\approx0\,, \quad f(\omega\approx\Delta)\approx1\,,$$ the reordering is smoothly captured by $$\nu_n(\omega) = \begin{cases} [1 - f(\omega)/2] n \, , & n \text{ even} \\ [1 - f(\omega)/2] n - 5 f(\omega)/2 \, , & n \text{ odd} \end{cases} \label{eq:order_transition}$$ which is tabulated in Table \[tab:order\_and\_truncation\_functions\]. For definiteness we use a logistic form inspired by the Fermi function $$f(\omega) = \left[1 + \exp(-4\ln 3 \cdot \frac{\omega - \omega_m}{\omega_2 - \omega_1})\right]^{-1}, \label{eq:fermi_interp}$$ where $\omega_1 = 180\MeV$ and $\omega_2 = 240\MeV$ are the locations where $f(\omega_1) = 1/10$ and $f(\omega_2) = 9/10$, and $\omega_m = (\omega_1 + \omega_2)/2\approx210\MeV$ is the midpoint $f(\omega_m) = 1/2$. The same form was already used in the plots of Ref. [@Griesshammer:2017txw] to parametrize the “gray mist” at high energies, but our framework puts this “mist” on a quantitative footing via the EFT-inspired theory error, Eqs. –. These choices are consistent with the estimate in Sec. \[sec:basic-compton-facts\] that the Delta resonance region starts around $230\MeV$. This form for $f(\omega)$ is only one of several possibilities. Other sensible models for $f(\omega)$ lead to results which are compatible with those presented below. [SlSlSl]{} Order & Transition with $\omega$ & Leading\ at $\omega \simeq m_\pi$& &Truncation Error\ 0 (LO) & $\nu_0(\omega) = 0$ & $\nu_{\delta 0}(\omega) = 2 - 3 f(\omega)$\ 2 () & $\nu_2(\omega) = 2 - f(\omega)$ & $\nu_{\delta 2}(\omega) = 3 - 4f(\omega)$\ 3 () & $\nu_3(\omega) = 3 - 4f(\omega)$ & $\nu_{\delta 3}(\omega) = 4 - 4f(\omega)$\ 4 () & $\nu_{4}(\omega) = 4 - 2f(\omega)$ & $\nu_{\delta 4}(\omega) = 5 - 5f(\omega)$\ 5 () & $\nu_5(\omega) = 5 - 5f(\omega)$ & $\nu_{\delta 5}(\omega) = 6 - 4f(\omega)$ ![The power counting transitions from the $m_\pi$ regime to the regime around the Delta resonance. Solid lines corresponding to $\nu_n$ capture the most relevant reordering of diagrams, as described in the text. The shaded region is the approximate order up to which the $\NkLO{4}^+$ EFT is complete. The dashed line $\nu_{\delta 4}$ is one unit above the shaded boundary and represents the first order to be included in the EFT truncation error. []{data-label="fig:eft_order_transition"}](eft_order_transition) In Fig. \[fig:eft\_order\_transition\], we translate the first column of Table \[tab:order\_and\_truncation\_functions\] and the logistic function  to a graphical representation of the re-ordering of contributions. It is straightforward to read off the dominant theory uncertainty that an amplitude which is complete up to $\calO(Q^n)$ in the $\omega \simeq m_\pi$ regime has in the $\omega \simeq \Delta$ regime. This defines $\nu_{\delta n}(\Delta)$. The resulting form of the leading truncation error $\nu_{\delta n}(\omega)$ as function of $\omega$ is given in the third column of Table \[tab:order\_and\_truncation\_functions\]. We observe that starting with the full amplitude up to and including $\mathcal{O}(Q^4)$ () for $\omega \simeq m_\pi$ only yields an amplitude that is complete at $\calO(Q^{-1})$ (LO) for $\omega \simeq \Delta$. However, there are only a small number of diagrams that are missing at $\calO(Q^0)$ for $\omega \simeq \Delta$. These were identified, computed and added to the amplitude in both regimes in Ref. [@McGovern:2012ew]. This produces an amplitude that is complete up to $\calO(Q^0)$ for $\omega \simeq \Delta$ and is “$\NkLO{4}^+$”, [*i.e.*]{}, more than but not fully , for $\omega \simeq m_\pi$. Since the truncation error must include all orders that do not contain a complete set of diagrams, we therefore identify $$\begin{aligned} \label{eq:n3loplus_truncation_power} \nu_{\delta 4}^+(\omega) = 5 - 4 f(\omega)\end{aligned}$$ as the first omitted power of our $\NkLO{4}^+$ EFT. Experimental Design ------------------- The process of designing an experiment must begin with defining a goal. For example, this goal could be to make an accurate prediction of some future measurement, to discriminate between competing models, or to precisely constrain parameters of the theory. The goal could even be designed with a compromise between several different experimental aims in mind. It could also incorporate time and cost constraints. But in this work, we simply take constraining the nucleon polarizabilities as the goal of the experiments we are designing—although time and cost constraints will be assessed indirectly when we define different scenarios for the experimental accuracy. {width="\textwidth"} ![Exact vs linearized predictions of the differential cross section $\diffcs$ for proton (red) and neutron (blue) from the $\NkLO{4}^+$ EFT, with points as in Fig. \[fig:true\_vs\_linearized\_predictions\]. []{data-label="fig:true_vs_linearized_dsg"}](dsg_true_vs_linearized_both_trans2) The next step is to encode as mathematical objects the experimental goal and all uncertainties. Once encoded, our goal is known as a utility function, or design criterion, $U(\design, \lecs, \genobsset)$, that depends on the design points[^6] $\design$ in the design space $D$ from which experimental data $\genobsset$ is then measured, and the theory parameters $\lecs$. But, of course, $\genobsset$ will not be known until the experiment is conducted, and $\lecs$ is exactly the quantity we have constructed our experiment to find. Hence the optimal design $\design^\star$ is that which maximizes the *expected* utility $U(\design) = \E[U(\design, \lecs, \genobsset)]$. That is, $$\begin{aligned} \design^\star & = \argmax_{\design \in D} U(\design) \notag \\ & = \argmax_{\design \in D} \int U(\design, \lecs, \genobsset) \pr(\lecs, \genobsset \given \design) \dd{\lecs} \dd{\genobsset} \label{eq:expected_utility} \\ & = \argmax_{\design \in D} \int \Big\{ U(\design, \lecs, \genobsset) \pr(\lecs \given \genobsset, \design) \dd{\lecs} \Big\} \pr(\genobsset \given \design) \dd{\genobsset}. \notag $$ These integrals are usually intractable for nonlinear theories such as the observable predictions from , but we show in Figs. \[fig:true\_vs\_linearized\_predictions\] and \[fig:true\_vs\_linearized\_dsg\] that linearizing  predictions around the best known $\lecs$ is a very good approximation, and we employ it from here on. Equation says that the process of experimental design requires a theory $\genobs(\kinparvec; \lecs)$ and a probabilistic model relating data to theory parameters, $\pr(\lecs, \genobsset \given \design) = \pr(\genobsset \given \lecs, \design) \pr(\lecs)$. This is where our truncation error model from Eq.  comes into play. If a Gaussian prior is placed on the polarizabilities,[^7] $$\begin{aligned} \label{eq:polarizability_prior} \lecs \sim \normal(\vec{\mu}_0, V_0) \, ,\end{aligned}$$ then under the assumption that $\genobs(\kinparvec;\lecs)$ is linear in $\lecs$, one can show that the posterior is given by $$\begin{aligned} \label{eq:polarizability_posterior} \lecs \given \genobsset, \design \sim \normal(\vec{\mu}, V) \, ,\end{aligned}$$ where $\vec{\mu}(\genobsset, \design)$ and $V(\design)$ take into account both the truncation error and the experimental errors, and both depend on the values of the GP hyperparameters that have already been tuned to the EFT convergence pattern (see Sec. \[sec:truncation\_distribution\_compton\] and Appendix \[sec:experimental\_design\_details\]). The linearization point is chosen to be $\vec{\mu}_0$ and the prior for each nucleon is given in Table \[tab:polarizability\_info\]. We will discuss these priors momentarily. Our goal is to constrain polarizabilities, so the optimal design is that which is likely to provide the most information about $\lecs$. It is reasonable then to choose the utility to be the gain in Shannon information for $\lecs$ based on the experiment $(\design, \genobsset)$. This is equivalent to the Kullback-Leibler (KL) divergence between the prior and posterior for $\lecs$, followed by marginalizing over $\genobsset$: $$\begin{aligned} U_{\text{KL}}(\design) & = \int \!\!\left\{\! \ln\!\!\left[\frac{\pr(\lecs \given \genobsset, \design)}{\pr(\lecs)}\right]\! \pr(\lecs \given \genobsset, \design) \dd{\lecs}\right\}\! \pr(\genobsset \given \design) \dd{\genobsset}\!. \label{eq:expected_utility_kl}\end{aligned}$$ The assumptions of Eqs.  and  allow to be computed exactly, with the result $$\begin{aligned} \label{eq:utility_kl_analytic} U_{\text{KL}}(\design) = \frac{1}{2} \ln \frac{|V_0|}{|V(\design)|} \equiv \ln\shrinkage(\design) \geq 0 \, ,\end{aligned}$$ where we have defined the posterior shrinkage factor $\shrinkage \geq 1$. Consider the hyperellipsoids defined by given confidence levels for the $\lecs$ prior and posterior, and . Then $\shrinkage$ is the factor by which the volume of the prior ellipsoid shrinks as it is updated to the posterior, with larger values of $\shrinkage$ (or $U_{KL}$) being more informative than smaller values. An experiment yielding $\shrinkage = 1$ (or $U_{KL} = 0$) is then completely uninformative. The utility of an experiment designed to constrain any subset of $\lecs$, without regard to the others, can be assessed by simply computing Eq.  with the corresponding submatrices of $V_0$ and $V$. [l@d[4.2]{}d[0.3]{}@l@d[2.1]{}@d[1.1]{}@l]{} & &\ $\lecs$ & & &Ref.& & & Ref.\ $\alpha_{E1}+\beta_{M1}$ & 14.0 & 0.2&[@Gryniuk:2015eza] & 15.2 & 0.4&[@Levchuk:1999zy]\ $\alpha_{E1}-\beta_{M1}$ & 7.5 & 0.9&[@McGovern:2012ew]& 7.9 & 3.0&[@Myers:2015aba; @Myers:2014ace]\ $\gamma_{0}$ & -0.929 & 0.015&[@Gryniuk:2016gnm]& 0.4 & 2.2&\ $\gamma_{\pi}$ & 5.5 & 1.9& & 7.8 & 2.2&\ $\gamma_{E-}$ & -0.7 & 2.0& & -3.9 & 2.0&\ $\gamma_{M-}$ & 0.3 & 0.9& & -1.1 & 0.9&\ Although the posterior shrinkage has the benefit of being strictly non-negative and increasing with increasing information, it is unbounded, making it difficult to compare plots on different scales. Thus, we choose to show the percent decrease in uncertainty $$\begin{aligned} \label{eq:percent_decrease} \text{\% Decrease} & = \frac{|V_0|^{\frac{1}{2}} - |V|^{\frac{1}{2}}}{|V_0|^{\frac{1}{2}}} \times 100\% \notag \\ & = {\left(1 - \frac{1}{\shrinkage}\right)} \times 100 \% \, .\end{aligned}$$ This shares the beneficial aspects of $\shrinkage$, but is bounded in the range of 0–100%. Our assumptions lead to a form of the expected utility that is analytic, easy to understand, and quick to compute. This makes Eq.  very attractive. It allows quick assessment of both: - Optimal designs for various assumptions, such as experimental noise levels and truncation error forms. - Which polarizability subsets will have their constraints improved by a particular experiment—and by how much. Constraints from previous experiments are built in naturally via the prior on the polarizabilities. For example, a large utility in a previously well-measured observable or region of kinematic space means that there is still valuable constraining information to be gained there. Furthermore, Eq.  is invariant under any linear transformation of $\lecs$, meaning, [*e.g.*]{}, that the choice of units for $\lecs$ is irrelevant, and that this analysis would be consistent if we had instead used $\lecs = \{\alpha_{E1}, \beta_{M1}, \gamma_{E1E1}, \gamma_{M1M1}, \gamma_{E1M2}, \gamma_{M1E2}\}$, so long as $V_0$ were transformed accordingly (see Sec. \[sec:basic-compton-facts\]). Choice of Priors \[sec:priors\] ------------------------------- The priors summarized in Table \[tab:polarizability\_info\] are the uncertainties to which the polarizabilities are known at present. As we base our design on the results of the  variant of Refs. [@Griesshammer:2017txw; @Griesshammer:2015ahu; @McGovern:2012ew; @Griesshammer:2012we], it is natural to resort to Table 1 of Ref. [@Griesshammer:2015ahu] for the central values and uncertainties for all polarizabilities which are not well-determined by other means. Of these, $\alpha_{E1}+\beta_{M1}$ is best known, not from Compton scattering experiments directly, but from evaluations of the Baldin Sum Rule for the proton [@Gryniuk:2016gnm; @Gryniuk:2015eza] and neutron [@Levchuk:1999zy; @Levchuk:1999zy]. This recasts it as an energy-weighted integral over photoproduction cross sections. Likewise, the GDH Sum Rule provides a highly precise value for the proton’s $\gamma_0$ [@Gryniuk:2016gnm; @Gryniuk:2015eza]. The values of $\alpha_{E1}-\beta_{M1}$, on the other hand, were determined in the  variant we employ here from Compton scattering data on the proton [@McGovern:2012ew] and, for the neutron values, on the deuteron [@Myers:2014ace; @Myers:2015aba]. For the spin-polarizability $\gamma_\pi$ of the proton and neutron, some information is available from back-scattering Compton experiments, and for the neutron-$\gamma_0$ again from a GDH-sum-rule, but these are often under dispute, see, [*e.g.*]{}, the extended discussion in Ref. [@Schumacher:2005an]. In general, these are not of higher accuracy than the predictions inferred from the  results for the spin polarizabilities in Table 1 of Ref. [@Griesshammer:2015ahu]. Other values for the spin polarizabilities with overall similar uncertainty estimates are available [@Lensky:2015awa; @Babusci:1998ww; @Hildebrandt:2003fm; @Holstein:1999uu], as well as some from recent data analyses [@Martel:2014pba; @Sokhoyan:2016yrc; @Paudyal:2019mee; @MartelPhD]; see the summary in Table 1 of Ref. [@Griesshammer:2015ahu]. Other recent extractions of polarizabilities from unpolarized data should also be mentioned [@Pasquini:2017ehj; @Krupina:2017pgr]. Therefore, we derive the values and uncertainties for the neutron-$\gamma_0$, as well as for $\gamma_\pi$, $\gamma_{E-}$ and $\gamma_{M-}$ on both the proton and neutron from Table 1 of Ref. [@Griesshammer:2015ahu]. We add all uncertainties (theory and, as applicable, statistical) in quadrature. That publication derived theory uncertainties from the progression as more terms in the EFT expansion are considered [@Griesshammer:2015ahu], as advocated in Ref. [@Furnstahl:2015rha]. For these, probability distribution functions with reasonable priors are therefore available, but we found that the difference between convolving these and simple addition in quadrature is negligible. Results {#sec:results_experimental_design} ======= We start this section with the customary word of caution in mathematical statistics. The predictions which form the output of this formalism should be understood as likely outcomes, not as guarantees. They carry “errors on the errors.” Details depend on our input choices (priors) and model assumptions, and it is an advantage of the Bayesian approach that these must be discussed explicitly. We found our results to be robust against other reasonable choices, though reasonable people can make other reasonable choices, which then leads to scientific progress by discussion. Overall, the choices we explored led to different outcomes in details, but not to substantially different outcomes. We would therefore not label one design’s superiority as significant if its decrease in uncertainty \[see Eq. \] is within a few percentage points of others. But we are confident that a difference of, say, ten percent indicates a clear preference of one design over others. The guidance we provide for observables and kinematic locations is documented in a publicly available Jupyter notebook [@BUQEYEgithub]. We hope this will facilitate improvements on this analysis, which is meant to be the first, not the last, word in the ongoing conversation regarding the best way to improve the constraints on the nucleon polarizabilities. Precision Levels and Constraints of Compton Experiments {#sec:experiments} ------------------------------------------------------- We attempt to choose estimates of experimental input which are realistic for modern accelerators and detectors, but also realize that the specifics depend on experimental details. For a first take, we focus on a scenario which is not optimized to a *particular* facility but should be at least of some use for planning and design at *any* facility. Therefore, we consider three levels of detector precision given in Table \[tab:experimental\_precision\_levels\] to provide a range of plausibly achievable experimental uncertainties for measurements on the proton. The “standard” scenario assumes uncertainties in the cross section of $\pm5\%$ (systematic and statistical combined in quadrature), and an absolute uncertainty in spin observables of $\pm0.10$. This is state-of-the-art for proton Compton experiments for the cross section and those spin observables that have already been measured [@Martel:2014pba; @Sokhoyan:2016yrc; @Paudyal:2019mee; @Martel:2019tgp; @Martel:2017pln; @Ahmed:2020hux; @privcomm]. A second scenario lists experimental error bars $\Delta\!\diffcs\approx\pm4.0\%$ and an absolute $\Sigma_i$ error of $\pm0.06$ which are deemed “doable” nowadays without excessive improvements. The “aspirational” scenario assumes considerable but realistic new resources and possibly new equipment. Our choices were informed by discussions with our experimental colleagues who work on Compton scattering at MAMI and HI$\gamma$S, for whose input we are very grateful [@privcomm]. Unless otherwise stated, all results in figures assume the “doable” level of experimental precision, with the remaining levels reserved for [the Supplemental Material.]{} {width="\textwidth"} \[fig:utility\_grid\_compare\_subsets\_and\_truncation\_proton\] [ld[2.2]{}d[2.2]{}]{} Level & &\ Standard & 5.00 & 0.10\ Doable & 4.00 & 0.06\ Aspirational & 3.00 & 0.03\ It should be noted that achieving even “standard” errors of $\pm0.10$ for some of the hitherto-unmeasured spin observables is not simple. Especially for the spin-polarization transfer observables, the experimental challenges of detecting recoil spin polarizations are considerable. In that case, the estimate can serve as benchmark, with the “standard” scenario already an “aspirational goal.” Due to the absence of quasi-stable free-neutron targets, a “standard” uncertainty for neutron Compton scattering is of course well beyond “aspirational.” We nonetheless chose to use the same error bars for the neutron, to ease comparison. We search for the optimal one-point design and the optimal five-point design, [*i.e.*]{}, a search over all accessible combinations of five unique angles at a given $\omegalab$ (“5-point design”). In line with experimentalists’ constraints on the placement of bulky detectors, we require that the angles be at least ${10^\circ}$ apart. The focus on one photon energy and multiple angles mirrors the capabilities of “monochromatic-beam” facilities like HI$\gamma$S which measure several angles at one energy simultaneously, but many other choices could be made. The assessment can easily be extended to “bremsstrahlung facilities,” where a number of both angles and energies can be measured simultaneously. In that case, a typical spacing between the central energy of each energy bin of about $10$ to $20\MeV$ appears realistic, given that a sufficient number of events must be collected in each “bin” for meaningful statistics [@Martel:2014pba; @Sokhoyan:2016yrc; @Paudyal:2019mee; @Martel:2019tgp; @Martel:2017pln; @Ahmed:2020hux; @privcomm]. Hence, our results attempt to be as realistic as possible given the choices above, and are a proof of principle for further, more specific research. While the plots show a full range of energies and angles, we also indicate on them regions defined by $\omega\le{60}\MeV$ or $\theta\le{40}^\circ$ or $\theta\ge{150}^\circ$ in which experiments are unlikely to be conducted, because forward and backward angles are physically hard to access, or because sensitivity to polarizabilities at very low energies is negligible. Therefore, we do not elaborate on designs that involve these kinematic regions. Only the cross section and $\Sigma_{3}$ are non-vanishing as $\omega\to0$, the physics of both being governed by the Thomson limit, with polarizability corrections very small. Our LO result provides the correct Thomson limit for each observable automatically, and we constrained the unknown higher-order corrections so that they do not change this; see Appendix \[sec:truncation\_model\_details\_compton\]. Indeed, we find a typical energy correlation length of $\ell_\omega\approx{50}\MeV$ for the proton; see Table \[tab:truncation\_details\_observables\], so that such a constraint becomes less important around and above $60\MeV$. In addition, due to the coordinate singularity at $\theta=0^\circ$ and $180^\circ$, observables or their derivatives with respect to $\theta$ must be zero there. We implemented these constraints as described in Appendix \[sec:truncation\_model\_details\_compton\]. As the angular correlation lengths from Table \[tab:truncation\_details\_observables\] are all smaller than $\ell_\theta\approx{55^\circ}$, this is not a strong constraint on observables at intermediate angles where experiments are most feasible. Our design model does not include the constraint that spin observables $\Sigma_i$ can only have values between $-1$ and $1$. This is a reasonable omission because the mean value of most $\Sigma_i$ and their  uncertainties are mostly well contained within these bounds (except maybe at the largest $\omega$), see Appendix \[sec:truncation\_model\_details\_compton\] for details. Finally, we reiterate that around the pion-production threshold ($\omega_\pi\pm20\MeV$ or so, with $\omega_\pi\approx150\MeV$ marked by a vertical line in plots), the  truncation error estimates for the spin observables $\Sigma_i$ are less understood, and experimental conditions are difficult as well; see Sec. \[sec:truncation\_distribution\_compton\] and Appendix \[sec:truncation\_model\_details\_compton\]. First Discussion and Impact of Accounting for Theory Uncertainties: The Cross Section {#sec:crosssection} ------------------------------------------------------------------------------------- The unpolarized differential cross section is the most extensively studied nuclear Compton scattering observable. Therefore we begin by showing the expected utility of further measurements, with the goal of constraining various subsets of the polarizabilities. Our results focus on proton observables unless otherwise stated, due to the difficulty of performing experiments on neutrons; see [the Supplemental Material]{} for the corresponding neutron design results. We start by considering the following subsets: all polarizabilities simultaneously, only $\alpha_{E1}+\beta_{M1}$, only $\alpha_{E1}-\beta_{M1}$, and only the spin polarizabilities $\{\gamma_i\} \equiv \{\gamma_{0}, \gamma_{\pi}, \gamma_{E-}, \gamma_{M-} \}$, or each of them separately. Figure \[fig:utility\_grid\_compare\_subsets\_and\_truncation\] shows the expected utility of future proton experiments, with and without an estimate of the truncation error. Without truncation, the utility of an experiment to measure $\alpha_{E1}\pm\beta_{M1}$ mirrors the sensitivity analysis of Fig. 8 in Ref. [@Griesshammer:2017txw]. There, the derivative of the observable with respect to a particular polarizability was plotted, and the truncation error of the EFT was only accounted for indirectly by casting a “gray mist” over the plot which thickens into the Delta resonance region, starting at $\omega\gtrsim210\MeV$, [*i.e.*]{}, where our transition region starts. In fact, the constraining power of truncation-free 1-point measurements (as measured by $U_{KL}$) on each individual polarizability follow exactly the patterns of the local sensitivities for all observables and polarizabilities. This can be verified by comparing the remainder of our zero-truncation-error results in [the Supplemental Material]{} with the appropriate subplots of Figs. 9–20 in Ref. [@Griesshammer:2017txw]. However, when truncation-error estimates are included, the optimal designs are pushed to lower $\omegalab$, with a particularly dramatic shift for $\alpha_{E1} - \beta_{M1}$. This is expected because the  uncertainty $\delta\genobs_k$ increases with energy. Still, the optimal locations for constraining the spin polarizabilities often remain at or above the pion-production threshold. One of the benefits of our Bayesian analysis over a purely derivative-based approach—like that of Ref. [@Griesshammer:2017txw]—is that we can examine the collective gain in information for multiple polarizabilities. The second, third, and fourth panels in the lower row of Fig. \[fig:utility\_grid\_compare\_subsets\_and\_truncation\] provide the optimal kinematics at which to constrain either of the scalar polarizabilities, or the spin-polarizabilities collectively. Looking across them reveals that the collective information gain in the first panel is approximately the sum of the information gains of each subset. In this study, we found that the correlations between these linear combinations of polarizabilities that are induced by fitting are rather small; see the extended discussion of Fig. \[fig:shrinkage\_per\_subset\]. Equation  then says that, to the extent that the covariance matrix $V$ is diagonal, the total utility is the sum of the individual utilities. The feature seen here is thus generic in the absence of correlations: the amount of benefit derived from collectively constraining $\lecs$ is related to how much the utilities for individual components of $\lecs$ overlap in kinematic space. All Observables: Discussion {#sec:observables} --------------------------- {width="\textwidth"} {width="\textwidth"} We now extend our analysis of the differential cross section to the spin observables $\Sigma_i$. For the remainder of this work, we include truncation error estimates, because otherwise the constraining power of any measurement would be overstated; see [the Supplemental Material]{} for corresponding results without truncation errors included. Figures \[fig:utility\_grid\_subset\_polarizabilities\_observable\_set\_1\] and \[fig:utility\_grid\_subset\_polarizabilities\_observable\_set\_2\] show heat maps of the expected utility of all proton observables, with truncation-error estimates included. Note that the scale has changed dramatically, as can be seen by comparing the results in the bottom row of Fig. \[fig:utility\_grid\_compare\_subsets\_and\_truncation\] to the same results repeated in the top row of Fig. \[fig:utility\_grid\_subset\_polarizabilities\_observable\_set\_1\]. Figures \[fig:utility\_grid\_subset\_polarizabilities\_observable\_set\_1\] and \[fig:utility\_grid\_subset\_polarizabilities\_observable\_set\_2\] contain a wealth of information about the relative information gain between observables at various kinematics, but for more readily interpretable statements about potential constraining power, we turn first to Fig. \[fig:shrinkage\_per\_subset\]. Here we can see the largest percent decrease in uncertainty \[Eq. \] of all optimal 5-point designs for each observable, with utilities split up based on the polarizabilities one might be interested in measuring. Thus, given a decision about which polarizability is of most interest—a decision which we do not encode mathematically—our approach provides a quantitative method for evaluating the worth of future experiments. Any *set* of utilities, such as “All” or “$\{\gamma_i\}$” are guaranteed to be greater than or equal to the optimal utility of any individual polarizability that contributes to it. How much more is learned by considering multiple polarizabilities, depends on how much their optimal designs overlap in kinematic space—because we have found that in this case only small correlations are induced in the covariance matrix $V$ by fitting. {width="\textwidth"} For the proton, the combinations $\alpha_{E1}+\beta_{M1}$ and $\gamma_0$ are well-constrained by sum rules [@Gryniuk:2015eza; @Gryniuk:2016gnm]; see the small error bars in Table \[tab:polarizability\_info\]. In Compton scattering, these are the only two linear combinations of polarizabilities which enter the cross section as $\theta\to0$. Figure \[fig:shrinkage\_per\_subset\] reveals that indeed little information on them can be gained from direct Compton experiments. The other four combinations, $\alpha_{E1}-\beta_{M1}$, $\gamma_\pi$, $\gamma_{E-}$ and $\gamma_{M-}$, will therefore dominate our discussion. One observable that stands out in Fig. \[fig:shrinkage\_per\_subset\] is $\Sigma_{2x}$ (circularly polarized photons on a transversely polarized target). Its overall information gain of around $50\%$ stems near-exclusively from the gain in $\gamma_{E-}$. Combining this with Fig. \[fig:utility\_grid\_subset\_polarizabilities\_observable\_set\_1\], we see that the gain occurs in a quite robust and large region which extends from about $\omega_\pi$ to above $200\MeV$, at angles between $\theta \approx 30^\circ$ and $90^\circ$. The other polarizabilities are optimally constrained in a similar kinematic region, but according to Fig. \[fig:shrinkage\_per\_subset\], their contribution to the overall utility from such an experiment is negligible ($\lesssim5\%$) compared to what would be learned about $\gamma_{E-}$. This means that measurements of $\Sigma_{2x}$ in that region allow for an extraction of $\gamma_{E-}$ which is highly insensitive to the particular values of $\alpha_{E1}$, $\beta_{M1}$ and the other spin-polarizabilities used. This observable was already explored in a pioneering experiment at MAMI for $\omega\approx290\dots330\MeV$ [@Martel:2014pba], where unfortunately the information content of an EFT interpretation is not very high. (Remember that the color scale in Fig. \[fig:utility\_grid\_subset\_polarizabilities\_observable\_set\_1\] changes *rapidly* with decreasing utility.) This analysis implies that much more information can be gained from an experiment at $\omega\lesssim200\MeV$. A similarly large constraint on the polarizabilities (gain $\approx50\%$) comes from the analogous polarization-transfer observable $\Sigma_{2x'}$ (incident circularly photon on unpolarized target, transverse spin polarization of recoil proton detected). Now, $\gamma_{E-}$ is somewhat less constrained on its own (gain $\approx40\%$). Compared to $\Sigma_{2x}$, the utilities of $\alpha_{E1}+\beta_{M1}$ and of $\gamma_{M-}$ are slightly increased to $\lesssim 15\%$ each, indicating that some limited information can also be gained about these polarizabilities at the same time. Such an experiment would need to be made near—or a few dozen MeV above—$\omega_\pi$ and towards forward angles. The polarization-transfer $\Sigma_{2z^\prime}$ (incident circularly polarized photon on unpolarized target with detection of longitudinal recoil polarization) provides a gain of about $30\%$ overall, at similar energies but slightly smaller angles. With about $25\%$, most of the gain is again in $\gamma_{E-}$, followed by a gain of a bit less than $15\%$ in $\alpha_{E1}-\beta_{M1}$. Decent information gain on $\gamma_{E-}$ (about $20\%$) can also be found from measuring the beam asymmetry $\Sigma_3$ (linearly polarized beam on unpolarized target) at intermediate angles in two narrow corridors, namely close to the pion-production threshold and slightly higher, $\omega\approx200\MeV$. Some data is actually available there [@Blanpied:2001ae; @Sokhoyan:2016yrc; @Martel:2017pln; @CollicottPhD] but has not yet been analyzed in EFT. It has thus not entered in the determination of the error bars on the priors in Table \[tab:polarizability\_info\]; these results suggest such an analysis could be valuable. Measurements of $\Sigma_3$ at lower $\omega$ have been used in attempts to constrain the scalar polarizability $\beta_{M1}$ [@Sokhoyan:2016yrc]. But we see that even in the most sensitive kinematics its impact on $\beta_{M1}$ amounts to just a few percent. Instead, the combination $\alpha_{E1}-\beta_{M1}$ can be measured with an estimated information gain of about $20\%$ from the cross section in a region somewhat above $100\MeV$ at back-angles. Qualitatively, this angle regime is not surprising since is well known that this particular linear combination enters the cross section as $\theta\to180^\circ$, as does $\gamma_\pi$. Interestingly, the next-largest information gain for $\alpha_{E1}-\beta_{M1}$ appears to be found in $\Sigma_{2z}$ (circularly polarized beam on longitudinally polarized target) and the corresponding polarization-transfer, $\Sigma_{2z^\prime}$, but these amount to only slightly more than $10\%$. However, the region of greatest sensitivity lies in both these cases right at the pion-production threshold, where experiments are particularly challenging and where our  uncertainties may be less accurate (see Sec. \[sec:truncation\_distribution\_compton\]). According to Fig. \[fig:shrinkage\_per\_subset\], $\gamma_{M-}$ is quite elusive. Only the differential cross section shows appreciable information gain (about $20\%$), while the next-largest gains, in $\Sigma_{2z}$ and $\Sigma_{2x^\prime}$, hardly exceed $10\%$. In all three observables, the region of largest sensitivity to $\gamma_{M-}$ is right at the $\omega_\pi$ cusp, where the $\diffcs$ convergence pattern is well behaved (see Fig. \[fig:coefficients\_dsg\_slices\]). This makes us more confident in our design predictions for $\diffcs$ than for the spin observables, which fluctuate more strongly. Taking all this into account, we find that a measurement of the cross section in a broad band around $\omega_\pi$ and at intermediate angles is the best chance to constrain $\gamma_{M-}$. As a bonus, such a measurement would concurrently constrain other polarizabilities “for free.” Optimal 5-point measurements of the differential cross section $\diffcs$ can decrease the collective uncertainty of all polarizability combinations by about $40\%$, but the information gain is spread out amongst individual polarizabilities: about $20\%$ for $\alpha_{E1}-\beta_{M1}$ and $\gamma_{M-}$, $15\%$ for $\gamma_\pi$, less than $10\%$ for $\alpha_{E1}+\beta_{M1}$ and $\gamma_{E-}$ each, and no perceptible information gain for $\gamma_0$. In part, different kinematic regions are sensitive to individual combinations, so measurements across a wide array of energies and angles can be used to disentangle individual contributions. We pause here to highlight an important point. While the experimental design does not explicitly trace the kinematics and quality of available data, it is actually “aware” of the experimental information available at present. That there is a great number of proton Compton cross section data of widely varying quality, enters via the priors on $\alpha_{E1}$, $\beta_{M1}$ and $\gamma_{M1M1}$ in Table \[tab:polarizability\_info\] as the experimental (statistical plus systematic) uncertainties in the fits of those quantities. The fact that additional information can be gained from more high-quality data in specific kinematic regions implies that the quantity and, most importantly, the quality of future Compton data in that region can provide important information gains on the polarizabilities, even if that region appears at first glance to be already saturated. On the other hand, the available data for both $\Sigma_{2x}$ [@Martel:2014pba; @MartelPhD] and $\Sigma_{2z}$ [@Martel:2017pln; @Paudyal:2019mee] did not enter in the priors of Table \[tab:polarizability\_info\], but both data were taken in the Delta resonance region, where the sensitivity of these two observables to any polarizability is minuscule, according to Fig. \[fig:utility\_grid\_subset\_polarizabilities\_observable\_set\_1\]. Adding their information to the priors in any form will therefore not change our conclusions or improve polarizability error bars. For $\gamma_\pi$, any information gain can only be found in the cross section and does not exceed $15\%$, with other polarizabilities contributing equal or larger amounts; see discussion above. A dedicated $\theta=180^\circ$ experiment, like in Ref. [@Zieger:1992jq], may be able to resolve it and $\alpha_{E1}-\beta_{M1}$, but needs a special design. No other polarizability combination enters at that angle. The relatively steep differences in information-gain reflect to a large extent the fact that the uncertainties for the spin polarizabilities and for $\alpha_{E1}-\beta_{M1}$ are substantially larger than for the sum-rule constrained combinations $\alpha_{E1}+\beta_{M1}$ and $\gamma_0$. In particular, the very small error bar on $\gamma_0$ makes the probability to gain information via Compton scattering very small; and Fig. \[fig:shrinkage\_per\_subset\] shows that it is indeed close to zero for all observables. However, the size of a polarizability’s error bar is by itself not a reliable criterion for strong information gain. The error bar of $\gamma_{M-}$ is about half of that of $\gamma_\pi$ or $\gamma_{E-}$, so one might expect the information gain in measuring it to be about half of that for $\gamma_\pi$ or $\gamma_{E-}$. Instead, a measurement with substantial information gain is much more elusive than that, as explained above. At first glance, an approximate $10\%$ information gain for $\gamma_{M-}$ can also be found in $\Sigma_{1z^\prime}$. However, its favored kinematics is at $\omega\approx\Delta\approx300\MeV$, [*i.e.*]{}, right at the border of the region under consideration. This is not the only observable for which this happens. The biggest sensitivity of $\Sigma_y$, $\Sigma_{3y^\prime}$ and $\Sigma_{1z^\prime}$ to both $\gamma_{E-}$ and $\gamma_{M-}$ is likewise pushed to the maximum considered energies $\omega\approx300\MeV$. On the one hand, such behavior might be interpreted as in apparent tension with the fact that  is significantly less reliable in the Delta resonance region than at lower energies. On the other hand, the uncertainty of  is accounted for in our experimental design; see discussion in Sec. \[sec:crosssection\]. Possibly, the 5-point design in that region probes a sensitivity of the correlated angular dependence at high energies, rather than on individual values/rates at a particular angle. If so, and if  predicts these correlations more robustly than overall sizes of an observable, then the phenomenon would be explained and measurements of the functional dependence of observables on angle at such high energies could provide determinations of $\gamma_{E-}$ and $\gamma_{M-}$. However, optimal 1-point designs would not be sensitive to correlations and still appear sometimes at very high energies in these same observables. Apparently, the sensitivity is so strong at such kinematics as to win over the decreased theory uncertainties. As we did not find an intuitively obvious resolution, this merits further study. Equally as notable as these powerful information gains on polarizabilities are those observables that seem to provide almost no information about the polarizabilities at this level of experimental and theoretical precision. The most prominent such example is $\Sigma_{1x'}$ (total gain $<5\%$). Given our truncation error estimates for this quantity, there is little information on the polarizabilities to be gained from any 5-point experiment. Measuring it, or indeed any observable, in a region where the information gain for polarizabilities is negligible, can still be useful though. It provides information about how accurately  describes the Compton process, independent of the polarizabilities. This is an important cross-check of , even though it is not part of the utility used in this work. ![The percent decrease in $\lecs$ uncertainties, as in Fig. \[fig:shrinkage\_per\_subset\], applied to decide on the trade-off between different allocations of experimental resources (exploration vs. exploitation). Larger values imply that the measurement is more informative. The 1-point (5-point) optimal design is denoted by a circle (cross), and the experimental precision levels are given in Table \[tab:experimental\_precision\_levels\]. The decision to increase precision or measure at more kinematic points (or neither) can vary significantly by observable. []{data-label="fig:shrinkage_fixed_photons"}](fvr_n-pts_and_precision_subset-all_nucleon-proton) Such an analysis raises a further question: if experimental resources are limited, does it make sense to measure 1 point very precisely or many points less precisely? Our framework can supply answers to this and many other such questions. By comparing the optimal designs of 1- and 5-point experiments at both the “doable” and “aspirational” level of experimental precision (see Table \[tab:experimental\_precision\_levels\]) we get an idea of how to design the most effective experiment. The results are given in Fig. \[fig:shrinkage\_fixed\_photons\]. Again, it is clear that the details depend on the observable, which proves the usefulness of our approach: one need not rely on heuristics when a quantitative scheme is readily available. For example, the differential cross section does not appear to benefit as much from an increase in precision (red circle) as it would from more data across $\theta$ (blue cross). In other cases, such as $\Sigma_{2x}$, $\Sigma_{2x'}$ or $\Sigma_{2z'}$, the gain in information an “aspirational” 1-point experiment is about the same as 5 measurements from a “doable” experiment. Surprisingly, other observables, such as $\Sigma_{1z'}$, benefit very little from either increased precision or an increased number of data points: one “doable" measurement in the right spot already realizes most of the information gain to be had from them. For completeness, we show the utility of performing neutron Compton scattering experiments, with more plots for the neutron in [the Supplemental Material.]{} Figure \[fig:dsg\_neutron\_utility\] shows the 1-point profile of the differential cross section with truncation error, which is similar to the corresponding utility in Fig. \[fig:utility\_grid\_compare\_subsets\_and\_truncation\]. Such measurements are notoriously difficult, so the interpretation of these results should proceed with caution. More realistically, our analysis should be applied to  predictions of light incident on the deuteron, $^3$He or other few-nucleon targets for which calculations of Compton scattering are available in the same  formulation [@Margaryan:2018opu; @Hildebrandt:2005iw; @Griesshammer:2013vga; @Griesshammer:2012we]. ![The expected utility from measuring the differential cross section for the neutron while including truncation error. All neutron polarizabilities are included in this analysis. Its profile resembles that of the analogous proton observable, see Fig. \[fig:utility\_grid\_compare\_subsets\_and\_truncation\], though the color scales differ. []{data-label="fig:dsg_neutron_utility"}](utilities_dsg_neutron_trunc_level-doable_subset-all) Summary {#sec:summary_design} ======= We have proposed a powerful and versatile framework to help plan experiments which rely on EFT to extract or check parameters. Using the example of Compton experiments in order to constrain nucleon polarizabilities, this method quantifies the expected gain in information from an experiment: it maximizes shrinkage of the posterior. The framework solves the problems of theoretical errors conflicting with experimental considerations, and finds a compromise between the two. Under reasonable assumptions, we obtain an algorithm that is analytic, easy to understand, and quick to compute. Furthermore, we employed a Bayesian machine learning algorithm for estimating EFT truncation errors whose power counting varies across the domain. This is a novel extension of the model introduced in Ref. [@Melendez:2019izc]. Gaussian processes efficiently and accurately account for correlations in the EFT truncation error, and impose the symmetry constraints on observables and their derivatives that must vanish, [*e.g.*]{}, at $\theta = 0^\circ$ or $180^\circ$. This physically motivated model is crucial to the study of experimental design with EFTs, as otherwise errors will be underestimated. To facilitate reproduction and extension of our results, we provide all of the codes and data that generated our results [@BUQEYEgithub]. Our Bayesian experimental design framework has the following benefits: 1. It can incorporate the effects of both experimental and theoretical uncertainties. 2. Its output contains both the optimal design and an estimate of the gain in information for that design which can be understood quite easily. 3. It can include the effects from measuring multiple kinematic points and can assess the interaction of multiple polarizabilities at once. 4. It permits a quantitative analysis of competing choices, [*e.g.*]{}, one can answer the question: should an experiment measure one point with high precision or many points with less precision? 5. Bayesian statistics mandates us to clearly specify our assumptions. Those who disagree with any assumption (size of error bars, priors on GP hyperparameters, the power counting in the transition region, design constraints, etc.) can readily modify our calculations, provided at [@BUQEYEgithub], to their own specifications, thereby facilitating an ongoing dialog regarding the robustness of our experimental design results. We also make the obvious point that while we have focused on Compton scattering experiments here, our EFT-based Bayesian approach to experimental design is easily adapted to other experiments informed by EFT calculations. We tried to make a realistic assessment of experimental specifications in this work, but realize that experiments can differ greatly. In the future, we could apply our framework to a specific experiment at, [*e.g.*]{}, MAMI or HI$\gamma$S with fine details accounted for. Bayesian experimental design could answer specific questions for the design of future experiments, such as how to determine the expected amount or quality of measurements that are required to reach a given level of precision. This framework can be extended to sequential designs, where experimental campaigns are split into a sequence of parts and the design of future experiments depends on the results of the initial experiments [@Ryan:2015aaa]. Although we have found the assumption of linearity to be good in this case, one could perform a full Bayesian experimental design if this assumption no longer holds [@Ryan:2015aaa; @jacksonDesignPhysicalSystem2018]. Our theoretical truncation estimates are the most comprehensive to date, but further study of chiral EFT convergence for Compton observables should be performed. These are all tasks for future work. We thank Ian Vernon for useful discussions, and M. Ahmed, E. Downie, G. Feldman, P. P. Martel, as well as the MAMI-A2/CB and Compton@HI$\gamma$S teams for their patience in discussing experimental constraints. We gratefully acknowledge the stimulating atmosphere created by organizers and participants of the workshops <span style="font-variant:small-caps;">Uncertainty Quantification at the Extremes (ISNET-6)</span> at T.U. Darmstadt (Germany) and <span style="font-variant:small-caps;">Bayesian Inference in Subatomic Physics - A Wallenberg Symposium (ISNET-7)</span> at Chalmers U. (Göteborg, Sweden), which triggered and expanded these investigations. H.W.G. gratefully acknowledges the warm hospitality and financial support of the A2/Crystall Ball Collaboration Meeting 2020 at MAMI (U. Mainz, Germany), of both Ohio University and the Ohio State University, and of the University of Manchester, where part of this work was conducted. The work of R.J.F. and J.A.M. was supported in part by the National Science Foundation under Grant Nos. PHY–1614460 and PHY–1913069 and the NUCLEI SciDAC Collaboration under US Department of Energy MSU subcontract RC107839-OSU. The work of D.R.P. was supported by the US Department of Energy under contract DE-FG02-93ER-40756 and by the ExtreMe Matter Institute EMMI at the GSI Helmholtzzentrum für Schwerionenphysik, Darmstadt, Germany. The work of H.W.G. was supported in part by the US Department of Energy under contract DE-SC0015393, by the High Intensity Gamma-Ray Source [HI$\gamma$S]{} of the Triangle Universities Nuclear Laboratory TUNL in concert with the Department of Physics of Duke University, and by The George Washington University: by the Dean’s Research Chair programme and an Enhanced Faculty Travel Award of the Columbian College of Arts and Sciences; and by the Office of the Vice President for Research and the Dean of the Columbian College of Arts and Sciences. His work was conducted in part at GW’s Campus in the Closet. The work of J.McG. was supported by the UK Science and Technology Facilities Council grant ST/P004423/1. The work of M.T.P. was supported in part by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award No. OSR-2018-CRG7-3800.3. Experimental Design Details {#sec:experimental_design_details} =========================== Suppose that our theoretical model $y_k(\kinparvec;\lecs)$ is related to measurements $\genobsexp(\kinparvec)$ via additive theoretical and experimental noise, as in Eq. . We can linearize $y_k(\kinparvec;\lecs)$ about some point $\lecs_\star$ by keeping only the first order terms in its Taylor expansion, [*i.e.*]{}, $$\begin{aligned} y_k(\kinparvec;\lecs) & \approx y_k(\kinparvec; \lecs_\star) + \sum_i b_i(\kinparvec) [\lecs_i - \lecs_\star] \notag \\ & = c(\kinparvec; \lecs_\star) + \vec{b}(\kinparvec) \cdot \lecs \, ,\end{aligned}$$ where $\vec{b}(\kinparvec) \equiv \partial \genobs_k(\kinparvec;\lecs)/\partial \lecs$ evaluated at $\lecs_\star$ are our basis functions and $c(\kinparvec;\lecs_\star) \equiv y_k(\kinparvec;\lecs_\star) - \vec{b} \cdot \lecs_\star$ is constant with respect to the polarizabilities $\lecs$ but depends on the kinematic point $\kinparvec$. Thus, the vector of $N$ measurements $\genobsset$ is related to the polarizabilities via the likelihood $$\begin{aligned} \label{eq:likelihood_linear} \genobsset \given \lecs \sim \normal[B\lecs + \mathbf{c}, \Sigma]\end{aligned}$$ where $B \equiv \vec{b}(\kinparvecset)$ is an $N \times 6$ matrix, $\mathbf{c} \equiv c(\kinparvecset; \lecs_\star)$ is a length $N$ vector, and $\Sigma$ is the $N \times N$ covariance matrix due to theoretical and experimental error. That is, given some experimental covariance $\Sigma_{\rm exp}$ and a theoretical covariance $\sdth^2 \discrcorr{k}$ from Eqs.  and , then $$\begin{aligned} \Sigma = \sdth^2 \discrcorr{k} + \Sigma_{\rm exp} \, .\end{aligned}$$ Note that $\discrcorr{k}$ depends on the values of the tuned $\ell_\omega$ and $\ell_\theta$, whose estimates from the order-by-order convergence pattern are given in Table \[tab:truncation\_details\_observables\]. The linear model of Eq.  is well known in the statistics literature [@gelman2013bayesian; @o1994bayesian], so here we will simply state the relevant results. If a Gaussian prior with mean $\vec{\mu}_0$ and covariance $V_0$ is placed on $\lecs$ as in Eq. , then the resulting posterior is also Gaussian, with mean and covariance given by $$\begin{aligned} \vec{\mu} & = V\left[V_0^{-1}\vec{\mu}_0 + B^\trans \Sigma^{-1} (\genobsset - \mathbf{c})\right] \, , \label{eq:pol_posterior_mean_linear} \\ V & = (V_0^{-1} + B^\trans \Sigma^{-1} B)^{-1} \, . \label{eq:pol_posterior_cov_linear}\end{aligned}$$ Importantly to our study of experimental design, the posterior covariance $V$ depends on the kinematic points $\kinparvecset$ where the experiment is performed, and on the specifics of the observable through $\Sigma$, but not on the exact results of the experiment $\genobsset$. Given that we choose to maximize the expected information gain in the polarizabilities, then the integrals of Eq.  must still be performed. The integral over $\lecs$ splits into the difference of two terms: the differential entropy of the prior $\pr(\lecs)$ and of the posterior $\pr(\lecs \given \genobsset, \design)$. The differential entropy of a Gaussian $\normal(\mu, \Sigma)$ is well known to be $\frac{1}{2}\ln{|2\pi e\Sigma|}$. Therefore $$\begin{aligned} U_{\text{KL}}(\design) & = - \int \ln[\pr(\lecs)] \pr(\lecs) \dd{\lecs} \notag \\ & ~~~ + \int \ln[\pr(\lecs\given \genobsset, \design)]\pr(\lecs\given \genobsset, \design) \dd{\lecs} \pr(\genobsset \given \design)\dd{\genobsset} \notag \\ & = \frac{1}{2}\ln{|2\pi e V_0|} - \frac{1}{2}\ln{|2\pi e V|} \int \pr(\genobsset \given \design)\dd{\genobsset} \notag\\ & = \frac{1}{2} \ln \frac{|V_0|}{|V|} \, ,\end{aligned}$$ where we used the fact that $V$ does not depend on $\genobsset$ and then performed the trivial integration over all possible measurements $\genobsset$. Observable Constraints and EFT Truncation Model Details {#sec:truncation_model_details_compton} ======================================================= Constraints on Compton observables are discussed in detail in Ref. [@Griesshammer:2017txw]. Some of this is reproduced here, with particular attention paid to $n$th-order chiral *corrections* to observables $\Delta y_n$ rather than the value of the observable $y$ itself. The $\Delta y_n$ impact the distribution for the  uncertainty $\delta \genobs_k$, but because we restrict the “experimentally accessible regime” in this study from small $\omega$, and forwards/backwards angles, these constraints are not as important as they otherwise would be. These constraints are summarized for particular $\omegalab$ and $\thetalab$ values in Table \[tab:compton\_observable\_symmetry\_constraints\]. All observables that are nonzero below $\omega_\pi$ approach the Thomson limit as $\omega \to 0$ [@Griesshammer:2012we]. Thus, higher-order corrections must vanish there , and approach $\omega = 0$ as at least $\omega^2$. Therefore, at least the first derivative of all corrections must vanish there as well. The remaining observables must vanish for $\omega \leq \omega_\pi$, but there is no constraint on the the derivative of corrections at $\omega = \omega_\pi$. We have found that the corrections approach 0 *very* quickly, so that imposing the constraint $\Delta \genobs_n(\omega_\pi,\theta) = 0$ for all higher order terms is actually a worse approximation than not imposing the constraint at all; see, [*e.g.*]{}, Fig. \[fig:coefficients\_1x\_and\_1xp\_slices\]. This comes back to the large cusps in the spin-observable $c_n$ found near $\omega_\pi$, discussed in Sec. \[sec:truncation\_distribution\_compton\], which remain an unresolved aspect of this model. Due to the coordinate singularity at $\theta=0^\circ$ and $180^\circ$, observables or their derivative with respect to $\theta$ must vanish there [@Griesshammer:2017txw]. But this does not preclude *both* the value and their derivatives from vanishing there. These constraints can be deduced by symmetry arguments, and are summarized in Table \[tab:compton\_observable\_symmetry\_constraints\]. The hyperparameters $\sdth^2$ and $\ell_i$, shown in Table \[tab:truncation\_details\_observables\], are tuned to coefficients $c_n$ at the best known $\lecs$ (see Table \[tab:polarizability\_info\]) for $\Lambda_b = 650\MeV$. The training data is on a grid with $\thetalab = \{30^\circ, 50^\circ, 70^\circ, 90^\circ, 110^\circ, 130^\circ\}$ and $\omegalab = \{200, 225, 250\}\MeV$ for observables which are zero below $\omega_\pi$. For observables that are non-zero below $\omega_\pi$, the additional training points $\omegalab = \{50, 75, 100, 125\}\MeV$ are included, and common $\ell_\omega$ and $\ell_\theta$ are used between the two regions. The training region is well outside the kinematic endpoints where additional constraints arise on observables or their derivatives, and excludes the pion-production threshold region. Because the first nonzero order often behaves differently than the corrections, we do not use it for induction on the $c_n$; that is we only train the hyperparameters on *corrections*. Hence, we train on $c_2$–$c_4$ for $\diffcs$ and $\Sigma_3$, but otherwise we train on $c_3$ and $c_4$. The coefficients for various observable slices are shown in Figs. \[fig:coefficients\_dsg\_slices\]–\[fig:coefficients\_3y\_and\_3yp\_slices\]. These plots also include uncertainty bands for higher order coefficients, with the symmetry constraints given in Table \[tab:compton\_observable\_symmetry\_constraints\] included. These constraints on both the coefficient functions and their derivatives propagate directly to the truncation error $\delta\genobs_k$ by replacing $r(x,x';\ell_\omega, \ell_\theta)$ in Eq.  by its *conditional* form $\tilde r(x,x';\ell_\omega, \ell_\theta)$, see Refs. [@rasmussen2006gaussian; @Melendez:2019izc]. For example, if the value of $c_n$ is known at the set of points $\mathbf{x}$, then one can compute its conditional GP, with covariance kernel given by $$\begin{aligned} \tilde r(x, x') = r(x, x') - r(x, \mathbf{x}) r(\mathbf{x}, \mathbf{x})^{-1} r(\mathbf{x}, x') \, .\end{aligned}$$ See Refs [@Rasmussen:2003gphmc; @Solak:2003dgpds; @Eriksson:2018scaling] for details about adding derivative observations to GPs. Because the RBF kernel \[Eq. \] is separable in $\omega$ and $\theta$, these constraints can simply be applied to each one-dimensional kernel separately, and multiplied to yield the total constrained kernel. We employ the `gptools` python package for easily implementing derivative constraints [@Chilenski_2015_gptools]. For completeness, we also provide the profile for the truncation error standard deviation (up to factors of $\cbar$, which vary by observable); see Fig. \[fig:truncation\_error\_stdv\_compton\]. It assumes the form of $Q$ provided in Eq.  along with the first omitted  order given in Eq. . This allows us to return to the discussion of the omitted constraints $\Sigma_i \in [-1, 1]$ on the spin observables in Sec. \[sec:experiments\]. Over the physically interesting kinematic range, the actual value of most spin observables lies in the much more narrow interval $[-0.6,0.6]$; see Fig. 5 in Ref. [@Griesshammer:2017txw]. So, then the question becomes: are the mean prediction *and* its theory uncertainty contained in $[-1,1]$ with a high degree of probability? From Eq. , one can see that the $1\sigma$ interval for the truncation error $\delta\genobs_k$ is $\genobsref\sdth$ times another factor $Q^{\nu_{\delta k}(\omega)}/\sqrt{1-Q^2(\omega)}$. Here $\genobsref=1$ and $\sdth \lesssim 0.6$ for most spin observables (see Table \[tab:truncation\_details\_observables\]). The third factor is plotted in Fig. \[fig:truncation\_error\_stdv\_compton\] and does not exceed $0.55$ unless $\omega>260\MeV$. Therefore, even for the spin observables with the largest magnitudes, at the highest energies, the $1\sigma$ upper range of a GP will only give values about $0.3$ larger than the established maximum of $0.6$, namely about $0.9$ in total. This is close but still below $|\Sigma_i|=1$. Therefore, a majority of our test functions in the GP will not probe, let alone exceed, the strict bounds on those spin observables. Furthermore, if observables and their truncation errors vanish at $\theta = 0^\circ$ or $180^\circ$, this will make the constraint even more trivially satisfied near these regions. We are therefore confident that implementing the constraint $\Sigma_i\in[-1, 1]$ would not impact our results for $\omega\lesssim220\MeV$, and cautiously optimistic that the impact would be small even at higher energies. Though we likewise do not constrain the cross-section to be non-negative, we are confident that within our constrained angle range, corrections are highly unlikely to be large enough to for this to be a worry. According to Fig. 4 in Ref. [@Griesshammer:2017txw], the cross section is small ($<10\,\mathrm{nb/sr}$) in a narrow region at forward angles around $\omega_\pi$. Figure \[fig:truncation\_error\_stdv\_compton\] shows that the expansion parameter is small, and Fig. \[fig:coefficients\_dsg\_slices\] shows that the coefficients $c_i$ are natural-sized. Therefore, the GP corrections are highly unlikely to exceed the size of the predicted cross section and create negative (unphysical) values. [SlScScScSc]{} & &\ & $\Delta y = 0$ & $\Delta y' = 0$ & $\Delta y = 0$ & $\Delta y' = 0$\ $\diffcs$ & — & 0, 180 & 0 & 0\ $\Sigma_{1x}$ & 0, 180 & — & $\dagger$ & $\dagger$\ $\Sigma_{1z}$ & 0, 180 & 0, 180 & $\dagger$ & $\dagger$\ $\Sigma_{2x}$ & 0, 180 & — & 0 & 0\ $\Sigma_{2z}$ & — & 0, 180 & 0 & 0\ $\Sigma_{3}$ & 0, 180 & 0, 180 & 0 & 0\ $\Sigma_{y}$ & 0, 180 & — & $\dagger$ & $\dagger$\ $\Sigma_{3y}$ & 0, 180 & — & $\dagger$ & $\dagger$\ $\Sigma_{3y'}$ & 0, 180 & — & $\dagger$ & $\dagger$\ $\Sigma_{1x'}$ & 0, 180 & 0 & $\dagger$ & $\dagger$\ $\Sigma_{1z'}$ & 0, 180 & 180 & $\dagger$ & $\dagger$\ $\Sigma_{2x'}$ & 180 & 0 & 0 & 0\ $\Sigma_{2z'}$ & 0 & 180 & 0 & 0\ ![A component of the standard deviation due to  uncertainty at $\NkLO{4}^+$, see Eq. . The factor of $\cbar$ is unique to each observable, and is not included. See Table \[tab:truncation\_details\_observables\]. []{data-label="fig:truncation_error_stdv_compton"}](truncation_error_stdv) ![Coefficients for the differential cross section $\diffcs$.[]{data-label="fig:coefficients_dsg_slices"}](coeffs_slices_dsg){width="49.50000%"} {width="49.50000%"} {width="49.50000%"} {width="49.50000%"} {width="49.50000%"} {width="49.50000%"} {width="49.50000%"} {width="49.50000%"} {width="49.50000%"} {width="49.50000%"} {width="49.50000%"} {width="49.50000%"} {width="49.50000%"} Supplemental Material {#sec:extra_compton_figures} ===================== Proton Observables with Other Experimental Precision ---------------------------------------------------- ![Comparison of the shrinkage power of the optimal designs for each precision level. Note that the optimal design likely differs between precision levels. []{data-label="fig:shrinkage_per_level"}](fvr_per_precision_level_subset-all_nucleon-proton) Figure \[fig:shrinkage\_per\_level\] compares the maximal information gain in each observable for “standard”, “doable” and “aspirational” experiments. Not surprisingly, data with aspirational experimental error bars are far superior to those with only standard ones. If theory errors were absent, one would naively assume the information gain of the scenarios to scale roughly like $1/\sqrt{\Delta\Sigma_i}$. This appears to be largely fulfilled, except for $\Sigma_{2x^\prime}$ and, less noticeably, $\Sigma_{3y^\prime}$. Figures \[fig:utility\_grid\_subset\_polarizabilities\_observable\_set\_1\_doable\_no-truncation\] and \[fig:utility\_grid\_subset\_polarizabilities\_observable\_set\_2\_doable\_no-truncation\] are the analogs of Figs. \[fig:utility\_grid\_subset\_polarizabilities\_observable\_set\_1\] and \[fig:utility\_grid\_subset\_polarizabilities\_observable\_set\_2\] in the main text, but this time without any  truncation error included. Figures \[fig:utility\_grid\_subset\_polarizabilities\_observable\_set\_1\_standard\], \[fig:utility\_grid\_subset\_polarizabilities\_observable\_set\_2\_standard\], and \[fig:shrinkage\_per\_subset\_standard\] are the analogs of Figs. \[fig:utility\_grid\_subset\_polarizabilities\_observable\_set\_1\] and \[fig:utility\_grid\_subset\_polarizabilities\_observable\_set\_2\], but for the “standard” level of precision, rather than the “doable” one employed for results in the main text. Figures \[fig:utility\_grid\_subset\_polarizabilities\_observable\_set\_1\_aspirational\], \[fig:utility\_grid\_subset\_polarizabilities\_observable\_set\_2\_aspirational\], and \[fig:shrinkage\_per\_subset\_aspirational\] show the corresponding results for the “aspirational” precision level. {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} Neutron Observables ------------------- Here we show the corresponding results for the neutron observables. Because such experiments are difficult, only the “standard” level of precision is used (see Table \[tab:experimental\_precision\_levels\]). Even this is likely optimistic for such measurements, as discussed in the main text. {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} [^1]: In the  we are using, the lowest-lying nucleonic resonance, the $\Delta(1232)$, is retained as an explicit degree of freedom. [^2]: In that regime, the relation between the expansion used here and the notation of Refs. [@Pascalutsa:2002pi; @McGovern:2012ew; @Griesshammer:2017txw; @Griesshammer:2015ahu] is $Q^n=e^2\delta^n$; the simpler symbol leads to more compact formulae later. The LO defined in this presentation corresponds to performing the  power counting on the structure part of the nucleon Compton amplitude, [*i.e.*]{}, what remains after the (relativistic) nucleon and pion Born terms are subtracted. [^3]: In Compton scattering, these include the six nucleon polarizabilities in the combinations defined in Sec. \[sec:basic-compton-facts\]: $\alpha_{E1}\pm\beta_{M1}$ and $\gamma_0,\,\gamma_\pi,\,\gamma_{E-},\,\gamma_{M-}$. [^4]: \[note:stat\_notation\_compton\] The notation $z \sim \dots$ in Eq.  is statistical shorthand for “$z$ is distributed as $\dots$,” and $z\given g$ is read as “$z$ given $g$.” Furthermore $\GP[m(x), \kernel(x,x';\param)]$ denotes a GP with mean function $m(x)$ and covariance function $\kernel(x,x';\param)$. The hyperparameters $\param$ of the GP are often tuned to data. [^5]: Interestingly, the spikes disappear when considering the $c_n$ for rate-differences $\diffcs \times \Sigma_i$, but a Gaussian uncertainty in these and in $\diffcs$ does not lead to a simple Gaussian uncertainty in $\Sigma_i$. Further study of the convergence patterns of $\Sigma_i$ near $\omega_\pi$ is needed. [^6]: A single *design* $\design$ in this work is specified by an observable and a set of kinematic points at which to measure it, and possibly the experimental noise levels. The space $D$ is the set of all considered experiments over which the utility is optimized, [*e.g.*]{}, all possible 5-angle measurements at a given energy. [^7]: The notation $\normal(\vec{\mu}_0, V_0)$ denotes a Gaussian with mean $\vec{\mu}_0$ and covariance $V_0$. Also, see footnote \[note:stat\_notation\_compton\].

--- abstract: 'Starburst regions with multiple powerful winds of young massive stars and supernova remnants (SNRs) are favorable sites for high-energy cosmic ray (CR) acceleration. A supernova (SN) shock colliding with a fast wind from a compact cluster of young stars allows the acceleration of protons to energies well above the standard limits of diffusive shock acceleration in an isolated SN. The proton spectrum in such a wind-SN [pevatron]{} accelerator is hard with a large flux in the high-energy-end of the spectrum producing copious [$\gamma$-rays]{} and neutrinos in inelastic nuclear collisions. We argue that SN shocks in the Westerlund 1 () cluster in the Milky Way may accelerate protons to $\gsim 40$PeV. Once accelerated, these CRs will diffuse into surrounding dense clouds and produce neutrinos with fluxes sufficient to explain a fraction of the events detected by IceCube from the inner Galaxy.' author: - | A.M.Bykov$^{1,2}$[^1], D.C.Ellison$^{3}$[^2], P.E.Gladilin$^{1}$[^3] and S.M.Osipov$^{1}$[^4]\ $^{1}$Ioffe Institute of the Russian Academy of Sciences, Saint-Petersburg, Russia,\ $^{2}$Saint-Petersburg State Polytechnical University, Saint-Petersburg, Russia,\ $^{3}$North Carolina State University, Department of Physics, Raleigh, NC 27695-8202, USA bibliography: - 'Biblio\_mn.bib' title: Ultrahard spectra of PeV neutrinos from supernovae in compact star clusters --- \[firstpage\] neutrinos — acceleration of particles — ISM: cosmic rays — ISM: supernova remnants — magnetohydrodynamics (MHD) — shock waves Introduction ============ Gamma-ray observations with both space- and ground-based telescopes have shown that supernova remnants are the main sources of CRs at least up to 100 TeV. Galactic sources of PeV CRs, however, are still to be identified [see e.g. @blandford14; @amato14] although it has been argued that type IIb supernovae, a subclass which comprise about 3% of the observed core collapse SNe, may be able to produce CRs with energies beyond 100 PeV [e.g., @ptu10]. Cosmic neutrino observations are well suited for identifying galactic [pevatrons]{}[see e.g. @HH02; @aharonian04; @Becker08; @pevatrons14]. The (IceCube) has detected 37 neutrino events above the expected atmospheric neutrino background in a 988-day sample [@Aartsen14], with three PeV neutrinos, $1.041^ {+132}_{-144}$ PeV, $1.141^{+143}_{-133}$ PeV and $2.004^{+236}_{-262}$ PeV, being the most energetic neutrino events in history. While significant spatial or time clustering of the events has not yet been reported, a possible association of some events with galactic center sources was proposed [e.g., @Razzaque13]. The [*ANTARES*]{} neutrino telescope [@ANTARES_GC_14], using six years of data collected near the galactic centre, reported 90% confidence level upper limits on the muon neutrino flux to be between 3.5 and 5.1 $\times 10^{-8}$ GeV $\flux4$, depending on the exact location of the source. They excluded a single point source as the origin of 7 neutrinos observed by IceCube in the vicinity of galactic centre. However, an extended source of a few degrees is not excluded. Since the most likely high-energy neutrino producing mechanisms are the inelastic $p-$nuclei and $p-\gamma$ collisions of protons, where the reaction kinematics result in the energy of the neutrinos to be $\sim 0.05$ that of the protons, the energy of the parent protons should exceed $4 \cdot10^{16}$ eV to explain the IceCube observations. The [$\gamma$-rays]{} produced in these reactions have $\sim 0.1$ of the proton energy [e.g., @halzen13]. Furthermore, the proton accelerators must be very efficient to produce the high-energy neutrino flux of $\nu F_{\nu}\approx 10^{-8}\enfnu$ per flavor in the 0.1-1PeV range detected by IceCube. Neutrinos from photo-meson $p-\gamma$ interactions in compact particle accelerators, like the cores of active galactic nuclei (AGN) [e.g., @stecker13] and [$\gamma$-ray]{} bursts (GRBs) [e.g., @waxman97; @MW01], along with other models [e.g., @ahlers2013; @Fox13; @Kashiyamaea13; @He2013; @muraseea13; @2014PhRvD..89h3004L; @CRs_mergers_ApJ14; @starburst_neutrinos14; @neutrino_starburst14], have been proposed to explain the origin of these first-ever IceCube neutrinos. It has been suggested by @neronovea14 that IceCube neutrinos and Fermi/LAT [$\gamma$-rays]{} are both produced in interactions of CRs which have a hard spectrum with a power-law index harder than 2.4 and a cut-off above $\sim 10$PeV. This assumes the CRs are interacting with the interstellar medium in the Norma arm and/or in the galactic bar. The role of isolated SNRs in producing PeV CRs is uncertain. The [$\gamma$-ray]{} spectra from most shell-type galactic SNRs observed by Cherenkov telescopes show a spectral cutoff well below PeV energies [see e.g. @acero15]. While future [*Cherenkov Telescope Array*]{} observations may be needed to confirm these results [see e.g. @fa13], a population of [pevatrons]{} is needed to explain the observed spectrum of galactic CRs at and above the spectral knee region. Supernovae with transrelativistic shocks [@budnik08; @Chakraborti2011] and type IIb supernovae [@ptu10] were proposed to accelerate CRs above PeV energies but the statistics of these potential sources remain to be established. Here, we present a model of a galactic [pevatron]{} which produces hard CR spectra with a high efficiency of conversion of SN kinetic energy into the highest energy CRs. We further argue that this PeV CR source has the properties required to explain a number of the IceCube neutrinos detected from the direction of the inner Galaxy. Core-collapse supernovae in associations of massive OB stars are certain to produce a fraction of galactic CRs, as demonstrated by isotopic measurements by the [*Advanced Composition Explorer*]{} [see e.g., @binns_SB07]. Active star-forming regions may comprise extended associations of massive young stars like Cyg OB2 [see e.g., @cygOB2_mnras14], as well as compact dense clusters of young massive stars like Westerlund 1 (). These extended and compact cluster types are distinct in both spatial and temporal scales [e.g., @clust_assoc_mnras11]. Both type of clusters of massive stars are expected to be efficient PeV CR accelerators in starburst and normal galaxies [see @b14 for a review]. The compact massive cluster , with an estimated age of $3.5\!-\!5$Myr, contains more than 50 post main sequence stars [e.g., @Clark_Wd1_AA05], including at least 24 Wolf-Rayet (WR) stars of both flavours (representing about 8% of the observed galactic population of WR stars) in about a parsec-scale core [e.g., @Clark_Wd1_AA05; @Wd1_WR_stars_06]. These massive stars have strong individual winds with an estimated total cluster kinetic power exceeding 10$^{38} \ergs$. In the small, dense cluster core these individual winds should combine and drive a fast cluster-scale wind by the mechanisms studied by @chev_clegg85 and @stevens03. The magnetar CXOU J1647-45, discovered by @munoea06 using high resolution [*Chandra*]{} X-ray observations, has been associated with . This magnetar was likely produced about 10,000 years ago by a supernova with a progenitor star of mass $\geq$ 40 [@munoea06; @mereghetti08] and remains the only direct evidence of supernova activity in . In our [pevatron]{} model, a SN blast wave collides with the termination shock of a strong wind generated by the collective action of many massive stars in a compact cluster. Both shocks are assumed to propagate in a homogeneous upstream plasma. We show that proton energies well above a PeV may be produced with a hard spectrum where the CR spectrum at PeV energies is most sensitive to the shock speeds and amplification of magnetic turbulence associated with CR driven instabilities. The model provides a high efficiency of conversion of SN shock–cluster wind ram pressure into PeV CRs which is needed to explain at least some of the IceCube neutrino events. We don’t consider a SN interacting with its own wind since individual stellar winds will be unimportant in the compact cluster environment where dozens of massive stars are located within a few parsecs and the wind cavity is smoothed out on this scale. We show that diffusive shock acceleration (DSA) in systems with colliding shock flows (CSFs) can provide maximum particle energies, CR fluxes, and energy conversion efficiencies well above those produced in an isolated SNR shock of the same velocity [@MNRAS_BGO13]. While CSFs are expected to occur in colliding stellar winds, the most powerful events should happen when a supernova shock impacts the extended fast wind of a nearby young massive star cluster. The important features of shock acceleration in are: (i) the production of a piece-wise power-law particle distribution with a very hard spectrum of confined particles at the high-energy end just before a break, and (ii) an increase in the maximum energy of the accelerated particles, and the acceleration efficiency, compared to that obtained with DSA at an isolated SNR shock of the same speed. These two properties imply that a substantial fraction of the flow ram pressure is converted into relativistic particle pressure. The high-energy CRs, therefore, must modify the dynamics of the CSF system. To model the spectra of accelerated particles in CSFs, a [nonlinear]{}, time-dependent model was constructed in @MNRAS_BGO13. The maximum energy and absolute fluxes of the accelerated particles, both inside the CSFs, and those escaping the acceleration site, depend on the shock velocities, the number densities, and the magnetic fields in the flows. We show below that sub-PeV and PeV neutrinos from colliding shocks in galactic and extragalactic compact clusters of young stars with reasonable parameters, such as those expected in the  compact cluster, can explain a fraction of the IceCube neutrino events. ![Colliding flow geometry where the stellar wind (left) was approximated from the analytic model of [@wilkin96], while the SNR shock was assumed to be spherical. The star, or cluster center, approaches the SN explosion center with speed $V_0$. The spectrum of the Fermi accelerated CRs was derived taking into account effects of the flow velocity projection on the curved shock surfaces. []{data-label="fig:3D"}](f1.pdf){width="3.4in"} Particle acceleration in colliding shock flows {#sec:CSF} ============================================== In order to model the proton acceleration in a compact stellar cluster, we used the [nonlinear]{}, time-dependent model of CSFs presented in @MNRAS_BGO13. The main modification is that here we allow for different parameters for the wind termination shock and the SNR blast wave. We have also introduced an approximation to account for the shape of the shock (as illustrated in Fig. \[fig:3D\]) using an analytic expression by @wilkin96. Other approximations, such as the Bohm diffusion in the shock vicinity, and a parametrization of the magnetic field amplification (MFA) due to CR-driven instabilities, are the same as in @MNRAS_BGO13. The reader is referred to that paper for full details. Our results for neutrino and [$\gamma$-ray]{} production in , given in Section \[sec:WestI\], use our [nonlinear]{}, time-dependent model. However, to describe the general characteristics of , we shall start in this section with simple analytic estimates from a linear model in a plain-parallel case [e.g., @bgo11]. Considering a SNR expanding in a compact OB-association; at some expansion phase the distance $L$ between the SNR blast wave and a stellar wind shock is less than the mean free path of the highest energy CRs in the SNR shock precursor. At this point, the CR distribution function around the two shocks, indicated by $i = 1,2$, can be approximated as $$\label{SolveDC} \begin{array}{l} f_i \left( {x,p,t} \right)= A p^{-3}\exp \left( {-\frac{u_i }{D_i }\left| x \right|} \right)\times \\ \times H\left( {p-p_0 } \right)H\left( {t- {t_\mathrm{acc}}} \right) \ , \end{array}$$ where the CR acceleration time is $$\label{TauAc} {t_\mathrm{acc}}=\int\limits_{p_0 }^p {\frac{3}{\left( {u_1 +u_2 } \right)}\left( {\frac{D_1 }{u_1 }+\frac{D_2 }{u_2 }} \right)} \frac{dp}{p}.$$ It is important to note that, besides only applying to high-energy particles with mean free paths larger than $L$, these equations are qualitatively different from those of [test-particle]{} DSA in an isolated shock: the spectrum $f_i$ below the exponential break is harder and the acceleration time is shorter. Our model assumes a high level of CR-driven magnetic instabilities and Bohm diffusion for CRs in the close vicinity of the shocks [see, e.g., @bell04; @schureea12; @beov14]. However, once high-energy particles obtain mean free paths on the order or larger than the distance between the shocks, $L$, scattering will become much weaker. A specific feature of the simulation is that the highest energy CRs, with ${p_\mathrm{max}}\geq p \gsim p_{\star}$, propagate with little scattering. Despite the weak scattering between the shocks, their momenta are still nearly isotropic since they scatter for long periods in the extended regions downstream from the shocks. Here, $p_{\star}$ is the momentum such that the proton mean free path $\Lambda(p_{\star}) \gsim L$. Since the particle distributions are nearly isotropic even for high-energy particles with $\Lambda(p) \gsim L$, the kinetic equation reduces to the so-called telegrapher equation [e.g., @earl74] and this allows a smooth transition between the diffusive and the scatter-free propagation regimes. The time-dependent nature of our simulation means that protons escape the accelerator at different stages of the system evolution (i.e., as the SNR blast wave approaches the stellar wind termination shock) producing pions and neutrinos with varying hardness and maximum energy and these effects are included in our results. Of course, the complex evolution of the source and some unknown details of the mass distribution in the outer ISM region (e.g., the presence of dense shells or clouds) will also influence the results. However, we believe the general properties of our simulation are robust. Assuming Bohm diffusion with $D_1(p) =D_2(p) = D(p) = cR_g(p)/3$, due to CR-driven, amplified magnetic instabilities in the CR accelerator, one obtains $$\label{TauAc1} {t_\mathrm{acc}}\approx \frac{c R_g(p)}{u_s\,u_w},$$ where $u_1=u_s$ is the SNR shock velocity, $u_2=u_w$ is the stellar wind speed, and $R_g(p)$ is the momentum dependent proton gyroradius. Then, using the scaling $B \approx \sqrt{4\pi \eta_b \rho}\,u_1$ for the amplified magnetic field, the acceleration time can be estimated as ${t_\mathrm{acc}}\approx 2\,\cdot10^{10}\, \epsilon_{\rm PeV}\, (\eta_b n)^{-0.5}\,u_{s3}^{-2}\,u_{w3}^{-1}$ s. In the above expressions, $\rho=m_p n$, $\eta_b$ is the acceleration efficiency, i.e., the fraction of ram kinetic energy in the plasma flows converted into accelerated particles, the energy is in PeV, and the speeds are in units of $10^3$[km s$^{-1}$]{}. While the ejecta speed in the free expansion SNR phase can have a wide range of values, we use a mean ejecta speed of $\sim 10^4 (M/{\mbox{$M_{\odot}\;$}})^{-1/2}E_{51}^{1/2}$[km s$^{-1}$]{} and take the mean duration of the free expansion phase to be $\sim 200 (M/{\mbox{$M_{\odot}\;$}})^{5/6} n^{-1/2} E_{51}^{-1/2}$yr. In the specific case of a young SN shock propagating through the winds of massive stars in the compact cluster , where $n \sim 0.6$[cm$^{-3}$]{}  [see @munoea06], and assuming $\eta_b \sim$ 0.1, we find ${t_\mathrm{acc}}\approx$ 400 yr for a proton accelerated to $\epsilon_{\rm PeV}\sim$ 40, when $u_{s3} \sim 10$ and $u_{w3}\sim 3$. The high SN shock velocity $u_{s3} \sim 10$ is expected in the free expansion stage if the fast ejecta mass is about one solar mass. Therefore, for SNRs with ages less than ${t_\mathrm{SNR}}\sim 400$yr, one can get ${t_\mathrm{acc}}< {t_\mathrm{SNR}}$ for 40 PeV protons with standard parameters. In this case, at ${t_\mathrm{SNR}}\sim 400$yr, the SNR radius is $\sim 3-4$pc. Furthermore, the hard spectrum expected from CSFs puts most of the energy into the highest energy protons and one can estimate the power in the highest energy neutrinos produced in inelastic [$p\!-\!p$]{}–collisions by the decay of charged pions ($\pi^{\pm}\to e \nu_e\,\nu_\mu\,\bar\nu_\mu $) as $$\begin{aligned} &&L_{\nu} \approx 8\times 10^{33}\, \left( \frac{f_{\nu}}{0.15}\right) \left(\frac{\eta_p}{0.1}\right)\, \left(\frac{n}{1 \cmc}\right)^2 \times \nonumber \\ && \left(\frac{S}{10^{38} \cm2}\right)\left(\frac{u_s}{5,000\,{\mathrm{km\,s}^{-1}}}\right)^3 \left(\frac{\tau_c}{10^{10}\, \rm{s} }\right)\, \ergs , \label{eq:Lnu}\end{aligned}$$ where $f_{\nu}$ is the fraction of energy in the inelastic [$p\!-\!p$]{}–collisions which is deposited in the high-energy neutrinos, $S$ is the cross section of the colliding flows, and $\tau_c$ is the confinement time for protons in the emission region where the target density is $r_s \geq 4$ times the ambient density $n$ due to the shock compression, $r_s$. In Eq. (\[eq:Lnu\]), we take the inelastic [$p\!-\!p$]{} collision cross section to be $\sim 70$mb above a proton energy of 10 PeV. Then the proton cooling time, $n\,t_{pp}$, can be estimated as $n\,t_{pp} \approx 2.5 \cdot 10^{14}$ s assuming that two inelastic collisions are needed to convert most of the proton energy into secondaries. The fraction of the proton energy deposited into neutrinos of all types, $f_{\nu} \sim$ 0.15, was derived using both the analytical parameterizations for the inelastic [$p\!-\!p$]{}–collisions presented by [@kelner2006; @cross_sect_PRD14] and the [*GEANT4*]{} package simulations. At the energies of photons and neutrinos considered in Figs. \[fig:spectra\_src\], \[fig:gam1m\] and \[fig:NeuOnly\], the neutrino fluxes calculated with the [@kelner2006] or [@cross_sect_PRD14] cross sections differ by less than 20%. We note that the observed CR energy density and the galactic SN statistics require that $\eta_p \gsim 10$% over the lifetimes of isolated SNRs to power galactic CRs. However, the instantaneous efficiency may be much larger during early stages of the SNR evolution [@blasiAARv13]. In the case of a SNR colliding with a wind, we expect the efficiency to be even higher than in a young isolated SNR and assume it can reach $\eta_p \gsim$ 0.5 for the CSF stage lasting for a few hundred years [@MNRAS_BGO13]. Furthermore, since CSFs produce very hard spectra when ${t_\mathrm{acc}}\ll {t_\mathrm{esc}}$, i.e., $N(\gamma) \propto \gamma^{-1}$, most of the CR energy lies in the high-energy tail just below the upper break which occurs when ${t_\mathrm{acc}}$ is greater than the escape time, ${t_\mathrm{esc}}$. The essential properties of CSF acceleration in young stellar clusters are high overall acceleration efficiency, spectra harder than $N(\gamma) \propto \gamma^{-2}$, and maximum proton energies $\gsim 40$PeV achieved in a few hundred years. ![The small dashed circles (not to scale) indicate the inner CR acceleration region in the massive young cluster . The inner CR acceleration region has a radius of 3-4pc (shown in a Chandra image) and the blow-up of the inner region shows the position of the magnetar CXOU J1647-45 found by @munoea06 with a high angular resolution Chandra observation of . The outer region is 30-40pc centered around   and indicated by the large dashed circle. This is overlaid on a H.E.S.S. map of TeV emission adapted from @hessWd1. The angular resolution of IceCube is larger than the outer circle. In Fig. \[fig:Galactic\] we show the larger neutrino emitting ISM volume of radius $\sim$ 140 pc around . The volume is filled over $\sim 10^4$yr with CRs accelerated during a short CR acceleration phase $\sim$ 400 years right after the supernova explosion in which produced the magnetar. []{data-label="fig:West"}](f2.pdf){width="250"} Colliding flows and 3D geometry effects {#3D} --------------------------------------- While an exact treatment of non-linear CSF with a significant backreaction from accelerated CRs is unfeasible in 3D, we have introduced an approximation to account for effects from 3D geometry in our plane-parallel model. Instead of taking the cluster wind termination shock and the SNR blast wave as plane, we account for aspects of the curved shock surfaces at positions parameterized by the angles $\theta$ or $\phi$ in Fig. \[fig:3D\]. We still assume planar shocks for the DSA calculation but with varying projected velocities at distances $d(\theta)>L_{12}$ along the shock surfaces away from the symmetry axis. Here, $L_{12}$ is the time-dependent minimum distance between the colliding shocks. For each position determined by $d(\theta)$ for $0<\theta<90^0$ we calculate the non-linear particle distribution using the projected speeds $V_w \cos\theta$ and $V_\mathrm{snr} \cos\Phi$ as parallel flow speeds. That is, we assume the wind and the SNR shocks are locally plane with converging parallel flows set by the projected speeds. For this approximation, we use the bow shock wind model of @wilkin96 for $R(\theta)$ and $V_w$, assume the SNR shock is spherical, and restrict our calculations to $3$ arc surfaces: $0-30^0$, $30-60^0$, $60-90^0$ which produce accelerated particle densities with the weights $0.84$, $0.15$ and $0.01$, respectively. Then the weighted CR distribution function is used to calculate the [$\gamma$-ray]{} and neutrino emissivities both in the accelerator (see §\[sec:trap\]) and in the surrounding ISM from the CRs escaped the accelerator (see §\[sec:esc\]). ![Model predictions of [$\gamma$-rays]{}  (solid curves) and neutrinos (dashed curves) from [$p\!-\!p$]{}-interactions calculated in a CSF source of age $400$yr. The dotted curve is the inverse Compton emission from primary and secondary electrons accelerated directly in this source. The extreme upward curvature in the neutrino spectrum above $\sim 10$TeV reflects the transition from CR acceleration in the single SNR shock for low-energy particles to the more efficient acceleration for high-energy particles as they scatter back and forth between the SNR shock and the cluster wind. The data points for the H.E.S.S. source, and the five Ice Cube events explained in Fig. \[fig:NeuOnly\], are presented to illustrate how they compare to our model predictions when the source is $\sim 400$ yr old and point-like, i.e.,  about 10$^4$ years ago. The simulated [$\gamma$-ray]{} and neutrino emission at the present time from CRs that escaped the accelerator in  $\sim 10^4$ yr ago are summarized in Figs. \[fig:gam1m\] and \[fig:NeuOnly\].[]{data-label="fig:spectra_src"}](f3.pdf){width="300pt"} Neutrinos and Gamma rays from the Westerlund 1 Cluster {#sec:WestI} ====================================================== Emission from Trapped CRs {#sec:trap} ------------------------- To explain the IceCube neutrinos, we combine the [nonlinear]{}, time-dependent model of particle acceleration in CSFs with a propagation model applied to the  compact cluster (Fig. \[fig:West\]). The time-dependent simulations provide the evolving energy spectra of CR protons and electrons as the SNR shock approaches the strong wind of a nearby early-type star. For relativistic electrons/positrons we account for the energy losses due to synchrotron and inverse Compton (IC) radiation. The particle acceleration is combined with a propagation model relevant to . As described in Section \[sec:CSF\], high-energy particles will obtain a hard spectrum when their mean free path is large enough so they scatter back and forth between the two converging shocks. Lower energy particles however, will be confined to, and accelerated by a single shock and obtain a softer spectrum. This phenomenon is illustrated in Fig. 3 of @MNRAS_BGO13. It is important to note that even though the CSF acceleration is efficient and the CR population modifies the structure of the plasma flows, the spectrum of high-energy CRs scattering between the two shocks remains hard until they gain enough energy to escape from the system. The shock modification can cause ${t_\mathrm{acc}}$ to increase, and the energy where the exponential turnover in Eq. (\[SolveDC\]) starts to dominate drop, but below the turnover, the spectrum remains close to $N(\gamma) \propto \gamma^{-1}$. Thus, the CSF system will have two spectral regimes for trapped CRs: a low-energy region ($\lsimX 1$ TeV) from particles accelerated in a single SNR shock (produced both before and after the start of the two-shock acceleration period), and a high-energy, hard spectrum region ($\gsimX 1$ TeV) from particles accelerated in the converging flows. The transition between these two regimes of acceleration occurs when the energy of a particle, $E_T$, is large enough so it can easily travel between the two shocks. This transition, occurring at $\sim 20$TeV, is seen as a bend in the neutrino spectrum shown in Fig. \[fig:spectra\_src\]. This figure shows neutrinos (solid curve) and [$\gamma$-rays]{} (dashed curve) from [$p\!-\!p$]{}-decay from CR protons that are still trapped near the SNR and stellar wind, along with IC from trapped CR electrons. This is the emission expected $\sim 400$yr after the SN explosion in a region of radius $\sim 3\!-\!4$pc. Note that the [$\gamma$-ray]{} fluxes of the sources in Fig. \[fig:spectra\_src\] are presented in ${\rm erg~cm^{-2}~s^{-1}~sr^{-1}}$ in order to be compared with the observed diffuse neutrino fluxes, while the fluxes of the sources in Fig. \[fig:gam1m\] are measured in ${\rm erg~cm^{-2}~s^{-1}}$ as usual. In Fig. \[fig:West\] we have overlaid a schematic of this inner CSF region on a map of . At later times, these “trapped" CRs will escape the accelerator and diffuse beyond the inner CSF region into a much larger outer ISM region where they may encounter dense clouds producing neutrinos and [$\gamma$-rays]{} for an extended period of time, as we discuss in Section \[sec:esc\]. For clarity, the regions are not drawn to scale in Fig. \[fig:West\]. To derive the gamma-emissivity of PeV CRs we accounted for the Breit-Wheeler effect of pair production by energetic photons interacting with the interstellar radiation field as well as the extragalactic light background [see e.g. @aharonian04; @Dwekea13]. This interaction leads to a significant suppression of the [$\gamma$-ray]{} flux at the high-energy end of the spectrum for distant ($\geq10$ kpc) sources. For the emission from , with an estimated distance of $\sim$ 4 kpc, this effect leads to a relatively small suppression of the [$\gamma$-ray]{} flux at $1-10$ PeV, as can be seen in Figs. \[fig:spectra\_src\] and \[fig:gam1m\]. In addition to the [$\gamma$-rays]{}  produced by pion decay, we have also included [$\gamma$-rays]{} produced by the IC radiation of the secondary $e^{\pm}$ pairs produced [*in situ*]{} by the same [$p\!-\!p$]{}-interactions (see the IC curve in Fig. \[fig:gam1m\]). In this calculation, we accounted for the pair energy losses in a mean magnetic field of magnitude 10 $\mu$G in the extended cloud of number density 25$\cmc$ [see e.g. @strongea14]. Emission from Escaping CRs {#sec:esc} -------------------------- The initial acceleration stage lasts a few hundred years after the SN explosion producing high-energy CRs that escape the CSF system and diffuse through the ambient ISM. However, the estimated age of the supernova which produced the magnetar CXOU J1647-45 in  is $\sim 10^4$yr [@NS_WesterlundI_muno06; @mereghetti08]. If the CSF acceleration occurred $\sim 10^4$yrs ago, these CRs will have produced pions over a much longer time span as the TeV-PeV particles diffuse away, fill a region of about 140 pc radius, and interact with the ambient ISM (the outer ISM region indicated in Fig. \[fig:West\] is about 30 pc). In Figs. \[fig:gam1m\] and \[fig:NeuOnly\] we show simulated [$\gamma$-ray]{} and neutrino spectra produced by the CRs that escaped from the accelerator and diffused into the surrounding cloudy medium over $10^4$yr. At this point the acceleration responsible for the emission in Fig. \[fig:spectra\_src\] has ceased long ago. The diffusion model used to propagate the CRs has three regions. Within the accelerator of 3-4 pc radius we assume a Bohm diffusion coefficient $D_B = 3 \times10^{27} E_\mathrm{PeV} \diff$. Outside of the dense cloud region we use an ISM value $D_B = 3 \times10^{29} E^{0.33}_\mathrm{PeV} \diff$, which is consistent with the standard models of CR propagation in the Galaxy [see e.g. @Strong2007]. Between these two regions we assume a transition coefficient $D_\mathrm{tran} = D_B(R/3 pc)^2$, where $R$ is the distance from the core. We assume a mean cloud density $\sim 25$[cm$^{-3}$]{}, cloud radius $\sim 30$pc, and have set, at 1 PeV, $D_B = D_\mathrm{ISM}$ at $R = $ 30 pc. A cloud of radius 30 pc with mean density 25 $\cmc$ would have a mass $\sim 10^5 M_{\odot}$. Outside the cloud we assumed an ISM density of $\sim 1$[cm$^{-3}$]{}. We note that the densities we assume for the clouds match available [$\gamma$-ray]{} observations of  [@hessWd1; @ohmea13]. The IceCube data points in Fig. \[fig:NeuOnly\] show the neutrino energy flux consistent with the position of  considering the angular resolution of the instrument. The CSF model applied to  with reasonable parameters can explain a subset of the observed Ice Cube neutrinos. ![Gamma-ray emission from inelastic [$p\!-\!p$]{}-interactions in the CSF source at $\sim 10^4$yr after the SN explosion when CR protons produced in the short-lived accelerator have propagated into a nearby cloud of $\sim 30$pc size. The magnetic field amplified by the CR-driven instabilities in the vicinity of the fast shock in the CSF accelerator were parameterized as 0.8 mG (c), 0.9 mG (b), and 1 mG (a), all below 10% of the ram pressure. The IC curve is inverse Compton emission from the secondary electrons produced by the inelastic [$p\!-\!p$]{}-interactions in the cloud. Only the [$\gamma$-rays]{} from the H.E.S.S. field of view are included. The gas number density of the nearby cloud is 25[cm$^{-3}$]{}, except for the light-weight solid curve where it is 30[cm$^{-3}$]{} with $B=0.9$mG.[]{data-label="fig:gam1m"}](f4.pdf){width="300pt"} ![Neutrino emission from an extended ($\sim$ 140 pc radius) source $\sim 10^4$yr after the SN explosion when CRs produced in the short-lived accelerator have propagated into the surrounding material. The amplified magnetic fields are 0.8 mG (c), 0.9 mG (b), and 1 mG (a). We note that the [$\gamma$-rays]{} in Figs. \[fig:gam1m\] and the neutrinos here originate from different volumes: only the [$\gamma$-rays]{} from the H.E.S.S. field of view are shown in Fig. \[fig:gam1m\], while the neutrinos are from a larger region of radius 140 pc. The neutrino data points (1$\sigma$ energy flux error bars) are a subset from all 37 IceCube events [@Aartsen14]). These five events are within 2-$\sigma$ contours from  based on Fig. \[fig:Galactic\]. Two PeV events (14, “Bert") and 35 (“Big Bird"), as well as three sub-PeV (2, 15, 25) events, are included in the subset. Note that the sub-PeV event 25 has a very large position uncertainty in Fig. \[fig:Galactic\] and have to be considered with some care.[]{data-label="fig:NeuOnly"}](f5.pdf){width="280pt"} Neutrino PeV and sub-PeV events =============================== The median angular error values for the IceCube neutrino events are given in the Supplementary Material for @Aartsen14. Using these median angular errors, and assuming Gaussian statistics [see @Aartsen14a], we have produced a sky map with $2\!-\!\sigma$ contours (corresponding to about $86\%$ confidence in 2D Gaussian statistics) for the neutrino events in the vicinity of  (see Fig. \[fig:Galactic\]). As seen in this map, a source of neutrinos with a radius of $\sim 140$pc around  (black circle) can be associated with five neutrino events, including two PeV events. These are $2$, $14$ (PeV event “Bert"), $15$, $25$ and $35$ (PeV event “Big Bird"). Based on this sky map, we compared our model neutrino spectra from the  source with the fluxes in five IceCube energy bins corresponding to the five neutrinos within $2\!-\!\sigma$ of the source. This is shown in the Fig. \[fig:NeuOnly\]. A more precise comparison will require both more sophisticated models of the cloud distribution within a few hundred parsecs of  and a more accurate determination of the event positions. ![Map showing 2-$\sigma$ contours for a subset of IceCube events associated with the inner Galaxy as determined from 2D Gaussian statistics and the median angular errors and positions given by [@Aartsen14a]. Two PeV events (14, “Bert") and 35 (“Big Bird") as well as three sub-PeV (2, 15, 25) events are within 2-$\sigma$ from . []{data-label="fig:Galactic"}](f6.pdf){width="280pt"} Discussion ========== Explaining the origin of the recently detected PeV neutrinos by IceCube is a fundamental challenge for models of particle acceleration. The observations imply a source that can produce substantial fluxes of protons with energies considerably higher than those expected from isolated SNRs. Since isolated SNRs remain the most likely source of CRs below a few PeV, the neutrino source must produce high-energy protons without conflicting with the observed properties of galactic CRs. The underlying protons producing the neutrinos in the CSF model described above have a hard spectrum, and are few in number, avoiding any conflict with low-energy CR population measurements. The CSFs may contribute to the high-energy end of the CR population produced by isolated SNRs and superbubbles [see e.g. @binns_SB07] and, we believe the strong plasma flows in compact clusters of young stars, such as , contain the energy and specific properties needed to explain a significant fraction of the IceCube neutrinos. Compact clusters contain massive stars with strong winds and recent SN activity. It is inevitable that occasions will occur when a SN blast wave collides with the termination shock from the strong wind of a nearby massive star, or with an extended cluster wind from several massive stars. We have developed a model of the Fermi acceleration expected from such colliding shock flows and, using realistic parameters, obtained simultaneous fits to the H.E.S.S. [$\gamma$-ray]{} observations (Fig. \[fig:gam1m\]) and the fraction of IceCube neutrinos expected from  (Fig. \[fig:NeuOnly\]). Multi-wavelength signatures of CSF scenario ------------------------------------------- A unique property of our CSF model is that the acceleration, while producing hard proton spectra to multi-PeV energies with high efficiencies, only lasts a small fraction of the SNR lifetime, just the time when the SNR is colliding with the nearby stellar wind. Because of the hard spectra of accelerated CRs most of the energy is in the highest energy regime. In Fig. \[fig:spectra\_src\] we show the predictions for high-energy photon and neutrino emission of the source at the end of the acceleration stage (which was supposedly $\sim$ 10$^4$ years ago). The synchrotron radio emission then is estimated to be $\gsim$ 10Jy at 2.2 GHz, a value well above the current level. This is consistent with our scenario where the brief CR acceleration stage ended $\sim 10^4$ years ago. Indeed, the total radio flux from  as measured with the [*Australia Telescope Compact Array*]{} (ATCA) interferometer by @Wd1_radio_AA10 is 422, 461, 523, and 669 mJy at 8.6, 4.8, 2.2, and 1.4 GHz, respectively, and after subtracting the radio emission from stellar sources, they derived diffuse emission fluxes of 307, 351, and 426 mJy at 8.6, 4.8, and 2.2 GHz. Colliding winds in massive binary star systems were proposed by @eu93 to accelerate relativistic particles and produce non-thermal radio and GeV regime [$\gamma$-ray]{} emission. Some of the stellar radio sources detected in  with the ATCA by @Wd1_radio_AA10 exhibited composite spectra of both non-thermal and thermal emission potentially indicating particle acceleration in colliding wind binaries. ![The spectral energy distribution of the synchrotron (dashed lines), inverse Compton (dot-dashed line) emission from secondary electrons and positrons, as well as photons produced by pion-decay (solid line) from the inelastic [$p\!-\!p$]{}-interactions in the nearby clouds which are H.E.S.S. sources shown in Fig. \[fig:West\]. The CRs were accelerated at  and diffused into the clouds. The cloud gas number density is 25[cm$^{-3}$]{}  with magnetic field $B=10$$\mu$G. The [$\gamma$-rays]{} detected by H.E.S.S. are indicated in the figure. The upper dashed synchrotron curve is the result with no energy threshold for CR protons to penetrade into the cloud. The lower synchrotron curves correspond to threshold values 10 GeV and 20 GeV respectively. The threshold values are low enough to not influence the pion-decay emission.[]{data-label="fig:SED"}](f7.pdf){width="280pt"} In contrast to the binary wind model, CRs in CSFs are generated in the violent environment of a SN blast wave colliding with a cluster wind. While the acceleration stage is brief, CRs will escape the source and interact with the nearby ISM clouds for long periods ($\gsimX 10^4$yr). During this time pions will be produced in inelastic [$p\!-\!p$]{}-interactions resulting in relativistic secondary electrons and positrons from $\pi^{\pm}$ decay and photons from $\pi^{0}$ decay. The hard CR spectra from CSFs will result in prominent peaks in the spectral energy distribution produced by synchrotron, IC, and pion-decay emission, as shown in Fig. \[fig:SED\]. For an extended cloud of size $\sim 30$pc and number density $\sim 25$[cm$^{-3}$]{}, the peaks correspond to keV, TeV, and PeV energy bands. For Fig. \[fig:SED\] we assumed the cloud magnetic field to be $B=10$$\mu$G consistent with Zeeman splitting measurements of a number of molecular clouds compiled recently by @strongea14. Since the penetration of CR nuclei into the cloud may be reduced at low energies [e.g., @cv78; @protheroe08], we show in Fig. \[fig:SED\] spectra for the case with no proton penetration energy threshold (upper dashed curve) and for proton threshold energies $E_{\ast} = 10$ and $20$ GeV. The radio fluxes are sensitive to $E_{\ast}$ but the X-ray synchrotron fluxes corresponding to the peak of the synchrotron SED are not. The cloud associated with the H.E.S.S. source in this model is a diffuse, low-surface-brightness ($\lsimX 0.1\,\mu$Jy arcsec$^{-2}$ at 1.4 GHz), flat-spectrum, synchrotron source of polarized radio emission with bright spots of brightness $\sim\,$ 1 mJy arcsec$^{-2}$ at 1.4 GHz, corresponding to local strong enhancements of the magnetic field in dense clumps. The future [*Square Kilometre Array*]{}, with a sensitivity of $\sim\,$ 1 $\mu$Jy/beam, may allow detection of radio emission from clouds irradiated by CRs [see e.g. @strongea14]. The synchrotron peak in the CSF model is at keV X-ray energies and a source with a half-degree extension and total flux of $\sim 10^{-13}\, {\rm erg~cm^{-2}~s^{-1}}$ may be detectable with the future [*eROSITA*]{} (extended ROentgen Survey with an Imaging Telescope Array) instrument on the [*Spectrum-Roentgen-Gamma*]{} [see e.g. @SRG] and [*ASTRO-H*]{} satellites . The search for synchrotron X-ray emission from the cold clouds located near these powerful CR sources may be conducted with the next generation of X-ray sky surveys. In Fig. \[fig:gam1m\] we show simulated spectra of [$\gamma$-ray]{} emission from CRs that escaped the accelerator and diffused into the surrounding cloudy medium over $10^4$ yr. The models are compared to H.E.S.S. data for the source associated with   [@hessWd1; @ohmea13] and we have reduced the [$\gamma$-ray]{} flux to correspond to the H.E.S.S. field of view at . At $10^4$ yr, the acceleration responsible for the emission has long ceased and the emission comes only from CRs accelerated in the source and propagating through the ISM . We note that an analysis of 4.5 yr of [*Fermi-LAT*]{} data by @ohmea13 found extended emission offset from   by about 1 degree. This study concluded that acceleration of electrons in a pulsar wind nebula could provide a natural explanation of the observed GeV emission. However, @ohmea13 found that the pulsar wind nebula could not explain the TeV emission observed by [H.E.S.S.]{}. As seen in Fig. \[fig:gam1m\], the CSF model can satisfactorily explain the TeV [$\gamma$-ray]{} emission. There is an apparent excess of neutrino events, including two of the three PeV neutrinos in the IceCube map presented in [@Aartsen14], within a radian from . In Fig. \[fig:Galactic\] we show this set in a map with the positions of the events and 2-$\sigma$ contours as determined from 2D Gaussian statistics and the median angular errors of the IceCube telescope. Westerlund 1 is indicated with a contour (black circle) corresponding to a 140 pc radius region - a few degrees - where neutrinos are produced in our CSF model as escaping CRs diffuse out from the compact accelerator for $\sim 10^4$yr. The neutrino energy flux corresponding to the solid curve in Fig. \[fig:NeuOnly\] is $\sim 3.7{\!\times\!10^{-8}}$GeV$\flux4$ This is well below the 90% confidence level upper limits imposed by [*ANTARES*]{} observations [@ANTARES_GC_14] for a source of width $>0.5$ degrees at the   declination. CSFs in the starburst galaxies ------------------------------ Another issue concerns how hard spectrum PeV neutrino sources contribute to starburst galaxy radiation. @loeb_waxman06 suggested that CR interactions in starburst galaxies may efficiently produce high energy neutrinos and contribute cumulatively into the neutrino background. The population of star forming galaxies with AGNs and the starburst galaxies peaked at a redshift $z \gsim$ 1, with a wide tail of the distribution revealed by [*Herschel*]{} [@Herschel_gal13] up to $z \sim$ 4.5. These objects most likely contribute to both the isotropic diffuse [$\gamma$-ray]{} background measured by [*Fermi-LAT*]{} [@Fermi_EGB15] between 100 MeV and 820 GeV, and the diffuse flux of high-energy neutrinos measured by IceCube. Assuming that a CR spectral index for all the starburst-like galaxies is 2.1–2.2 at the high-energy part of the spectrum, @starburst_neutrinos14 were able to provide a reasonable fit to both the [*Fermi*]{} and IceCube data. Larger indices failed to explain the observed diffuse neutrino flux. That CR spectra harder than in normal galaxies like the Milky Way were required, may reflect a different population of CR sources and/or different CR propagation in the starburst galaxies. Superclusters of young massive stars are likely much more abundant in starburst galaxies compared to the Milky Way since mergers and interactions of galaxies result in abundant supercluster formation [see e.g. @Conti_book]. The hard spectra and high efficiencies from CSFs make them an attractive way to produce CRs well beyond PeV in starburst galaxies where the collective contribution from many CSFs sources might extend the CR knee to higher energies compared to the Milky Way. High resolution radio observations of the star forming galaxies M82, Arp 220, NGC 253, M31, M33 and others provide information on the magnetic field structure and leptonic CRs [@Tabatabaeiea13; @m82_radio; @pr14], and [$\gamma$-ray]{} telescopes have observed some starburst galaxies up to $\sim 10$TeV [see @acero_SFgalaxies_Sci09a; @HESS_NGC253_ApJ12a; @Lacki11; @Fermi_SF_ApJ12a and the references therein]. However, this is still well below PeV energies where CSF CRs are expected to be dominant and the [*Cherenkov Telescope Array*]{} [@CTA11] would be needed to constrain the gamma-ray spectra in PeV energy regime. More quantitative models of the CSF contribution to the diffuse [$\gamma$-ray]{} and neutrino backgrounds will require better statistics of CSF SNe in starburst regions, accurate models of CR escape from the sources, as well as realistic models of CR propagation in starburst regions. Conclusions =========== We have presented a colliding shock flow model for CR production in compact stellar clusters that efficiently produces hard CR spectra and neutrinos. Acceleration in colliding plasmas is a very efficient version of Fermi acceleration given the strong confinement of CRs in the converging flows. The mechanism is strongly nonlinear and time-dependent. We simulated the acceleration process in a simplified geometry. Protons escape the accelerator with varying hardness and maximum energy as the SN shock approaches the wind. Furthermore, there is uncertainty in details of the mass distribution in the complicated outer ISM region (e.g., dense clouds) which will influence the results. Nevertheless, we believe our simulations include enough essential physics to estimate the neutrino and [$\gamma$-ray]{} emission from the galactic cluster Westerlund 1 and we show it is a likely source for IceCube events detected from the inner galaxy. Our [$\gamma$-ray]{} predictions are consistent with H.E.S.S observations as well. While the relatively large angular uncertainty in the arrival directions of PeV neutrinos precludes an exact identification, we believe some PeV IceCube events may result from $\geq$ 10 PeV CR protons accelerated in  (see Fig. \[fig:Galactic\]). This cluster is a good candidate because it is one of the most massive clusters in the local group of galaxies and has an observed 10$^4$ yr old magnetar, allowing enough time to spread PeV CRs over a few hundred parsecs scale. Future work that is critical for determining if a galactic supercluster can explain the apparent clump of 4-5 IceCube events includes developing a more accurate model of multi-PeV CR diffusion on kpc scales. This requires a careful treatment of the matter distribution within a few degrees of  and will result in a more accurate determination of the neutrino flux, as well as radio to [$\gamma$-ray]{} emission, from CR interactions. Acknowledgments {#acknowledgments .unnumbered} =============== We thank the anonymous referee for careful reading of our paper and useful comments. A.M.B. thanks Markus Ackermann for a useful discussion. A.M.B. and D.C.E. wish to thank the International Space Science Institute in Bern where part of this work was done. D.C.E. acknowledges support from NASA grant NNX11AE03G. [^1]: E-mail: byk@astro.ioffe.ru [^2]: E-mail: don\_ellison@ncsu.edu [^3]: E-mail: peter.gladilin@gmail.com [^4]: E-mail: osm2004@mail.ru

--- author: - | Christian Szegedy\ Google Inc.\ [szegedy@google.com]{} - | Vincent Vanhoucke\ [vanhoucke@google.com]{} - | Sergey Ioffe\ [sioffe@google.com]{} - | Jonathon Shlens\ [shlens@google.com]{} - | Zbigniew Wojna\ University College London\ [zbigniewwojna@gmail.com]{} bibliography: - 'references.bib' title: Rethinking the Inception Architecture for Computer Vision ---

--- abstract: 'While deep learning makes significant achievements in Artificial Intelligence (AI), the lack of transparency has limited its broad application in various vertical domains. Explainability is not only a gateway between AI and real world, but also a powerful feature to detect flaw of the models and bias of the data. Local Interpretable Model-agnostic Explanation (LIME) is a widely-accepted technique that explains the prediction of any classifier faithfully by learning an interpretable model locally around the predicted instance. As an extension of LIME, this paper proposes an high-interpretability and high-fidelity local explanation method, known as Local Explanation using feature Dependency Sampling and Nonlinear Approximation (LEDSNA). Given an instance being explained, LEDSNA enhances interpretability by feature sampling with intrinsic dependency. Besides, LEDSNA improves the local explanation fidelity by approximating nonlinear boundary of local decision. We evaluate our method with classification tasks in both image domain and text domain. Experiments show that LEDSNA’s explanation of the back-box model achieves much better performance than original LIME in terms of interpretability and fidelity.' author: - bibliography: - 'refs.bib' title: An Extension of LIME with Improvement of Interpretability and Fidelity --- Introduction ============ In recent years, people have witnessed the fast development of Artificial Intelligence (AI) [@AI01; @AI02; @AI03]. Compared to traditional machine learning methods, deep learning has achieved superior performance in many challenging tasks. There has been an increasing interest in leveraging deep learning methods to aid decision makers in critical domains such as healthcare and criminal justice. However, because of the nested complicated structure, deep learning models remain mostly black boxes, which are extremely weak in explaining the reasoning process and prediction results. This makes it challenging for decision makers to understand and trust their functionality. Therefore, the explainability and transparency of deep learning models are essential to ensure their broad applications in various vertical domains. Recently, the development of techniques on explainability and transparency of deep learning models has recently received much attention in the research community [@EX01; @EX02; @EX03]. Among them, the post-hoc techniques for explaining black-box models in a human-understandable manner have received much attention [@POST01; @POST02; @POST03], which generate perturbed samples of a given instance in the feature space and observe the effect of these perturbed samples on the output of the black-box classifier. Due to the generality, these techniques have been used to explain neural networks and complex ensemble models in various domains ranging from medicine, law and finance [@App01] [@App02]. The most representative system in this category is LIME[@POST01]. As LIME assumes the local area of the classification boundary near the input instance is linear, it uses a linear regression model which is self-explanatory to locally represent the decision and pinpoint important features based on the regression coefficients. It is found relevant works [@Other01; @Other02; @Other03] proposed to use other models such as decision tree to approximate the target detection boundaries. There are two drawbacks in current existing local explanations such as LIME. Perturbed samples are generated from a uniform distribution, ignoring the intrinsic correlation between features. This may lead to lose much useful information to learn the local explanation models. Proper sampling operation is especially essential in natural language processing and image recognition. Moreover, most existing methods assume the decision boundary is local linearity, which may produce serious errors as in most complex networks, the local decision boundary is non-linear. In this paper, we design and develop a novel, high-interpretability and high-fidelity local explanation method to address the above challenges. First, we design a unique local sampling process which incorporate the feature clustering method to handle the feature dependency problems. Then, we adopt Support Vector Regression (SVR) with a kernel function to approximate locally nonlinear boundary. In this way, by simultaneously preserving feature dependency and local non-linearity, our method produces high-interpretability and high-fidelity explanation. For convenience, we refer to our method as LEDSNA “Local Explanation using feature Dependency Sampling and Nonlinear Approximation”. Method ====== In this section, we first introduce the two core characteristics of the local explanation method: interpretability and fidelity. Then we introduce the feature sampling with intrinsic dependency and nonlinear boundary of local decision. Finally, we present the framework of LEDSNA algorithm. An explainable model with good interpretability should be faithful to the original model, understandable to the observer, and graspable in a short time so that the end-user can make wise decisions. Local explanation method learns a model from a set of data samples which is sampled around the instance being explained. The dissimilarity between the true label and predicted label is defined as the loss function $L(f(x),g(x))$ which is a measure of how unfaithful $g(x)$ is in approximating $f(x)$. In order to ensure both local fidelity and understandability, we add regularization term to loss function: $$J(\theta)=argmin{L(f(x), g_{\theta}(x)) + \lambda\Omega(\theta)}. %J(\theta)=argmin{L(f(x), g_{\theta}(x)) + {\lVert\theta\rVert}_0}.$$ The regularisation term is a measure of complexity of the explainable model $g(x)$. The smaller the regularisation term is, the better the sparsity of model $g(x)$, which leads to better understandability. This is the general framework of LIME [@POST01]. Feature Sampling with Intrinsic Dependency ------------------------------------------ In current existing local explanations, the original sampling procedure is made on each feature independently, ignoring the intrinsic correlation between features. Proper sampling operation is essential as the independent sampling process may lead to lose much useful information to learn the local explanation models. In some cases, when most uniformly generated samples are unrealistic about the actual distribution, false information contributors lead to poorly fitting of the local explanation model. In this section, we design an unique local sampling process which incorporate the feature clustering method to activate a subset of features for better local exploration. ### Feature Dependency Sampling for Image Proper sampling operation is especially essential in natural image recognition because the visual features of natural objects exhibit a strong correlation in the spacial neighborhood. For image classification, we adopt a superpixel based interpretable representation. Each superpixel segment is the primary processing unit, which is a group of connected pixels with similar colors or gray levels. We denote $x\in{\mathbb{R}^d}$ be the original representation of an image, and binary vector $x'\in{\{{0,1}\}^{d'}}$ be its interpretable representation, which indicating the ¡®presence¡¯ or ¡®absence¡¯ of a superpixel segment. There $d$ is the number of pixels and $d'$ is the number of superpixel. For the images, especially natural images, superpixel segments often correspond to the coherent regions of visual objects, showing strong correlation in a spacial neighborhood. In order to learn the local behavior of image classifier $f$, we generate a group of perturbed samples of a given instance, $x$, by activating a subset of superpixels in $x$. Firstly, we convert the superpixel segments into an undirected graph. the superpixel segments are represented as vertices of a graph whose edges connect to only those adjacent segments. Considering a graph $G=(V, E)$, where $V$ and $E$ are the sets of vertices and undirected edges, with cardinalities $|V|=d'$ and $|E|$, a subset of $V$ can be represented by a binary vector $z'\in{\{{0,1}\}^{d'}}$, where $1$ indicates that vertice is in the subset. The perturbed sampling operation is formalized as finding the clique $C$ ($C\subseteq V$), where every two vertices are adjacent. We use the Depth-First Search (DFS) method to get the clique $C$. Some samples in the clique are shown in Fig. \[fig:2\]. Since there is a strong correlation between the adjacent superpixel image segments, the clique $C$ set construction can take into full account the various types of neighborhood correlation. ![[]{data-label="fig:1"}](figures/image){width="100.00000%"} ![[]{data-label="fig:1"}](figures/image_p){width="100.00000%"} ![[]{data-label="fig:1"}](figures/figure){width="100.00000%"} ![[]{data-label="fig:2"}](figures/figure_11){width="100.00000%"} ![[]{data-label="fig:2"}](figures/figure_21){width="100.00000%"} ![[]{data-label="fig:2"}](figures/figure_31){width="100.00000%"} ![[]{data-label="fig:2"}](figures/figure_32){width="100.00000%"} ### Feature Dependency Sampling for Text It is also essential for natural language processing to have a proper sampling operation. For text classification, we let the interpretable representation be a bag of words. Similar to image, $x\in{\mathbb{R}^d}$ denotes the original representation of a text, and binary vector $x'\in{\{{0,1}\}^{d'}}$ denotes its interpretable representation. In order to learn the local behavior of text classifier, we generate a group of perturbed samples of a given instance by activating a subset of features. Fig. \[fig:3\] shows two natural language in Chinese and English, we can find there are strong semantic dependency between words especially in Chinese. If the activated features are get by using a sampling process where features are independent to each other, we may loss much useful information to learn the local explanation models. In sampling process, the semantic dependent words correspond to adjacent superpixels in the image. Semantic dependent words should be selected or unselected at the same time. There are many methods to analyze semantic dependency of natural language. There, we incorporate the Stanford CoreNLP [@CoreNLP] tools into sampling process to get the perturbed samples. ![[]{data-label="fig:3"}](figures/chi-seg){width="100.00000%"} ![[]{data-label="fig:3"}](figures/eng-seg){width="100.00000%"} Nonlinear Boundary of Local Decision ------------------------------------- Most existing local explanation methods assume the decision boundary is local linearity. Those explanation methods may produce serious errors as in most complex networks, the local decision boundary is non-linear. Experiments show a simple linear approximation will significantly degrade the explanation fidelity. In this section, we adopt Support Vector Regressor (SVR) with kernel function to approximate nonlinear boundary. In approximation processing, when data are not distributed linearly in the current feature space, we use kernel function to project data points into higher dimensional feature space and find the optimal hyperplane. The perturbed samples of a given instance are impossible to be fitted by a linear model. Our way to tackle this problem is to apply a kernel function mapping to bring data to a higher dimensional feature space. The formula to transfrom the data is as follow: $$g(\boldsymbol{x},\boldsymbol{w})=\sum_{i=1}^N{{w_i}k(x-x')}. %g(x,\boldsymbol{w})=\sum_{i=1}^N{{w_i}e^{-(x-x')^2/\sigma^2}}=\boldsymbol{w}^\top\Phi(x).$$ After project data point into higher dimensional feature space. We search for a hyperplane by using hinge error measure. Specifically, we introduce slack variables for data points that violate $\varepsilon-$insensitive error: $$\nonumber err(f(x_i),(g(x_i,\boldsymbol{w}))=$$ $$\begin{cases} 0,& \|f(x_i)-g(x_i,\boldsymbol{w})\|\leqslant{\varepsilon}\\ \|f(x_i)-g(x_i,\boldsymbol{w})\|-{\varepsilon}, & \|f(x_i)-g(x_i,\boldsymbol{w})\|>{\varepsilon} \end{cases}$$ ![[]{data-label="fig:5"}](figures/epsilon){width="80.00000%"} For each data point $x_i$, two slack variables, $\xi_i, \hat{\xi_i}$ are required to measure whether $g(x_i)$ is above or below the tube. $$\begin{cases} \xi_i=f(x_i)-(g(x_i,\boldsymbol{w})+\varepsilon),& if f(x_i)>g(x_i,\boldsymbol{w})+\varepsilon\\ \xi_i=0,& otherwise \end{cases}$$ $$\begin{cases} \hat{\xi_i}=(g(x_i,\boldsymbol{w})-\varepsilon)-f(x_i),& if f(x_i)<g(x_i,\boldsymbol{w})-\varepsilon\\ \hat{\xi_i}=0,& otherwise \end{cases}$$ The learning is by the optimization: $$\begin{aligned} \nonumber &\mathop{\min}_{\boldsymbol{w},\xi_i,\hat{\xi_i}}{\sum_{i=1}^N(\xi_i+\hat{\xi_i})+\lambda{\lVert\boldsymbol{w}\rVert}^2}\\ \nonumber s.t:\\ \nonumber & f(x_i)-g(x_i,\boldsymbol{w})\geqslant{\varepsilon+\xi_i}; \\ \nonumber &f(x_i)-g(x_i,\boldsymbol{w})\leqslant{\varepsilon+\hat{\xi_i}};\\ & \xi_i\geqslant0; \hat{\xi_i}\geqslant0, i=1,2,...,N.\end{aligned}$$ This is the famous support vector regression method which can be solved by building Lagrangian functions. Algorithm 1 shows a simplified workflow diagram of LEDSNA. Firstly, LEDSNA incorporates the feature clustering method into sampling process to activate a subset of features. Then, LEDSNA uses kernel function to project data points into higher dimensional feature space. Finally, LEDSNA use the support vector regression to search for a hyperplane and get the coefficient of important feature. Classifier $f$, Instance $x$, get interpretable presentation of $x'$ (e.g. superpixel image for image and bag of word for text) get $f(x')$ by classifier $f$ incorporate the feature clustering method into sampling process to activate a subset of features initial $Z \leftarrow \{\}$ get $z$ by recovering $z'$ $Z \leftarrow Z \cup (z'_i,f(z_i),\pi_x(z_i))$ use kernel function to project data points into higher dimensional feature space: $g(\boldsymbol{x},\boldsymbol{w})=\sum_{i=1}^N{{w_i}k(x-x')}.$; use the support vector regression to search for a hyperplane feature coefficient Experiments =========== In this section, we first introduce the evaluation criterion of explanation methods. Then, we perform experiments in natural language processing in Chinese. Finally, we perform experiment to explain the Google’s pre-trained Inception neural network [@Inception] on imagenet database. Experiment results show the flexibility of LEDSNA. Evaluation criterion -------------------- A good explainable model requires same characteristics. One of the essential criterion is interpretability. The explanation must appear as a certain form understandable to the observer, i.e., providing visual explanations which lists most significant features contributed to the prediction. Another essential criterion is local fidelity. The explanation must be faithful to the model in the vicinity of the instance being predicted. Local Approximation Error ($Err$) and R-squared ($R^2$) are two important measurements of the accuracy of our local approximation with respect to the original decision boundary. Local Approximation Error can reflect the prediction accuracy: $$Err=|f(x_0)-g(x_0)|,$$ where $f(x_0)$ is a single prediction obtained from a target deep learning classifier, $g(x_0)$ is the predicted value by explanation model. $R^2$ is the “percent of variance explained” by the explanation model. That is to say that $R^2$ is the fraction by which the variance of the errors is less than the variance of the dependent variable. $R^2$ is calculated by Total Sum of Squares ($SST$) and Error Sum of Squares ($SSE$): $$\begin{aligned} \nonumber & R^2=1-SSE/SST\\ \nonumber& SSE=\sum_{i=1}^n(f(x_i)-g(x_i))^2\\ & SST=\sum_{i=1}^n(f(x_i)-f_{mean})^2,\end{aligned}$$ where $f(x_i)$ is the label of perturbed sample $x_i$, obtained from a target deep learning classifier. $g(x_i)$ is the predicted value and $f_{mean}$ is the mean value of $f(x_i)$. Moreover, $R^2$ can be expressed by Mean Square Error ($MSE$) and Variance ($Var$) which are familiar to us: $$\begin{aligned} % \nonumber & R^2=1-{{\frac{1}{n}}\sum_{i=1}^n(f(x_i)-g(x_i))^2}/{{\frac{1}{n}}\sum_{i=1}^n(f(x_i)-f_{mean})^2}\\ \nonumber & R^2=1-\frac{{\frac{1}{n}}\sum_{i=1}^n(f(x_i)-g(x_i))^2}{{\frac{1}{n}}\sum_{i=1}^n(f(x_i)-f_{mean})^2}\\ & =1-MSE/Var %& =1-\frac{MSE}{Var}\end{aligned}$$ $R^2$ is a relative measure which is conveniently scaled between 0 and 1. The best $R^2$ is $1.0$. The closer the score is to $1.0$, the better the performance of fidelity is to explainer. Experiment on Image Classifiers ------------------------------- In this section, LEDSNA and LIME explain image classification predictions made by Google’s pre-trained Inception neural network [@Inception]. Fig. \[fig:3\] shows two original image to be processed. Fig. \[fig:4\] and Fig. \[fig:5\] lists some visual explanations of LEDSNA and LIME: the first row shows the superpixels explanations by LIME (K=1,2,3,4) respectively, the second row shows the superpixels explanations by LEDSNA (K=1,2,3,4) respectively. The explanations highlight the top K superpixel segments, which have the most considerable positive weights towards the predictions. We can see LEDSNA can effectively get the correlation between the adjacent superpixel segments, which provide a better understanding to users. ![[]{data-label="fig:3"}](figures/image2){width="100.00000%"} ![[]{data-label="fig:3"}](figures/image2_p){width="100.00000%"} ![[]{data-label="fig:3"}](figures/image1){width="100.00000%"} ![[]{data-label="fig:3"}](figures/image1_p){width="100.00000%"} ![[]{data-label="fig:4"}](figures/li_img2_1){width="100.00000%"} ![[]{data-label="fig:4"}](figures/li_img2_2){width="100.00000%"} ![[]{data-label="fig:4"}](figures/li_img2_3){width="100.00000%"} ![[]{data-label="fig:4"}](figures/li_img2_4){width="100.00000%"} ![[]{data-label="fig:4"}](figures/cli_img2_1){width="100.00000%"} ![[]{data-label="fig:4"}](figures/cli_img2_2){width="100.00000%"} ![[]{data-label="fig:4"}](figures/cli_img2_3){width="100.00000%"} ![[]{data-label="fig:4"}](figures/cli_img2_4){width="100.00000%"} ![[]{data-label="fig:5"}](figures/li_img1_1){width="100.00000%"} ![[]{data-label="fig:5"}](figures/li_img1_2){width="100.00000%"} ![[]{data-label="fig:5"}](figures/li_img1_3){width="100.00000%"} ![[]{data-label="fig:5"}](figures/li_img1_4){width="100.00000%"} ![[]{data-label="fig:5"}](figures/cli_img1_1){width="100.00000%"} ![[]{data-label="fig:5"}](figures/cli_img1_2){width="100.00000%"} ![[]{data-label="fig:5"}](figures/cli_img1_3){width="100.00000%"} ![[]{data-label="fig:5"}](figures/cli_img1_4){width="100.00000%"} In addition, Table \[tab:1\] lists some instances of the local approximation error and $R^2$ of two algorithm. Comparing to LIME, we can see LEDSNA provides better predictive accuracy than LIME. Besides, $R^2$ of LEDSNA is much bigger than LIME. By comparing the two criterion, we conclude that LEDSNA has better fidelity than LIME. Compared with LIME in term of interpretability and fidelity, LEDSNA has better performance in explaining classification. f(x) g(x) $Err$ $R^2$ -------- ------ -------- -------- -------- LIME 0.8129 0.2053 0.4662 LEDSNA 0.6066 0.001 0.9803 LIME 0.9857 0.2211 0.3219 LEDSNA 0.7633 0.0012 0.896 LIME 0.5133 0.2248 0.4644 LEDSNA 0.288 0.0005 0.5890 LIME 1.5995 0.6194 0.5939 LEDSNA 0.9025 0.0010 0.8407 LIME 1.2854 0.2655 0.3602 LEDSNA 0.945 0.0010 0.7955 LIME 1.2422 0.2753 0.6341 LEDSNA 0.9657 0.0012 0.8414 \[tab:1\] Experiment on Sentiment Analysis of Text ---------------------------------------- ### Experiment on Chinese Natural Language Databse Simplified Chinese Text Processing (SnowNLP) is a sentiment analysis tool especially for Chinese natural language. This section we use LEDSNA and LIME to explain the predictions made by SnowNLP on Public Comment Dataset. As there is a strong semantic dependency between words in Chinese, we incorporate the Stanford Word Segmenter [@CoreNLP] into sampling process to get the perturbed samples. In nonlinear approximating, we use Gaussian kernel function to compute the similarity between the data points in a much higher dimensional space. ![[]{data-label="fig:6"}](figures/chi3){width="100.00000%"} ![[]{data-label="fig:6"}](figures/chi4){width="100.00000%"} ![[]{data-label="fig:7"}](figures/chi1){width="100.00000%"} ![[]{data-label="fig:7"}](figures/chi2){width="100.00000%"} Fig. \[fig:6\] and Fig. \[fig:7\] shows visual explanations of LEDNSA and LIME, we can see the explanation of LEDNSA can offer more useful information than that of LIME. Table \[tab:2\] lists the local approximation error and $R^2$ of six instances. Comparing to LIME, we find LEDSNA achieves better performance across the board, and by average a magnitude of local approximation error than LIME. For $R^2$, similar observation is obtained. f(x) g(x) $R^2$ $Err$ -------- ------ -------- -------- -------- LIME 0.1755 0.4795 0.0033 LEDSNA 0.1765 0.9973 0.0023 LIME 0.1136 0.4969 0.0088 LEDSNA 0.1209 0.8710 0.0016 LIME 0.3082 0.4823 0.0784 LEDSNA 0.2283 0.9790 0.0015 LIME 0.3526 0.5876 0.1313 LEDSNA 0.4756 0.9822 0.0083 LIME 0.6901 0.4449 0.0419 LEDSNA 0.6482 0.9473 0.0008 LIME 0.8717 0.5779 0.0335 LEDSNA 0.9050 0.9533 0.0001 \[tab:2\] ![[]{data-label="fig:8"}](figures/R21){width="100.00000%"} ![[]{data-label="fig:8"}](figures/R22){width="100.00000%"} ![[]{data-label="fig:9"}](figures/Err1){width="100.00000%"} ![[]{data-label="fig:9"}](figures/Err2){width="100.00000%"} Moreover, we randomly selected 1000 data samples to constitute testing database. For each testing data sample, we use LEDSNA and LIME to explain SnowNLP and compute the $Err$ and $R^2$. Results show for LEDSNA, the Err of $95\%$ of test data samples are smaller than LIME. Similarly, the $R^2$ of $98.4\%$ of test data samples are bigger than LIME. In conclusion, LEDSNA exhibits strong interpretability and fidelity over LIME Conclusion ========== There are two drawbacks in current existing local explanations. Perturbed samples are generated from a uniform distribution, ignoring the complicated correlation between features. This may lead to lose much useful information to learn the local explanation models. Moreover, most existing methods assume the decision boundary is local linearity, which may produce serious errors as in most complex networks, the local decision boundary is non-linear. In this paper, we design and develop a novel, high-fidelity local explanation method to address the above challenges. First, we design a unique local sampling process which incorporate the feature clustering method to handle the feature dependency problems. Then, we adopt SVR to approximate locally nonlinear boundary. In this way, by simultaneously preserving feature dependency and local non-linearity, our method produces high-fidelity and high-interpretability explanation.

**PLANETARY AND LIGHT MOTIONS FROM NEWTONIAN THEORY: AN AMUSINGEXERCISE** K.K. Nandi$^{1}$ Department of Mathematics, University of North Bengal, Siliguri, WB 734430, India N.G. Migranov$^{2}$ Department of Physics, Bashkir State Pedagogical University, 3-A, October Revolution Street, Ufa 450000, Bashkortostan, Russia J.C. Evans$^{3}$ Department of Physics, University of Puget Sound, Tacoma, WA 98416, USA M.K. Amedeker$^{4}$ Department of Physics, University of Education, Winneba, Ghana ————————————————————————– $^{1}$E-mail address: kamalnandi1952@yahoo.co.in $^{2}$E-mail address: migranov@bspu.ru $^{3}$E-mail address: jcevans@ups.edu $^{4}$E-mail address: mawuden@yahoo.com **Abstract**  *We attempt to see how closely we can *formally* obtain the planetary and light path equations of General Relativity by employing certain operations on the familiar Newtonian equation. This article is intended neither as an alternative to nor as a tool for grasping Einstein’s General Relativity. Though the exercise is understandable by readers at large, it is especially recommended to the teachers of Relativity for an appreciative understanding of its peculiarity as well as its pedagogical value in the teaching of differential equation*s. —————————————————- **** Everyone knows Newton’s theory of gravity and some know Einstein’s theory of General Relativity (GR). Undoubtedly, GR is one of the most beautiful self-consistent modern creations in the realm of theoretical physics. It has wonderfully tested against various astronomical observations to date including those in the Solar system. However, at a popular level, a naīve question is often asked as to whether the GR effects could have been interpreted using a more mundane theory than the abstract theory of GR in which gravity - which is as real a force as any other - has been geometrized". For instance, some ask the question: What is the difference between the bending of light rays in GR with that occuring in a refractive optical medium? The answer lies in the well known fact that the propagation of light rays in a gravity field *a la* GR can be exactly *rephrased* as propagation in an equivalent optical refractive medium with appropriate constitutive equations \[1\]. The refractive index can be employed in a new set of optical-mechanical equations so that a single equation covers motions of both massive and massless particles in a spherically symmetric field \[2-4\]. An approach of this kind provides a useful and interesting window to look at familiar observed GR results but, by no means, implies a replacement of GR. The whole point of the above paragraph is that one inevitably needs to know the metric solutions of GR *in advance*. Only after knowing them, one can derive appropriate refractive indices and the method of optical-mechanical analogy in terms of these indices then exactly reproduces the GR geodesic equations. That is to say, we might employ different working methods but the physical content remains essentially that of GR. (There have been attempts to set aside GR altogether and propose alternative physics by introducing a variable test mass \[5\], or even assuming variable speeds of light in flat space \[6\]. These ideas have their own values and we are not going to discuss them here.) The object of the present article is somewhat different: We are not going to suggest any working method of the kind described above, but present an interesting calculation. (However, it must not be weighed against the grand edifice of GR). Using PPN-like approximations on the Newtonian theory, we shall *formally* obtain planetary and light path equations. They resemble the path equations of GR only fortuitously and this is the amusing part. Apart from this, the contents could be instructive in exemplifying the role of numerically smaller terms in the differential equations. To begin with, one recalls an earlier discussion of MØller \[7\] that has shown that the bending of light rays is due partly to the geometrical curvature of space and partly to the variation of light speed in a Newtonian potential. In fact, the ratio is exactly 50:50. The GR null trajectory equations can be integrated, once assuming a Euclidean space with a variable light speed and again a curved space with a constant light speed. This analysis and arguments clearly elucidate the complementary roles of curved space and Newtonian theory in the best possible manner. This complementarity motivates us to examine how far, if at all, we are able to introduce curvature effects in the path equations of the Newtonian theory. That is: We try to obtain, from the familiar Newtonian theory itself, the form of the known GR path equations of motion *without* geometrizing gravity. (It is known that the gravitational redshift is a prediction of GR, but it is also known that it can be predicted from the Equivalence Principle without using GR equations \[8\]. Hence we shall not address this result here.) Let us start from the usual Kepler problem of a massive test particle moving around a spherical gravitating mass $M$ under the Newtonian inverse square law. Let $T$ and $V$ denote the kinetic and potential energies respectively. Then $T+V=$ constant $=$ $\frac{E_{0}}{2}$ (say) implies in relativistic units$$\frac{1}{2}[\overset{.}{r}^{2}+r^{2}\overset{.}{\varphi }^{2}]-mc_{0}^{2}r^{-1}=\frac{E_{0}}{2}$$where $m=GMc_{0}^{-2}$ and a dot denotes differentiation with respect to Newtonian time $t$, $c_{0}$ is the speed of light in vacuum. The central nature of the force implies constancy of the angular momentum (the Lagrangian is independent of $\varphi $) such that$$r^{2}\overset{.}{\varphi }=h_{0}.$$With $u=\frac{1}{r}$, we can rewrite Eq.(1) as$$h_{0}^{2}\left[ u^{2}+\left( \frac{du}{d\varphi }\right) ^{2}\right] -2muc_{0}^{2}=E_{0}$$where the constant $E_{0}$ has the dimension of $c_{0}^{2}$. For bound material orbits $E_{0}$ $<0$. Customarily, by differentiating again with respect to $\varphi $, one finds a second order differential equation that yields a Keplerian ellipse given by$$u=\frac{1}{p}(1+e\cos \varphi )$$where $e$ is the eccentricity, $p=\frac{h_{0}^{2}}{GM}$ is the semi-latus rectum. Let us redefine the radial variable $u\rightarrow u^{\prime }$ through the equations$$u^{\prime }=u\Phi (u)$$ $$\Phi (u)=\left( 1+\frac{mu}{2}\right) ^{-2}$$ $$u^{\prime }=\frac{1}{r^{\prime }}.$$ (Aside: These transformations are not unfamiliar to those conversant with GR.) After some straightforward algebra, we get$$du^{\prime }=\Phi (u)\Omega (u)du$$where$$\Omega (u)=\left( 1+\frac{mu}{2}\right) ^{-1}\left( 1-\frac{mu}{2}\right)$$ $$\Omega (u^{\prime })=\left( 1-2mu^{\prime }\right) ^{\frac{1}{2}}$$ $$\Phi (u^{\prime })=\frac{1}{4}\left[ 1+(1-2mu^{\prime })^{\frac{1}{2}}\right] ^{2}.$$ Note that $\Phi (u)$ of Eq.(6) is numerically the same as $\Phi (u^{\prime }) $ of Eq.(11). The same applies between $\Omega (u)$ of Eq.(9) and $\Omega (u^{\prime })$ of Eq.(10). The following expansions can also be directly verified:$$2mu=2mu^{\prime }+2m^{2}u^{\prime 2}+5m^{3}u^{\prime 3}+...=2mu^{\prime }+O(m^{2}u^{\prime 2}).$$This implies that, to first order, $r\simeq r^{\prime }$. Also,$$\Phi ^{2}(u^{\prime })\Omega ^{2}(u^{\prime })=1-4mu^{\prime }+O(m^{2}u^{\prime 2}).$$Let us now express Eq.(3) in terms of the new variable $u^{\prime }$. Multiplying both sides of Eq.(3) by $\Phi ^{2}\Omega ^{2}$ and using Eqs.(5)-(13), we get$$h_{0}^{2}\left[ \Omega ^{2}u^{\prime 2}+\left( \frac{du^{\prime }}{d\varphi }\right) ^{2}\right] =c_{0}^{2}\left[ E_{0}c_{0}^{-2}+2mu^{\prime }+O(m^{2}u^{\prime 2})\right] \Phi ^{2}\Omega ^{2}.$$Simplifying further using Eqs.(10) and (13), we have$$h_{0}^{2}\left[ u^{\prime 2}+\left( \frac{du^{\prime }}{d\varphi }\right) ^{2}-2mu^{\prime 3}\right] =c_{0}^{2}\left[ E_{0}c_{0}^{-2}+2mu^{\prime }(1-2E_{0}c_{0}^{-2})+O(m^{2}u^{\prime 2})\right] .$$Apply this equation to a practical situation, the Solar system. At the site of Mercury, the planet nearest to the Sun, $mu\simeq mu^{\prime }\simeq 2.5\times 10^{-8}$. Let us ignore the terms $O(m^{2}u^{\prime 2})$ in comparison to the $mu^{\prime }$ term. Then Eq.(15) reduces to$$h_{0}^{2}\left[ u^{\prime 2}+\left( \frac{du^{\prime }}{d\varphi }\right) ^{2}-2mu^{\prime 3}\right] =E_{0}+2mu^{\prime }c_{0}^{2}(1-2E_{0}c_{0}^{-2}).$$Differentiating with respect to $\varphi $, we get$$u^{\prime }+\frac{d^{2}u^{\prime }}{d\varphi ^{2}}=\frac{1}{p^{\prime }}+3mu^{\prime 2}$$where$$\frac{1}{p^{\prime }}=\frac{mc_{0}^{2}}{h^{\prime 2}},h^{\prime }=\frac{h_{0}}{(1-2E_{0}c_{0}^{-2})^{\frac{1}{2}}}$$is a rescaled constant. The final Eq.(17) seems suggestive with the usual perturbation term $3mu^{\prime 2}$ appearing: It is exactly of the same form as the GR path equation! One notes that the constant $h^{\prime }$ involves the test particle energy $E_{0}$ similar to what one finds in the GR treatment. To see this, compare with Eq.(17) the corresponding GR equation given by (Take henceforth $G=1$):$$u+\frac{d^{2}u}{d\varphi ^{2}}=\frac{1}{p}+3mu^{2}$$in which $p$ is given by $p=\frac{U_{3}^{2}}{Mm_{0}^{2}c_{0}^{4}}$ where $m_{0}$ is the test particle rest mass, $J=-\frac{U_{3}}{U_{0}}$ is the constant angular momentum rescaled by the energy at infinity $U_{0}=\frac{m_{0}c_{0}^{2}}{\sqrt{1-\overset{.}{r}_{\infty }^{2}/c_{0}^{2}}}$ and the constant $U_{3}=r^{2}\frac{d\varphi }{d\lambda }$, $\lambda $ being the affine pararneter \[9\]. As usual, considering low velocity, we can take $U_{0}=m_{0}c_{0}^{2}$ and identifying the asymptotic value of $J$ as $h_{0}$, we have$$p\simeq \frac{h_{0}^{2}}{M}.$$With this value of $p$, the GR perturbation term $3mu^{2}$ then gives the well known perihelion advance of the Keplerian ellipse. In our case, the parallel of $p$ from Eq.(17) is:$$p^{\prime }:=\left( \frac{mc_{0}^{2}}{h^{\prime 2}}\right) ^{-1}=\frac{h_{0}^{2}}{M(1-2E_{0}c_{0}^{-2})}.$$Its asymptotic value can be computed using Eq.(1). For near circular orbits, the kinetic and potential energies are roughly of the same order of magnitude such that the velocity is $v^{2}\sim \frac{M}{r}=muc_{0}^{2}$. Then, from Eq.(1), and noting that $u\simeq u^{\prime }$ asymptotically, we can write $E_{0}=\alpha mu^{\prime }c_{0}^{2}$ where $-1<\alpha <1.$ Then the denominator becomes $M(1-2\alpha mu^{\prime })$. The term $2\alpha mu^{\prime }\simeq 10^{-8}$ can be easiliy ignored compared to unity and we are left with$$p^{\prime }\simeq \frac{h_{0}^{2}}{M}.$$just as in Eq.(20). So we can replace $p^{\prime }$ in Eq.(17) by its asymptotic value $p$ given either by Eq.(20) or (22). For the motion of light, the situation is different: the dimensionless quantity $E_{0}c_{0}^{-2}$ must be fixed to the value $\frac{1}{2}$ so that $p^{\prime }\rightarrow \infty $. Recall that only a nonzero value for light ($E_{0}\neq 0$) in Newtonian theory is consistent with the zero value in GR \[10\]. (The zero rest mass of photons is a Special Relativistic or GR concept but is not a Newtonian concept). Consequently, we have the equation of the light ray trajectory exactly as in GR:$$u^{\prime }+\frac{d^{2}u^{\prime }}{d\varphi ^{2}}=3mu^{\prime 2}$$ Thus Eqs.(17) and (23), respectively, seem to provide the same GR results as far as the weak field tests for the perihelion advance and the bending of light are concerned. To examine the situation more closely, recall what steps were involved. The first step is the radial rescaling $u\rightarrow u^{\prime }$ which has no physical import. The second step is that, in arriving at Eq.(16), we have ignored terms like $O(m^{2}u^{\prime 2})$ on numerical grounds. Note that it is only Eq.(15) *per se,* and *not* Eq.(17), that inverts exactly to the original Eq.(3) in the ($u$, $\varphi$) coordinates describing the inverse square law. As we see, Eq.(17) produces an additional $3mu^{\prime 2} $ term! Strictly speaking, Eq.(17) is approximate to the extent we ignored the smaller terms compared to unity (of the order of $10^{-16}$ and less!) in arriving at it. Treating this Eq.(17) as an *exact* equation means that we are retaining the cubic additional term as the only perturbation while disregarding the remaining smaller perturbations. This is the only *nontrivial* step we have adopted in the above computation. If we had retained the smaller terms in Eq.(15), then it could tell the original situation: the exact Newtonian orbits. It is our nontrivial, but numerically justified, omission of the smaller terms that has brought forth equations similar to those in GR. Thus the exact solution of Eq.(15) is still a Keplerian ellipse but its expression does not *look* as familiar as in Eq.(4). Instead, in the primed coordinates, it looks like$$u^{\prime }=u\Phi (u)=\frac{\Phi (u)}{p}(1+e\cos \varphi ).$$where $u$ is given by Eq.(4). Expressions might differ in looks depending on the choice of coordinates, but the orbital shapes do not change. One might think that though Eq.(17) looks different from Eq.(15), it still represents a Keplerian ellipse in the ($u^{\prime }$,$\varphi$) coordinates. This is not the case since Eq.(17) is now nonlinear. We can find its solution by standard procedures starting with the zeroth order solution $u_{0}^{\prime }=\frac{1}{p}(1+e\cos \varphi )$ which is the solution of $u^{\prime }+\frac{d^{2}u^{\prime }}{d\varphi ^{2}}=\frac{1}{p}$. Eq.(17) then gives the observed perihelion advance as $\frac{6\pi M}{p}$. \[Note that if one starts with the same $u_{0}^{\prime }$ in Eq.(15) or its second derivative form, one would eventually end up with Eq.(24) as the final solution\]. Likewise, the exact equation for a straight line is$$u^{\prime }=\frac{1}{R}\Phi (u)\cos \varphi$$where $R$ is the distance from the origin. To zeroth order, $u_{0}^{\prime }=\frac{\cos \varphi }{R}$  is a solution of $u^{\prime }+\frac{d^{2}u^{\prime }}{d\varphi ^{2}}=0$. By usual methods again with Eq.(23), one finds a total observed bending of light rays $\triangle \varphi \simeq \frac{4M}{R}$. The procedure leading to Eq.(17) has some similarity with that in GR. In the curved spacetime of GR, one needs to consider coordinate independent proper length $l$ instead of the radial coordinate $r$. Thus, in the Schwarzschild metric, $l$ is given by $$\begin{aligned} l=\int\frac{dr}{\sqrt{1-\frac{2m}{r}}}=\end{aligned}$$ $$\begin{aligned} \frac{\sqrt{r}(-2m+r)+2m\sqrt{2m-r}\arctan\sqrt{r/(2m-r)}}{\sqrt{r(1-\frac{2m}{r}})}\end{aligned}$$ In terms of ($l$,$\varphi$) coordinates, the GR Eq.(19) can not maintain its form or assume another exact closed form due to the fact that $r$ can not be expressed in terms of $l$ in a closed form. However, in the weak field region, $r\simeq l$, and we can maintain the form of Eq.(19) as it is, while ignoring higher order terms in $l$. In the present calculation, the background is Euclidean and so we can express $l$, using Eq.(8), as $l$=$\int{dr}$=$\int{\Phi(r^{\prime })\Omega^{-1}(r^{\prime }) dr^{\prime}}$. In our calculation, we have ignored higher order terms in $u^{\prime}$in the weak field region so that $r\simeq r^{\prime}$and we ended up with Eq.(17). Can we physically interpret our nontrivial step as a modification of the Newtonian force law? In this context, it is to be noted that, historically, Newton himself attempted to modify his force law to explain some phenomenon (for details, see Ref. \[5\]). One might also recall other efforts, for instance, Sommerfeld’s calculation \[11\] for the precession of an electron in a Coulomb potential due to a proton ($Z=1$):$$\frac{d}{dt}\left( \frac{m_{0}\overrightarrow{v}}{\sqrt{1-v^{2}}}\right) =\frac{Ze^{2}}{r^{2}}\widehat{r}$$where $\widehat{r}$ is a unit vector in the radial direction and $e$ is the electronic charge. However, it produces only (1/6)th of the observed perihelion advance of planets if the Coulomb potential on the right is replaced by the Newtonian potential. One could try the above special relativistic equation with another kind of force law on the right \[12\]$$\frac{d}{dt}\left( \frac{m_{0}\overrightarrow{v}}{\sqrt{1-v^{2}}}\right) =\frac{Mm_{0}}{r^{2}(1-v^{2})^{\frac{5}{2}}}\widehat{r}$$where $v^{2}=\overset{.}{r}^{2}+r^{2}\overset{.}{\varphi }^{2}$ does produce the observed perihelion advance, but the difficulty is that its first integral does not produce the conserved relativistic energy. This is understandable because the potential is velocity dependent. Coming back to our calculation, one might say that Eq.(17) \[which is the same as Eq.(19)\] corresponds to a potential $V(r)=-\frac{M}{r}-\frac{M}{r^{3}}$ but then the last term leads to a dimensional mismatch (see ref.\[5\]). Because of this, our procedure can not be interpreted as a modification of the Newtonian force law. Also, there was absolutely no use of the concept of geometric curvature in the calculation; it was completely Euclidean. Thus, we conclude that the similarity between Eqs.(17) and (19) is only a fortuitous though amusing coincidence; it is just a mirage resulting from the choice of coordinates. There is *absolutely* no reason to prefer ($u^{\prime }$,$\varphi$) coordinates over others and in this case, the formal coincidence will be lost. Nonetheless, the procedure illustrates something of pedagogical importance in the treatment of differential equations: One should be watchful with smaller terms! Their removal can *nonlinearize* a given linear equation \[like going from Eq.(15) to (17)\] and conversely, their restoration can *linearize* a known nonlinear equation \[like returning from Eq.(17) to (15)\]. It is a pleasure to thank Guzel Kutdusova and Arunava Bhadra for useful discussions. **References** \[1\] F. de Felice, On the gravitational field acting as an optical medium", Gen. Rel. Grav. **2**, 347-357 (1971). \[2\] A recent useful reformulation of the historical optical-mechanical analogy has been conceived by: J. Evans and M. Rosenquist, $F=ma$ optics“, Am. J. Phys. **54**, 876-883 (1986). Newton’s laws of motion are obtained directly from Fermat’s principle in: M. Rosenquist and J. Evans, The classical limit of quantum mechanics from Fermat’s principle and the de Broglie relation”, Am. J. Phys. **56**, 881-882 (1988). For application to gradient-index lenses, see: J. Evans, Simple forms for equations of rays in gradient-index lenses“, Am. J. Phys. **58**, 773-778 (1990). A short yet very illuminating discussion of the limitations and significance of the analogy may be found in: J. A. Arnaud, Analogy between optical rays and non-relativistic particle trajectories: A comment”, Am. J. Phys. **44**, 1067-1069 (1976). \[3\] The above reformulation has been applied in the GR context by: K.K. Nandi and A. Islam, On the optical-mechanical analogy in general relativity“, Am. J. Phys. **63**, 251-256 (1995). The method has been extended to rotating bodies by: P.M. Alsing, The optical-mechanical analogy for stationary metrics in general relativity”, Am. J. Phys. **66**, 779-790 (1998). The medium approach yields a possible observable effect in a new setting. This is discussed in: K. K. Nandi, Yuan-Zhong Zhang, P. M. Alsing, J. C. Evans, and A. Bhadra, Analogue of the Fizeau effect in an effective optical medium", Phys. Rev. D **67**, 025002 (1-11) (2003). Historically, Einstein himself conjectured the idea of an equivalent optical medium (This is reported in Ref.\[1\]). However, to our knowledge, Sir A.S. Eddington seems to be the first to have calculated the bending of light rays by assuming an approximate index $n(r)\simeq 1-\frac{2M}{r}$. This can be found in: *Space, Time and Gravitation* (Cambridge University, Cambridge, 1920), reissued in the Cambridge Science Classic Series, 1987, p.109. \[4\] For a further extension of the analogy that covers both massive and massless particles as well as applications to Cosmology, see: J. Evans, K.K. Nandi, and A. Islam, The optical-mechanical analogy in general relativity: New methods for the paths of light and of the planets“, Am. J. Phys. **64**, 1404-1415 (1996). For a semiclassical application, interested readers may have a look at: J. Evans, P.M. Alsing, S. Giorgetti, and K.K. Nandi, Matter waves in a gravitational field: An index of refraction for massive particles in general relativity”, Am. J. Phys. **69**, 1103-1110 (2001). \[5\] G. Maneff, La gravitation et le principe de l’egalité de l’action et de la réaction", Comptes Rendus Acad. Sci. Paris **178**, 2159-2161 (1924). Maneff assumed a variable test mass, viz., $m_{0}=m_{0}^{\prime }\exp (\frac{M}{r})$ where $m_{0}^{\prime }$ is an invariant. This led to a force law: $F=\frac{Mm_{0}}{r^{2}}\left( 1+\frac{3M}{r}\right) $. For a complete reference of the works by Maneff, see the interesting article by  R.I. Ivanov and E.M. Prodanov \[Arxiv: gr-qc/0504025\]. The authors also mention that Newton modified his potential law to $V(r)=A\frac{M}{r}+B\frac{M^{2}}{r^{2}}$ where $A$ and $B$ are constants, to explain the deviation of Moon’s motion from the Keplerian laws. (Note the dimensional consistencies in both modifications.) \[6\] F.R. Tangherlini, Particle approach to the Fresnel coefficients“, Phys. Rev. A **12**, 139-147 (1975). See also: R. Tian and Z. Li, The speed and apparent rest mass of photons in a gravitational field”, Am. J. Phys. **58**, 890-892 (1990). \[7\] C. MØller, *The Theory of Relativity*, 2nd Ed. (Oxford University, Oxford, 1972), pp 498-501. \[8\] W. Rindler, *Essential Relativity*, 2nd Ed. (Springer-Verlag, New York, 1977), p.143. \[9\] S.K. Bose, *An Introduction to General Relativity* (Wiley Eastern, New Delhi, 1980), pp. 37-40. \[10\] See, for instance, the treatise by S. Weinberg, *Gravitation and Cosmology* (John Wiley, New York, 1972), pp.186-187. If we start with the usual GR geodesic equations, then, in the low velocity, weak field limit, they reduce to $r^{2}\overset{.}{\varphi }\simeq h_{0}$ and $\frac{1}{2}[\overset{.}{r}^{2}+\frac{h_{0}^{2}}{r^{2}}]-\frac{M}{r}\simeq \frac{1-E}{2}$. For photons, $E=0$ so that for the total energy, we are left with a nonzero value $\frac{1}{2.}$. If we start with the Newtonian equations, we instead get for light motion the value $\frac{1}{4}$ from Eq.(1) because $E_{0}=\frac{1}{2}$. The discrepant factor of $2$ is actually a contribution from GR but it makes no difference to us as we have essentially started from the Newtonian theory. It is the *nonzero* value on the right of Eq.(1) for light that is consistent with the zero value in GR. \[11\] Sommerfeld’s calculation is discussed in: P.G. Bergmann, *Introduction to the Theory of Relativity* (Prentice-Hall, Englewood Cliffs, New Jersey, 1942). \[12\] T.K. Ghoshal, K.K. Nandi, and S.K. Ghosal, On the precession of the perihelion of Mercury", Indian J. Pure & Appl. Math. **18**, 194-199 (1987).

--- abstract: 'The classical dynamics for a charged point particle with intrinsic spin is governed by a relativistic Hamiltonian for the orbital motion and by the Thomas-Bargmann-Michel-Telegdi equation for the precession of the spin. It is natural to ask whether the classical Hamiltonian (with both the orbital and spin parts) is consistent with that in the relativistic quantum theory for a spin-$1/2$ charged particle, which is described by the Dirac equation. In the low-energy limit, up to terms of the 7th order in $1/E_g$ ($E_g=2mc^2$ and $m$ is the particle mass), we investigate the Foldy-Wouthuysen (FW) transformation of the Dirac Hamiltonian in the presence of homogeneous and static electromagnetic fields and show that it is indeed in agreement with the classical Hamiltonian with the gyromagnetic ratio being equal to 2. Through electromagnetic duality, this result can be generalized for a spin-$1/2$ dyon, which has both electric and magnetic charges and thus possesses both intrinsic electric and magnetic dipole moments. Furthermore, the relativistic quantum theory for a spin-$1/2$ dyon with arbitrary values of the gyromagnetic and gyroelectric ratios can be described by the Dirac-Pauli equation, which is the Dirac equation with augmentation for the anomalous electric and anomalous magnetic dipole moments. The FW transformation of the Dirac-Pauli Hamiltonian is shown, up to the 7th order again, to be also in accord with the classical Hamiltonian.' author: - 'Tsung-Wei Chen' - 'Dah-Wei Chiou' title: 'Foldy-Wouthuysen transformation for a Dirac-Pauli dyon and the Thomas-Bargmann-Michel-Telegdi equation' --- Introduction {#sec:introduction} ============ The relativistic quantum theory for a spin-$1/2$ point particle is described by the Dirac equation [@Dirac28]. The wavefunction used for the Dirac equation is the Dirac bispinor, which is composed of two Weyl spinors corresponding to the particle and antiparticle parts. Rigorously, the Dirac equation is self-consistent only in the context of quantum field theory, in which the particle-antiparticle pairs can be created. In the low-energy limit, if the relevant energy (the particle’s energy interacting with electromagnetic fields) is much smaller than the Dirac energy gap $E_g=2mc^2$ ($m$ is the particle mass), the probability of creation of particle-antiparticle pairs is negligible and the Dirac equation is adequate to describe the relativistic quantum dynamics of the spin-$1/2$ particle without taking into account the field-theory interaction to the antiparticle. The Foldy-Wouthuysen (FW) transformation is one of the methods developed to investigate the low-energy limit of the Dirac equation [@Foldy50].[^1] In the FW method, $1/E_g$ is treated as the small parameter; the Dirac Hamiltonian in the Dirac bispinor representation is block diagonalized up to a certain order of $1/E_g$ and the remaining off-diagonal matrices, which correspond to the particle-antiparticle interactions, are brought into the next order of $1/E_g$ and thus neglected. This is achieved by a series of successive unitary transformations performed on the Dirac Hamiltonian. Furthermore, a series of successive transformation in FW method can be reduced into one single transformation by the use of the Löwding partitioning method [@Lowdin51]. For a charged spin-$1/2$ particle subject to a non-explicitly time-dependent field, an exact FW transformation has been found by Eriksen [@Eriksen58], and the validity of the transformation is studied in [@Vries68]. Alternatively, the Dirac Hamiltonian can also be expanded in powers of Plank constant $\hbar$ [@Silenko03]. In this approach, the small parameter is not the particle’s energy (divided by $E_g$), but its wave length. A diagonalization procedure based on the expansion in powers of $\hbar$ has been constructed in [@Bliokh05; @Goss07]. In this procedure, the Berry phase correction can also be taken into account. Furthermore, the semiclassical $\hbar$-expansion enables us to describe the quantum corrections on the classical expression in strong fields [@Silenko08]. On the other hand, the classical (non-quantum) dynamics for a relativistic point particle endowed with charge and intrinsic spin in static and homogeneous electromagnetic fields is well understood. The orbital motion is govern by the relativistic Hamiltonian and the precession of the spin by the Thomas-Bargmann-Michel-Telegdi (TBMT) equation [@BMT59]. The relativistic Hamiltonian for the orbital motion plus the Hamiltonian obtained from the TBMT equation (called TBMT Hamiltonian) is expected to provide a low-energy description of the relativistic quantum theory. The conjecture that the low-energy limit of the Dirac Hamiltonian reduces to the classical orbital Hamiltonian plus the TBMT Hamiltonian has been suggested but remains to be affirmed. In order to investigate the consistency between the low-energy limit of the Dirac equation and the classical dynamics, we perform a series of FW transformations and expand the Dirac Hamiltonian up to terms of the 7th order in $1/E_g$. The electromagnetic fields are assumed to be static and homogeneous. Taking care of the relation between the kinematic momentum used in the Dirac Hamiltonian and the boost velocity used in the TBMT Hamiltonian, we show that the FW transformation of the Dirac Hamiltonian is in agreement with the classical orbital Hamiltonian plus the TBMT Hamiltonian for the case of the gyromagnetic ratio equal to 2. Through electromagnetic duality, this result can be generalized for a spin-$1/2$ dyon [@Shnir], which has both electric and magnetic charges and thus possesses both intrinsic electric and magnetic dipole moments (with both gyromagnetic and gyroelectric ratios equal to 2). To affirm the consistency to a broader extent, we need to show that the relativistic quantum theory of a spin-$1/2$ dyon with arbitrary values of the gyromagnetic and gyroelectric ratios also reduces to the classical counterparts as a low-energy limit. The relativistic quantum theory of a spin-$1/2$ dyon with the inclusion of anomalous magnetic dipole moment (AMM) and anomalous electric dipole moment (AEM) can be described by the Dirac-Pauli equation [@Silenko08; @Pauli41], which is the Dirac equation with augmentation for AMM and AEM. The FW transformation is performed on the Dirac-Pauli Hamiltonian, again up the 7th order in $1/E_g$, and the result confirms that it remains in agreement with the classical orbital Hamiltonian plus the TBMT Hamiltonian for arbitrary values of the gyromagnetic and gyroelectric ratios. This paper is organized as follows. In Sec. \[sec:dipoles\], we investigate the tensorial structure of the orbital and intrinsic dipole moments. In Sec. \[sec:TBMT\], we briefly review the classical orbital Hamiltonian and the TBMT equation. In Sec. \[sec:Dirac\], we perform the FW transformation on the Dirac Hamiltonian for a spin-$1/2$ dyon and show that it agrees with the TBMT equation for the case of the gyromagnetic and gyroelectric ratios equal to 2. Later in Sec. \[sec:Dirac Pauli\], we perform the FW transformation on the Dirac-Pauli Hamiltonian and show that it again agrees with the TBMT equation even with the inclusion of AMM and AEM. Finally, the conclusions are summarized and discussed in Sec. \[sec:conclusions\]. Some calculational details are supplemented in Appendices \[sec:FW transform\] and \[sec:derivation\]. Orbital and intrinsic dipole moments {#sec:dipoles} ==================================== For a general Lorentz transformation from the primed (boosted) frame to the unprimed (laboratory) frame, the transformation of a 4-vector $\mathrm{k}^\mu=(\mathrm{k}^0,\mathbf{k})$ is given by [@Jackson]: $$\label{LTboost} \begin{split} &\mathrm{k}^{0}=\gamma(\mathrm{k}'^{0}+\boldsymbol{\beta}\cdot\mathbf{k'}),\\ &\mathbf{k}=\mathbf{k'}+\frac{\gamma-1}{\beta^2}(\boldsymbol{\beta}\cdot\mathbf{k'})\boldsymbol{\beta}+\gamma \mathrm{k}'^0\boldsymbol{\beta},\\ \end{split}$$ where $\mathbf{v}=c\boldsymbol{\beta}$ is the boost velocity of the primed frame relative to the unprimed frame, $\gamma$ is the Lorentz factor $\gamma=1/\sqrt{1-\beta^2}$ and $\beta=|\boldsymbol{\beta}|$. In the primed system, let us consider the case that charge and current densities satisfy the conditions: $$\label{Ncondition} \int_{V'}d^3x'\rho'=0, \qquad\int_{V'}d^3x'\mathbf{J}'=0.$$ The vanishing of the total charge means that the system is *neutral*, and the vanishing of the total current is a consequence of the *static* condition: $\partial\rho'/\partial t'=-\nabla'\cdot\mathbf{J}'=0$.[^2] Because the charge density and current density form a 4-vector $J^\mu=(c\rho,\mathbf{J})$, it can be shown that the same conditions also hold in the unprimed system: $$\int_Vd^3x\rho=0, \qquad \int_Vd^3x\mathbf{J}=0.$$ In the unprimed frame, the magnetic dipole moments $\mathbf{m}$ is defined as $$\mathbf{m}=\int_Vd^3x\boldsymbol{\mu}_{m},$$ where $$\label{mdmd} \boldsymbol{\mu}_m=\frac{1}{2c}(\mathbf{x}\times\mathbf{J})$$ is the magnetic dipole moment density, and the electric dipole moment $\mathbf{p}$ is defined as $$\mathbf{p}=2\int_Vd^3x\boldsymbol{\mu}^c_p,$$ where $$\label{cedmd} \boldsymbol{\mu}^c_p=\frac{1}{2}\mathbf{x}\rho$$ is the *canonical* electric dipole moment density (the extra factor of 2 is introduced for later convenience). In the primed system, the definitions of both dipole moments are the same as those in the unprimed system. Using Eq. (\[LTboost\]), Eq. (\[mdmd\]) can be written as $$\begin{split} \boldsymbol{\mu}_m=&\boldsymbol{\mu}'_m+\frac{\gamma-1}{\beta^2}\boldsymbol{\beta}\times(\boldsymbol{\mu}'_m\times\boldsymbol{\beta})\\ &+\frac{1}{2}\gamma(\mathbf{x}'c\rho'-x'^0\mathbf{J}')\times\boldsymbol{\beta}, \end{split}$$ where $\boldsymbol{\mu}'_m=\mathbf{x}'\times\mathbf{J}'/2$ is the magnetic dipole density in the primed system. It is interesting to note that if we integrate the term $\mathbf{x}'c\rho'-x'^0\mathbf{J}'$ in the primed system, we obtain $$\int_{V'}d^3x'(\mathbf{x}'c\rho'-x'^0\mathbf{J}')=\int_{V'}d^3x'\mathbf{x}'c\rho' =c\mathbf{p}',$$ where the neutral condition \[Eq. (\[Ncondition\])\] has been used. This suggests that we can define the *tensorial* electric dipole moment as $$\boldsymbol{\mu}_p=\frac{1}{2c}\left(\mathbf{x}J^0-x^0\mathbf{J}\right)$$ so that the dipole moment can be defined as a second rank antisymmetric tensor: $$\label{DipoleTensor} M^{\mu\nu}=\frac{1}{2c}(x^{\mu}J^{\nu}-x^{\nu}J^{\mu}).$$ The canonical and tensorial dipole moments densities yield the same (integrated) dipole moments, because the neutral condition ensures that the integration of the second term $x^0\mathbf{J}$ vanishes in the unprimed system. The components of the second rank tensor $M^{\mu\nu}$ are $$\begin{split} &M^{0i}=\frac{1}{2c}(x^0J^i-x^iJ^0)=-\mu_p^i,\\ &M^{ij}=\frac{1}{2c}(x^iJ^j-x^jJ^i)=\epsilon_{ijk}\mu_m^k. \end{split}$$ Consequently, the Lorentz transformation between $(\boldsymbol{\mu}_p,\boldsymbol{\mu}_m)$ and $(\boldsymbol{\mu}'_p,\boldsymbol{\mu}'_m)$ is of the form $$\label{LTdipole} \begin{split} &\boldsymbol{\mu}_p=\gamma(\boldsymbol{\mu}'_p+\boldsymbol{\beta}\times\boldsymbol{\mu}'_m)-\frac{\gamma^2}{\gamma+1}\boldsymbol{\beta}(\boldsymbol{\beta}\cdot\boldsymbol{\mu}'_p),\\ &\boldsymbol{\mu}_m=\gamma(\boldsymbol{\mu}'_m-\boldsymbol{\beta}\times\boldsymbol{\mu}'_p)-\frac{\gamma^2}{\gamma+1}\boldsymbol{\beta}(\boldsymbol{\beta}\cdot\boldsymbol{\mu}'_m). \end{split}$$ The transformation \[Eq. (\[LTdipole\])\] is exactly the same as that for the electric and magnetic fields if we take the replacement rules: $\mathbf{E}\leftrightarrow \boldsymbol{\mu}_p$ $\mathbf{B}\leftrightarrow-\boldsymbol{\mu}_m$. The corresponding transformation for the (integrated) dipole moments is given by $$\label{LTdipole2} \begin{split} &\frac{\mathbf{p}}{2}=\gamma^2\left[\frac{\mathbf{p}'}{2}+\boldsymbol{\beta}\times\mathbf{m}'-\frac{\gamma}{\gamma+1}\boldsymbol{\beta}(\boldsymbol{\beta}\cdot\frac{\mathbf{p}'}{2})\right],\\ &\mathbf{m}=\gamma^2\left[\mathbf{m}'-\boldsymbol{\beta}\times\frac{\mathbf{p}'}{2}-\frac{\gamma}{\gamma+1}\boldsymbol{\beta}(\boldsymbol{\beta}\cdot\mathbf{m}')\right], \end{split}$$ where $d^3x=\gamma d^3x'$ is used. Since the extra factor $\gamma$ arises in the right hand side of Eq. (\[LTdipole2\]) due to the spatial integral, $\mathbf{p}/2$ and $\mathbf{m}$ do not transform covariantly and thus do not form a second rank tensor unlike $\boldsymbol{\mu}_p$ and $\boldsymbol{\mu}_m$. In the case with only proper electric dipole moment and no proper magnetic dipole moment (i.e. $\mathbf{p}'\neq0$ and $\mathbf{m}'=0$), when boosted, the electric dipole will result in a magnetic dipole moment $\mathbf{m}=-\gamma^2(\boldsymbol{\beta}\times\mathbf{p}'/2)$ in the unprimed system. This can be understood as follows: If we think $\boldsymbol{p}'$ as two endpoints separated by a short distance and charged with $+q$ and $-q$, in the unprimed system, the positive and negative charges acquire a velocity and give rise to currents in opposite directions, thus resulting in a magnetic dipole moment. In an inhomogeneous magnetic field, a moving object with only proper electric dipole moment can feel the magnetic force $\mathbf{F}=(\mathbf{m}\cdot\nabla)\mathbf{B}$. On the other hand, in the case with only proper magnetic dipole moment and no proper electric dipole moment (i.e. $\mathbf{m}'\neq0$ and $\mathbf{p}'=0$), when boosted, the magnetic dipole will result in an electric dipole moment $\mathbf{p}=2\gamma^2(\boldsymbol{\beta}\times\mathbf{m}')$ in the unprimed system. This is due to the fact that in the unprimed system the charge density $\rho$ arises through the Lorentz transformation even if the charge density is zero in the primed system ($\rho'=0$). The charge density in the primed system $c\rho=\gamma\boldsymbol{\beta}\cdot\mathbf{J}'$ is positive (negative) when the current is parallel (anti-parallel) to the boost velocity; therefore, as a magnetic dipole can be thought as a small current loop, the small current loop in the primed system gives rise to opposite charges separated by a short distance in the unprimed system, thus resulting in an electric dipole moment. In an inhomogeneous electric field, a moving object with only proper magnetic dipole moment can feel the electric force $\mathbf{F}=(\mathbf{p}\cdot\nabla)\mathbf{E}$. The dipole moments considered above are *orbital* in the sense that they are sourced by the orbital distribution of $J^\mu(x)$. On the other hand, a point particle can give rise to an *intrinsic* dipole moment if it is charged and endowed with intrinsic spin. The fact that $\boldsymbol{\mu}_p$ and $\boldsymbol{\mu}_m$ form an antisymmetric tensor $M^{\mu\nu}$ suggests that the intrinsic spin $\mathbf{s}$ can be generalized to a second-rank antisymmetric tensor $S^{\mu\nu}$, which gives the intrinsic dipole moments as $$\label{M and S} M^{\mu\nu}=\frac{g_ee}{2mc}\,S^{\mu\nu},$$ where $e$ is the electric charge of the particle, $m$ the mass and $g_e$ the *gyromagnetic ratio*. The spin has only three independent components; thus $S^{\mu\nu}$ is dual to an axial 4-vector $S^\alpha=(S^0,\mathbf{S})$ via $$S^{\mu\nu}=\frac{1}{c}\,\epsilon^{\mu\nu\alpha\beta}U_\alpha S_\beta$$ and conversely $$S^\alpha=\frac{1}{2c}\,\epsilon^{\alpha\beta\gamma\delta}U_\beta S_{\gamma\delta},$$ where $U^\alpha$ is the particle’s 4-velocity. The 4-vector $S^\alpha$ reduces to the spin $\mathbf{s}$ in the particle’s rest frame; i.e., $S'^\alpha=(S'^0,\mathbf{S}')=(0,\mathbf{s})$. The vanishing of the time-component in the particle’s rest frame is imposed by the covariant constraint: $$U_\alpha S^\alpha=0.$$ In the particle’s rest frame, $U^{\prime\alpha}=(c,0,0,0)$ and Eq. (\[M and S\]) yields $$\boldsymbol{\mu}'_m=\frac{g_e e}{2mc}\,\mathbf{s}, \qquad \boldsymbol{\mu}'_p=0.$$ Therefore, the intrinsic spin gives only the proper intrinsic magnetic dipole and no proper intrinsic electric dipole. In order to have both proper intrinsic magnetic and electric dipoles, we consider a *dyon* particle [@Shnir], which possesses both electric charge $e$ and magnetic charge $\tilde{e}$, and Eq. (\[M and S\]) is generalized as $$\label{M and S S} M^{\mu\nu}=M^{\mu\nu}_e+M^{\mu\nu}_{\tilde{e}}=\frac{g_ee}{2mc}\,S^{\mu\nu} +\frac{g_{\tilde{e}}\tilde{e}}{2mc}\,\tilde{S}^{\mu\nu},$$ where $g_{\tilde{e}}$ is the *gyroelectirc ratio* and $$\tilde{S}^{\mu\nu}:=\frac{1}{2}\,\epsilon^{\mu\nu\alpha\beta}S_{\alpha_\beta}$$ is the dual of $S^{\mu\nu}$. In the rest frame, Eq. (\[M and S S\]) yields both magnetic and electric dipoles: $$\boldsymbol{\mu}^{\prime e}_m=\frac{g_e e}{2mc}\,\mathbf{s}, \qquad \boldsymbol{\mu}^{\prime \tilde{e}}_p=-\frac{g_{\tilde{e}} \tilde{e}}{2mc}\,\mathbf{s}.$$ The Thomas-Bargmann-Michel-Telegdi equation {#sec:TBMT} =========================================== Consider a relativistic point particle endowed with electric charge and intrinsic spin subject to static and homogeneous electromagnetic fields. The orbital motion of the particle is described by $$\label{covariant eom 1} \frac{dU^\alpha}{d\tau}=\frac{e}{mc}F^{\alpha\beta}U_\beta$$ and the precession of the spin is govern by the TBMT equation [@BMT59]: $$\label{covariant eom 2} \frac{dS^\alpha}{d\tau}=\frac{e}{mc} \left[ \frac{g_e}{2}\,F^{\alpha\beta}S_\beta +\frac{1}{c^2}\left(\frac{g_e}{2}-1\right)U^\alpha \left(S_\lambda F^{\lambda\mu} U_\mu\right) \right].$$ Equation (\[covariant eom 1\]) in the covariant form can be shown to be equivalent to the Hamilton’s equations : $$\begin{split} \frac{d\mathbf{x}}{dt}&=\{\mathbf{x},H_\mathrm{oribt}\},\\ \frac{d\mathbf{p}}{dt}&=\{\mathbf{p},H_\mathrm{orbit}\} \end{split}$$ in the unprimed frame, where $\mathbf{p}$ is the conjugate momentum to $\mathbf{x}$ and the Hamiltonian $H_\mathrm{orbit}$ governing the orbital motion is given by $$\label{H orbit} H_\mathrm{orbit}(\mathbf{x},\mathbf{p}) =\sqrt{\left(c\,\mathbf{p}-e\mathbf{A}(\mathbf{x})\right)^2+m^2c^4}\, +e\,\phi(\mathbf{x})$$ with $A^\alpha=(\phi,\mathbf{A})$ being the 4-vector potential for the electromagnetic field $F^{\mu\nu}$. (See Sec. 12.1 in [@Jackson] for more details.) On the other hand, Eq. (\[covariant eom 2\]) leads to $$\label{Thomas} \frac{d\mathbf{s}}{dt}=\frac{e}{mc}\,\mathbf{s}\times\mathbf{F}(\mathbf{x})$$ with $$\label{ThomasF} \begin{split} \mathbf{F}&=\left(\frac{g_e}{2}-1+\frac{1}{\gamma}\right)\mathbf{B}-\left(\frac{g_e}{2}-1\right)\frac{\gamma}{\gamma+1}(\boldsymbol{\beta}\cdot\mathbf{B})\boldsymbol{\beta}\\ &~~-\left(\frac{g_e}{2}-\frac{\gamma}{\gamma+1}\right)\boldsymbol{\beta}\times\mathbf{E}, \end{split}$$ which gives the spin precession with respect to the time of the unprimed frame. (See Sec. 11.11 in [@Jackson] for more details.) Because $\{s_i,s_j\}=\epsilon_{ijk}s_k$, Eq. (\[Thomas\]) can be recast as the Hamilton’s equation: $$\frac{d\mathbf{s}}{dt}=\{\mathbf{s},H_\mathrm{spin}\}$$ with $$\label{H dipole} H_\mathrm{spin}(\mathbf{x},\mathbf{s})=-\frac{e}{mc}\,\mathbf{s}\cdot \mathbf{F}(\mathbf{x})$$ called the TBMT Hamiltonian, which governs the precession of the electric dipole subject to a static and homogeneous field. In the low-speed limit ($\beta\ll 1$), we have $\gamma\approx1$ and Eq. (\[H dipole\]) gives $$\begin{aligned} \label{low speed H dipole} H_\mathrm{spin} &\approx&-\frac{e}{2mc}\,\mathbf{s}\cdot \bigg[ g_e\mathbf{B} -\left(\frac{g_e}{2}-1\right)(\boldsymbol{\beta}\cdot\mathbf{B})\boldsymbol{\beta}\nonumber\\ &&\qquad\qquad-(g_e-1)\boldsymbol{\beta}\times\mathbf{E} \bigg].\end{aligned}$$ The first term in Eq. (\[low speed H dipole\]) is the interaction energy of the magnetic moment $\boldsymbol{\mu}^{\prime e}_m$ in the magnetic field, which accounts for the anomalous Zeeman effect. The second term corresponds to the change rate of the longitudinal polarization, which vanishes in the case of $g_e=2$. The third term is the spin-orbit interaction (the interaction of the boosted electric dipole $\boldsymbol{\mu}^{e}_p\approx\boldsymbol{\beta}\times\boldsymbol{\mu}^{\prime e}_m$ coupled to the electric field) plus the correction for the Thomas precession. By treating $\mathbf{x}$, $\mathbf{p}$ and $\mathbf{s}$ as independent phase space variables, the total Hamiltonian is given by[^3] $$\label{H total} H(\mathbf{x},\mathbf{p},\mathbf{s})=H_\mathrm{orbit}(\mathbf{x},\mathbf{p}) +H_\mathrm{spin}(\mathbf{x},\mathbf{s}).$$ If the particle has both electric charge $e$ and magnetic charge $\tilde{e}$ (i.e. the particle is a *dyon*), Eq. (\[H orbit\]) and Eq. (\[H dipole\]) are modified with the inclusion of the dual counterparts; i.e. $$\begin{aligned} \label{H orbit dyon} H_\mathrm{orbit}(\mathbf{x},\mathbf{p}) &=&\sqrt{\left(c\,\mathbf{p}-e\mathbf{A}(\mathbf{x})-\tilde{e}\tilde{\mathbf{A}}(\mathbf{x}) \right)^2+m^2c^4}\nonumber\\ &&+\,e\phi(\mathbf{x})+\tilde{e}\tilde{\phi}(\mathbf{x})\end{aligned}$$ and $$\label{H dipole dyon} H_\mathrm{spin}(\mathbf{x},\mathbf{s})=-\frac{e}{mc}\,\mathbf{s}\cdot \mathbf{F}(\mathbf{x}) -\frac{\tilde{e}}{mc}\,\mathbf{s}\cdot \tilde{\mathbf{F}}(\mathbf{x})$$ with $$\label{ThomasF dual} \begin{split} \tilde{\mathbf{F}}&=\left(\frac{g_{\tilde{e}}}{2}-1+\frac{1}{\gamma}\right)\tilde{\mathbf{B}} -\left(\frac{g_{\tilde{e}}}{2}-1\right) \frac{\gamma}{\gamma+1}(\boldsymbol{\beta}\cdot\tilde{\mathbf{B}})\boldsymbol{\beta}\\ &~~-\left(\frac{g_{\tilde{e}}}{2}-\frac{\gamma}{\gamma+1}\right) \boldsymbol{\beta}\times\tilde{\mathbf{E}}, \end{split}$$ where $\tilde{A}=(\tilde{\phi},\tilde{\mathbf{A}})$ is the dual 4-vector potential which gives $\tilde{F}^{\mu\nu}=\partial^\mu \tilde{A}^\nu-\partial^\nu \tilde{A}^\mu$ and $\tilde{F}^{\mu\nu}:=1/2\,\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$ is the dual field strength (i.e. $\tilde{\mathbf{B}}=-\mathbf{E}$ and $\tilde{\mathbf{E}}=\mathbf{B}$). Equation (\[covariant eom 1\]) and the TBMT equation given in Eq. (\[covariant eom 2\]) are derived as the requirement of covariant is considered. They are classical (non-quantum) equations and we wonder whether the Hamiltonian given in Eq. (\[H total\]) is consistent with that in the relativistic quantum theory for a charged point particle with intrinsic spin. The relativistic quantum theory of a spin-$1/2$ particle is described by the Dirac equation. The Dirac bispinor however has both the particle and antiparticle components, which are entangled by the Dirac equation. In order to compare with the TBMT equation, we consider the low-energy limit in which the relevant energy is much smaller than the Dirac energy gap $E_g$ and the FW transformation is used to block-diagonalize the Dirac Hamiltonian. In Sec. [\[sec:Dirac\]]{}, we will show that the FW transformation of the Dirac Hamiltonian indeed agrees perfectly with the TBMT equation up to the 7th order of $1/E_g$ with the intrinsic spin given by $\mathbf{s}=\hbar\,\boldsymbol{\sigma}/2$ ($\sigma_i$ are the Pauli matrices) and the gyromagnetic ratio given by $g_e=2$. This can be easily generalized for a Dirac dyon by adding the magnetic charge (and we will have $g_e=g_{\tilde{e}}=2$). As the Dirac equation always yields $g_e=2$, we will not see the second term in Eq. (\[ThomasF\]), which accounts for change of the longitudinal polarization. In order to see that the quantum theory is in accord with the TBMT equation even for the case of $g_e\neq2$, we study the Dirac-Pauli equation in Sec. \[sec:Dirac Pauli\] with the inclusion of anomalous dipole moments. The results again affirms the consistency between the FW transformation of the Dirac-Pauli Hamiltonian and the TBMT equation up to the 7th order of $1/E_g$. Foldy-Wouthuysen transformation for the Dirac Hamiltonian {#sec:Dirac} ========================================================= The relativistic quantum theory of a Dirac particle is described by the Dirac equation $$i\hbar\frac{\partial}{\partial t}|\psi\rangle=H|\psi\rangle,$$ where the Dirac bispinor $|\psi\rangle=(\chi,\varphi)^T$ is composed of two 2-component Weyl spinors $\chi$ and $\varphi$ corresponding to the particle and antiparticle parts. The Dirac Hamiltonian is given by $$\label{H} H=mc^2\matrixbeta+c\,\matrixboldalpha\cdot\boldsymbol{\Pi}+V,$$ where the $4\times4$ matrices $\matrixbeta=\sigma_z\otimes\mathbf{1}$ and $\matrixalpha_i=\sigma_x\otimes\sigma_i$ are given in the Pauli-Dirac representation [@Dirac28] and satisfy[^4] $$\begin{split} &\{\matrixbeta,\matrixalpha_i\}=0,\\ &\{\matrixalpha_i,\matrixalpha_j\}=2\delta_{ij},\\ &\matrixalpha_i^2=\matrixbeta^2=\mathbf{1}. \end{split}$$ The mass of the particle is $m$ and $\boldsymbol{\Pi}$ is the kinetic momentum. In order to have the proper intrinsic electric dipole moment and the proper intrinsic magnetic dipole moment at the same time, we consider a Dirac dyon (i.e. a Dirac particle with both electric charge $e$ and magnetic charge $\tilde{e}$). For a dyon [@Shnir], the kinetic momentum is given by $$\boldsymbol{\Pi}=\mathbf{p}-\frac{e}{c}\mathbf{A}-\frac{\tilde{e}}{c}\widetilde{\mathbf{A}},$$ and the scalar potential $V$ is composed of electric and magnetic monopole potentials: $$V=e\phi+\tilde{e}\widetilde{\phi}\,.$$ In the following, we will first perform the successive FW transformations of the Dirac Hamiltonian up to the 7th order of $1/E_g$ in Sec. \[sec:FW\], and later show that the results agree with the TBMT equation in Sec. \[sec:relation to TBMT\]. Foldy-Wouthuysen transformation {#sec:FW} ------------------------------- In order to perform the FW transformation [@Foldy50], we have to rewrite the Dirac Hamiltonian to the form: $$\label{H_D2} H=E_g\frac{\matrixbeta}{2}+\Omega_o+\Omega_E,$$ where $E_g=2mc^2$ is the Dirac energy gap, and the odd matrix $\Omega_o$ and the even matrix $\Omega_E$ are defined as $$\label{OE} \{\matrixbeta,\Omega_o\}=0,\qquad[\matrixbeta,\Omega_E]=0.$$ In the case of Eq. (\[H\]), we have $$\label{OE1} \Omega_o=c\,\matrixboldalpha\cdot\boldsymbol{\Pi}, \qquad\Omega_E=V.$$ The resulting effective hamiltonian $H_\mathrm{FW}$ can be obtained by the successive unitary transformations which partitioning off the odd matrices to a higher order. In general, we can use the FW matrix $U_\mathrm{FW}$ as a single transformation, and expand the exponent of the matrix in powers of $1/E_g$; this is the well-known Löwdin partitioning method [@Lowdin51]. It can be shown that the FW transformation of Eq. (\[H\_D2\]), namely the transformed Hamiltonian denoted as $H_\mathrm{FW}=U_\mathrm{FW}HU_\mathrm{FW}^{-1}$, up to terms of the 7th order in $1/E_g$ can be written as (see Appendix \[sec:FW transform\]) $$\label{HFW} H_\mathrm{FW}=\frac{\matrixbeta E_g}{2}+\Omega_E+\sum_{\ell=1}^{6}H^{(\ell)}_\mathrm{FW}+o(1/E_g^7),$$ where the first four terms $H_\mathrm{FW}^{(\ell)},~\ell=1,2,3,4$ are given by \[HFW\_list\] $$\begin{aligned} H^{(1)}_\mathrm{FW}&=\frac{\matrixbeta\Omega_o^2}{E_g}\label{HFW_list1},\\ H^{(2)}_\mathrm{FW}&=\frac{1}{E_g^2}\left(\frac{\mathcal{W}}{2}\right)\label{HFW_list2},\\ H^{(3)}_\mathrm{FW}&=\frac{1}{E_g^3}\left\{-\matrixbeta\Omega_o^4+\matrixbeta \left(\matrixbeta\mathcal{D}\right)^2\right\}\label{HFW_list3},\\ H^{(4)}_\mathrm{FW}&=\frac{1}{E_g^4}\left(\frac{1}{24}[[\Omega_o,\mathcal{W}] \Omega_o]-\frac{4}{3}[\mathcal{D},\Omega_o^3]\right)\label{HFW_list4},\end{aligned}$$ and the operators $\mathcal{W}$ and $\mathcal{D}$ are defined as \[DandW\] $$\begin{aligned} &\mathcal{D}=[\Omega_o,\Omega_E],\\ &\mathcal{W}=[\mathcal{D},\Omega_o].\end{aligned}$$ By using Eq. (\[OE1\]), it can be shown the three Hamiltonians $H_\mathrm{FW}^{(\ell=1,2,3)}$ are in agreement with the previous results [@Foldy50; @Froh93]. The term of the 5th order is given by $$\label{HFW_list5} \begin{split} E_g^5H^{(5)}_\mathrm{FW}&=\frac{1}{144}[(\matrixbeta\Omega_o)_{(5)},\Omega_o]+\frac{1}{2}\mathop{\sum_{\ell,m=1}^{3}}\limits_{(\ell+m=4)}[\matrixbeta\mathcal{O}^{(\ell)},\mathcal{O}^{(m)}]\\ &~~+\frac{1}{2}\mathop{\sum_{\ell,m=1}^{2}\sum_{n=0}^{1}}\limits_{(\ell+m+n=3)}[\matrixbeta\mathcal{O}^{(\ell)},[\matrixbeta\mathcal{O}^{(m)},h^{(n)}]], \end{split}$$ where the subscript $(5)$ in the commutator $[(\matrixbeta\Omega_o)_{(5)},\Omega_o]$ indicates that the commutation of $\matrixbeta\Omega_o$ with $\Omega_o$ is performed successively by five times; i.e., $[(\matrixbeta\Omega_o)_{(5)},\Omega_o] = [\matrixbeta\Omega_o,[\matrixbeta\Omega_o,[\matrixbeta\Omega_o,[\matrixbeta\Omega_o[\matrixbeta\Omega_o,\Omega_o]]]]]$. The term of the 6th order is $$\label{HFW_list6} \begin{split} E_g^6H^{(6)}_\mathrm{FW}&=\frac{1}{720}[(\matrixbeta\Omega_o)_{(6)},\Omega_E]+\frac{1}{2}\mathop{\sum_{\ell,m=1}^{4}}\limits_{(\ell+m=5)}[\matrixbeta\mathcal{O}^{(\ell)},\mathcal{O}^{(m)}]\\ &~~+\frac{1}{2}\mathop{\sum_{\ell,m=1}^{3}\sum_{n=0}^{2}}\limits_{(\ell+m+n=4)}[\matrixbeta\mathcal{O}^{(\ell)},[\matrixbeta\mathcal{O}^{(m)},h^{(n)}]], \end{split}$$ where the odd matrices $\mathcal{O}^{(\ell)}$ for $\ell=1,2,3,4$ are given by $$\label{listO} \begin{split} &\mathcal{O}^{(1)}=\matrixbeta\mathcal{D},\quad\mathcal{O}^{(2)}=-\frac{4}{3}\Omega_o^3,\\ &\mathcal{O}^{(3)}=\frac{1}{6}\matrixbeta[\Omega_o,\mathcal{W}],\quad \mathcal{O}^{(4)}=\frac{8}{15}\Omega_o^5, \end{split}$$ and the even matrices $h^{(n)}$ for $n=0,1,2$ are $$\begin{split} &h^{(0)}=\Omega_E,\quad h^{(1)}=\matrixbeta\Omega_o^2,\\ &h^{(2)}=\frac{\mathcal{W}}{2}. \end{split}$$ To obtain the FW transformed Hamiltonian $H_\mathrm{FW}^{(\ell)}$ up to the 7th order of $1/E_g$, we need only three successive transformations $U_\mathrm{FW}=\exp(S_3)\exp(S_2)\exp(S_1)$ (see Appendix \[sec:FW transform\]), and it can be shown that $S_1$, $S_2$ and $S_3$ are all anti-hermitian matrices. To simplify the calculation, some restrictions and assumptions are made. The electromagnetic field is assumed to be static, as has been used in obtaining Eq. (\[HFW\]). In order to demonstrate the equivalence clearly between the TBMT Hamiltonian and $H_\mathrm{FW}$, we further assume that the external fields $\mathbf{E}$ and $\mathbf{B}$ are homogeneous, and thus the field gradient vanishes. Furthermore, the terms proportional to products of field strengths, such as $E_iE_j$, $E_iB_j$ and $B_iB_j$, are all neglected as a good approximation for weak fields. We now evaluate each term of $H^{(\ell)}_\mathrm{FW}$. The kinetic term $\Omega_o^2/E_g$ in Eq. (\[HFW\_list1\]) can be written as $$\label{Omega2} \begin{split} \frac{\Omega_o^2}{E_g}&=\frac{(c\matrixboldalpha\cdot\boldsymbol{\Pi})^2}{2mc^2}\\ &=\frac{1}{2m}\left\{|\boldsymbol{\Pi}|^2+i\boldsymbol{\Sigma}\cdot\left(\boldsymbol{\Pi}\times\boldsymbol{\Pi}\right)\right\}, \end{split}$$ where $[\matrixalpha_i,\matrixalpha_j]=2i\epsilon_{ijk}\Sigma_k$ is used. By using the definition of magnetic field $\mathbf{B}=\nabla\times\mathbf{A}$ and dual magnetic field $\widetilde{\mathbf{B}}=\nabla\times\widetilde{\mathbf{A}}$, we have $c\,\boldsymbol{\Pi}\times\boldsymbol{\Pi}=i\hbar(e\mathbf{B}+\tilde{e}\widetilde{\mathbf{B}})$. By applying the duality $\widetilde{\mathbf{B}}=-\mathbf{E}$ in Eq. (\[Omega2\]), Eq. (\[HFW\_list1\]) then gives $$\label{HFW1} H^{(1)}_\mathrm{FW}=\frac{\matrixbeta|\boldsymbol{\Pi}|^2}{2m} -\matrixbeta\left(\frac{e\hbar}{2mc}\boldsymbol{\Sigma}\right)\cdot\mathbf{B} -\matrixbeta\left(-\frac{\tilde{e}\hbar}{2mc}\boldsymbol{\Sigma}\right)\cdot\mathbf{E}.$$ The second term of Eq. (\[HFW1\]) is the Zeeman Hamiltonian for an electron (with $e=-|e|$) [@Sakurai], and the third term is its duality. It is interesting to note that $\tilde{e}\hbar\boldsymbol{\Sigma}/2mc$ plays the role of electric dipole moment because it couples to the electric field. In this sense, we can define the (proper) intrinsic electric dipole moment ($\boldsymbol{\mu}_p^{\prime\tilde{e}}$) and (proper) intrinsic magnetic dipole moment ($\boldsymbol{\mu}_m^{\prime e}$) as \[Intrinsicdipoles\] $$\begin{aligned} \boldsymbol{\mu}_m^{\prime e}&=\frac{e\hbar}{2mc}\boldsymbol{\Sigma}\label{IMDM},\\ \boldsymbol{\mu}_p^{\prime\tilde{e}}&= -\frac{\tilde{e}\hbar}{2mc}\boldsymbol{\Sigma}\label{IEDM}.\end{aligned}$$ Equation (\[IMDM\]) implies that the dyon’s intrinsic gyromagnetic ratio $g_e=2$ for the Dirac Hamiltonian. We also find the same gyroelectric ratio $g_{\tilde{e}}=2$ for the dyon’s intrinsic electric dipole moment. We now focus on the 2nd-order Hamiltonian $H_\mathrm{FW}^{(2)}$ in which the spin-orbit coupled term is included. It can be shown that $\mathcal{W}$ is given by $\mathcal{W}=-2c^2\hbar\boldsymbol{\Sigma}\cdot ((e\mathbf{E}+\tilde{e}\widetilde{\mathbf{E}})\times\boldsymbol{\Pi})$ and Eq. (\[HFW\_list2\]) can be written as $$\label{HFW2} H_\mathrm{FW}^{(2)} =-\frac{1}{2}\mathbf{E}\cdot \left(\frac{\boldsymbol{\Pi}}{mc}\times\boldsymbol{\mu}_m^{\prime e}\right) -\frac{1}{2}\mathbf{B}\cdot \left(-\frac{\boldsymbol{\Pi}}{mc}\times\boldsymbol{\mu}_p^{\prime \tilde{e}}\right),$$ where the duality $\tilde{\mathbf{E}}=\mathbf{B}$ is used. The first term of Eq. (\[HFW2\]) is the spin-orbit interaction for an electron [@Sakurai]. On the other hand, we neglect the terms proportional to products of field strengths in evaluating the two terms in Eq. (\[HFW\_list3\]), and thus we can obtain $$\label{HFW3} \begin{split} H_\mathrm{FW}^{(3)} %&=-\frac{|\boldsymbol{\Pi}|^4}{8m^3c^2}\\ %&~~~~+\frac{1}{4}\left[\left(\frac{|\boldsymbol{\Pi}|}{mc}\right)^2\left(\boldsymbol{\mu}_m^e\cdot\mathbf{B}\right)+\left(\boldsymbol{\mu}_m^e\cdot\mathbf{B}\right)\left(\frac{|\boldsymbol{\Pi}|}{mc}\right)^2\right]\\ %&~~~~+\frac{1}{4}\left[\left(\frac{|\boldsymbol{\Pi}|}{mc}\right)^2(\boldsymbol{\mu}_p^{\tilde{e}}\cdot\mathbf{E})+(\boldsymbol{\mu}_p^{\tilde{e}}\cdot\mathbf{E})\left(\frac{|\boldsymbol{\Pi}|}{mc}\right)^2\right]\\ &\approx-\frac{\matrixbeta|\boldsymbol{\Pi}|^4}{8m^3c^2} +\frac{1}{2}\matrixbeta\left(\frac{|\boldsymbol{\Pi}|}{mc}\right)^2 \left(\boldsymbol{\mu}_m^{\prime e}\cdot\mathbf{B}\right)\\ &~~~~+\frac{1}{2}\matrixbeta \left(\frac{|\boldsymbol{\Pi}|}{mc}\right)^2 (\boldsymbol{\mu}_p^{\prime \tilde{e}}\cdot\mathbf{E}), \end{split}$$ where the assumption of homogeneous electromagnetic fields is used, and thus the operator $|\boldsymbol{\Pi}|^2$ commutes with the magnetic field $\mathbf{B}$. If the magnetic charge $\tilde{e}$ vanishes, the first term of Eq. (\[HFW3\]) is the relativistic mass correction that contributes to the spectrum of fine structure [@Sakurai]. The second term of Eq. (\[HFW3\]) is the relativistic correction to the Zeeman Hamiltonian appearing in $H_\mathrm{FW}^{(1)}$ \[Eq. (\[HFW1\])\]. For the 4th order $H^{(4)}_\mathrm{FW}$, it can be shown that $$\label{HFW_4_1} \begin{split} &\frac{1}{24}[[\Omega_o,\mathcal{W}],\Omega_o]-\frac{4}{3}[[\Omega_o,\Omega_e],\Omega_o^3]\\ &=-\frac{11}{8}(\Omega_o^2\mathcal{W}+\mathcal{W}\Omega_o^2) -\frac{5}{4}\Omega_o\mathcal{W}\Omega_o. \end{split}$$ If we neglect all terms proportional to products of electromagnetic fields, one can show that (see Appendix \[sec:derivation\]): \[HFW\_4\_2\] $$\begin{aligned} &(\Omega_o^2\mathcal{W}+\mathcal{W}\Omega_o^2)\approx 2c^2|\boldsymbol{\Pi}|^2\mathcal{W},\label{HFW_4_2(1)}\\ &\Omega_o\mathcal{W}\Omega_o\approx-c^2|\boldsymbol{\Pi}|^2\mathcal{W}\label{HFW_4_2(2)}.\end{aligned}$$ Note that there is a minus sign in Eq. (\[HFW\_4\_2(2)\]). The 4th-order term Eq. (\[HFW\_list4\]) with substitution of Eqs. (\[HFW\_4\_1\]) and (\[HFW\_4\_2\]) can be written as $$\label{HFW4} H^{(4)}_\mathrm{FW}\approx\frac{c^2}{E_g^4} \left(-\frac{3}{2}\right)|\boldsymbol{\Pi}|^2\mathcal{W} =-\frac{3}{4}\left(\frac{|\boldsymbol{\Pi}|}{mc}\right)^2H^{(2)}_\mathrm{FW}.$$ It is interesting to note that the 4th order Hamiltonian $H_\mathrm{FW}^{(4)}$ is in relation to the 2nd order Hamiltonian $H_\mathrm{FW}^{(2)}$ by a relativistic correction $-3(|\boldPi|/mc)^2/4$. For those terms in the 5th order, it can be sown that each term corresponding to Eq. (\[HFW\_list5\]) is given by $$\label{HFW_5_1} \begin{split} &\quad[(\matrixbeta\Omega_o)_{(5)},\Omega_o]=32\matrixbeta\Omega_o^6,\\ &\mathop{\sum_{\ell,m=1}^{3}}\limits_{(\ell+m=4)} [\matrixbeta\mathcal{O}^{(\ell)},\mathcal{O}^{(m)}] =-\frac{1}{3}\matrixbeta\{\mathcal{D},[\Omega_o,\mathcal{W}]\} +\frac{32}{9}\matrixbeta\Omega_o^6,\\ &\mathop{\sum_{\ell,m=1}^{2}\sum_{n=0}^{1}}\limits_{(\ell+m+n=3)} [\matrixbeta\mathcal{O}^{(\ell)},[\matrixbeta\mathcal{O}^{(m)},h^{(n)}]]\\ &\qquad=-\frac{7}{3}\matrixbeta\{\mathcal{D},[\Omega_o,\mathcal{W}]\} +\frac{18}{3}\{\mathcal{D},\Omega_o\mathcal{D}\Omega_o\}. \end{split}$$ We note that the operator $\mathcal{D}$ is proportional to $\matrixboldalpha\cdot\mathbf{E}$, which is of the 1st order of the electric field as well as the operator $\mathcal{W}$. We find that the terms $\{\mathcal{D},[\Omega_o,\mathcal{W}]\}$ and $\{\mathcal{D},\Omega_o\mathcal{D}\Omega_o\}$ in Eq. (\[HFW\_5\_1\]) are proportional to the product of only electric field, and thus will be neglected. On the other hand, the magnetic field in $\Omega_o^2$ is also of the 1st order \[see Eq. (\[Omega2\])\]. If we further neglect those terms proportional to the products of magnetic field and consider the homogeneous field, $\Omega_o^6$ becomes $$\label{HFW_5_2} \begin{split} \frac{\Omega_o^6}{E_g^5}&=\frac{(\Omega_o^2)^3}{E_g^5}\\ %=c^6\left\{|\boldsymbol{\Pi}|^2-\boldsymbol{\Sigma}\cdot\frac{\hbar}{c}(e\mathbf{B}+\tilde{e}\widetilde{\mathbf{B}})\right\}^3\\ &\approx\frac{1}{32}mc^2\left(\frac{|\boldsymbol{\Pi}|}{mc}\right)^6 -\frac{3}{16}\left(\frac{|\boldsymbol{\Pi}|}{mc}\right)^4\cdot(\boldsymbol{\mu}_p^{\prime e}\mathbf{B} +\boldsymbol{\mu}_m^{\prime\tilde{e}}\mathbf{E}). \end{split}$$ Therefore, $H_\mathrm{FW}^{(5)}$ with substitution of Eqs. (\[HFW\_5\_1\]) and (\[HFW\_5\_2\]) becomes $$\label{HFW5} \begin{split} H_\mathrm{FW}^{(5)}&\approx\left(\frac{32}{144}+\frac{32}{18}\right)\frac{\matrixbeta\Omega_o^6}{E_g^5}=\frac{2\matrixbeta\Omega_o^6}{E_g^5}\\ &=\frac{1}{16}mc^2\left(\frac{|\boldsymbol{\Pi}|}{mc}\right)^6 -\frac{3}{8}\left(\frac{|\boldsymbol{\Pi}|}{mc}\right)^4(\boldsymbol{\mu}_p^{\prime e}\cdot\mathbf{B} +\boldsymbol{\mu}_m^{\prime \tilde{e}}\cdot\mathbf{E}). \end{split}$$ The first and second terms of Eq. (\[HFW5\]) contribute to the relativistic mass correction and the Zeeman effect, respectively. For the 6th-order term $H^{(6)}_\mathrm{FW}$, it can be shown that the commutators of the form $[\matrixbeta\mathcal{O}^{\ell},\mathcal{O}^{(m)}]$ are given by $$\label{HFW_6_1} \begin{split} [\matrixbeta\mathcal{O}^{(1)},\mathcal{O}^{(4)}]&=\frac{8}{15}(\Omega_o^4\mathcal{W}+\Omega_o^3\mathcal{W}\Omega_o+\Omega_o^2\mathcal{W}\Omega_o^2+\Omega_o\mathcal{W}\Omega_0^3+\mathcal{W}\Omega_o^4),\\ [\matrixbeta\mathcal{O}^{(2)},\mathcal{O}^{(3)}]&=-\frac{2}{9}(-\Omega_o^4\mathcal{W}+\Omega_o^3\mathcal{W}\Omega_o+\Omega_o\mathcal{W}\Omega_o^3-\mathcal{W}\Omega_o^4),\\ \end{split}$$ where we also have $[\matrixbeta\mathcal{O}^{(3)},\mathcal{O}^{(2)}]=[\matrixbeta\mathcal{O}^{(2)},\mathcal{O}^{(3)}]$ and $[\matrixbeta\mathcal{O}^{(4)},\mathcal{O}^{(1)}]=[\matrixbeta\mathcal{O}^{(1)},\mathcal{O}^{(4)}]$. On the other hand, the commutators of the form $[\matrixbeta\mathcal{O}^{(\ell)},[\matrixbeta\mathcal{O}^{(m)},h^{(n)}]]$ in $H^{(6)}_\mathrm{FW}$ are given by $$\label{HFW_6_2} \begin{split} &[(\matrixbeta\Omega_o)_{(6)},h^{(0)}]=\Omega_0^4\mathcal{W}-4\Omega_o^3\mathcal{W}\Omega_o+6\Omega_o^2\mathcal{W}\Omega_o^2-4\Omega_o\mathcal{W}\Omega_o^3+\mathcal{W}\Omega_o^4,\\ &[\matrixbeta\mathcal{O}^{(1)},[\matrixbeta\mathcal{O}^{(3)},h^{(0)}]]=\frac{1}{6}[\mathcal{D},[[\Omega_o,\mathcal{W}],\Omega_E]],\\ &[\matrixbeta\mathcal{O}^{(2)},[\matrixbeta\mathcal{O}^{(2)},h^{(0)}]]=\frac{16}{9}(\Omega_0^4\mathcal{W}+2\Omega_o^3\mathcal{W}\Omega_o+3\Omega_o^2\mathcal{W}\Omega_o^2+2\Omega_o\mathcal{W}\Omega_o^3+\mathcal{W}\Omega_o^4),\\ &[\matrixbeta\mathcal{O}^{(1)},[\matrixbeta\mathcal{O}^{(2)},h^{(1)}]]=\frac{8}{3}(\Omega_0^4\mathcal{W}+\Omega_o^3\mathcal{W}\Omega_o+\Omega_o^2\mathcal{W}\Omega_o^2+\Omega_o\mathcal{W}\Omega_o^3+\mathcal{W}\Omega_o^4),\\ &[\matrixbeta\mathcal{O}^{(2)},[\matrixbeta\mathcal{O}^{(1)},h^{(1)}]]=\frac{4}{3}(\Omega_0^4\mathcal{W}+\Omega_o^3\mathcal{W}\Omega_o+2\Omega_o^2\mathcal{W}\Omega_o^2+\Omega_o\mathcal{W}\Omega_o^3+\mathcal{W}\Omega_o^4),\\ &[\matrixbeta\mathcal{O}^{(1)},[\matrixbeta\mathcal{O}^{(1)},h^{(2)}]]=\frac{1}{2}[\mathcal{D},[\mathcal{D},\mathcal{W}]].\\ \end{split}$$ Because $\mathcal{W}$ is of the 1st order of an electric field as well as $\mathcal{D}$, we can use Eq. (\[HFW\_4\_2\]) to reduce these equations into a form with only fields of the 1st order. For example, the term $\Omega_o^3\mathcal{W}\Omega_o$ becomes $\Omega_o^3\mathcal{W}\Omega_o=\Omega_o^2(\Omega_o\mathcal{W}\Omega_o)\approx c^4|\boldsymbol{\Pi}|^2(-|\boldsymbol{\Pi}|^2\mathcal{W})$. On the other hand, $[\mathcal{D},[\mathcal{D},\mathcal{W}]]$ and $[\mathcal{D},[[\Omega_o,\mathcal{W}],\Omega_E]]$ are neglected because they are at least of the 2nd order of fields. In that sense, by the use of Eqs. (\[HFW\_6\_1\]) and (\[HFW\_6\_2\]), one can obtain $$\label{HFW_6_3} \begin{split} &[(\matrixbeta\Omega_o)_{(6)},\Omega_E]\approx 16c^4|\boldsymbol{\Pi}|^4\mathcal{W},\\ &\mathop{\sum_{\ell,m=1}^{4}}\limits_{(\ell+m=5)}[\matrixbeta\mathcal{O}^{(\ell)},\mathcal{O}^{(m)}]\approx\frac{128}{45}c^4|\boldsymbol{\Pi}|^4\mathcal{W},\\ &\mathop{\sum_{\ell,m=1}^{3}\sum_{n=0}^{2}}\limits_{(\ell+m+n=4)}[\matrixbeta\mathcal{O}^{(\ell)},[\matrixbeta\mathcal{O}^{(m)},h^{(n)}]]\approx\frac{64}{9}c^4|\boldsymbol{\Pi}|^4\mathcal{W}. \end{split}$$ Therefore, Eq. (\[HFW\_list6\]) with substitution of Eq. (\[HFW\_6\_3\]) becomes $$\label{HFW6} \begin{split} H_\mathrm{FW}^{(6)}&\approx\frac{1}{E_g^6}\left[\frac{16}{720}+\frac{1}{2}\left(\frac{128}{45}\right)+\frac{1}{2}\left(\frac{64}{9}\right)\right]|\boldsymbol{\Pi}|^4\mathcal{W}\\ &=\frac{5c^4}{E_g^6}|\boldsymbol{\Pi}|^4\mathcal{W}\\ &=\frac{5}{8}\left(\frac{|\boldsymbol{\Pi}|}{mc}\right)^4\left(\frac{\mathcal{W}}{2E_g^2}\right). \end{split}$$ Eq. (\[HFW6\]) is the relativistic correction to the spin-orbit interaction in $H_\mathrm{FW}^{(2)}$. Therefore, $H_\mathrm{FW}^{(1)}$ and $H_\mathrm{FW}^{(3)}$ and $H_\mathrm{FW}^{(5)}$ are composed of kinetic energy, interaction energy of Zeeman effect and their relativistic corrections. On the other hand, $H_\mathrm{FW}^{(2)}$ and $H_\mathrm{FW}^{(4)}$ and $H_\mathrm{FW}^{(6)}$ contain only spin-orbit interaction and its relativistic corrections. To simplify the expression of $H_\mathrm{FW}^{(\ell)}$, we can define a scaled kinetic momentum operator $\boldxi$ as[^5] $$\label{scaledKM} \boldxi=\frac{\boldsymbol{\Pi}}{mc}.$$ By replacing $|\boldsymbol{\Pi}|/mc$ with Eq. (\[scaledKM\]), Eq. (\[HFW\]) with substitution of Eqs. (\[HFW1\]), (\[HFW2\]), (\[HFW3\]), (\[HFW4\]), (\[HFW5\]) and (\[HFW6\]) becomes a sum of two terms: $$\label{HFW_dipole2} H_\mathrm{FW}\approx H_\mathrm{orbit}+H_\mathrm{spin},$$ where $H_\mathrm{orbit}$ is the kinetic energy plus the potential energy, namely $$\label{HFW_kinetic} H_\mathrm{orbit}=\matrixbeta mc^2\left(1+\frac{1}{2}|\boldxi|^2-\frac{1}{8}|\boldxi|^4+\frac{1}{16}|\boldxi|^6\right)+V,$$ and $H_\mathrm{spin}$ is the energy of intrinsic dipole moments placing in electromagnetic fields, namely, $$\label{HFW_dipole3} \begin{split} H_\mathrm{spin}&=-\mathbf{E}\cdot \left[\matrixbeta\boldsymbol{\mu}_p^{\prime\tilde{e}} +\frac{1}{2}\left(\boldxi\times\boldsymbol{\mu}_m^{\prime e}\right)\right] -\mathbf{B}\cdot\left[\matrixbeta\boldsymbol{\mu}_m^{\prime e} -\frac{1}{2}\left(\boldxi\times\boldsymbol{\mu}_p^{\prime\tilde{e}}\right)\right]\\ &+\matrixbeta\left(\frac{1}{2}|\boldxi|^2-\frac{3}{8}|\boldxi|^4\right) \left(\boldsymbol{\mu}_m^{\prime e}\cdot\mathbf{B} +\boldsymbol{\mu}_p^{\prime\tilde{e}}\cdot\mathbf{E}\right) +\left(-\frac{3}{4}|\boldxi|^2+\frac{5}{8}|\boldxi|^4\right) \left\{-\frac{1}{2}\mathbf{E}\cdot\left(\boldxi\times\boldsymbol{\mu}_m^{\prime e}\right) -\frac{1}{2}\mathbf{B}\cdot\left(-\boldxi\times\boldsymbol{\mu}_p^{\prime\tilde{e}}\right) \right\}. \end{split}$$ In Sec. \[sec:relation to TBMT\], we will focus on the dipole Hamiltonian \[Eq. (\[HFW\_dipole3\])\]. We will show that Eq. (\[HFW\_dipole3\]) is in agreement with TBMT equation, provided that the proper transformation of $\boldxi$ and Lorentz boost velocity $\boldsymbol{\beta}$ is taken care of. In relation to TBMT equation {#sec:relation to TBMT} ---------------------------- We will show that the FW transformation of the Dirac Hamiltonian of a dyon is equivalent to the Hamiltonian obtained from TBMT equation with $g_e=g_{\tilde{e}}=2$. That is, Eq. (\[HFW\_kinetic\]) is equivalent to Eq. (\[H orbit dyon\]) and Eq. (\[HFW\_dipole3\]) to Eq. (\[H dipole dyon\]) with Eq. (\[ThomasF\]) and Eq. (\[ThomasF dual\]) for $g_e=g_{\tilde{e}}=2$. However, we must first find the boost velocity in order to compare them. It must be emphasized that $\boldsymbol{\beta}$ in TBMT equation is the boost velocity but $\boldxi$ in $H_\mathrm{FW}$ is not. One has to define the boost operator $\operatorboost$ via $$\label{transform} \boldxi=\frac{\operatorboost}{\sqrt{1-|\operatorboost|^2}}, \qquad \operatorgamma\equiv\frac{1}{\sqrt{1-|\operatorboost|^2}},$$ because the kinetic momentum $\boldsymbol{\Pi}\equiv mc \boldxi=m\mathbf{U}$ and the 4-velocity $U^\alpha=(\gamma c,\gamma\boldsymbol{\beta})$. By using Eq. (\[transform\]), the kinetic energy operator Eq. (\[HFW\_kinetic\]) behaves like $mc^2\left(1+\frac{1}{2}|\boldxi|^2-\frac{1}{8}|\boldxi|^4+\frac{1}{16}|\boldxi|^4\right)=mc^2(1+\frac{1}{2}|\operatorboost|^2+\frac{3}{8}|\operatorboost|^4+\frac{5}{16}|\operatorboost|^6+o(8))$. On the other hand, the expansion of Lorentz factor $\gamma={1}/{\sqrt{1-\beta^2}}$ with respect to small boost $\beta$ is $\gamma=1+\frac{1}{2}\beta^2+\frac{3}{8}\beta^4+\frac{5}{16}\beta^6+o(8)$. This implies that the the kinetic energy operator corresponds to the classical relativistic energy $\gamma mc^2$, as expected. The boost operator $\operatorboost$ plays an important role on showing the equivalence between $H_\mathrm{spin}$ and TBMT Hamiltonian. For an electron, Eq. (\[HFW\_dipole3\]) without a magnetic charge ($\tilde{e}=0$) becomes $$\label{HD_electron} \begin{split} H_\mathrm{spin}^{(\tilde{e}=0)}&=-\mathbf{E}\cdot \left[\frac{1}{2}\left(1-\frac{3}{4}|\boldxi|^2+\frac{5}{8}|\boldxi|^4\right) \left(\boldxi\times\boldsymbol{\mu}_m^{\prime e}\right)\right]-\mathbf{B}\cdot \left[\matrixbeta\left(1-\frac{1}{2}|\boldxi|^2+\frac{3}{8}|\boldxi|^4\right) \boldsymbol{\mu}_m^{\prime e}\right]\\ &=-\boldsymbol{\mu}_m^{\prime e}\cdot\left[\matrixbeta\left(1-\frac{1}{2}|\boldxi|^2 +\frac{3}{8}|\boldxi|^4\right)\mathbf{B}-\frac{1}{2}\left(1-\frac{3}{4}|\boldxi|^2 +\frac{5}{8}|\boldxi|^4\right)\boldxi\times\mathbf{E}\right]. \end{split}$$ The first term in the right hand side of the first equality of Eq. (\[HD\_electron\]) is an effective electric dipole moment caused by the boosted intrinsic spin magnetic moment, which is the spin-orbit interaction. The second one is the Zeeman term. Nevertheless, Eq. (\[HD\_electron\]) provides the relativistic correction to the Zeeman and spin-orbit interactions. For the Zeeman term, the non-relativistic limit up to $1/mc$ is $$H_\mathrm{Zeeman}=-\matrixbeta\boldsymbol{\mu}_m^{\prime e}\cdot\mathbf{B},$$ which is the same as the interaction of a classical magnetic moment and a magnetic field. To the 4th order of $\boldxi$, the relativistic correction to $H_\mathrm{Zeeman}$ is $$H_\mathrm{Zeeman}=-\matrixbeta\left(1-\frac{1}{2}|\boldxi|^2+\frac{3}{8}|\boldxi|^4\right) \boldsymbol{\mu}_m^{\prime e}\cdot\mathbf{B}.$$ On the other hand, the spin-orbit interaction denoted as $H_\mathrm{so}$ is $$\begin{split} H_\mathrm{so}&=\frac{1}{2}\boldsymbol{\mu}_m^{\prime e}\cdot\boldxi\times\mathbf{E}\\ &=\frac{|e|\hbar}{4m^2c^2}\boldsymbol{\Sigma}\cdot\mathbf{E}\times\boldPi, \end{split}$$ where $e=-|e|$ is used in the second equality, and the relativistic correction to this term is $$H_\mathrm{so}=\frac{1}{2}\left(1-\frac{3}{4}|\boldxi|^2+\frac{5}{8}|\boldxi|^4\right) \boldsymbol{\mu}_m^{\prime e}\cdot\boldxi\times\mathbf{E}.$$ We now go back to the discussion of $H_\mathrm{spin}^{\tilde{e}=0}$ and TBMT equation. In order to compare Eq. (\[HD\_electron\]) with TBMT Hamiltonian, we have to transform $\boldxi$ in Eq. (\[HD\_electron\]) to $\operatorboost$. Using Eq. (\[transform\]), we have $$\begin{aligned} &\left(1-\frac{1}{2}|\boldxi|^2+\frac{3}{8}|\boldxi|^4\right)=1-\frac{|\operatorboost|^2}{2}-\frac{|\operatorboost|^4}{8}+o(6),\label{expansion2}\\ &\frac{1}{2}\left(1-\frac{3}{4}|\boldxi|^2+\frac{5}{8}|\boldxi|^4\right)|\boldxi|=\frac{|\operatorboost|}{2}-\frac{|\operatorboost|^3}{8}-\frac{|\operatorboost|^5}{16}+o(7).\label{expansion1}\end{aligned}$$ The effective spin magnetic moment in TBMT Hamiltonian \[Eq. (\[H dipole\])\] transforms like $(1/\gamma)\boldsymbol{\mu}_m^{\prime e}$, and we have $$\frac{1}{\gamma}=1-\frac{\beta^2}{2}-\frac{\beta^4}{8}+o(6),$$ which is exactly the same as Eq. (\[expansion2\]) up to terms of the 4th order in $\beta$. On the other hand, the effective electric dipole moment transforms like $(g_e/2-\gamma/(\gamma+1))$, and $g_e$ factor in the Dirac Hamiltonian is always 2. We obtain $$\left(1-\frac{\gamma}{1+\gamma}\right)\beta=\frac{\beta}{2}-\frac{\beta^3}{8}-\frac{\beta^5}{16}+o(7),\\$$ which is exactly the same as Eq. (\[expansion1\]) up to terms of the 5th order in $\beta$. Because $g_e$ factor equals 2, the longitudinal term $\boldsymbol{\mu}_m^{\prime e}\cdot\boldxi$ disappears in both TBMT Hamiltonian and $H_\mathrm{spin}^{(\tilde{e}=0)}$. In the following, we will use the following two approximations directly: $$\label{transgamma} \begin{split} &\left(1-\frac{1}{2}|\boldxi|^2+\frac{3}{8}|\boldxi|^4\right)\approx \frac{1}{\operatorgamma}\,,\\ &\frac{1}{2}\left(1-\frac{3}{4}|\boldxi|^2+\frac{5}{8}|\boldxi|^4\right)|\boldxi|\approx\left(1-\frac{\operatorgamma}{\operatorgamma+1}\right). \end{split}$$ Therefore, we show that up to the fifth order of boost velocity $\beta$, the Dirac Hamiltonian of an electron is equivalent to the TBMT Hamiltonian which is obtained from the requirement of covariance form of classical spin. This implies that in the FW representation, after summing over all infinite expansion terms, the Dirac Hamiltonian of an electron would be of the form $$\label{HD_electron2} H_\mathrm{spin}^{\tilde{e}=0}=-\boldsymbol{\mu}_m^{\prime e}\cdot\left[\matrixbeta\frac{1}{\operatorgamma}\mathbf{B}-\left(1-\frac{\operatorgamma}{1+\operatorgamma}\right)\operatorboost\times\mathbf{E}\right]$$ for the spin part, and of the form $$H_\mathrm{orbit}=\operatorgamma\matrixbeta mc^2+V$$ for orbital part. The effective magnetic field in Eq. (\[HD\_electron2\]) is the same as Eq. (\[H dipole\]) with Eq. (\[ThomasF\]) for $g_e=2$. Furthermore, for the FW transformation of the Dirac Hamiltonian \[Eq. (\[HFW\_dipole2\])\], the TBMT equation can be generalized to include an effective spin magnetic moment resulting from the boosted intrinsic electric dipole moment. To the 1st order in $|\boldxi|=|\boldsymbol{\Pi}|/mc$, we find that the effective dipole moments transform like $$\label{effdipoles} \begin{split} (\boldsymbol{\mu}_p^{\tilde{e}})_\mathrm{eff}&\approx \matrixbeta\boldsymbol{\mu}_p^{\prime\tilde{e}}+\frac{1}{2} \left(\boldxi\times\boldsymbol{\mu}_m^{\prime e}\right),\\ (\boldsymbol{\mu}_m^e)_\mathrm{eff}&\approx\matrixbeta\boldsymbol{\mu}_m^{\prime e} -\frac{1}{2}\left(\boldxi\times\boldsymbol{\mu}_p^{\prime\tilde{e}}\right). \end{split}$$ This means that an intrinsic electric dipole moment can result in an effective magnetic dipole moment when it is moving. Nevertheless, a moving spin magnetic moment can also intrinsically induce an effective electric dipole moment. Consider higher orders of the boost velocity, we rewrite Eq. (\[HFW\_dipole3\]) as $$\begin{split} H_\mathrm{spin}&=-\mathbf{E}\cdot \left[\matrixbeta\left(1-\frac{1}{2}|\boldxi|^2 +\frac{3}{8}|\boldxi|^4\right)\boldsymbol{\mu}_p^{\prime\tilde{e}}+\frac{1}{2} \left(1-\frac{3}{4}|\boldxi|^2+\frac{5}{8}|\boldxi|^4\right) \left(\boldxi\times\boldsymbol{\mu}_m^{\prime e}\right)\right]\\ &~~~~-\mathbf{B}\cdot\left[\matrixbeta \left(1-\frac{1}{2}|\boldxi|^2+\frac{3}{8}|\boldxi|^4\right) \boldsymbol{\mu}_m^{\prime e}-\frac{1}{2}\left(1-\frac{3}{4}|\boldxi|^2 +\frac{5}{8}|\boldxi|^4\right) \left(\boldxi\times\boldsymbol{\mu}_p^{\prime\tilde{e}}\right)\right]. \end{split}$$ It is shown that intrinsic dipole moments transform like $\left(1-\frac{1}{2}|\boldxi|^2+\frac{3}{8}|\boldxi|^4\right)\approx 1/\operatorgamma$ and the boosted dipole moments transform as $\frac{1}{2}\left(1-\frac{3}{4}|\boldxi|^2+\frac{5}{8}|\boldxi|^4\right)\approx(1-\frac{\operatorgamma}{\operatorgamma+1})$ (see Eq. (\[transgamma\])). Therefore, $(\boldsymbol{\mu}_p^{\prime\tilde{e}})_\mathrm{eff}$ and $(\boldsymbol{\mu}_m^{\prime e})_\mathrm{eff}$ do not form a second rank tensor in the sense that their transformation is not a covariant form like Eq. (\[LTdipole\]), but the following form: $$\begin{split} &(\boldsymbol{\mu}_p^{\tilde{e}})_\mathrm{eff} \approx\matrixbeta\frac{1}{\operatorgamma}\boldsymbol{\mu}_p^{\prime\tilde{e}} +\left(1-\frac{\operatorgamma}{\operatorgamma+1}\right) \left(\operatorboost\times\boldsymbol{\mu}_m^{\prime e}\right),\\ &(\boldsymbol{\mu}_m^e)_\mathrm{eff} \approx\matrixbeta\frac{1}{\operatorgamma}\boldsymbol{\mu}_m^{\prime e} -\left(1-\frac{\operatorgamma}{\operatorgamma+1}\right) \left(\operatorboost\times\boldsymbol{\mu}_p^{\prime\tilde{e}}\right).\\ \end{split}$$ This implies that an energy caused by dipole moments in the description of Dirac Hamiltonian is not simply the contraction of tensorial dipole density and field tensor: $H_\mathrm{spin}\neq-\boldsymbol{\mu}_p\cdot\mathbf{E}-\boldsymbol{\mu}_m\cdot\mathbf{B}$, in which $\boldsymbol{\mu}_p$ and $\boldsymbol{\mu}_m$ transform as in Eq. (\[LTdipole\]). As a result, $H_\mathrm{spin}$ is not a Lorentz scalar. In short, in this section we have shown that up to terms of the 7th order in $1/E_g$, the FW transformation of the Dirac Hamiltonian of an electron is in agreement with TBMT Hamiltonian \[Eq. (\[H dipole\])\] with $g_e=2$. The result can be generalized to a particle with an intrinsic electric dipole moment. Because of the duality of electromagnetic fields, a Dirac dyon would manifest this feature. Furthermore, we also find the relativistic corrections to the Zeeman term and spin-orbit interaction Foldy-Wouthuysen transformation for the Dirac-Pauli Hamiltonian {#sec:Dirac Pauli} =============================================================== In Sec. \[sec:Dirac\], we have shown that, up to the 7th order in $1/E_g$, the FW transformation of the Dirac Hamiltonian for a dyon is in agreement with Eq. (\[H orbit\]) and Eq. (\[Thomas\]) for $g_e=g_{\tilde{e}}=2$. Since the Dirac Hamiltonian automatically yields $g_e=2$ and $g_{\tilde{e}}=2$, the second term in Eq. (\[ThomasF\]) and Eq. (\[ThomasF dual\]) vanishes and thus the longitudinal polarization does not change. In order to see that the relativistic quantum theory of a spin-$1/2$ particle is in accord with the TBMT equation even when the change rate of the longitudinal polarization is concerned, we have to study the spin-$1/2$ particle with anomalous magnetic dipole moment (AMM) and anomalous electric dipole moment (AEM). The relativistic quantum theory of a spin-$/2$ dyon with the inclusion of AMM and AEM can be described by the Dirac-Pauli equation [@Silenko08; @Pauli41] $$i\hbar\frac{\partial}{\partial t}|\psi\rangle=\mathcal{H}|\psi\rangle,$$ where the Dirac-Pauli Hamiltonian $\mathcal{H}$ is the Dirac Hamiltonian $H$ \[given in Eq. (\[H\])\] augmented with the corrections for the AMM and AEM: $$\label{H_ADM} \mathcal{H}=H+\mu'(-\matrixbeta\boldsymbol{\Sigma}\cdot\mathbf{B} +i\matrixboldgamma\cdot\mathbf{E})+d'(\matrixbeta\boldsymbol{\Sigma}\cdot\mathbf{E} +i\matrixboldgamma\cdot\mathbf{B}).$$ The coefficients $\mu'$ and $d'$ are defined as follows $$\label{mu-and-d} \mu'=\left(\frac{g_e}{2}-1\right)\frac{e\hbar}{2mc}, \quad d'=\left(\frac{g_{\tilde{e}}}{2}-1\right)\frac{\tilde{e}\hbar}{2mc},$$ which measures the AMM and AEM, respectively (note $\mu'=0$ for $g_e=2$ and $d'=0$ for $g_{\tilde{e}}=2$). The $4\times4$ matrices $\matrixbeta\boldsymbol{\Sigma}$ and $\matrixboldgamma$ are defined as $$\label{matrices beta and gamma} \matrixbeta\boldsymbol{\Sigma}=\left(\begin{array}{cc} \boldsymbol{\sigma}&0\\ 0&-\boldsymbol{\sigma} \end{array}\right), \qquad \matrixboldgamma=\left(\begin{array}{cc} 0&\boldsymbol{\sigma}\\ -\boldsymbol{\sigma}&0 \end{array}\right),$$ where $\boldsymbol{\sigma}=(\sigma_x,\sigma_y,\sigma_z)$ are Pauli matrices. We will see that the Dirac-Pauli Hamiltonian given in Eq. (\[H\_ADM\]) is compatible to generic values of $g_e$ and $g_{\tilde{e}}$ and thus can accommodate AMM and AEM. In order to obtain the FW transformation of Eq. (\[H\_ADM\]), we have to rewrite Eq. (\[H\_ADM\]) in terms of odd and even matrices. According to Eq. (\[OE\]), we have $$\label{AH} \mathcal{H}=\matrixbeta mc^2+\Omega_E^A+\Omega_o^A,$$ where the superscript $A$ indicates the inclusion of anomalous dipole moments. We note that $\mu'$ and $d'$ are of the 1st order of $1/E_g$. Therefore, $\Omega_E^A$ and $\Omega_o^A$ can be written as $$\begin{split} &\Omega_E^A=\Omega_E+\frac{\Omega_E^f}{E_g},\\ &\Omega_o^A=\Omega_o+\frac{\Omega_o^f}{E_g}, \end{split}$$ where $\Omega_E$ and $\Omega_o$ are given in Eq. (\[OE1\]) and $$\begin{split} &\Omega_E^f=\matrixbeta\boldsymbol{\Sigma}\cdot(-\mu''\mathbf{B}+d''\mathbf{E}),\\ &\Omega_o^f=i\matrixboldgamma\cdot(\mu''\mathbf{E}+d''\mathbf{B}),\\ &\mu''=E_g\mu', \quad d''=E_gd'. \end{split}$$ The superscript $f$ indicates that these terms are of the 1st order of electromagnetic fields. Because we consider only those terms proportional to the 1st order of fields, the products of fields will be neglected. The validity of Eq. (\[Appendix\_mainresult\]) is still true provided that the odd term of the second FW transformation denoted as $\mathcal{O}'$ in Eq. (\[AH\]) starts from $1/E_g^3$. This can be seen as follows. After the first FW transformation, $S_1=\matrixbeta\Omega_o^A/E_g$, Eq. (\[AH\]) becomes $$\mathcal{H}'=\frac{\matrixbeta}{2}E_g+h+\mathcal{O},$$ where $h$ and $\mathcal{O}$ are even and odd terms, respectively. It can be shown that the odd term $\mathcal{O}$ can be written as $$\label{AH2} \mathcal{O}=\frac{\mathcal{O}^{(1)}}{E_g}+\frac{\mathcal{O}^{(2)}}{E^2_g}+\frac{\mathcal{O}^{(3)}}{E^3_g}+\frac{\mathcal{O}^{(4)}}{E^4_g}+o(\frac{1}{E_g^5}),$$ where the corresponding $\mathcal{O}^{(n)},~n=1,2,3,4$, are given by $$\label{AO} \begin{split} &\mathcal{O}^{(1)}=\matrixbeta[\Omega_o,\Omega_E],\\ &\mathcal{O}^{(2)}=2c\matrixbeta\matrixeta(-\mu''\mathbf{B}+d''\mathbf{E})\cdot\boldPi-\frac{4}{3}\Omega_o^3,\\ &\mathcal{O}^{(3)}=-\frac{4}{3}\left(\Omega_o\{\Omega_o,\Omega_o^f\}+\Omega_o^f\Omega_o^2\right)+\frac{1}{6}\matrixbeta[\Omega_o,\mathcal{W}],\\ &\mathcal{O}^{(4)}=-\frac{4}{3}c^3\matrixbeta\matrixeta(-\mu''\mathbf{B}+d''\mathbf{E})\cdot\boldPi|\boldPi|^2+\frac{8}{15}\Omega_o^5, \end{split}$$ where the matrix $\matrixeta$ is defined as $$\label{matrix eta} \matrixeta=\left(\begin{array}{cc} 0&-\mathbf{1}\\ \mathbf{1}&0 \end{array}\right).$$ If $g_e$ and $g_{\tilde e}$ are both equal to 2, then Eq. (\[AO\]) goes back to Eq. (\[listO\]). Using the second FW transformation $S_2=\matrixbeta\mathcal{O}/E_g$ on Eq. (\[AH2\]), we can obtain the other odd term denoted as $\mathcal{O}'$. The 1st order term $\mathcal{O}'^{(1)}$ is zero because $\mathcal{O}$ starts form the 1st order of $1/E_g$ at least. $\mathcal{O}'^{(2)}$ can be written as $\mathcal{O}'^{(2)}=[\matrixbeta\mathcal{O}^{(1)},h^{(0)}]$. Nevertheless, we have $\mathcal{O}^{(1)}=\matrixbeta[\Omega_o,\Omega_E]=\matrixbeta i\hbar\matrixboldalpha\cdot(e\mathbf{E}+\tilde{e}\mathbf{B})$ and $h^{(0)}=\Omega_E=V$ is a scalar that commutes with $\mathcal{O}^{(1)}$, and thus $\mathcal{O}'^{(2)}$ vanishes. This implies that Eq. (\[Appendix\_mainresult\]) is still valid in this case. The matrix $h$ in Eq. (\[AH2\]) is given by $$\label{AMMAEM_h} \begin{split} h=&\Omega_E^A+\frac{1}{2E_g}[\matrixbeta\Omega_o^A,\Omega_o^A]+\frac{1}{2E_g^2}[(\matrixbeta\Omega_o^A)_{(2)},\Omega_E^A]\\ &+\frac{1}{8E_g^3}[(\matrixbeta\Omega_o^A)_{(3)},\Omega_o^A]+\frac{1}{24}[(\matrixbeta\Omega_o^A)_{(4)},\Omega_E^A]\\ &+\frac{1}{144E_g^5}[(\matrixbeta\Omega_o^A)_{(5)},\Omega_o^A]+\frac{1}{720E_g^6}[(\matrixbeta\Omega_o^A)_{(6)},\Omega_E^A]. \end{split}$$ To obtain $h'$, we need extra corrections to $h$, as shown in Eq. (\[Appendix\_h’expand\]). The resulting Hamiltonian can be written as $$\mathcal{H}_\mathrm{FW}=H_\mathrm{FW}+\mathcal{H}_\mathrm{FW1}+\mathcal{H}_\mathrm{FW2}.$$ The Hamiltonian $H_\mathrm{FW}$ is given in Eq. (\[HFW\_dipole2\]), $\mathcal{H}_\mathrm{FW1}$ contains those terms proportional to $(-\mu''\mathbf{B}+d''\mathbf{E})$, and $\mathcal{H}_\mathrm{FW2}$ contains those terms proportional to $(\mu''\mathbf{E}+d''\mathbf{B})$. Focusing on the term proportional to $(-\mu'\mathbf{B}+d'\mathbf{E})$, namely, $[(\matrixbeta\Omega_o^A)_{(n=2,4)},\Omega_E^A]$ in Eq. (\[AMMAEM\_h\]). For $n=2$, we have $$\begin{split} [(\matrixbeta\Omega_o^A)_{(2)},\Omega_E^A]&=[\matrixbeta\Omega_o^A,[\matrixbeta\Omega_o^A,\Omega_E^A]]\\ &=[[\Omega_o^A,\Omega_E^A],\Omega_o^A]\\ &=\mathcal{W}+\frac{\mathcal{W}^f}{E_g}\,, \end{split}$$ where $\mathcal{W}$ is given in Eq. (\[DandW\]) and $\mathcal{W}^f$ is defined as $$\label{Wf} \begin{split} \mathcal{W}^f&=[[\Omega_o,\Omega_E^f],\Omega_o]\\ &=-4c^2\matrixbeta\boldsymbol{\Sigma}\cdot\boldPi(-\mu''\mathbf{B}+d''\mathbf{E})\cdot\boldPi, \end{split}$$ which is proportional to electric and magnetic fields. By using Eq. (\[Wf\]), the term $[(\matrixbeta\Omega_o^A)_{(4)},\Omega_E^A]$ can be written as $$\label{Wf2} \begin{split} &[(\matrixbeta\Omega_o^A)_{(4)},\Omega_E^A]\\ &=[\matrixbeta\Omega_o^A,[\matrixbeta\Omega_o^A,\mathcal{W}+\frac{\mathcal{W}^f}{E_g}]]\\ &=[[\Omega_o,\mathcal{W}],\Omega_o]-\frac{4c^2}{E_g}\mathcal{W}^f|\boldPi|^2. \end{split}$$ The first term of Eq. (\[Wf2\]) is just one of the 4th order terms of $H_\mathrm{FW}^{(4)}$ shown in Sec. \[sec:dipoles\]. It is important to note that the second term of Eq. (\[Wf2\]) is collected in $h^{(5)}$, not $h^{(4)}$. It can be shown that the only term that contributes to $\boldsymbol{\Sigma}\cdot\boldPi(-\mu''\mathbf{B}+d''\mathbf{E})\cdot\boldPi|\boldPi|^2$ is $[\matrixbeta\mathcal{O}^{(2)},\mathcal{O}^{(2)}]$ that is the correction term in $h'^{(5)}$ (see Eq. (\[Appendix\_h’expand\])). Using $\mathcal{O}^{(2)}$ in Eq. (\[AO\]), we have $$\begin{split} &\frac{1}{2}[\matrixbeta\mathcal{O}^{(2)},\mathcal{O}^{(2)}]\\ &=\matrixbeta(\mathcal{O}^{(2)})^2\\ &=\matrixbeta\left(2c\matrixbeta\matrixeta(-\mu''\mathbf{B}+d''\mathbf{E})\cdot\boldPi-\frac{4}{3}\Omega_o^3\right)^2\\ &=\matrixbeta\frac{16}{9}\Omega_o^6-\frac{8}{3}c^4(-\mu''\mathbf{B}+d''\mathbf{E})\cdot\boldPi|\boldPi|^2[\matrixeta,\matrixalpha_{\ell}]\Pi_{\ell}. \end{split}$$ It can be shown that $[\matrixeta,\matrixalpha_{\ell}]=-2\matrixbeta\Sigma_{\ell}$, and thus we obtain $$\begin{aligned} \label{Wf3} &&\frac{1}{2}[\matrixbeta\mathcal{O}^{(2)},\mathcal{O}^{(2)}]\\ &=&\matrixbeta\frac{16}{9}\Omega_o^6+\frac{16}{3}c^4\matrixbeta\boldsymbol{\Sigma}\cdot\boldPi(-\mu''\mathbf{B}+d''\mathbf{E})\cdot\boldPi|\boldPi|^2.\nonumber\end{aligned}$$ Consider those terms proportional to $(-\mu''\mathbf{B}+d''\mathbf{E})$: one comes form Eq. (\[Wf\]) in the corresponding term and the other is obtained form $\Omega_E^f$, Eq. (\[Wf2\]) and Eq. (\[Wf3\]). Using the definitions of $\boldxi=\boldPi/mc$ (see Eq. (\[transform\])), $\mu''=E_g\mu'$ and $d''=E_gd'$, one obtain $$\label{H_FW1_1} \begin{split} \mathcal{H}_\mathrm{FW1}&=\frac{1}{2E_g^3}\mathcal{W}^f+\matrixbeta\boldsymbol{\Sigma}\cdot(-\mu'\mathbf{B}+d'\mathbf{E})\\ &~~~~+\frac{c^4}{E_g^5}\left(\frac{16}{3}+\frac{16}{24}\right)\matrixbeta\boldsymbol{\Sigma}\cdot\boldPi(-\mu''\mathbf{B}+d''\mathbf{E})\cdot\boldPi|\boldPi|^2\\ &=\left(-\frac{1}{2}+\frac{3}{8}|\boldxi|^2\right)\matrixbeta\boldsymbol{\Sigma}\cdot\boldxi(-\mu'\mathbf{B}+d'\mathbf{E})\cdot\boldxi\\ &~~~~+\matrixbeta\boldsymbol{\Sigma}\cdot(-\mu'\mathbf{B}+d'\mathbf{E}). \end{split}$$ For $g_e=2$ and $g_{\tilde{e}}=2$, $\mathcal{H}_\mathrm{FW1}$ vanishes. We now transform $\boldxi$ in Eq. (\[H\_FW1\_1\]) in terms of the boost velocity $\operatorboost$. By using the transformation between $\boldxi$ and $\operatorboost$ \[see Eq. (\[transform\])\], it can be shown that $\left(\frac{1}{2}-\frac{3}{8}|\boldxi|^2\right)|\boldxi|^2\approx\frac{\operatorgamma}{1+\operatorgamma}|\operatorboost|^2$. Substituting AMM coefficient $\mu'=\left(\frac{g_e}{2}-1\right)\frac{e\hbar}{2mc}$ and AEM coefficient $d'=\left(\frac{g_{\tilde{e}}}{2}-1\right)\frac{\tilde{e}\hbar}{2mc}$ into Eq. (\[H\_FW1\_1\]), we find that $\mathcal{H}_\mathrm{FW1}$ can be written as $$\begin{split} \mathcal{H}_\mathrm{FW1}& =-\frac{\operatorgamma}{\operatorgamma+1}\matrixbeta\boldsymbol{\Sigma} \cdot\operatorboost(-\mu'\mathbf{B}+d'\mathbf{E})\cdot\operatorboost +\matrixbeta\boldsymbol{\Sigma}\cdot(-\mu'\mathbf{B}+d'\mathbf{E})\\ &=-\frac{\operatorgamma}{\operatorgamma+1}\matrixbeta \left[-\left(\frac{g_e}{2}-1\right)\boldsymbol{\mu}^{\prime e}_m\cdot \operatorboost\mathbf{B}\cdot\operatorboost-\left(\frac{g_{\prime\tilde{e}}}{2} -1\right)\boldsymbol{\mu}^{\prime\tilde{e}}_p\cdot\operatorboost\mathbf{E} \cdot\operatorboost\right]+\matrixbeta\left[-\left(\frac{g_e}{2}-1\right) \boldsymbol{\mu}_m^{\prime e}\cdot\mathbf{B}-\left(\frac{g_{\tilde{e}}}{2}-1\right) \boldsymbol{\mu}_p^{\prime \tilde{e}}\cdot\mathbf{E}\right]. \end{split}$$ On the other hand, the terms proportional to $(\mu''\mathbf{E}+d''\mathbf{B})$ correspond to $\{\Omega_o,\Omega_o^f\}=2c\matrixbeta\boldsymbol{\Sigma}\cdot\boldPi\times(\mu''\mathbf{E}+d''\mathbf{B})$. Similar to the derivation for $\mathcal{H}_\mathrm{FW1}$, we collect all terms proportional to $\{\Omega_o,\Omega_o^f\}$ and obtain $$\mathcal{H}_\mathrm{FW2}=\matrixbeta\{\Omega_o,\Omega_o^f\}\frac{1}{E_g^2}\left(1-\frac{1}{2}|\boldxi|^2+\frac{3}{8}|\boldxi|^4\right).$$ Using $\mu''=E_g\mu'$ and $d''=E_gd'$, we have $$\mathcal{H}_\mathrm{FW2}=\left(1-\frac{1}{2}|\boldxi|^2+\frac{3}{8}|\boldxi|^4\right)\boldsymbol{\Sigma}\cdot\boldxi\times(\mu'\mathbf{E}+d'\mathbf{B}).$$ By using Eq. (\[transform\]), we find that $$\begin{aligned} \mathcal{H}_\mathrm{FW2}&=& \boldsymbol{\Sigma}\cdot\operatorboost\times(\mu'\mathbf{E}+d'\mathbf{B})\\ &=&\left(\frac{g_e}{2}-1\right)\boldsymbol{\mu}^{\prime e}_m \cdot(\operatorboost\times\mathbf{E}) -\left(\frac{g_{\tilde{e}}}{2}-1\right)\boldsymbol{\mu}^{\prime\tilde{e}}_p\cdot(\operatorboost\times\mathbf{B}).\nonumber\end{aligned}$$ To focus on the interaction of dipole moments and external fields, we have to combine $H_\mathrm{spin}$, $\mathcal{H}_\mathrm{FW1}$ and $\mathcal{H}_\mathrm{FW2}$ together. After a straightforward calculations, we find that $$\label{mainresult} \begin{split} &H_\mathrm{spin}+\mathcal{H}_\mathrm{FW1}+\mathcal{H}_\mathrm{FW2}\\ &=-\boldsymbol{\mu}_m^{\prime e}\cdot\left\{\left(\frac{g_e}{2}-1+\frac{1}{\operatorgamma}\right)\matrixbeta\mathbf{B}-\left(\frac{g_e}{2}-\frac{\operatorgamma}{\operatorgamma+1}\right)\operatorboost\times\mathbf{E}-\frac{\operatorgamma}{\operatorgamma+1}\left(\frac{g_e}{2}-1\right)\operatorboost(\mathbf{B}\cdot\operatorboost)\right\}\\ &~~~-\boldsymbol{\mu}_p^{\prime\tilde{e}}\cdot\left\{\left(\frac{g_{\tilde{e}}}{2}-1+\frac{1}{\operatorgamma}\right)\matrixbeta\mathbf{E}+\left(\frac{g_{\tilde{e}}}{2}-\frac{\operatorgamma}{\operatorgamma+1}\right)\operatorboost\times\mathbf{B}-\frac{\operatorgamma}{\operatorgamma+1}\left(\frac{g_{\tilde{e}}}{2}-1\right)\operatorboost(\mathbf{E}\cdot\operatorboost)\right\}.\\ \end{split}$$ Eq. (\[mainresult\]) is in agreement with Eq. (\[H dipole dyon\]) when the replacement $\gamma\rightarrow\operatorgamma$ and the duality transformation for electromagnetic fields are used. Without magnetic charge, Eq. (\[mainresult\]) coincides with Eq. (\[ThomasF\]) for arbitrary values of $g_e$. The dual part of the TBMT equation for spin is also obtained. In short, the whole derivations in this section have assumed that electromagnetic fields are static and homogeneous. Therefore, we show that up to terms of the 7th order in $1/E_g$, the FW transformation including anomalous dipole moments coincides with the TBMT equation for the spin-$1/2$ particle with arbitrary $g_e$ and $g_{\tilde e}$. Conclusions and discussion {#sec:conclusions} ========================== To investigate the low-energy limit of the relativistic quantum theory for a spin-$1/2$ charged particle, which is described by the Dirac equation, we perform a series of successive FW transformations on the Dirac Hamiltonian up to terms of the 7th order in $1/E_g$. Assuming the electromagnetic fields are static and homogeneous, and taking care of the relation between the kinematic momentum $\boldsymbol{\Pi}$ used in the Dirac Hamiltonian and the boost velocity $\boldsymbol{\beta}$ used in the TBMT equation, we show that the resulting FW transformation of the Dirac Hamiltonian is in agreement with the classical orbital Hamiltonian $H_\mathrm{orbit}$ plus the TBMT Hamiltonian $H_\mathrm{spin}$ with the gyromagnetic ratio $g_e$ being equal to 2. Through electromagnetic duality, this can be generalized for a spin-$1/2$ dyon, which has both electric and magnetic charges and thus possesses both intrinsic magnetic dipole moment $\boldsymbol{\mu}_m^{\prime e}$ and intrinsic electric dipole moment $\boldsymbol{\mu}_p^{\prime\tilde{e}}$. To affirm the consistency between the low-energy limit of the relativistic quantum theory and the classical counterpart to a broader extent, we consider the relativistic quantum theory for a spin-$1/2$ dyon with arbitrary values of the gyromagnetic and gyroelectric ratios, which is described by the Dirac-Pauli equation, namely, the Dirac equation with augmentation for AMM and AEM. Up the 7th order in $1/E_g$ again, we show that the FW transformation of the Dirac-Pauli Hamiltonian is also in accord with $H_\mathrm{orbit}+H_\mathrm{spin}$. Many phenomena regarding spin dynamics have been observed and can be explained by the TBMT equation. These include the anomalous Zeeman effect, spin-orbit interaction, Thomas precession and change rate of the longitudinal polarization (see Sec. 11.11 of [@Jackson] for a brief review). The TBMT equation is however derived merely by the requirement of covariance without invoking any first principles. By studying the FW transformation of the Dirac Hamiltonian and the Dirac-Pauli Hamiltonian, we have shown that the TBMT equation as a phenomenological formula is in fact supported by the first principle of the fundamental relativistic quantum theory as a low-energy limit. (The relativistic quantum theory further requires the spin to be quantized as $\mathbf{s}=\hbar\,\boldsymbol{\sigma}/2$; this result cannot be obtained at the phenomenological level.) Therefore, the correspondence principle is again shown to be established. By far, the agreement between the Dirac/Dirac-Pauli equation and the orbital equation plus the TBMT equation is only proven up to the 7th order in $1/E_g$. Further research is needed to investigate the FW transformation to higher orders and a generic expression for the FW transformation at an arbitrary order could be obtained by mathematical induction. If this is the case, performing successive FW transformations ad infinitum is expected to yield the result in precise agreement with the orbital equation plus the TBMT equation. Furthermore, the assumption of static and homogeneous fields can be released. In time-varying and/or inhomogeneous fields, the TBMT equation has to be generalized to allow gradient force terms like $(\boldsymbol{\mu}_m\cdot\boldsymbol{\nabla})\mathbf{B}$ and the FW transformation of the Dirac/Dirac-Pauli Hamiltonian shall yield the corresponding terms accordingly. The gradient force terms should not be missing, as $(\boldsymbol{\mu}_m\cdot\boldsymbol{\nabla})\mathbf{B}$ is used in the Stern-Gerlach experiment to separate spin-up and spin-down particles. Furthermore, the detailed investigation for the dipole moments interacting with the time-variation of electromagnetic fields may predict new physics. However, the calculation for the FW transformation will be much more complicated if the fields are non-static and inhomogeneous. ACKNOWLEDGMENTS {#acknowledgments .unnumbered} =============== The authors would like to thank Chih-Wei Chang for valuable discussions and D.W.C. is grateful to Jiun-Huei Wu for the warm hospitality during his visit at National Taiwan University. T.W.C. is supported by the financial support from the National Science Council and NCTS of Taiwan; D.W.C. is supported by the NSFC Grant No. 10675019 and the financial support of Grants No. 20080440017 and No. 200902062 from the China Postdoctoral Science Foundation. Foldy-Wouthuysen transformation {#sec:FW transform} =============================== In this appendix, we expand the Dirac Hamiltonian up to terms of the 7th order in $1/E_g$ with $E_g=2mc^2$. The Dirac Hamiltonian can be separated in to two parts. One is the even operator denoted as $\Omega_E$, which commutes with $\matrixbeta$, and the other is odd operator $\Omega_o$, which anti-commutes with $\matrixbeta$: $$\begin{split} [\matrixbeta,\Omega_E]=0,\\ \{\matrixbeta,\Omega_o\}=0. \end{split}$$ The Dirac Hamiltonian can be written as $$H=\frac{\matrixbeta}{2}E_g+\Omega_E+\Omega_o,$$ where $\Omega_E=e\phi+\tilde{e}\tilde{\phi}$ and $\Omega_o=c\matrixboldalpha\cdot\boldPi$. The kinetic momentum $\boldPi$ is $\boldPi=\mathbf{p}-\frac{e}{c}\mathbf{A}-\frac{\tilde{e}}{c}\widetilde{\mathbf{A}}$. The first transformation operator can be written as $$U_{1}=e^{S_1}, \quad S_1=\matrixbeta\Omega_o/E_g.$$ The Dirac Hamiltonian under the unitary transformation $U_1$ can be written as $$\label{Appendix_HFW1} H_{1}=U_1HU^{-1}_1=\frac{\beta}{2}E_g+\Omega_E+\sum_{n=1}^{\infty}\frac{1}{E_g^n}\left\{\left(\frac{1}{n!}-\frac{1}{(n+1)!}\right)[(\matrixbeta\Omega_o)_{(n)},\Omega_o]+\frac{1}{n!}[(\matrixbeta\Omega_o)_{(n)},\Omega_E]\right\},$$ where the subscript $n$ at $(\matrixbeta\Omega)_{(n)}$ is defined as, for example, $[(\matrixbeta\Omega_o)_{(3)},\Omega_o]=[\matrixbeta\Omega_o,[\matrixbeta\Omega_o,[\matrixbeta\Omega_o,\Omega_E]]]$. Equation (\[Appendix\_HFW1\]) can be again separated into odd and even parts. The even part of Eq. (\[Appendix\_HFW1\]) denoted as $h$ can be written as $$h=h^{(0)}+\sum_{n=1}^{\infty}\frac{h^{(n)}}{E_g^n},$$ where $h^{(0)}=\Omega_E$ and $$\label{Appendix_h} \begin{split} &h^{(n=1,3,5,\cdots)}=\left(\frac{1}{n!}-\frac{1}{(n+1)!}\right)[(\matrixbeta\Omega_o)_{(n)},\Omega_o],\\ &h^{(n=2,4,6,\cdots)}=\frac{1}{n!}[(\matrixbeta\Omega_o)_{(n)},\Omega_E]. \end{split}$$ The odd part of Eq. (\[Appendix\_HFW1\]) denoted as $\mathcal{O}$ can be written as $$\mathcal{O}=\sum_{n=1}^{\infty}\frac{\mathcal{O}^{(n)}}{E_g^n},$$ where $$\label{Appendix_unprimeO} \begin{split} &\mathcal{O}^{(n=1,3,5,\cdots)}=\frac{1}{n!}[(\matrixbeta\Omega_o)_{(n)},\Omega_E],\\ &\mathcal{O}^{(n=2,4,6,\cdots)}=\left(\frac{1}{n!}-\frac{1}{(n+1)!}\right)[(\matrixbeta\Omega_o)_{(n)},\Omega_o]. \end{split}$$ Therefore, Eq. (\[Appendix\_HFW1\]) becomes $$H_1=\frac{\matrixbeta}{2}E_g+h+\mathcal{O},$$ where $h$ contains those terms with only even matrices and $\mathcal{O}$ contains only odd matrices. The second transformation denoted as $U_2=\exp(S_2)$, where $S_2$ is $S_2=\matrixbeta\mathcal{O}/E_g$. We have $$\label{Appendix_HFW2} \begin{split} H_2=U_2H_1U^{-1}_2&=\frac{\beta}{2}E_g+h+\sum_{n=1}^{\infty}\frac{1}{E_g^n}\left\{\left(\frac{1}{n!}-\frac{1}{(n+1)!}\right)[(\matrixbeta\mathcal{O})_{(n)},\mathcal{O}]+\frac{1}{n!}[(\matrixbeta\mathcal{O})_{(n)},h]\right\}\\ &=\frac{\beta}{2}E_g+h'+\mathcal{O}', \end{split}$$ where $h'$ and $\mathcal{O}'$ are the new even and odd parts, respectively, of the right hand side of the first equality. The even part of Eq. (\[Appendix\_HFW2\]) denoted as $h'$ can be written as $$\label{Appendix_h'} \begin{split} h'&=h+\left\{\begin{array}{c} \displaystyle \sum_{n=1,3,5\cdots}\frac{1}{E_g^n}\left(\frac{1}{n!}-\frac{1}{(n+1)!}\right)[(\matrixbeta\mathcal{O})_{(n)},\mathcal{O}]\\ \\ \displaystyle \sum_{n=2,4,6\cdots}\frac{1}{E_g^n}\frac{1}{n!}[(\matrixbeta\mathcal{O})_{(n)},h] \end{array}\right.\\ &=h'^{(0)}+\sum_{m=1}^{\infty}\frac{h'^{(m)}}{E_g^{m}}. \end{split}$$ The odd term $\mathcal{O}'$ in Eq. (\[Appendix\_HFW2\]) is given by $$\label{Appendix_primeO} \begin{split} \mathcal{O}'&=\left\{\begin{array}{c} \displaystyle \sum_{n=2,4,6\cdots}\frac{1}{E^n_g}\left(\frac{1}{n!}-\frac{1}{(n+1)!}\right)[(\matrixbeta\mathcal{O})_{(n)},\mathcal{O}]\\ \\ \displaystyle \sum_{n=1,3,5\cdots}\frac{1}{E_g^n}\frac{1}{n!}[(\matrixbeta\mathcal{O})_{(n)},h] \end{array}\right.\\ &=\sum_{m=3}^{\infty}\frac{\mathcal{O}'^{(m)}}{E_g^m}. \end{split}$$ Firstly, we note that $\mathcal{O}'^{(1)}$ and $\mathcal{O}'^{(2)}$ in Eq. (\[Appendix\_primeO\]) vanish as we have $m$ starting from $3$. For the former result, the reason is that the lowest order of the unprimed odd term $\mathcal{O}$ is 1, and thus in the second line of the first equality in Eq. (\[Appendix\_primeO\]), the primed odd term $\mathcal{O}'$ is at least of the 2nd order. On the other hand, the explicit form of $\mathcal{O}'^{(2)}$ can be written as $\mathcal{O}'^{(2)}=[\matrixbeta\mathcal{O}^{(1)},h^{(0)}]$. However, $\mathcal{O}^{(1)}$ is given by Eq. (\[Appendix\_unprimeO\]) for $n=1$, namely, $\mathcal{O}^{(1)}=\matrixbeta[\Omega_o,\Omega_E] =ic\hbar\matrixboldalpha\cdot(e\mathbf{E}+\tilde{e}\widetilde{\mathbf{E}})$, which commutes with $h^{(0)}=\Omega_E=e\phi+\tilde{e}\tilde{\phi}$. If we perform the third transformation which is $S_3=\matrixbeta\mathcal{O}'/E_g$ at $H_2=\frac{\matrixbeta E_g}{2}+h'+\mathcal{O}'$, we will obtain a new even term $h''$ as well as the new odd term $\mathcal{O}''$: $H_3=U_{3}H_2U^{-1}_3=\frac{\matrixbeta}{2}E_g+h''+\mathcal{O}''$. The even term $h''$ is given by $$\label{Appendix_h''} \begin{split} h''&=h'+\left\{\begin{array}{c} \displaystyle \sum_{n=1,3,5\cdots}\frac{1}{E_g^n}\left(\frac{1}{n!}-\frac{1}{(n+1)!}\right)[(\matrixbeta\mathcal{O}')_{(n)},\mathcal{O}']\\ \\ \displaystyle \sum_{n=2,4,6\cdots}\frac{1}{E_g^n}\frac{1}{n!}[(\matrixbeta\mathcal{O}')_{(n)},h'] \end{array}\right.\\ &=h''^{(0)}+\sum_{m=1}^{\infty}\frac{h''^{(m)}}{E_g^{m}}. \end{split}$$ The odd term $\mathcal{O}''$ is given by $$\label{Appendix_prime2O} \begin{split} \mathcal{O}''&=\left\{\begin{array}{c} \displaystyle \sum_{n=2,4,6\cdots}\frac{1}{E^n_g}\left(\frac{1}{n!}-\frac{1}{(n+1)!}\right)[(\matrixbeta\mathcal{O}')_{(n)},\mathcal{O}']\\ \\ \displaystyle \sum_{n=1,3,5\cdots}\frac{1}{E_g^n}\frac{1}{n!}[(\matrixbeta\mathcal{O}')_{(n)},h'] \end{array}\right.\\ &=\sum_{m=4}^{\infty}\frac{\mathcal{O}''^{(m)}}{E_g^m}. \end{split}$$ The odd term $\mathcal{O}''$ starts form $m=4$. Obviously, $\mathcal{O}^{(1)}$ is zero because $\mathcal{O}'^{(m)}$ starts from $m=3$. This can also be seen as follows. The explicit form of the next three terms of $\mathcal{O}''^{(m)}$ are $$\begin{split} &\mathcal{O}''^{(2)}=[\matrixbeta\mathcal{O}'^{(1)},h'^{(0)}],\\ &\mathcal{O}''^{(3)}=[\matrixbeta\mathcal{O}'^{(2)},h'^{(0)}]+[\matrixbeta\mathcal{O}'^{(1)},h'^{(1)}],\\ &\mathcal{O}''^{(4)}=[\matrixbeta\mathcal{O}'^{(1)},h'^{(2)}]+[\matrixbeta\mathcal{O}'^{(2)},h'^{(1)}]+[\matrixbeta\mathcal{O}'^{(3)},h'^{(0)}].\\ \end{split}$$ Because $\mathcal{O}'^{(1)}$ and $\mathcal{O}'^{(2)}$ are zero, $\mathcal{O}''^{(2)}$ and $\mathcal{O}''^{(3)}$ vanish. It can be shown that $\mathcal{O}''^{(4)}$ does not vanish. Using Eq. (\[Appendix\_h”\]), $h''^{(m)}$ for $m=1,2,\cdots 6$ are given by $$\begin{split} &h''^{(0)}=h'^{(0)},\\ &h''^{(1)}=h'^{(1)},\\ &h''^{(2)}=h'^{(2)},\\ &h''^{(3)}=h'^{(3)}+\left(1-\frac{1}{2!}\right)[\matrixbeta\mathcal{O}'^{(1)},\mathcal{O}'^{(1)}],\\ &h''^{(4)}=h'^{(4)}+\left(1-\frac{1}{2!}\right)\left([\matrixbeta\mathcal{O}'^{(1)},\mathcal{O}'^{(2)}]+[\matrixbeta\mathcal{O}'^{(2)},\mathcal{O}'^{(1)}]\right)+\frac{1}{2!}[\matrixbeta\mathcal{O}'^{(1)},[\matrixbeta\mathcal{O}'^{(1)},h'^{(0)}]],\\ &h''^{(5)}=h'^{(5)}+\left(1-\frac{1}{2!}\right)\mathop{\sum_{\ell,m=1}^{3}}\limits_{(\ell+m=4)}[\matrixbeta\mathcal{O}'^{(\ell)},\mathcal{O}'^{(m)}]+\mathop{\frac{1}{2!}\sum_{\ell,m=1}^2\sum_{n=0}^1}\limits_{(\ell+m+n=3)}[\matrixbeta\mathcal{O}'^{(\ell)},[\matrixbeta\mathcal{O}'^{(m)},h'^{(n)}]],\\ &h''^{(6)}=h'^{(6)}+\left(1-\frac{1}{2!}\right)\mathop{\sum_{\ell,m=1}^{4}}\limits_{(\ell+m=5)}[\matrixbeta\mathcal{O}'^{(\ell)},\mathcal{O}'^{(m)}]+\mathop{\frac{1}{2!}\sum_{\ell,m=1}^3\sum_{n=0}^2}\limits_{(\ell+m+n=4)}[\matrixbeta\mathcal{O}'^{(\ell)},[\matrixbeta\mathcal{O}'^{(m)},h'^{(n)}]].\\ \end{split}$$ Because $\mathcal{O}'^{(1)}$ and $\mathcal{O}'^{(2)}$ are zero, we have $h''^{(3)}=h'^{(3)}$ and $h''^{(4)}=h'^{(4)}$, since the constraints $\ell+m=4$ and $\ell+m=5$ imply $(\ell,m)=\{(1,3),(2,2),(3,1)\}$ and $(\ell,m)=\{(1,4),(2,3),(3,2),(4,1)\}$, respectively. The commutator $[\matrixbeta\mathcal{O}'^{(\ell)},\mathcal{O}'^{(m)}]$ in $h''^{(5)}$ and $h''^{(6)}$ vanishes. Consider the term $[\matrixbeta\mathcal{O}'^{(\ell)},[\matrixbeta\mathcal{O}'^{(m)},h'^{(n)}]]$ in $h''^{(5)}$ and $h''^{(6)}$ subject to the constraints $\ell+m+n=3$ and $\ell+m+n=4$, respectively. For $n=0$, we have $(\ell,m)=\{(1,2),(2,1)\}$ and $(\ell,m)=\{(1,3),(2,2),(3,1)\}$ and thus $[\matrixbeta\mathcal{O}'^{(\ell)},[\matrixbeta\mathcal{O}'^{(m)},h'^{(n=0)}]]$ vanishes in $h''^{(5)}$ and $h''^{(6)}$. For $n=1$ and $n=2$, the term $[\matrixbeta\mathcal{O}'^{(\ell)},[\matrixbeta\mathcal{O}'^{(m)},h'^{(n)}]]$ still vanishes. Therefore, up to terms of the 7th order in $\frac{1}{E_g}$, we obtain an important result $$\label{Appendix_mainresult} h''^{(n)}=h'^{(n)}, \quad n=1,2,\cdots 6,$$ and $h'^{(n)}$ (i.e., Eq. (\[Appendix\_h’\])) is given by $$\label{Appendix_h'expand} \begin{split} &h'^{(0)}=h^{(0)},\\ &h'^{(1)}=h^{(1)},\\ &h'^{(2)}=h^{(2)},\\ &h'^{(3)}=h^{(3)}+\left(1-\frac{1}{2!}\right)[\matrixbeta\mathcal{O}^{(1)},\mathcal{O}^{(1)}],\\ &h'^{(4)}=h^{(4)}+\left(1-\frac{1}{2!}\right)\left([\matrixbeta\mathcal{O}^{(1)},\mathcal{O}^{(2)}]+[\matrixbeta\mathcal{O}^{(2)},\mathcal{O}^{(1)}]\right)+\frac{1}{2!}[\matrixbeta\mathcal{O}^{(1)},[\matrixbeta\mathcal{O}^{(1)},h^{(0)}]],\\ &h'^{(5)}=h^{(5)}+\left(1-\frac{1}{2!}\right)\mathop{\sum_{\ell,m=1}^{3}}\limits_{(\ell+m=4)}[\matrixbeta\mathcal{O}^{(\ell)},\mathcal{O}^{(m)}]+\mathop{\frac{1}{2!}\sum_{\ell,m=1}^2\sum_{n=0}^1}\limits_{(\ell+m+n=3)}[\matrixbeta\mathcal{O}^{(\ell)},[\matrixbeta\mathcal{O}^{(m)},h^{(n)}]],\\ &h'^{(6)}=h^{(6)}+\left(1-\frac{1}{2!}\right)\mathop{\sum_{\ell,m=1}^{4}}\limits_{(\ell+m=5)}[\matrixbeta\mathcal{O}^{(\ell)},\mathcal{O}^{(m)}]+\mathop{\frac{1}{2!}\sum_{\ell,m=1}^3\sum_{n=0}^2}\limits_{(\ell+m+n=4)}[\matrixbeta\mathcal{O}^{(\ell)},[\matrixbeta\mathcal{O}^{(m)},h^{(n)}]].\\ \end{split}$$ Therefore, in order to obtain the FW transformation up to $1/E_g^7$, we need to know four odd terms $\mathcal{O}^{(n)}$ for $n=1,2,3,4$. On the other hand, if we perform the transformation again by using $S_4=\matrixbeta\mathcal{O}''/E_g$, since $\mathcal{O}''^{(m)}$ starts form $m=4$ (i.e. $\mathcal{O}''^{(m)}=0$ for $m=1,2,3$), the transformation $S_4$ does not change Eq. (\[Appendix\_mainresult\]). The resulting $\mathcal{O}'''^{(m)}$ will start from at least $m=5$. To bring the odd term to the 7th order, we need $S_5=\matrixbeta\mathcal{O}'''/E_g$ and $S_6=\matrixbeta\mathcal{O}''''/E_g$. However, the two transformations also do not change the validity of Eq. (\[Appendix\_mainresult\]). Therefore the resulting Hamiltonian can be written as $$\begin{split} H_\mathrm{FW}&=U_\mathrm{FW}HU^{-1}_\mathrm{FW}\\ &=\frac{\matrixbeta}{2}E_g+\sum_{n=0}^{6}H^{(n)}_\mathrm{FW}+o(1/E_g^7). \end{split}$$ By using Eqs. (\[Appendix\_h\]), (\[Appendix\_h’expand\]) and (\[Appendix\_mainresult\]), after straightforward calculations it can be shown that \[ALLHFW\] $$\begin{aligned} &H^{(1)}_\mathrm{FW}=\frac{\matrixbeta\Omega_o^2}{E_g},\\ &H^{(2)}_\mathrm{FW}=\frac{1}{E_g^2}\left(\frac{\mathcal{W}}{2}\right),\\ &H^{(3)}_\mathrm{FW}=\frac{1}{E_g^3}\left\{-\matrixbeta\Omega_o^4+\matrixbeta\left(\matrixbeta\mathcal{D}\right)^2\right\},\\ &H^{(4)}_\mathrm{FW}=\frac{1}{E_g^4}\left(\frac{1}{24}[[\Omega_o,\mathcal{W}],\Omega_o]-\frac{4}{3}[\mathcal{D},\Omega_o^3]\right),\\ &H^{(5)}_\mathrm{FW}=\frac{1}{E_g^5}\left\{\frac{1}{144}[(\matrixbeta\Omega_o)_{(5)},\Omega_o]+\frac{1}{2}\mathop{\sum_{\ell,m=1}^{3}}\limits_{(\ell+m=4)}[\matrixbeta\mathcal{O}^{(\ell)},\mathcal{O}^{(m)}]+\frac{1}{2}\mathop{\sum_{\ell,m=1}^{2}\sum_{n=0}^{1}}\limits_{(\ell+m+n=3)}[\matrixbeta\mathcal{O}^{(\ell)},[\matrixbeta\mathcal{O}^{(m)},h^{(n)}]]\right\},\\ &H^{(6)}_\mathrm{FW}=\frac{1}{E_g^6}\left\{\frac{1}{720}[(\matrixbeta\Omega_o)_{(6)},\Omega_o]+\frac{1}{2}\mathop{\sum_{\ell,m=1}^{4}}\limits_{(\ell+m=5)}[\matrixbeta\mathcal{O}^{(\ell)},\mathcal{O}^{(m)}]+\frac{1}{2}\mathop{\sum_{\ell,m=1}^{3}\sum_{n=0}^{2}}\limits_{(\ell+m+n=4)}[\matrixbeta\mathcal{O}^{(\ell)},[\matrixbeta\mathcal{O}^{(m)},h^{(n)}]]\right\},\\\end{aligned}$$ where $\mathcal{D}\equiv[\Omega_o,\Omega_E]$ and $\mathcal{W}\equiv[\mathcal{D},\Omega_o]$. Derivation of Equation (\[HFW\_4\_2(2)\]) {#sec:derivation} ========================================= The explicit form of $\Omega_o\mathcal{W}\Omega_o$ \[Eq. (\[HFW\_4\_2\])\] plays an important role in obtaining the correct coefficient of each term in $H_\mathrm{FW}^{(6)}$. By definition, $\Omega_o=c\,\matrixboldalpha\cdot\boldPi$ and $\mathcal{W}=[[\Omega_o,\Omega_E],\Omega_o] =[[c\matrixboldalpha\cdot\boldPi,V],c\,\matrixboldalpha\cdot\boldPi]$, where $V=e\phi+\tilde{e}\widetilde{\phi}$. In the derivation for Eq. (\[HFW\_4\_2\]), electromagnetic fields are assumed to be homogeneous and static. This means that the terms involving gradient of fields are neglected. Therefore, we have $$\label{A2_1} \frac{1}{c^2}\Omega_o\mathcal{W}\Omega_o=-2c^2\hbar\epsilon_{pqr}\mathcal{E}_q\Pi_i\Pi_r\Pi_j(\matrixalpha_i\Sigma_p\matrixalpha_j),$$ where $\mathcal{E}_q\equiv(eE_q+\tilde{e}\tilde{E}_q)$ and $$\mathcal{W}=-2c^2\hbar\boldsymbol{\Sigma}\cdot(\boldsymbol{\mathcal{E}}\times\boldsymbol{\Pi})$$ is used. Furthermore, it can be shown that $$\label{A2_PiPj} [\Pi_i,\Pi_j]=\frac{i\hbar}{c}\,\epsilon_{ijk}\mathcal{B}_k,$$ were $\mathcal{B}_k=eB_k+\tilde{e}\tilde{B}_k$. By using $[\matrixalpha_i,\sigma_p]=2i\epsilon_{ipm}\matrixalpha_m$ and $\epsilon_{pqr}\epsilon_{ipm}=\delta_{qm}\delta_{ri}-\delta_{qi}\delta_{rm}$, Eq. (\[A2\_1\]) can be written as $$\label{A2_2} \begin{split} &\frac{1}{c^2}\Omega_o\mathcal{W}\Omega_o\\ &=-2\hbar\mathcal{E}_q(2i\matrixalpha_q\matrixalpha_j|\boldPi|^2\Pi_j-2i\matrixalpha_r\matrixalpha_j\Pi_q\Pi_r\Pi_j\\ &~~+\epsilon_{pqr}\Sigma_p\Pi_i\Pi_r\Pi_i+i\epsilon_{pqr}\epsilon_{ijm}\Sigma_m\Sigma_p\Pi_i\Pi_r\Pi_j). \end{split}$$ By using $\matrixalpha_q\matrixalpha_j=\delta_{qj}+i\epsilon_{qj\ell}\Sigma_{\ell}$, the first term of Eq. (\[A2\_2\]) becomes $$\label{A2_3} 2i\matrixalpha_q\matrixalpha_j|\boldPi|^2\Pi_j=2i|\boldPi|^2\Pi_q-2\epsilon_{qj\ell}\Sigma_{\ell}|\boldPi|^2\Pi_j.$$ On the other hand, the second term of Eq. (\[A2\_2\]) can be written as $$\label{A2_4} -2i\matrixalpha_r\matrixalpha_j\Pi_q\Pi_r\Pi_j=-2i\Pi_q|\boldPi|^2+\frac{2i\hbar}{c}\Sigma_{\ell}\Pi_q\mathcal{B}_{\ell},$$ where Eq. (\[A2\_PiPj\]) is used. The third term of Eq. (\[A2\_2\]) can be written as $$\label{A2_5} \begin{split} &\epsilon_{pqr}\Sigma_p\Pi_i\Pi_r\Pi_i\\ &=\epsilon_{pqr}\Sigma_p\Pi_i\left(\frac{i\hbar}{c}\epsilon_{ri\ell}\mathcal{B}_{\ell}+\Pi_i\Pi_r\right)\\ &=\frac{i\hbar}{c}\left(\Sigma_i\Pi_i\mathcal{B}_q-\Sigma_{\ell}\Pi_{q}\mathcal{B}_{\ell}\right)+\epsilon_{pqr}\Sigma_p|\boldPi|^2\Pi_r. \end{split}$$ The fourth term of Eq. (\[A2\_2\]) becomes $$\label{A2_6} \begin{split} &i\epsilon_{pqr}\epsilon_{ijm}\Sigma_m\Sigma_p\Pi_i\Pi_r\Pi_j\\ &=i\epsilon_{pqr}\epsilon_{ijm}\Sigma_m\Sigma_p\Pi_i\left(\frac{i\hbar}{c}\epsilon_{rj\ell}\mathcal{B}_{\ell}+\Pi_j\Pi_r\right)\\ &=-\frac{\hbar}{c}\epsilon_{pqr}(2B_{\ell}\Sigma_{\ell}\Pi_r-\mathcal{B}_i\Pi_i\Sigma_r+\mathcal{B}_m\Sigma_m)\Sigma_p, \end{split}$$ where $\epsilon_{ijm}\Pi_i\Pi_j=\frac{i\hbar}{c}\mathcal{B}_m$ and $\epsilon_{ijm}\epsilon_{rj\ell}=\delta_{ir}\delta_{m\ell}-\delta_{i\ell}\delta_{mr}$ are used. We note that there is a field $\mathcal{E}_q$ in the Eq. (\[A2\_2\]). By neglecting the product of fields $\mathcal{E}_i\mathcal{B}_j$, Eq. (\[A2\_2\]) with substitutions of Eqs. (\[A2\_3\]), (\[A2\_4\]), (\[A2\_5\]) and (\[A2\_6\]) becomes $$\begin{split} \frac{1}{c^2}\Omega_o\mathcal{W}\Omega_o&\approx-2c^2\hbar\mathcal{E}_q(-2\epsilon_{qj\ell}\Sigma_{\ell}|\boldPi|^2\Pi_j+\epsilon_{pqr}\Sigma_p|\boldPi|^2\Pi_r)\\ &=-|\boldPi|^2(-2c^2\epsilon_{pqr}\Sigma_p\mathcal{E}_q\Pi_r)\\ &=-|\boldPi|^2\mathcal{W}, \end{split}$$ and we have $$\Omega_o\mathcal{W}\Omega_o\approx-c^2|\boldPi|^2\mathcal{W}.$$ [99]{} P. A. M. Dirac, Proc. R. Soc. London [**117**]{}, 610 (1928). L. L. Foldy and S. A. Wouthuysen, Phys. Rev. [**78**]{}, 29 (1950). P. O. Löwding, J. Chem. Phys. [**19**]{}, 1396 (1951); J. M. Luttinger and W. Kohn, Phys. Rev. [**97**]{}, 896 (1955). E. Eriksen, Phys. Rev. [**111**]{}, 1011 (1958). E. de. Vries and J. E. Jonker, Nucl. Phys. B [**6**]{}, 213 (1968). A. J. Silenko, J. Math. Phys. [**44**]{}, 2952 (2003). K. Yu. Bliokh, Europhys. Lett. [**72**]{}, 7 (2005). P. Gosselin, A. Berard and H. Mohrbach, Eur. Phys. J. B [**58**]{}, 137 (2007); Phys. Lett. A [**368**]{}, 356 (2007). A. J. Silenko, Phys, Rev. A [**77**]{}, 012116 (2008). L. H. Thomas, Phil. Mag. [**3**]{}, 1 (1927); V. Bargmann, L. Michel and V. L. Telegdi, Phys. Rev. Lett. [**2**]{}, 435 (1959). Ya. M. Shnir, [*Magnetic Monopole*]{} (Springer, 2005, First Edition). W. Pauli, Rev. Mod. Phys. [**13**]{}, 203 (1941). J. D. Jackson, [*Classical Electrodynamics*]{} (John Wiley & Sons, 1999, Third Edition). J. Fröhlich and U. M. Studer, Rev. Mod. Phys. [**65**]{}, 733 (1993). J. J. Sakurai, [*Modern Quantum Mechanics*]{} (Addison-Wesley, 1994, Revised Edition). [^1]: It is often said that FW method gives the nonrelativistic limit of the Dirac equation. The phrase “nonrelativistic” is somewhat misleading as it usually refers to “low-speed” limit. As we will show in this paper, the FW transformation (if performed to orders high enough) actually agrees with the relativistic classical dynamics even when the speed of the particle is large. The appropriate description is to say that the FW transformation yields “low-energy” limit. [^2]: Since the static condition gives $\nabla'\cdot\mathbf{J}'=0$, it can be shown $J'_i=\nabla'\cdot(x'_i\mathbf{J'})$. Consequently, $\int_{V'}d^3x'J'_i=\int_{V'}d^3x'\,\nabla'\cdot(x'_i\mathbf{J'}) =\int_{\partial V'}d\,\mathbf{a}'\cdot (x'_i\mathbf{J'})=0$ if the current $\mathbf{J}'$ is localized. [^3]: Note that, in order to add $H_\mathrm{orbit}$ and $H_\mathrm{spin}$ together, we have to consider $d\mathbf{s}/dt\equiv d\mathbf{S}'/dt$ in Eq. (\[Thomas\]), instead of $d\mathbf{S}/dt$, $d\mathbf{S}/d\tau$ or $d\mathbf{s}/d\tau$. This is because $s_i$ are degrees of freedom independent of $\mathbf{x}$ and $\mathbf{p}$, but $S_i$ are not. Furthermore, to be consistent with the orbital motion, the precession is cast with respect to $t$, instead of the proper time $\tau$ of the moving particle. [^4]: To avoid confusion with the boost velocity $\beta$, we use the checked notation $\matrixbeta$ to denote the $4\times4$ matrix. With the same style, the notations $\matrixalpha$, $\matrixboldgamma$ \[defined in Eq. (\[matrices beta and gamma\])\] and $\matrixeta$ \[defined in Eq. (\[matrix eta\])\] are checked as well. [^5]: It must be stressed that the operator $\boldxi$ does not directly correspond to the Lorentz boost velocity $\boldsymbol{\beta}$ given in the previous section. The appropriate transformation between $\boldxi$ and operator for the boost velocity is considered in Sec.\[sec:relation to TBMT\].

--- abstract: 'Suppose that a $d$-dimensional domain is filled with a gas of (in general, interacting) diffusive particles with density $n_0$. A particle is absorbed whenever it reaches the domain boundary. Employing macroscopic fluctuation theory, we evaluate the probability ${\mathcal P}$ that no particles are absorbed during a long time $T$. We argue that the most likely gas density profile, conditional on this event, is stationary throughout most of the time $T$. As a result, ${\mathcal P}$ decays exponentially with $T$ for a whole class of interacting diffusive gases in any dimension. For $d=1$ the stationary gas density profile and ${\mathcal P}$ can be found analytically. In higher dimensions we focus on the simple symmetric exclusion process (SSEP) and show that $-\ln {\mathcal P}\simeq D_0TL^{d-2} \,s(n_0)$, where $D_0$ is the gas diffusivity, and $L$ is the linear size of the system. We calculate the rescaled action $s(n_0)$ for $d=1$, for rectangular domains in $d=2$, and for spherical domains. Near close packing of the SSEP $s(n_0)$ can be found analytically for domains of any shape and in any dimension.' author: - Tal Agranov - Baruch Meerson - Arkady Vilenkin title: Survival of interacting diffusing particles inside a domain with absorbing boundary --- Introduction ============ Diffusive lattice gases serve as useful simplified models of many stochastic spatio-temporal systems in nature. Among them are diffusion-controlled chemical reactions: reactions which occur quickly once the diffusing reagent particles “find" each other in space. A simple but amazingly rich model of this process, due to Smoluchowski [@Smoluchowski], treats one of the two reacting species as an immobile large-size minority. The Smoluchowsi’s model allows one to calculate the expected reaction rate (that is, the expected rate of absorption of a random walker by a target). Statistics of *fluctuations* of this rate have been the subject of numerous studies [@R85; @OTB89; @bAH; @Rednerbook; @Paulbook]. When the majority molecules are treated as noninteracting random walkers, the calculation of the effective reaction rate and its fluctuation statistics boils down to calculating a single-particle probability. Recently some progress has been also made in the situation when the diffusing particles interact with each other [@MVK; @M15]. Here we extend this line of work by putting the walkers inside a domain which boundary is the “target". Suppose that a gas of diffusing and, in general, interacting particles with density $n_0$ fills a $d$-dimensional domain $\Omega$. Each particle is absorbed immediately whenever it reaches the domain boundary $\partial \Omega$. This simple setting is a caricature of a host of processes inside the living cell, where a molecule needs to reach the cell membrane [@Bresloff]. We assume that, on macroscopic length and time scales, the average gas density inside the domain, $n(\mathbf{x},t)$, evolves according to a diffusion equation, $$\label{difeq} \partial_t n = \nabla \cdot [D(n) \nabla n]$$ with diffusivity $D(n)$ that may depend on $n$. The boundary condition at the absorbing domain boundary is $$\label{bq} n(\mathbf{x}\in\partial \Omega,t) = 0.$$ Solving Eqs. (\[difeq\]) and (\[bq\]) for a given initial condition, such as $n(\mathbf{x},t=0)=n_0$, and calculating the diffusion flux into the domain boundary, one can find the *expected* number $\bar{N}(T)$ of absorbed particles during time $T$. In individual realizations of the underlying microscopic stochastic process, the number of absorbed particles fluctuates around $\bar{N}(T)$, and it is interesting to determine the fluctuation statistics. In this work we will deal with an extreme limit of this statistics, corresponding to the *survival* probability ${\mathcal P} (T)$: the probability that not a single particle hit the domain boundary by time $T$ which is long compared to the characteristic diffusion time through the domain. For non-interacting diffusing particles (we will call them Random Walkers, or RWs), $D(n)=D_0=\text{const}$. In this case one obtains [@Paulbook] $$\label{RWgeneral} -\ln {\mathcal P}_{\text{RW}} \simeq n_0 D_0 TV \mu_1^2,$$ where $V$ is the domain volume, and $\mu_1$ is the lowest positive eigenvalue of the eigenvalue problem $\nabla^2 u+\mu^2 u=0$ for the Laplace’s operator inside the domain with the boundary condition $u(\mathbf{x}\in\partial \Omega) = 0$. For a $d$-dimensional sphere of radius $R$, Eq. (\[RWgeneral\]) yields the well-known results $$\label{RWresult} -\ln {\mathcal P}_{\text{RW}} \simeq n_0 D_0T R^{d-2} f_d ,$$ where [f\_d ]{} , & $d=1$, \[survivaldecay1\]\ z\_1\^2, & $d=2$, \[survivaldecay2\]\ ,& $d=3$, \[survivaldecay3\] and $z_1=2.4048\dots$ is the first positive root of the Bessel function $J_0(z)$. The exponential decay of ${\mathcal P}_{\text{RW}}$ with time $T$, as described by Eqs. (\[RWgeneral\]) and (\[RWresult\]), reflects the fact that, at long times, the *single-particle* survival probability decays exponentially with time. Indeed, for a single RW, the survival probability distribution $\rho_1(\mathbf{x},t)$ inside the domain obeys the diffusion equation $\partial_t \rho_1 = D_0 \nabla^2 \rho_1$ with the absorbing boundary condition $\rho_1(\mathbf{x} \in \partial \Omega,t) = 0$ and a delta-function initial condition [@Rednerbook]. Finding $\rho_1(\mathbf{x},t)$ and integrating it over the domain, one obtains the single-particle survival probability as a function of time. Its long-time asymptotic describes an exponential decay with the decay rate corresponding to the lowest positive eigenvalue $\mu_1$ of the Laplace’s operator. The *gas* survival probability ${\mathcal P}_{\text{RW}}$ is given by the product of the survival probabilities of all independent particles inside the domain. What is left to arrive at Eq. (\[RWgeneral\]) is to go over to the continuum limit by replacing a discrete sum in the exponent of ${\mathcal P}_{\text{RW}}$ by an integral. For completeness, this procedure is presented in Appendix \[mic\]. As a result, the exponential long-time decay of the survival probability holds, for the independent RWs, in all spatial dimensions, as evidenced by Eq. (\[RWgeneral\]). An important finding that we report here is that the exponential-in-time decay of the survival probability $\mathcal P$ holds when the diffusing particles interact with each other, and the single-particle picture breaks down. This non-trivial result is a consequence of the fact that the optimal gas density history, conditional on the long-time survival of all particles, is almost stationary, in any spatial dimension. We show it here by employing the Macroscopic Fluctuation Theory (MFT) [@MFTreview]. For $d=1$ the stationary MFT problem is soluble for a whole family of interacting gases. In higher dimensions the solution is in general unavailable. Here we focus on a gas of particles interacting via exclusion, so as to describe e.g. diffusion-controlled chemical reactions in a crowded environment of a living cell [@crowded]. Specifically, we will study the symmetric simple exclusion process (SSEP). In the SSEP each particle can hop to a neighboring lattice site if that site is vacant. If it is occupied, the move is not allowed [@Spohn]. For the SSEP we obtain $-\ln {\mathcal P}\simeq D_0TL^{d-2} \,s(n_0)$, where $L$ is the linear size of the domain [@dimensions]. We calculate the rescaled action $s(n_0)$ for several domain shapes and in different dimensions. As we show, $s(n_0)$ increases with the density $n_0$ faster than linearly, see Fig. \[fff\] for $d=1$, Fig. \[s2d\] for a rectangle in $d=2$, and Fig. \[s3d\] for a sphere in $d=3$. In the MFT formalism $s(n_0)$ diverges as $n_0$ approaches unity, but this divergence is cured when $n_0$ is sufficiently close to unity where the MFT breaks down. The interior survival problem, considered here, has an exterior analog that is known by the name “survival of the target". In the exterior problem a gas of particles surrounds an absorbing domain from outside. As in the interior problem, one is interested in the probability that no particle hits the domain during time $T$. For the RWs the exterior problem was extensively studied in the past [@ZKB83; @T83; @RK84; @BZK84; @BKZ86; @BO87; @Oshanin; @BB03; @BMS13]. Recently, the theory has been extended to interacting diffusive gases: for the survival probability [@MVK] and in the more general context of full absorption statistics [@M15]. As our present work, Refs. [@MVK; @M15] employed the MFT formalism. In contrast to the interior problem, the optimal gas density history in the exterior case becomes almost stationary only for $d>2$. Furthermore, there is a subtle but important difference in the stationary MFT formulations of the interior and exterior problems, as we explain in the following. In the next Section we formulate the MFT in the context of the interior survival problem. In Sec. \[RWs\] we apply it to the non-interacting RWs in one dimension, where ${\mathcal P}$ is known, see Eqs. (\[RWresult\]) and (\[survivaldecay1\]). In this case we can solve the full time-dependent problem exactly. The solution shows that the optimal density profile for this ${\mathcal P}$ is time-independent for most of the time. The full time-dependent solution will help us identify the correct stationary formulation of the MFT problem. In Sec. \[generalstat\] we apply this stationarity ansatz to an arbitrary interacting diffusive gas in any dimension. This yields a stationary equation which can be simplified further upon a transformation of variable. In Sec. \[SSEP\] we apply this procedure to the SSEP. In subsection \[SSEP1d\] we present exact results for the SSEP in $d=1$: for the stationary optimal density profile and for the survival probability. We verify these results, in the same Section, by solving the full time-dependent MFT problem numerically. In subsection \[SSEP2d\] we solve the stationary MFT problem for a rectangular domain, $d=2$. In subsection \[SSEPspherical\] we study, analytically and numerically, the SSEP survival in spherical domains. In Sec. \[high\] we identify a universal behavior of the solution in the high-density limit of the SSEP: inside a domain of any shape and in any dimension. Finally, in Sec. \[otherlattice\] we present a general solution of the gas survival problem in $d=1$ which holds for a whole family of interacting diffusive gases. Section \[conclusion\] presents a brief discussion of our results. For completeness, in Appendix \[mic\] we calculate the survival probability of a gas of RWs from the microscopic perspective. Appendix \[rwnd\] extends the one-dimensional solution of the MFT equations for the RWs, presented in Sec. \[RWs\], to an arbitrary dimension and arbitrary domain shape. Macroscopic fluctuation theory of particle survival {#MFT} =================================================== The starting point of the macroscopic fluctuation theory (MFT) [@MFTreview] is fluctuational hydrodynamics: a Langevin equation for the fluctuating gas density $q(\mathbf{x},t)$: $$\label{Lang} \partial_t q = \nabla \cdot \left[D(q) \nabla q\right] +\nabla \cdot \left[\sqrt{\sigma(q)} \,\text{\boldmath$\eta$} (\mathbf{x},t)\right],$$ where $\text{\boldmath$\eta$} (\mathbf{x},t)$ is a zero-average Gaussian noise, delta-correlated both in space and in time. Equation (\[Lang\]) provides an asymptotically correct large-scale description of fluctuations in a broad family of diffusive lattice gases [@Spohn]. At the level of fluctuational hydrodynamics, a diffusive gas is fully characterized by the diffusivity $D(q)$ and additional coefficient, $\sigma(q)$, which comes from the shot noise and is equal to twice the gas mobility coefficient [@Spohn]. For example, for the non-interacting RWS one has $D(q)=D_0=\text{const}$ and $\sigma(q)=2 D_0 q$, whereas for the SSEP $D(q)=D_0=\text{const}$ and $\sigma(q)=2 D_0 q(1-q)$ [@Spohn; @dimensions]. The MFT equations are essentially the saddle-point equations of the path-integral formulation, corresponding to the weak-noise limit of Eq. (\[Lang\]) [@MFTreview; @Tailleur; @DG2009b]. The MFT theory employs the typical number of particles in the relevant region of space as a large parameter. It allows to calculate the optimal path of the system: the most probable density history conditional on a specified large deviation. If the large deviation is described in terms of a spatial integral constraint, this constraint can be accommodated via the Lagrange multiplier formalism and provides a problem-specific boundary condition in time [@DG2009b]. Suppose we are interested in the probability that $N$ particles are absorbed by the domain boundary by time $T$. (We will ultimately consider the limit of $N=0$.) This defines an integral constraint on the solution: $$\label{number} \int_{\Omega} d\mathbf{x}[n_0-q(\mathbf{x},T)] = N.$$ The same type of constraint appears in the exterior problem [@MVK; @M15]. A similar constraint also appears in the problem of statistics of integrated current through a lattice site during a specified time [@DG2009b; @KM_var; @MS2013; @MS2014; @MR]. Referring the reader to Ref. [@DG2009b] for a detailed derivation, we will only present here the resulting MFT equations and boundary conditions. The MFT equations can be written as two coupled partial differential equations for the optimal density field $q(\mathbf{x},t)$ (the “coordinate") and the conjugate “momentum" density field $p(\mathbf{x},t)$: $$\begin{aligned} \partial_t q &=& \nabla \cdot \left[D(q) \nabla q-\sigma(q) \nabla p\right], \label{d11} \\ \partial_t p &=& - D(q) \nabla^2 p-\frac{1}{2} \,\sigma^{\prime}(q) (\nabla p)^2, \label{d22}\end{aligned}$$ where the prime denotes the derivative with respect to the single argument. Equations  and are Hamiltonian, $$\partial_t q = \delta H/\delta p\,, \quad \partial_t p = -\delta H/\delta q\,.$$ Here $$\label{Hamiltonian} H[q(\mathbf{x},t),p(\mathbf{x},t)]= \int_{\Omega} d\mathbf{x}\,\mathcal{H}$$ is the Hamiltonian, and $$\label{Ham} \mathcal{H}(q,p) = -D(q) \nabla q\cdot \nabla p +\frac{1}{2}\sigma(q)\!\left(\nabla p\right)^2$$ is the Hamiltonian density. The absorbing boundary imposes zero boundary conditions for $q$: $$\label{bcgenq} q(\mathbf{x}\in{\partial\Omega},t)=0.$$ Since the values of $q$ are fixed at the boundary, the conjugate field $p$ must vanish there [@MFTreview; @Tailleur; @MS2011]: $$\label{bcgenp} p(\mathbf{x}\in{\partial\Omega},t)=0.$$ For the RWs and SSEP this boundary condition was derived from the microscopic models [@Tailleur; @MS2011]. Although a general proof of Eq. (\[bcgenp\]) is unavailable [@JLprivate], its validity has been verified in many examples, see Refs. [@MFTreview; @MVK; @M15; @MR]. The boundary conditions in time are the following. For the density $q$ we choose a deterministic initial condition $$\label{t0q} q(\mathbf{x}\in{\Omega},t=0)=n_0.$$ The boundary condition in time for $p(\mathbf{x},t=T)$ follows from the integral constraint (\[number\]), accounted for via a Lagrange multiplier [@DG2009b]: $$\begin{aligned} \label{t0p} p(\mathbf{x}\in{\Omega},t=T)&=&\lambda,\\\nonumber p(\mathbf{x}\in{\partial\Omega},t=T)&=&0, \\end{aligned}$$ where the Lagrange multiplier $\lambda$ is ultimately set by the constraint (\[number\]). The zero-absorption limit $N=0$, that we are interested in here, corresponds to the limit of $\lambda\to +\infty$ [@MR; @MVK; @M15]. In this limit the particle flux to the boundary vanishes at all times $0<t<T$. Once the MFT equations with the proper boundary conditions are solved, we can calculate the action $S$ that yields $-\ln {\mathcal P}$ up to a pre-exponential factor: $$\begin{aligned} \label{actionmainsection1} % \nonumber to remove numbering (before each equation) -\ln {\mathcal P} \simeq S &=& \int_0^T dt\, \int_\Omega d\mathbf{x}\left(p\partial_t q-\mathcal{H}\right) \nonumber \\ &=&\frac{1}{2}\,\int_0^T dt \int_\Omega d\mathbf{x}\, \sigma(q)\, (\nabla p)^2.\end{aligned}$$ MFT of Random Walkers in one dimension: time-dependent solution and stationary asymptotic {#RWs} ========================================================================================= A one-dimensional domain can be set to be an interval of length $2R$, centered at the origin. For the RWs the MFT equations become: $$\begin{aligned} % \nonumber to remove numbering (before each equation) \partial_t q &=& D_0\partial_{x}^2 q-2D_0\partial_x (q \partial_x p), \label{qt_rw} \\ \partial_t p &=& -D_0 \partial_{x}^2 p- D_0(\partial_x p)^2. \label{pt_rw}\end{aligned}$$ The boundary conditions in space are $$\begin{aligned} q(|x|=R,t)&=&0,\label{bcq}\\ p(|x|=R,t)&=&0.\label{bcp}\end{aligned}$$ The boundary conditions in time are $$\begin{aligned} q(x,t=0)&=&n_0,\label{incond}\\ p(x,t=T)&=&\lambda\,\theta(R-|x|),\label{fincond}\end{aligned}$$ where $\theta(\dots)$ is the Heaviside step function. As many other large deviation problems for the RWs, the problem (\[qt\_rw\])-(\[fincond\]) is exactly soluble using the Hopf-Cole transformation $Q=qe^{-p}$ and $P=e^p$, defined by the generating functional $$\label{ajenerating} \int_{-R}^R dx\,F_1(q,Q)=\int_{-R}^R dx\left[q\ln(q/Q)-q\right].$$ In the new variables the Hamiltonian density is $$\label{Hamrw} \mathcal{H}(q,p) = -D_0 \partial_x Q \partial_x P,$$ and the action can be expressed using only the initial and final states of the system (see the Appendix of Ref. [@MR]) $$\begin{aligned} \label{aaction} -\ln {\mathcal P} \simeq S &=&\int_0^T dt\int_{-R}^R dx D_0q\left(\partial_x p\right)^2\\ &=&\int_{-R}^R dx \left[Q\left(P\ln P -P+1\right)\right] \big|_0^T\,.\end{aligned}$$ \[1d solution\] The transformed MFT equations are fully decoupled: $$\begin{aligned} % \nonumber to remove numbering (before each equation) \partial_t Q &=& D_0\partial_{x}^2 Q, \label{aQt_rw} \\ \partial_t P &=& -D_0 \partial_{x}^2 P, \label{aPt_rw}\end{aligned}$$ and the transformed boundary conditions are: $$\begin{aligned} Q(|x|=R,t)=0,\label{abcQ}\\ P(|x|=R,t)=1,\label{abcP}\end{aligned}$$ and $$\begin{aligned} Q(x,t=0)&=&\frac{n_0}{P(x,t=0)},\label{aincondrw}\\ P(x,t=T)&=&1+(e^{\lambda}-1)\theta(R-|x|).\label{afcondrw}\end{aligned}$$ The solution of the antidiffusion equation (\[aPt\_rw\]) backward in time is obtained by integrating the “final" condition (\[afcondrw\]) with Green’s function $G(x,x^{\prime},T-t)$, where $$\begin{aligned} \label{green} G(x,x^{\prime},t)&=&\frac{1}{R}\sum_{n=1}^{\infty}\sin\left[\frac{\pi n(x+R)}{2R}\right]\nonumber\\ &\times&\sin\left[\frac{\pi n(x^{\prime}+R)}{2R}\right]e^{-\frac{\pi^2 n^2 D_0 t}{4R^2}}.\end{aligned}$$ The integration yields $$\begin{aligned} P(x,t)&=&1+(e^{\lambda}-1)\int_{-R}^{R} dx^{\prime} G(x,x^{\prime},T-t)\nonumber\\ &=&1+(e^{\lambda}-1)g(x,T-t), \label{aP}\end{aligned}$$ where $$\begin{split} % \nonumber to remove numbering (before each equation) g&(x,t)=\int_{-R}^{R} dx^{\prime} G(x,x^{\prime},t)\\ =&\sum_{n=0}^{\infty}\frac{4}{\pi (2n+1)}\sin\left[\frac{\pi (2n+1)(x+R)}{2R}\right]e^{-\frac{\pi^2 (2n+1)^2 D_0t}{4R^2}}. \label{g} \end{split}$$ Evaluating $P(x,t=0)$ from Eq. (\[aP\]) and using Eq. (\[aincondrw\]), we obtain the initial condition, $$Q(x,t=0)=\frac{n_0}{1+(e^{\lambda}-1)g(x,T)},$$ for the diffusion equation (\[aQt\_rw\]). The solution of the latter equation is $$Q(x,t)=n_0 \int_{-R}^Rdx^{\prime}\frac{G(x,x^{\prime},t)}{1+(e^{\lambda}-1)g(x^{\prime},T)}. \label{aQ}$$ Transforming back to $q$ and $p$, and taking the zero-absorption limit of $\lambda \to \infty$, we obtain: $$\begin{aligned} % \nonumber to remove numbering (before each equation) q(x,t)&=&n_0g(x,T-t)\int_{-R}^{R}dx^{\prime}\frac{G(x,x^{\prime},t)}{g(x^{\prime},T)}, \label{eqqrw1} \\ v(x,t)&=&\partial_{x}p=\partial_{x}\ln g(x,T-t), \label{eqVrw1}\\ -\ln {\mathcal P}&\simeq &-n_0\int_{-R}^{R} dx\ln g(x,T).\label{eqsrw}\end{aligned}$$ We are interested in the long-time limit, $T\gg R^2/D_0$. A close inspection of the exact relations and reveals an important feature that plays a crucial role in our further analysis. For $D_0t/R^2 \gg 1$ and $D_0(T-t)/{R^2}\gg 1$, that is, outside of narrow boundary layers (in time) of typical width $R^2/D_0$ around $t=0$ and $t=T$, the functions $G(x,x^{\prime},t)$ and $g(x,T-t)$ are well approximated by their lowest modes, $n=1$ and $n=0$, respectively: $$\label{aprox1} % \nonumber to remove numbering (before each equation) G(x,x^{\prime},t) \simeq \frac{1}{R}\cos\left(\frac{\pi x}{2R}\right)\cos\left(\frac{\pi x^{\prime}}{2R}\right) e^{-\frac{\pi^2 D_0t}{4R^2}},$$ $$\label{aprox2} g(x,T-t) \simeq \frac{4}{\pi }\cos\left(\frac{\pi x}{2R}\right) e^{-\frac{\pi^2 D_0(T-t)}{4R^2}}.$$ Plugging these approximations into Eqs. (\[eqqrw1\]) and (\[eqVrw1\]), we obtain *time-independent* expressions: $$\begin{aligned} % \nonumber to remove numbering (before each equation) q(x)&= &2n_0\cos ^2\left(\frac{\pi x}{2R}\right),\label{q_rw_st} \\ v(x)&= &-\frac{\pi}{2R}\tan\left(\frac{\pi x}{2R}\right).\label{v_rw_st}\end{aligned}$$ That the density profile $q(x,t)$ is stationary most of the time is clearly seen in Figure \[qrwprof\] which shows the time-dependent solution at different times. Notice also that it is the momentum density gradient $v(x,t)=\partial_{x}p$, and not the momentum density $p(x,t)$ itself, that stays almost stationary. This is in contrast to the exterior survival problem in $d>2$. There the density profile is also almost stationary, but it is the momentum density, and not only its gradient, that is almost stationary [@MVK]. Using Eqs. (\[aprox2\]) and (\[eqsrw\]), we obtain the leading-order term of the long-time survival probability: $$\begin{aligned} % \nonumber to remove numbering (before each equation) -\ln {\mathcal P}&\simeq &n_0\frac{\pi^2 D_0T}{2R},\label{s_rw_st}\end{aligned}$$ which is simply the action evaluated for the stationary solutions (\[q\_rw\_st\]) and (\[v\_rw\_st\]) on the entire interval $0<t<T$. This result agrees with Eqs. (\[RWresult\]) and (\[survivaldecay1\]) as to be expected. Note that the stationary approximation remains accurate even when $D_0t/R^2$ and $D_0(T-t)/R^2$ are of order unity. This is because the sub-leading terms in Eqs. (\[eqqrw1\])-(\[eqsrw\]) include a large factor $2\pi^2$ in the exponent. This explains the narrowness of the boundary layers on Fig. \[qrwprof\], where $D_0T/R^2=10$. Therefore, at sufficiently long times, the leading-order contribution to the survival probability of the RWs inside a domain with an absorbing boundary comes from stationary solutions for $q$ and $v=\partial_x p$. The stationary solution for $v$ solves the equation $$\partial_tv=\partial_x\left(-D_0\partial_x v -D_0v^2 \right), \label{mftstatVrw}$$ which is obtained by differentiating Eq. (\[pt\_rw\]) in $x$. The stationarity of the optimal density profile, which gives the leading-order contribution to the survival probability, is not unique to the one-dimensional case. We prove in Appendix \[rwnd\] that this property holds for the RWs in any dimension and in an arbitrary domains. Now let us revisit the problem for RWs in one dimension by directly looking for stationary solutions of Eqs.  and . Setting the mass flux in Eq. (\[qt\_rw\]) to zero, we end up with the following set of ordinary differential equations, which we will call stationary MFT equations: $$\begin{aligned} q^\prime(x)&=& 2q v, \label{eqqrw} \\ v^{\prime}(x) &=& -v^2-\Lambda, \label{eqVrw}\end{aligned}$$ where $\Lambda$ is a yet unknown integration constant. Note that, for the exterior problem, a similar constant vanishes [@MVK]. This mathematical difference between the internal and external problems is crucial. Although its effect on the solution is immediate, its physical interpretation is somewhat elusive. After plugging Eq. (\[eqqrw\]) into Eq. (\[eqVrw\]) we obtain a single second-order equation $$q^{\prime\prime} - \frac{(q^\prime)^2}{2q} + 2\Lambda q=0 \label{statqrw}$$ which needs to be solved subject to the boundary condition (\[bcq\]) and a normalization condition, following from mass conservation: $$\frac{\int_{-R}^R dx \,q(x)}{2R}=n_0. \label{normstatqrw}$$ To solve Eq. (\[statqrw\]) we make a transformation of variable $$q(x)= u^2(x) \label{transformrw}$$ and obtain harmonic oscillator equation for $u(x)$: $$u^{\prime\prime} + \Lambda u=0. \label{staturw}$$ The solution is $u=B\sin\left[\sqrt{\Lambda}\left(x+x_0\right)\right]$, where $B$, and $x_0$ are integration constants. The boundary conditions, $$u(|x|=R,t)=0,\label{bcu}\\$$ set $x_0=R$ and $\Lambda=m^2\pi ^2/\left(4R^2\right)$, where $m=1,2,\dots$. Imposing the mass conservation (\[normstatqrw\]), we obtain a family of solutions, parameterized by $m$: $$q_m(x)=2n_0\sin^2\left[\frac{m\pi(x+R)}{2R}\right].\label{eqqrwm} \\$$ The corresponding $v$-solutions, calculated from Eq (\[eqqrw\]), are the following: $$v_m(x)=\frac{m \pi}{2R}\cot\left[\frac{m\pi(x+R)}{2R}\right].\label{eqvrwm}$$ Now we can calculate the action from Eq. (\[actionmainsection1\]): $$\begin{aligned} -\ln {\mathcal P_m} \simeq S_m &=& \int_0^T dt\int_{-R}^{R}dx D_0q_m v_m^2 \nonumber \\ &=&m^2n_0\frac{\pi^2 D_0T}{2R} \label{actionm}.\end{aligned}$$ As we know from the full time-dependent solution, only the “fundamental mode", $m=1$ is selected by the actual dynamics, see Eqs. (\[q\_rw\_st\]) and (\[v\_rw\_st\]). Not surprisingly, this solution has the minimum action, see Eq. (\[actionm\]). We argue that the same feature (selection of the lowest stationary mode) holds for the RWs in all spatial dimensions, see Appendix \[rwnd\]. Furthermore, it also holds for a whole class of *interacting* diffusive gases. In the next section we derive stationary MFT equations for an arbitrary diffusive gas. We then solve them for the SSEP, and support our findings by a numerical solution of the full time-dependent MFT equations for the SSEP in $d=1$. Stationary MFT equations for an arbitrary diffusive gas in arbitrary dimension {#generalstat} ============================================================================== We start with the general MFT equations (\[d11\]) and (\[d22\]). Taking the gradient of Eq. (\[d22\]), we obtain $$\partial_t \mathbf{v} = \nabla\left[ - D(q) \nabla \cdot\mathbf{v}-\frac{1}{2} \,\sigma^{\prime}(q) \mathbf{v}^2\right], \label{Vt}$$ where $\mathbf{v}=\nabla p$. Now we look for time-independent solutions, $q(\mathbf{x})$ and $\mathbf{v}(\mathbf{x})$ of Eqs. (\[d11\]) and (\[Vt\]). Equation (\[Vt\]) yields $$\nabla \cdot\mathbf{v}=\frac{-\frac{1}{2} \,\sigma^{\prime}(q) \mathbf{v}^2-\Lambda}{D(q)} ,\label{Vstat}$$ where $\Lambda$ is an integration constant to be determined later. In its turn, Eq. (\[d11\]) yields a zero divergence of the mass flux, so that the mass flux is a solenoidal vector field. In the survival problem, this vector field must have zero normal component to the domain boundary. Using these two properties one can show (see Appendix A of Ref. [@void]), that the minimum of the action is achieved when this vector field vanishes identically. Therefore, we arrive at the equation $$\nabla q=\frac{\sigma(q) \mathbf{v}}{D(q)}.\label{qstat}$$ Essentially, this equation states that, for the optimal profile, the fluctuation contribution to the flux exactly counterbalances the deterministic flux. Expressing $\mathbf{v}$ from here and plugging it into Eq. (\[Vstat\]), we obtain a closed equation for $q$: $$\nabla\cdot \left[\frac{D(q)}{\sqrt{\sigma(q)}}\nabla q\right] + \frac{\Lambda\sqrt{\sigma(q)}}{D(q)}=0,\label{qstatnd2}$$ so that $D(q)$ and $\sigma(q)$ only enter through the combination $D/\sqrt{\sigma}$. Introduce a transformation $q=f(u)$ that satisfies the equation $$\frac{D\left[f(u)\right]}{\sqrt{\sigma \left[f(u)\right]}}f^{\prime}(u)=1.\label{transform}$$ Performing the integration (and assuming that the integral converges), we can define the function $f(u)$ implicitly: $$\int_0^f dz \frac{D(z)}{\sqrt{\sigma (z)}}=u .\label{transform1}$$ As a result of this transformation, Eq. (\[qstatnd2\]) becomes $$\nabla^2 u + \Lambda f^{\prime}(u)=0. \label{ustatnd}$$ The boundary condition for $u$ is the same as for $q=f(u)$: $$u(\mathbf{x}\in{\partial\Omega})=0.\label{bqu}$$ An additional constraint on $u(\mathbf{x})$ comes from the mass conservation: $$\label{massgen} \frac{\int_{\Omega} d\mathbf{x}f(u)}{V}=n_0,$$ where $V$ is the domain’s volume. Nonlinear equations similar to Eq. (\[ustatnd\]) appear in a host of physical problems. Probably the best known is the equation for stream function of an ideal incompressible fluid in two dimensions, where $-\Lambda f^{\prime}(u)$ is the vorticity [@Lamb; @Batchelor; @kaptsov]. Although the original time-dependent MFT problem has a unique solution, the stationary problem, defined by Eqs. (\[ustatnd\])-(\[massgen\]), may have multiple solutions, and we will need to address the ensuing selection problem. We already encountered this feature in the previous section, dealing with the RWs. The stationary formulation makes it possible to deduce the scaling of the optimal density profile, and of the survival probability, with the linear system size $L$. We note that Eq. (\[ustatnd\]) remains invariant upon rescaling of $\mathbf{x}$ by $L$, and of $\Lambda$ by $\sqrt{L}$. In the rescaled coordinates Eqs. (\[bqu\]) and (\[massgen\]) become $$\begin{aligned} u(\mathbf{x}\in{\partial\tilde{\Omega}})=0.\label{bqunorm}\\ \frac{\int_{\tilde{\Omega}} d\mathbf{x}f(u)}{\tilde{V}}=n_0,\label{massnorm}\end{aligned}$$ where $\tilde{\Omega}$ is the rescaled domain, and $\tilde{V}$ is its volume. Extracting $\mathbf{v}$ from Eq. (\[qstat\]), substituting it in Eq. (\[actionmainsection1\]), and using Eq. (\[transform\]), we obtain a simple expression for the action in terms of the new variable $u$: $$\begin{aligned} \label{statactionmainsection1} % \nonumber to remove numbering (before each equation) -\ln {\mathcal P} \simeq S =\frac{TL^{d-2}}{2}\int_{\tilde{\Omega}} d\mathbf{x}\left[\nabla u(\mathbf{x})\right]^2,\end{aligned}$$ where $u$ is the solution of the rescaled problem. As we can see, the $L^{d-2}$ scaling, previously observed for the RWs, see Eq. (\[RWresult\]), holds for a whole class of interacting gases. In particular, $\ln \mathcal{P}$ is independent of $L$ for $d=2$. We can also see how the survival probability depends on the diffusivity and mobility of the gas. Suppose that we can express $D(q)$ and $\sigma (q)$ as $D(q)=D_0\tilde{D}(q)$ and $\sigma(q)=D_0\tilde{\sigma}(q)$, where $D_0=D(n_0)$, and $\tilde{D}, \tilde{\sigma}$ are dimensionless functions of the dimensionless density $q$. Then, from Eq. (\[transform1\]) we have $u=\sqrt{D_0}\,F(q)$, where $F$ is a dimensionless function of $q$ determined solely by $\tilde{D}$ and $\tilde{\sigma}$. Using this relation in Eq. (\[statactionmainsection1\]), we obtain $$-\ln {\mathcal P} \simeq S = D_0TL^{d-2}s(n_0), \label{thesame}$$ where $s(n_0)$ is a dimensionless function determined by the domain shape, and specialized for each model only via $\tilde{D}$ and $\tilde{\sigma}$. Comparing Eq. (\[thesame\]) with Eq. (\[RWresult\]), we see that the particle interaction manifests itself only in the rescaled action $s(n_0)$. The same feature has been observed for the exterior survival problem [@MVK]. For the RWs Eq. (\[transform1\]) yields $f=u^2/(2D_0)$, while Eq. (\[ustatnd\]) becomes the Helmholtz equation $$\label{Helmholtz} \nabla^2 u+(\Lambda/D_0) \,u=0 ,$$ which admits analytical solutions for domains of simple shapes. In one dimension this equation coincides, up to a redefinition of $\Lambda$, with Eq. (\[staturw\]) of the previous section. Let us return to Eq. (\[ustatnd\]). For interacting diffusive gases the function $f(u)$ is nonlinear. Still, Eq. (\[ustatnd\]) can be solved analytically in one dimension, and we will exploit this fact in Sec. \[otherlattice\]. In higher dimensions such a general solution is unavailable. Quite a few particular solutions, in different geometries, have been found for special choices of nonlinear $f(u)$ [@Batchelor; @Stuart; @Shercliff; @kaptsov; @Alfimov]. Among them there is the case of $f(u)\sim \sin u$, when Eq. (\[ustatnd\]) becomes a stationary sine-Gordon equation. Fortunately, this particular case describes the well-known simple symmetric exclusion process (SSEP). As many other lattice gases, the SSEP behaves in its dilute limit as RWs, so that $f(u\to 0)\sim u$. Therefore, we will demand that the nonlinear solution for the SSEP cross over at low densities to the (fundamental mode) of the Helmholtz equation (\[Helmholtz\]). SSEP: a Stationary sine-Gordon Equation {#SSEP} ======================================= Substituting $D(q)=2D_0 q$ and $\sigma(q)=2D_0 q (1-q)$ into Eqs. (\[transform1\]) and (\[ustatnd\]), we arrive at the stationary sine-Gordon equation $$\nabla^2 U + C\sin\,U=0 , \label{ustatssepnd1}$$ where $U= \sqrt{2/D_0}\, u$, $C= \Lambda/D_0$, and $$q=f(u)=\sin^2\left(\frac{U}{2}\right).\label{transssep1}$$ The survival probability is given by $$\begin{aligned} \label{statactionmainsection2} % \nonumber to remove numbering (before each equation) -\ln {\mathcal P} \simeq S &=&\frac{TL^{d-2}}{2}\int_{\tilde{\Omega}} d\mathbf{x}\left[\nabla u(\mathbf{x})\right]^2 \nonumber\\ &=& D_0TL^{d-2}s(n_0).\end{aligned}$$ where $$\label{ssinG} s(n_0)= \frac{1}{4}\int_{\tilde{\Omega}} d\mathbf{x}\left[\nabla U(\mathbf{x})\right]^2 .$$ We will now solve Eq. (\[ustatssepnd1\]) in several geometries. SSEP survival on an interval {#SSEP1d} ---------------------------- As we show in section \[otherlattice\], the one-dimensional case is exactly soluble in quadratures for any diffusive gas for which a stationary solution exists. In this section we find the explicit solution for the SSEP. For $d=1$ Eq. (\[ustatssepnd1\]) coincides with the equation of mathematical pendulum: $$\frac{d ^2U}{d x^2} + C\sin\,U=0 . \label{ustatssep1d}$$ As for the RWs in section \[RWs\], we set our domain to be a segment of length $2R$ centered about the origin. With the coordinate rescaling, presented at the end of section \[generalstat\], we can set $L=R=1$, whereas Eqs. (\[bqunorm\]) and (\[massnorm\]) become $$\begin{aligned} &&U(|x|=1)=0 ,\label{bcqp1d}\\ && \int_{-1}^1dx\,\sin ^2\left(\frac{U}{2}\right)=2n_0 .\label{mass1d} \\\end{aligned}$$ The general solution of Eq. (\[ustatssep1d\]) can be written as $$U(x)=2\arcsin\left\{\sqrt{\nu} \,\text{sn}\left[\sqrt{C}(x+x_0),\nu\right]\right\}, \label{u1d}$$ where $\text{sn}(\dots)$ is the Jacobi elliptic sine function, see e.g. Ref. [@elliptic], whereas $\nu$ and $x_0$ are integration constants. The boundary condition (\[bcqp1d\]) sets $x_0=1$ and $C=m^2\text{K}^2(\nu)$, where $\text{K}(\nu)$ is the complete elliptic integral of the first kind, and $m=1,2\dots$. The parameter $\nu$ is uniquely determined by Eq. (\[mass1d\]) which gives, for any $m$, $$\label{n0} 1-\frac{\text{E}(\nu)}{\text{K}(\nu)} =n_0,$$ where $\text{E}(\nu)$ is the complete elliptic integral of the second kind. The plot of $n_0$ versus $\nu$ is shown in Fig. \[n0(nu)\]. As one can see, there is a one-to-one mapping between $0<n_0<1$ and $0<\nu<1$. What is left is to select the correct stationary solution out of the family of solutions parameterized by $m=1,2,\dots$. We note that, in the dilute limit of the SSEP the solution must coincide with that for the RWs. This argument, and the action minimization, select the fundamental mode $m=1$. Substituting $U$ in (\[transssep1\]), we arrive at the stationary $q$-profile: $$\label{q1} q(x)=\nu \,\text{sn}^2\,\left[\text{K}(\nu) (x+1),\nu\right]=\nu \,\frac{\text{cn}^2\,\left[\text{K}(\nu) x,\nu\right]}{\text{dn}^2\,\left[\text{K}(\nu) x,\nu\right]},$$ where $\text{cn}(\dots)$ and $\text{dn}(\dots)$ are Jacobi elliptic functions. This solution is shown by the solid line in the upper panel of Fig. \[3dthnum\]. We can also calculate $v(x)=\partial_x p$ from the one-dimensional version of Eq. (\[qstat\]). Going back to the physical (non-rescaled) coordinate $x$, we obtain $$\label{V2} v(x)=-\frac{\text{K}(\nu)\,\text{sn}\left[\frac{ \text{K}(\nu) (x)}{R},\nu\right]\text{dn}\left[\frac{ \text{K}(\nu) (x)}{R},\nu\right]}{R\,\text{cn}\left[\frac{ \text{K}(\nu) (x)}{R},\nu\right]}.$$ Notice that $v(x)\simeq \mp (R-|x|)^{-1}$ as $x\to \pm R$, in the same way as in the exterior survival problem [@MVK]. The $v(x)$-profile from Eq. (\[V2\]) is shown, by solid line, in Fig. \[vx\]. Having found the stationary profile of $q$ (or $U$), we can evaluate the survival probability from Eqs. (\[statactionmainsection2\]) and (\[ssinG\]): $$\begin{aligned} % \nonumber to remove numbering (before each equation) -\ln {\mathcal P} &\simeq & \frac{D_0 T}{4 R} \int_{-1}^{1} dx\, (U_x)^2 \nonumber\\ &=& \frac{D_0T}{R} \int_{-1}^{1} dx\, \text{K}^2(\nu) \,\nu \,\text{cn}^2 \,\left[\text{K}(\nu) (x+1),\nu\right] \nonumber\\ &=& \frac{D_0Ts(\nu)}{R}, \label{action2}\end{aligned}$$ where $$\label{s} s(\nu)= 2\text{K}^2(\nu) \left[\frac{\text{E}(\nu)}{\text{K}(\nu)}+\nu-1\right].$$ Equations (\[n0\]) and (\[s\]) determine the rescaled action $s=s(n_0)$ in a parametric form. The low- and high-density asymptotics of $s(n_0)$ are the following: [ s(n\_0)]{} , & $n_0\ll 1$, \[RW\]\ , & $1-n_0\ll 1$, \[divergence\] The low-density asymptotic coincides with that for the RWs, see Eqs. (\[RWresult\]) and (\[survivaldecay1\]). The high-density asymptotic formally diverges as $n_0$ approaches $1$. This divergence, however, is cured when $n_0$ becomes very close to $1$, as explained below in this subsection. Figure \[fff\] shows a plot of $s(n_0)$, alongside with the asymptotics (\[RW\]) and (\[divergence\]). Also shown in Fig. \[fff\] are numerical results obtained by solving the full time-dependent MFT equations (\[d11\]) and (\[d22\]) for the SSEP in one dimension, using the boundary conditions in time (\[t0q\]) and (\[t0p\]). The numerical solution was obtained with a modified version of the iteration algorithm used in Ref. [@MVK] for the exterior problem. Figures \[3dthnum\] and \[vx\] show the time-dependent numerical solutions for $q$ and $v$ respectively, at different times. Apart from narrow boundary layers at $t=0$ and $t=T$, the solutions are very close to the analytical stationary solution. The numerically evaluated rescaled action $s(n_0)$, shown in Fig. \[fff\], is also in very good agreement with the analytical results. Now let us return to the high-density limit where the MFT action (\[divergence\]) tends to diverge. As $1-n_0\ll 1$, or $1-\nu\ll 1$, we can approximate $\text{sn}\, z =\tanh z +\mathcal{O}(1-\nu)$, and $\text{K} (\nu)\simeq (1-n_0)^{-1}$. As a result, $$\label{u1das} U(x)\simeq 2\arcsin \left[\tanh\left(\frac{1-|x|}{\delta}\right)\right],$$ where $\delta=1-n_0\ll 1$. The resulting density profile $q=\sin^2\left(U/2\right)$ describes two kinks, with characteristic width $\delta=1-n_0\ll 1$, located close to the ends of the interval: $$\label{q1das2} q(x)\simeq \tanh^2\left(\frac{1-|x|}{\delta} \right) .$$ The action mostly comes from the kinks, and each kink contributes $\simeq 1/\delta$ to the action, leading to the asymptotic (\[divergence\]). Now we will see how the apparent divergence of the action (\[divergence\]) at $n_0\to 1$ is cured. The MFT is only expected to apply when the length scales that it describes are much greater than the lattice constant $a$. Restoring all units, we can express the kink width as $\delta \times R=(1-an_0)R$. The MFT is valid when this quantity is much greater than $a$, that is when $1-an_0\gg a/R$. On the other hand, exactly at close packing, $n_0=1/a$, the survival probability of the SSEP is equal to the product of probabilities $P_{1,2}$ that each of the particles adjacent to the boundary does not hit the boundary during time $T$. Each of these probabilities is $P_{1,2}=\exp(-D_0T/a^2)$, so the total survival probability is equal to $\exp(-2D_0T/a^2)$ and is of course finite. As one can see, the crossover between the macroscopic result, $\ln \mathcal{P}_{MFT} \simeq -2D_0 T/[aR(1-an_0)]$, and the microscopic result, $\ln \mathcal{P}=-2D_0T/a^2$ occurs at $1-an_0\sim a/R$, when the MFT theory breaks down. As we will see in the following sections, the kink solution (\[q1das2\]) plays an important role in the high-density behavior of the stationary solutions in higher dimensions, in domains of different shapes. An apparent divergence of $S$ at $n_0\to 1$ also appears there (see below), and it is also cured at the microscopic level. SSEP survival inside a rectangle {#SSEP2d} -------------------------------- Here we will solve the stationary sine-Gordon equation (\[ustatssepnd1\]) inside a rectangular domain with dimensions $L_x$ and $L_y$. After rescaling the coordinates, the dimensions of the rectangle become $1$ and $\Delta=L_y/L_x$, in the $x$ and $y$ directions, respectively. The equation must be solved with zero boundary conditions, whereas Eq. (\[massnorm\]) reads $$\label{mass2d} \frac{1}{\Delta} \int_0^{1}dx\int_0^{\Delta}dy \,q=\frac{1}{\Delta}\int_0^{1}dx\int_0^{\Delta}dy \sin ^2\left(\frac{U}{2}\right)= n_0.$$ As we explain shortly, an infinite family of solutions to this problem can be obtained by the method of “generalized separation of variables" [@kaptsov]. In the dilute limit, one of this solution coincides with the fundamental mode of the Helmholtz’s equation  (\[Helmholtz\]), $$\label{low2d} U=4\sqrt{n_0}\,\sin \left(\pi x\right) \sin \left(\frac{\pi y}{\Delta}\right),$$ obeying the zero boundary conditions and the normalization condition (\[mass2d\]), where one should replace $\sin (U/2)$ by $U/2$. We argue, therefore, that this solution yields the true stationary optimal density profile. The generalized separation of variables employs the ansatz $$\label{genansatz} U(x,y)=4\arctan\left[f(x)g(y)\right]$$ which yields two *uncoupled* equations for $f$ and $g$ (see Ref. [@kaptsov] for a detailed derivation): $$\begin{aligned} % \nonumber to remove numbering (before each equation) \left(f^{\prime}\right)^2&=&nf^4+mf^2+k ,\label{fxx}\\ \left(g^{\prime}\right)^2&=&kg^4-(m+C)g^2+n,\label{gyy}\end{aligned}$$ where $m,n$, and $k$ are arbitrary parameters. Each of Eqs. (\[fxx\]) and (\[gyy\]) describe conservation of energy of an effective classical particle in a potential. As one can see, the particle motion is confined, and the resulting solution exhibits the correct low-density asymptotic (\[low2d\]), if and only if $-C<m<0$, $n>0$, $k>0$, and $(m+C)^2>4kn$. In this regime of parameters the solutions for $f$ and $g$ are elliptic functions [@elliptic]: $$\begin{aligned} % \nonumber to remove numbering (before each equation) f &=& \sqrt{\frac{-m\nu_1}{n(\nu_1+1)}}\,\text{sn}\left[\sqrt{\frac{-m}{\nu_1+1}}(x+c_1),\nu_1\right] ,\label{fx}\\ g &=& \sqrt{\frac{(m+C)\nu_2}{k(\nu_2+1)}}\,\text{sn}\left[\sqrt{\frac{m+C}{\nu_2+1}}(y+c_2),\nu_2\right],\label{gy}\end{aligned}$$ where $c_1$ and $c_2$ are the integration constants of the first order equations (\[fxx\]) and (\[gyy\]), and the constants $\nu_1$ and $\nu_2$ are given by $m$, $n$, and $k$ via the relations $$\begin{aligned} % \nonumber to remove numbering (before each equation) \frac{\nu_1}{(1+\nu_1)^2} &=& \frac{kn}{m^2} ,\label{kn1}\\ \frac{\nu_2}{(1+\nu_2)^2} &=& \frac{kn}{(m+C)^2}.\label{kn2}\end{aligned}$$ Imposing the zero boundary condition we obtain $$\begin{aligned} % \nonumber to remove numbering (before each equation) \sqrt{\frac{-m}{(1+\nu_1)}} &=& 2m_1\text{K}(\nu_1), \label{bc1}\\ \sqrt{\frac{m+C}{(1+\nu_2)}} &=& \frac{2m_2\text{K}(\nu_2)}{\Delta},\label{bc2}\\ c_1=c_2&=&0\label{bc3},\end{aligned}$$ where $m_1$ and $m_2$ are positive integers. Similarly to the one-dimensional case, we must put $m_1=m_2=1$. Now we can solve Eqs. (\[kn1\])-(\[bc2\]) for $\nu_1$ and obtain an expression for $fg$ in terms of $\nu_1$ alone: $$\label{fg} fg=(\nu_1\nu_2)^{1/4}\,\text{sn}\left[2\text{K}(\nu_1)x,\nu_1\right] \text{sn}\left[\frac{2\text{K}(\nu_2)y}{\Delta},\nu_2\right] ,$$ where $\nu_1$ and $\nu_2$ are related by the equation $$\label{nu122} \text{K}(\nu_2)^4\nu_2=\left[\text{K}(\nu_1)\Delta\right]^4\nu_1.$$ Now we use mass conservation (\[mass2d\]), where we substitute $$\label{q} q=\sin ^2 \left(\frac{U}{2}\right)=\frac{4(fg)^2}{\left[1+(fg)^2\right]^2}.$$ Thus all the constants are determined implicitly. The survival probability is given by Eq. (\[statactionmainsection2\]), which we can rewrite as $$\begin{aligned} \label{action2d} -\ln {\mathcal P} \simeq S &=&\frac{D_0 T}{4}\int_0^{1}dx\int_0^{\Delta}dy\,\left[\left(\partial_x U\right)^2 + \left(\partial_yU\right)^2\right]\nonumber \\ &=&D_0 T s(n_0,\Delta).\end{aligned}$$ As to be expected from Eq. (\[thesame\]), the resulting probability is independent of the system size, but it strongly depends on the gas density. Figure \[recq\] shows a two-dimensional plot of $q(x,y)$ for $n_0 = 0.67$ and $\Delta=0.7$. Figure \[s2d\] depicts $s(n_0,\Delta=1)$ versus $n_0$, alongside with the low- and high-density asymptotics that we will now discuss. ### Dilute gas, $n_0\ll 1$ In the dilute limit, $n_0\ll 1$, our results coincide with those for the RWs. Indeed, by virtue of Eq. (\[mass2d\]), $U$ must be much smaller than $1$ in this limit. Therefore, $fg$ must be also small, and we can approximate $U\simeq 4fg$, and $q\simeq 4f^2g^2$. Now, from Eq. (\[mass2d\]), there must be $\nu_1,\nu_2 \ll 1$. Therefore, as in the one-dimensional case, we can put $\text{sn}(\dots,\nu)\simeq \sin(\dots)$ and $\text{K}(\nu)\simeq \pi/2$ in the expression for $U$. Further, Eq. (\[nu122\]) yields in this limit $\nu_2\simeq \nu_1 \Delta^4$. The remaining constant $\nu_1\simeq n_0/\Delta^2$ is found from Eq. (\[mass2d\]). As a result, we arrive at the correct asymptotic (\[low2d\]). The optimal density profile in the dilute limit, back in the original coordinates, is $$\label{densrect} q\simeq 4n_0\sin^2 \left(\frac{\pi x}{L_x}\right) \sin^2 \left(\frac{\pi y}{L_y}\right).$$ The survival probability is given by Eq. (\[action2d\]) with $$s(n_0,\Delta) = \pi^2 n_0\left(\Delta+\frac{1}{\Delta}\right). \label{lows2d}$$ The survival probability is maximum when the rectangle is a square. ### Near close packing, $1-n_0\ll 1$ When $n_0$ is close to $1$, the gas density is close to $1$ everywhere except in narrow boundary layers of the size $\mathcal{\delta}=1-n_0$ along the boundary. From Eq. (\[q\]), $fg$ is close to $1$. As $\nu_1$ and $\nu_2$ are also close to $1$, $\text{K}(\nu_1)$ and $\text{K}(\nu_2)$ diverge as $n_0\to 1$. In this limit Eq. (\[nu122\]) yields $\text{K}(\nu_2)\simeq\text{K}(\nu_1)\Delta$. The explicit $n_0$-dependence can be obtained with the help of Eq. (\[mass2d\]): $2\text{K}(\nu_1) \simeq (1+\Delta)(\Delta\delta)^{-1}$, and $2\text{K}(\nu_2) \simeq (1+\Delta)(\delta)^{-1}$. Using the asymptotic $\text{sn}\,z \simeq \tanh z$, we obtain in the leading order: $$\begin{aligned} \label{hyp} fg&\simeq&\tanh\left[\frac{(1+\Delta)x}{\Delta\delta}\right]\tanh\left[\frac{(1+\Delta)y}{\Delta\delta}\right] \nonumber\\ &=&\frac{\tanh\left[\frac{(1+\Delta)x}{\Delta\delta}\right]+\tanh\left[\frac{(1+\Delta)y}{\Delta\delta}\right]-1} {\tanh\left[\frac{(1+\Delta)(x+y)}{\Delta\delta}\right]} .\end{aligned}$$ This asymptotic is valid for $x<1/2$ and $y<\Delta/2$; it can be extended to the other three quarters of the rectangle by obvious reflections. Away from the domain corners we can replace $\tanh(\dots)$ in the denominator of Eq. (\[hyp\]) by unity. Using the resulting “approximate product rule" in Eq. (\[q\]), we obtain, after some algebra, the “kink" asymptotic of $q(x,y)$ away from the domain corners: $$\label{qaprox} q\simeq \tanh^2\left[\frac{2(1+\Delta)x}{\Delta\delta}\right]\tanh^2\left[\frac{2(1+\Delta)y}{\Delta\delta}\right],$$ where $x<1/2$ and $y<\Delta/2$, and reflected formulas in the other three rectangle quarters. These asymptotics describe kinks with the characteristic width $\ell=\Delta\delta/[2(1+\Delta)]$. By analogy with one dimension, the action per unit length along the boundary is, in the leading order, $1/\ell$. Multiplying this expression by the perimeter $2(1+\Delta)$, we extract the asymptotics $$\label{highs2d} s(n_0,\Delta)\simeq \frac{4}{1-n_0} \left(\Delta+\frac{1}{\Delta}+2\right).$$ Again, the survival probability is maximum when the rectangle is a square. ### Very long rectangle, $\Delta\gg 1$ Here, sufficiently far from the edges $y=0$ and $y=\Delta$, $U(x,y)$ is almost independent of $y$, and close to the one-dimensional solution (\[u1d\]). Therefore, the rescaled action $s(n_0,\Delta)$ is approximately equal to $$\label{asdelta} s(n_0,\Delta\gg 1)\simeq 2\Delta s_{1d}(n_0),$$ where $s_{1\text{d}}(n_0)$ is described by Eqs. (\[n0\]) and (\[s\]). The factor $\Delta$ is due to additional integration along $y$, and the factor $2$ appears because the one-dimensional result (\[s\]) was obtained for a segment of length $2$, not $1$. SSEP survival inside a sphere {#SSEPspherical} ----------------------------- Here the stationary optimal density profile depends only on the radial coordinate $r$, and Eq. (\[ustatssepnd1\]) becomes $$\nabla_r^2 U + C\sin U=0 , \label{ustatssepnd1r}$$ where $\nabla_r^2U(r)=U_{rr} + (d-1)U_r/r$ is the radial Laplacian in $d$ dimensions. Upon the coordinate rescaling $r \to r/R$, we need to solve the stationary sine-Gordon equation (\[ustatssepnd1r\]) inside a sphere of unit radius. The boundary conditions, and the normalization condition, are $$\begin{aligned} U^{\prime}(r=0)&=&U(r=1)=0 ,\label{bcqpnd}\\ d\int_0^1drr^{d-1}q(r)&=&d\int_0^1drr^{d-1}\sin ^2\left(\frac{U}{2}\right)=n_0 .\label{massnd} \nonumber\\\end{aligned}$$ Then, from Eqs. (\[statactionmainsection2\]) and (\[ssinG\]), we obtain $-\ln {\mathcal P} \simeq S=D_0TR^{d-2}s(n_0)$, where $$\label{actionmain} s(n_0)=\frac{\Omega_d}{4}\int_0^1drr^{d-1}(U_r)^2,$$ and $\Omega_d$ is the surface area of the $d$-dimensional unit sphere. In the absence of general analytic solution of Eq. (\[ustatssepnd1r\]) for $d>1$, one can solve this equation numerically, and also explore analytically the low- and high-density limits. The first-order term of the density expansion of $s(n_0)$ corresponds to the RWs, see Appendix \[rwnd\]. The next, $n_0^2$-term can be obtained by treating the $q^2$ term of the MFT Hamiltonian of the SSEP perturbatively, similarly to how it was done in the exterior problem [@MVK]. For example, for $d=3$ the resulting correction is $$\label{correction} \delta s(n_0) = 4 \pi \int_0^1 dr\,r^2 q_{\text{RW}}^2(r) v_{\text{RW}}^2 (r),$$ where $q_{\text{RW}}(r)$ and $v_{\text{RW}}(r)$ are the stationary optimal profiles for the RWs, given by Eqs. (\[q\_rw\_st3d\]) and (\[v\_rw\_st3d\]) of Appendix B. Evaluating the integral, we obtain $\delta s(n_0) = \alpha \,n_0^2$, where $$\alpha = \frac{8\pi^4}{27}\left[2\,\text{Si}\,(2\pi)-\,\text{Si}\,(4 \pi)\right]=38.7945\dots ,$$ and $$\text{Si}\,(z)=\int_0^z \frac{\sin t}{t} \,dt$$ is the sine integral function. The resulting low-density asymptotic, for $d=3$, is $$\label{lowdens} s(n_0)\simeq \frac{4\pi^3}{3}\, n_0+\alpha \,n_0^2.$$ Near close packing, $1-n_0\ll 1$, the gas density $q(r)$ drops from a value close to $1$ to zero in a narrow boundary layer, of width $\mathcal{O}(\delta)$ near the sphere $r=1$. Correspondingly, $U(r)=2\arcsin \sqrt{q(r)}$ rapidly drops from a value close to $\pi$ to zero. As a result, we can neglect the first-derivative term in the radial Laplacian. This brings us back to the equation $U_{rr} + C\sin U=0$ that we considered in Sec. \[SSEP1d\]. The boundary conditions (\[bcqpnd\]) are also the same as in the one-dimensional case, see Eq. (\[bcqp1d\]). The only difference is in the normalization condition, Eq. (\[massnd\]) which introduces the factor $d$. As a result, $$\label{qaproxnd2} U(r)\simeq 2\arcsin\left\{\tanh\,\left[\frac{d(1-r)}{1-n_0}\right]\right\},$$ and $$\label{qaproxnd22} q(r)=\sin^2 \left(\frac{U}{2}\right)\simeq \tanh^2\,\left[\frac{d(1-r)}{1-n_0}\right].$$ Using Eq. (\[qaproxnd2\]), we obtain the high-density asymptotic $$\label{actionaproxnd} % \nonumber to remove numbering (before each equation) s(n_0) =\frac{\Omega_d}{4}\int_0^1drr^{d-1}(U_r)^2\simeq \frac{d \Omega_d}{1-n_0},$$ In particular, for $d=3$, $$\label{divergence3d} s_{3d}(n_0)\simeq \frac{12\pi}{1-n_0}, \quad 1-n_0\ll 1.$$ For an arbitrary density $n_0$ Eq. (\[ustatssepnd1r\]) can be solved numerically: either by the shooting method or by artificial relaxation. Examples of such solutions for $d=3$ are shown in Fig. \[q3d\]. Figure \[s3d\] shows the numerically found $s_{3d}(n_0)$, alongside with the asymptotic (\[lowdens\]), its linear term only, and asymptotic (\[divergence3d\]). SSEP survival in arbitrary domains near close packing {#high} ----------------------------------------------------- Consider a domain of arbitrary shape, in any dimension. As $n_0$ approaches $1$, the stationary optimal density field $q$ stays very close to $1$ across most of the domain, and drops to $0$ in a narrow boundary layer of characteristic width $\delta =1-n_0$ along the domain boundary. The density derivatives in the directions parallel to the boundary are negligibly small compared to the density derivative across the boundary. Therefore, we can approximate the Laplacian in Eq. (\[ustatssepnd1\]) by a one-dimensional one, $\partial^2U/\partial r_{\perp}^2$, where $r_{\perp}$ is a local coordinate normal to the domain boundary. As a result, the problem becomes effectively one-dimensional, and the solution of Eq. (\[ustatssepnd1\]) is a one-dimensional kink, $q(\mathbf{x})\simeq \tanh^2\left(\sqrt{C}\,r_{\perp} \right)$ (we set $r_{\perp}$ to vanish at the boundary). The action, Eq. (\[statactionmainsection2\]), mostly comes from the boundary layer. In the leading order, the action per unit surface across the boundary is equal to $\sqrt{C}$. The total action is therefore $\tilde{A}\sqrt{C}$, where $\tilde{A}$ is the normalized domain’s surface area. $\sqrt{C}$ is determined by the mass conservation: $\sqrt{C}\simeq \tilde{A}/(\tilde{V}\delta)$, where $\tilde{V}$ is the normalized domain’s volume. The final result is $$\label{highs} -\ln {\mathcal P} \simeq S \simeq D_0TL^{d-2}\frac{\tilde{A}^2}{\tilde{V}\left(1-n_0\right)}=\frac{A^2 D_0T}{V\left(1-n_0\right)}.$$ This expression is in agreement with our high-density results (\[divergence\]), (\[highs2d\]), and (\[actionaproxnd\]). Survival of a general diffusive gas on an interval {#otherlattice} ================================================== For a general diffusive gas in one dimension the stationary density profile and the survival probability can be found in quadrature. Indeed, for arbitrary $D(q)$ and $\sigma(q)$ in one dimension, Eq. (\[ustatnd\]) reads $$u^{\prime\prime}+ \Lambda f^{\prime}(u)=0 , \label{ustat1d}$$ where $f(u)$ is defined by Eq. (\[transform1\]). By virtue of the scaling properties of the problem, it suffices to solve Eq. (\[ustat1d\]) on the interval $|x|<1$, with the same additional conditions for $u$ as stated in Eqs. (\[bcqp1d\]) and (\[mass1d\]) for $U$. Equation (\[ustat1d\]) describes the motion of an effective classical particle with unit mass ($u$ is the “particle coordinate", $x$ is “time") in the potential $V(u)=\Lambda f(u)$. Let us denote the particle energy by $\Lambda\nu$. Energy conservation yields a first-order equation: $$\label{Qprimegen} u^{\prime}=\pm\sqrt{2 \Lambda \left[\nu-f(u)\right]}$$ Solving it with the boundary condition $u(x=-1)=0$, we obtain: $$\label{ueq} \int_{0}^{u}\frac{dh}{\sqrt{\nu-f(h)}}=\sqrt{2\Lambda}\,(x+1).$$ Changing the integration variable to $z=f(h)$ and using Eq. (\[transform\]), we obtain $$\label{qeq} \int_{0}^{q}dz\,\frac{D(z)}{\sqrt{(\nu-z)\sigma(z)}}=\sqrt{2\Lambda}\,(x+1).$$ Now we demand that $q=\nu$ at $x=0$, and express $\Lambda$ through $\nu$: $$\label{A1} \Lambda=\frac{1}{2}\left[\int_0^{\nu}\frac{dz\,D(z)}{\sqrt{(\nu-z) \,\sigma(z)}}\right]^2.$$ An additional condition comes from mass conservation: $$\begin{aligned} \label{conservation} % \nonumber to remove numbering (before each equation) 2n_{0} &=& \int_{-1}^{1}q(x)dx=2\int_{0}^{\nu}q\,\frac{dx}{dq}\left(q\right)\,dq \nonumber \\ &=& 2\int_{0}^{\nu}dq\frac{q \,D(q)}{\sqrt{2\Lambda(\nu-q)\sigma(q)}}.\end{aligned}$$ Equations (\[A1\]) and (\[conservation\]) yield the dependence of $n_0$ on $\nu$: $$\label{n0gen} n_0(\nu)=\frac{\int_{0}^{\nu}dq\frac{q D(q)}{\sqrt{(\nu-q)\sigma(q)}}}{\int_{0}^{\nu}dq\frac{D(q)}{\sqrt{(\nu-q)\sigma(q)}}}.$$ Now we can evaluate the survival probability: $$\label{actn1} -\ln {\mathcal P} \simeq \frac{T}{2R}\int_{-1}^{1}dx \, \left(\frac{d u}{d x}\right)^2.$$ By virtue of Eq. (\[Qprimegen\]) and the definition $q=f(u)$, we obtain $$\label{actn2} -\ln {\mathcal P} \simeq \frac {\Lambda T}{R}\int_{-1}^{1} [\nu-q(x)] dx =\frac{2\Lambda T(\nu-n_{0})}{R},$$ where we have again used $\int_{-1}^{1} q(x)dx = 2n_{0}$. Using Eq. (\[A1\]), we can rewrite Eq. (\[actn2\]) as $$\label{actn3} -\ln {\mathcal P} \simeq \frac{T}{R}\,s(n_0),$$ where the rescaled action $s=s(n_0)$ is given in a parametric form by the equation $$\label{LDFgen} s(\nu)= \left[\int_0^{\nu}\frac{dq\,D(q)}{\sqrt{(\nu-q) \,\sigma(q)}}\right]^2 \left[\nu-n_0(\nu)\right]$$ and Eq. (\[n0gen\]). When specialized to the SSEP, Eqs. (\[qeq\]) and (\[LDFgen\]) yields (\[q1\]) and (\[s\]) respectively. With these general results at hand, we can investigate the survival properties of a whole class of diffusive gases with known $D(q)$ and $\sigma(q)$ in one dimension: on the condition that the integral in the denominator of Eq. (\[n0gen\]) converges. Conclusions and Discussion {#conclusion} ========================== We dealt in this work with the survival of a gas of interacting diffusive particles inside a domain with absorbing boundary. Employing the MFT formalism, we evaluated the long-time survival probability of the gas and its optimal density history conditional on the survival. We found that this optimal density history is stationary during most of the process. As a consequence, the survival probability decays exponentially in time: inside domains of any shape in all dimensions. As we showed, the solution of the long-time survival problem reduces to solving a nonlinear Poisson equation, where the nonlinear term is determined by $D(q)$ and $\sigma(q)$. In one dimension, this problem is soluble exactly for a whole class of diffusive gases. For the SSEP the nonlinear Poisson equation becomes a stationary sine-Gordon equation, and we solved it in different geometries and dimensions. The dilute limit of the SSEP corresponds to non-interacting random walkers (RWs), where the problem reduces to finding the lowest positive eigenvalue $\mu_1$ of the Laplace’s operator inside the domain [@Paulbook], see Eq. (\[RWgeneral\]). Near close packing the problem becomes effectively one-dimensional and can be approximately solved for any domain shape and any dimension. What is the optimal domain shape, for a fixed number of particles and fixed volume of the domain, that maximizes the chances of long-time survival? Interestingly, both in the dilute limit of the SSEP, and near close packing, the optimal domain shape is the sphere. Indeed, in the dilute limit the minimum value of $\mu_1$ is achieved for the sphere, as guaranteed by the Rayleigh-Faber-Krahn theorem [@ball; @Chavel]. In its turn, near the close packing, the sphere is the minimizer of the surface area $A$ at fixed volume, see Eq. (\[highs\]). A natural conjecture is that the sphere maximizes the survival probability of the SSEP at any gas density, but we do not have a proof. For other diffusive gases we do not know the minimizing domain shape. It would be interesting to apply our approach to the “narrow escape problem", where there is a small hole in the *reflecting* boundary of the domain. The survival probability [@hole] and the mean escape time [@Holcman] of a single RW in this system have been extensively studied. The MFT formalism can give an interesting insight into fluctuations in the escape of *interacting* particles. Another interesting extension would address the full absorption statistics: evaluating the probability that a specified number of particles are absorbed by time $T$. An exterior variant of this problem has been recently considered, for the SSEP, in Ref. [@M15]. Finally, we emphasize that stationarity of the optimal gas density profile is a major simplifying factor in the large-deviation problem we have considered here. Cases of time-independence of the optimal gas density history are also encountered in other large-deviation settings in lattice gases [@Bodineau; @Hurtado1; @Hurtado2; @MVK; @M15], and they are intimately related to the “additivity principle" put forward by Bodineau and Derrida [@Bodineau]. Acknowledgments {#acknowledgments .unnumbered} =============== We thank O.V. Kaptsov for sending us the book [@kaptsov] that he coauthored, and P.L. Krapivsky for a useful discussion of the Rayleigh-Faber-Krahn theorem. This research was supported by grant No. 2012145 from the United States–Israel Binational Science Foundation (BSF). Survival probability of Random Walkers from a microscopic perspective {#mic} ===================================================================== For completeness, we present here a brief microscopic derivation of the survival probability of a gas of non-interacting Random Walkers (RWs) inside a domain $\Omega$. This quantity can be obtained from a single-particle survival probability: $$\label{rwp} -\ln {\mathcal P(T)}_{\text{RW}} = - \sum_i \ln g(\mathbf{x}_i,T),$$ where $g(\mathbf{x}_i,T)$ is the survival probability up to time $T$ of a particle initially positioned at $\mathbf{x}_i$ , and the sum is over all particles. Therefore, one needs to calculate $g(\mathbf{x}_i,T)$ and perform the summation. Calculating $g(\mathbf{x}_i,T)$ ------------------------------- The single-particle survival probability $g(\mathbf{x}_i,T)$ is given by the expression $$\label{sp} g(\mathbf{x}_i,T)=\int_{\Omega} d\mathbf{x}\,\rho_1(\mathbf{x},T,\mathbf{x}_i),$$ where $\rho_1(\mathbf{x},t,\mathbf{x}_i)$ is the probability distribution of the particle position, given its (deterministic) initial position at $\mathbf{x}_i$. This probability distribution obeys the diffusion equation [@R85; @OTB89; @bAH; @Rednerbook]: $$\label{dif} \partial_t \rho_1 = D_0 \nabla^2 \rho_1$$ with the absorbing boundary condition: $$\label{bcrw} \rho_1(\mathbf{x} \in \partial \Omega,t) = 0.$$ The initial condition is $$\label{incon} \rho_1(\mathbf{x},t=0)=\delta^d(\mathbf{x}-\mathbf{x}_i),$$ where $\delta^d$ is the $d$-dimensional Dirac delta-function. The solution to Eq. (\[dif\]) is the Green’s function of the diffusion equation: $$\label{agreennd} \rho_1(\mathbf{x},t,\mathbf{x}_i)=G(\mathbf{x},\mathbf{x}_i,t)=\sum_{n=1}^{\infty}\Psi_n(\mathbf{x})\Psi_n(\mathbf{x}_i)e^{-\mu_n^2 D_0t}.$$ Here $\Psi_n$ and $\mu_n$ are the normalized eigenfunctions and eigenvalues of the eigenvalue problem $\nabla^2 u+\mu^2 u=0$ for the Laplace’s operator inside the domain with the boundary condition $u(\mathbf{x}\in\partial \Omega) = 0$. We order the eigenvalues by their magnitude $\mu_1<\mu_2<\mu_3<\dots$. Plugging Eq. (\[agreennd\]) into Eq. (\[sp\]) one obtains [@Paulbook]: $$\begin{aligned} % \nonumber to remove numbering (before each equation) g(\mathbf{x}_i,T)&=&\int_{\Omega} d\mathbf{x}\,G(\mathbf{x},\mathbf{x}_i,t) \nonumber\\ &=& \sum_{n=1}^{\infty}\Psi_n(\mathbf{x}_i)\int_{\Omega} d\mathbf{x}\,\Psi_n(\mathbf{x})e^{-\mu_n^2 D_0T} \label{fnd}\end{aligned}$$ Evaluating the sum in Eq. (\[rwp\]) ----------------------------------- When the number of particles in the domain $\Omega$ is very large, the sum in (\[rwp\]) can be approximated by the integral: $$-\ln {\mathcal P(T)}_{\text{RW}} \simeq -n_0\int_\Omega d\mathbf{x}^{\prime}\ln\left[g(\mathbf{x}^{\prime},T)\right],\label{aeqsrwnd}$$ Furthermore, at times much longer than the characteristic diffusion time, the infinite sum in Eq. (\[fnd\]) can be approximated by its first term: $$\label{gaprox} g(\mathbf{x},T) \simeq \Psi_1(\mathbf{x})\int_{\Omega} d\mathbf{x^{\prime}}\Psi_1(\mathbf{x^{\prime}})e^{-\mu_1^2 D_0T}.$$ Plugging this approximation into Eq. (\[aeqsrwnd\]), we obtain the long-time asymptotic of the survival probability presented in Eq. (\[RWgeneral\]). Solving the MFT equations for the RWs in higher dimensions {#rwnd} ========================================================== The one-dimensional solution, presented in section \[RWs\], can be generalized to any simply-connected domain in arbitrary dimension. Consider the MFT equations (\[d11\]) and (\[d22\]) for the RWs: $$\begin{aligned} \partial_t q &=& \nabla \cdot \left[D_0 \nabla q-2D_0q \nabla p\right], \label{ad1} \\ \partial_t p &=& - D_0 \nabla^2 p-D_0 (\nabla p)^2, \label{ad2}\end{aligned}$$ The Hamiltonian density is $$\label{aHam} \mathcal{H}(q,p) = -D_0 \nabla q\cdot \nabla p +D_0q\!\left(\nabla p\right)^2 .$$ The absorbing boundary conditions are described by Eq. (\[bcgenq\]) and (\[bcgenp\]). The boundary conditions in time are given by Eqs. (\[t0q\]) and (\[t0p\]). As in one dimension, we solve the problem, using the Hopf-Cole transformation $Q=qe^{-p}$ and $P=e^p$, with the generating functional $$\label{ajeneratingnd} \int_{\Omega} d\mathbf{x}F_1(q,Q)=\int_{\Omega} d\mathbf{x}\left[q\ln(q/Q)-q\right].$$ In the new variables the Hamiltonian density is $$\label{Hamrwnd} \mathcal{H}(q,p) = -D_0 \nabla Q\cdot \nabla P,$$ and, again, the action can be expressed through the initial and final states of the system: $$\begin{aligned} \label{aactionnd} -\ln {\mathcal P}_{\text{RW}} &\simeq& S =\int_0^T dt\int_\Omega d\mathbf{x} D_0q\left(\nabla p\right)^2\\ &=&\int_\Omega d\mathbf{x}\left[Q\left(P\ln P -P+1\right)\right]\big|_0^T\end{aligned}$$ The transformed MFT equations are decoupled: $$\begin{aligned} % \nonumber to remove numbering (before each equation) \partial_t Q &=& D_0\nabla^2 Q, \label{aQt_rwnd} \\ \partial_t P &=& -D_0 \nabla^2 P. \label{aPt_rwnd}\end{aligned}$$ The transformed boundary conditions, in space and in time, are: $$\begin{aligned} &&Q(\mathbf{x}\in{\partial\Omega,t})=0, \label{abcQnd}\\ &&P(\mathbf{x}\in{\partial\Omega,t})=1, \label{abcPnd}\\ &&Q(\mathbf{x},t=0)=\frac{n_0}{P(\mathbf{x},t=0)}, \label{aincondnd}\end{aligned}$$ and [P(,t=T)=]{} e\^, & $\mathbf{x}\in{\Omega}$\ 1. & $\mathbf{x}\in{\partial\Omega}$ Solving the anti-diffusion equation (\[aPt\_rwnd\]), we obtain $$P(\mathbf{x},t)=1+(e^{\lambda}-1)g(\mathbf{x},T-t), \label{aPnd}$$ where $g$ is defined in Eq. (\[fnd\]). Evaluating $P(\mathbf{x},t=0)$, we obtain the initial condition for the diffusion equation (\[aQt\_rwnd\]): $$Q(\mathbf{x},t=0)=\frac{n_0}{1+(e^{\lambda}-1)g(\mathbf{x},T)}.$$ The resulting solution of Eq. (\[aQt\_rwnd\]) is $$Q(\mathbf{x},t)=n_0 \int_{\Omega} d\mathbf{x^{\prime}} \frac{G(\mathbf{x},\mathbf{x}^{\prime},t)}{1+(e^{\lambda}-1)g(\mathbf{x}^{\prime},T)}. \label{aQnd}$$ Now we calculate the action using Eq. (\[aactionnd\]). After some algebra, and taking the zero-absorption limit of $\lambda \to \infty$, we arrive at Eq. (\[aeqsrwnd\]), which describes the continuum approximation of the exact microscopic result (\[rwp\]). Transforming back to $q$ and $p$, and taking the limit of $\lambda \to \infty$, we obtain: $$\begin{aligned} % \nonumber to remove numbering (before each equation) q(\mathbf{x},t)&=& n_0 g(\mathbf{x},T-t)\int_\Omega d\mathbf{x^{\prime}} \frac{G(\mathbf{x},\mathbf{x}^{\prime},t)}{g(\mathbf{x}^{\prime},T)}, \label{aeqqrwnd}\\ \mathbf{v}(\mathbf{x},t)&=&\nabla p=\nabla \ln g(\mathbf{x},T-t). \label{aeqVrwnd}\end{aligned}$$ Being interested in long times, we observe that, outside the boundary layers of width $L^2/D_0$ around $t=0$ and $t=T$, one can approximate expressions Eq. (\[fnd\]) and (\[agreennd\]) by the first terms of the corresponding series: $$\begin{aligned} % \nonumber to remove numbering (before each equation) G(\mathbf{x},\mathbf{x^{\prime}},t)\!\!& \simeq &\!\!\Psi_1(\mathbf{x})\Psi_1(\mathbf{x^{\prime}})e^{-\mu_1^2 D_0t},\\ g(\mathbf{x},T-t)\!\! & \simeq & \!\!\Psi_1(\mathbf{x})\int_{\Omega} d\mathbf{x^{\prime}}\Psi_1(\mathbf{x^{\prime}})e^{-\mu_1^2 D_0(T-t)}.\end{aligned}$$ This approximation yields the stationary solution $$\begin{aligned} % \nonumber to remove numbering (before each equation) \!\!\!q(\mathbf{x})&=& n_0 V\Psi_1^2(\mathbf{x})\label{aproxeqqrwnd}\\ \!\!\!\mathbf{v}(\mathbf{x})&=&\nabla p = \frac{\nabla \Psi_1(\mathbf{x})}{\Psi_1(\mathbf{x})}, \label{aproxeqVrwnd}\end{aligned}$$ whereas $-\ln {\mathcal P}$ is given by Eq. (\[RWgeneral\]). When $\Omega$ is a circle of radius $R$ ($d=2$), we obtain $$\begin{aligned} % \nonumber to remove numbering (before each equation) q (r)&= &\frac{n_0J_0^2\left(\frac{z_1 r}{R}\right)}{J_1^2 (z_1)},\label{q_rw_st2d} \\ \mathbf{v}(r)&=&-\frac{ z_1 J_1\left(\frac{z_1 r}{R}\right)}{R\,J_0\left(\frac{z_1 r}{R}\right)}\mathbf{\hat{r}},\label{v_rw_st2d}\end{aligned}$$ where $J_0(z)$ and $J_1(z)$ are Bessel functions, and $z_1=2.4048\dots$ is the first positive root of $J_0(z)$. The survival probability is described by Eqs. (\[RWresult\]) and (\[survivaldecay2\]). When $\Omega$ is a sphere of radius $R$ ($d=3$), the stationary solution is $$\begin{aligned} % \nonumber to remove numbering (before each equation) q (r)&= &\frac{2n_0R^2\sin ^2\left(\frac{\pi r}{R}\right)}{3r^2},\label{q_rw_st3d} \\ \mathbf{v}(r)&=& \left[\frac{\pi}{R}\cot\left(\frac{\pi r}{R}\right) -\frac{1}{r} \right] \mathbf{\hat{r}},\label{v_rw_st3d}\end{aligned}$$ and the survival probability is described by Eqs. (\[RWresult\]) and (\[survivaldecay3\]). [99]{} M.v. Smoluchowski, Z. Phys. **17**, 557 (1916). S. A. Rice, *Diffusion-Limited Reactions* (Elsevier, Amsterdam, 1985). A. A. Ovchinnikov, S. F. Timashev, and A. A. Belyi, *Kinetics of Diffusion Controlled Chemical Processes* (Nova, Hauppauge, 1989). D. ben-Avraham and S. Havlin, *Diffusion and Reactions in Fractals and Disordered Systems* (Cambridge University Press, Cambridge, 2000). S. Redner, *A Guide to First-Passage Processes* (Cambridge University Press, Cambridge, 2001). P. L. Krapivsky, S. Redner, and E. 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--- abstract: 'In this paper, we explicitly construct a series of projectors on integral noncommutative orbifold $T^2/Z_4$ by extended $GHS$ constrution. They include integration of two arbitary functions with $Z_4$ symmetry. Our expression possess manifest $Z_{4}$ symmetry. It is proved that the expression include all projectors with minimal trace and in their standard expansions, the eigen value functions of coefficient operators are continuous with respect to the arguments $k$ and $q$. Based on the integral expression, we alternately show the derivative expression in terms of the similar kernal to the integral one. Since projectors correspond to soliton solutions of the field theory on the noncommutative orbifold, we thus present a series of corresponding solitons.' author: - | Hui Deng $$\thanks{Email:hdeng@phy.nwu.edu.cn}, \hspace{5mm} Bo-Yu Hou$$ [^1], Kang-Jie Shi$$\thanks{Email:kjshi@phy.nwu.edu.cn}, \hspace{5mm} Zhan-Ying Yang$$ [^2], Rui-Hong Yue$$ [^3]\ Institute of Modern Physics, Northwest University,\ Xi’an, 710069, P. R. China title: '**Soliton Solutions on Noncommutative Orbifold $ T^2/Z_4 $**' ---       Soliton, Projection operators, Noncommutative orbifold. Introduction ============ String theory is a very promising candidate for an unified description of the fundamental interactions, including quantum gravity. It may provide a conceptual framework to resolve the clash between two of the greatest achievements of 20th century physics: general relativity and quantum mechanics. Noncommutative geometry is originally an interesting topic in mathematics [@1; @2; @3], In the past few years, it has been shown that some noncommutative gauge theories can be embedded in string theories [@4; @5; @6]and noncommutative geometry can also be applied to condensed matter physics. The currents and density of a system of electrons in a strong magnetic field may be described by a noncommutative quantum field theory \[7,8,9\]. The connection between a finite quantum Hall system and a noncommutative Chern-Simon Matrix model first proposed by [@hall20] was further elaborated in papers [@hall21; @hall22]. Many papers are concentrated on the research for the related questions about the quantum hall effect \[12-18\]. Since the noncommutative space resemble a quantum phase space, it exhibits an interesting spacetime uncertainty relation, which cause a $UV/IR$ mixing [@bib:uv1; @bib:uv2] and a teleological behavior. Noncommutative field theories can be regarded as highly constrained deformation of local field theory. Thus it may help us to understand non-locality at short distances in quantum gravity. Solitons in various noncommutative theories have played a central role in understanding the physics of noncommutative theories and certain situations of string theories. The quantum Hall effect practically provides a good illustration of the combination of the three theories [@bib:hall2; @bib:hall6; @bib:hall9; @bib:hall10]. The existence and form of these classical solutions are fairly independent of the details of the theory, making them useful to probe the string behavior. In fact these solitons are the (lower-dimensional) D-branes of string theory manifested in a field theory limit while still capturing many string features. Starting from the celebrated paper of Gopakumar, Minwalla and Strominger [13]{}, there are many works to study soliton solutions of noncommutative field theory and integrable systems in the background of noncommutative spaces \[23-30\]. Although Derrick’s theorem forbids solitons in ordinary 2+1 dimensional scalar field theory [@bib:33], solitons in noncommutative scalar field theory on the plane were constructed in terms of projection operators in [@13]. It was soon realized that noncommutative solitons represent D-branes in string field theory with a background $B$ field, and many of Sen’s conjectures [@18; @19] regarding tachyon condensation in string field theory have been beautifully confirmed using properties of noncommutative solitons. Gopakumar, Minwalla and Strominger makde an important finding that in a noncommutative space, a projector may correspond to a soliton in the field theory [@13], which proves the significance of the study of projection operators in various noncommutative space. Reiffel [@15] constructed the complete set of projection operators on the noncommutative torus $T^{2}$. On the basis Boca studied the projection operators on noncommutative orbifold [@16] obtaining many important results and showed the well-known example of projection operator for $T^{2}/Z_{4}$ in terms of the elliptic function. Soliton solutions in noncommutative gauge theory were introduced by Polychronakos in [@soliton]. Martinec and Moore in their important article deeply studied soliton solutions namely projectors on a wide variety of orbifolds, and the relation between physics and mathematics in this area [@11]. Gopakumar, Headrick and Spradlin have shown a rather apparent method to construct the multi-soliton solution on noncommutative integral torus with generic $\tau$[@10]. This approach can be generalized to construct the projection operators on the integral noncommutative orbifold $T^2/Z_N$ [@9]. In this paper, in the case of integral noncommutative orbifold $T^2/Z_4$ generated by $u_{1}$ and $u_{2}$ with $$u_{1}u_{2} =u_{2}u_{1}e^{2\pi i/A},~~~A=1,2,3,\cdots$$ we generalize the $GHS$ construction, presenting the explicit symmetric form of a series of projectors with manifest $Z_{4}$ symmetry. It includes all the solutions with minimal trace, and in the standard expansions for the projectors (see equation (\[eq:41\])) $$P=\sum_{s,t}u_1^s u_2^t\Psi_{s,t}(u_1^A,u_2^A)$$ where the eigen value function $\Psi_{s,t}(v_1^A,v_2^A)$ is continuous (where $v^{A}_{1}$ and $v^{A}_{2}$ are eigenvalues of $u^{A}_{1}$ and $u^{A}_{2}$). The solutions include two arbitrary complex functions with $Z_4$ symmetry. The kernels of the integrations are closed analytic functions of $u_{1}$ and $u_{2}$. In the simplest case, when $A$ is an even number, we reobtain the Boca’s classic result [@16] and obtain a new result when $A$ is an odd number. Moreover the above construction is also applicable to the integral $T^2/Z_N (N=3,6)$ cases. This paper is organized as following: In Section 2, we introduce operators on the noncommutative orbifold $T^2/Z_N$. In Section 3, we introduce the $% |k,q>$ representation and provide the matrix element relation for the projectors and deduce the relation between the eigen value functions of coefficients and the matrix elements of operators in the $|k,q>$ representation. In Section 4, we study the general projectors with minimal trace when the eigen value functions of coefficients are continuous. In Section 5, we present two kinds of explicit expressions for the projectors with elliptical functions as kernel. Noncommutative Orbifold $T^2/Z_N$ ================================= In this section, we introduce operators on the noncommutative orbifold $% T^{2}/Z_{N}$. First we introduce two hermitian operators $\hat{y_{1}}$ and $% \hat{y_{2}}$, which satisfy the following commutation relation: $$\lbrack \hat{y_{1}},\hat{y_{2}}]=i.$$The operators made up of $\hat{y_{1}}$ and $\hat{y_{2}}$ $$\hat{O}=\sum_{m,n}C_{mn}\hat{y}_{1}^{m}\hat{y}_{2}^{n}$$form the noncommutative plane $R^{2}$. All operators on $R^{2}$ which commute with $U_{1}$ and $U_{2}$ $$U_{1}=e^{-il\hat{y_{2}}},~~~~~~~~~~U_{2}=e^{il(\tau _{2}\hat{y_{1}}-\tau _{1}% \hat{y_{2}})},$$where $l,\tau _{1},\tau _{2}$ are all real numbers and $l,\tau _{2}>0,\tau =\tau _{1}+i\tau _{2}$, constitute the noncommutative torus $T^{2}$. We have $$\begin{aligned} U_{1}^{-1}\hat{y_{1}}U_{1} &=&\hat{y_{1}}+l,~~~~~~~U_{2}^{-1}\hat{y_{1}}% U_{2}=\hat{y_{1}}+l\tau _{1}, \nonumber \\ U_{1}^{-1}\hat{y_{2}}U_{1} &=&\hat{y_{2}},~~~~~~~~~~~U_{2}^{-1}\hat{y_{2}}% U_{2}=\hat{y_{2}}+l\tau _{2}.\end{aligned}$$The operators $U_{1}$ and $U_{2}$ are two different wrapping operators around the noncommutative torus and their commutation relation is $% U_{1}U_{2}=U_{2}U_{1}e^{-2\pi i\frac{l^{2}\tau _{2}}{2\pi }}$. When $A=\frac{% l^{2}\tau _{2}}{2\pi }$ is an integer, we call the noncommutative torus integral.Introduce two operators $u_{1}$ and $u_{2}$: $$\begin{aligned} \label{eq:40} u_{1} &=&e^{-il\hat{y_{2}}/A},~~~~~~~~~~u_{2}=e^{-il(\tau _{2}\hat{y_{1}}% -\tau _{1}\hat{y_{2}})/A}, \nonumber \label{eq:14} \\ u_{1}u_{2} &=&u_{2}u_{1}e^{2\pi i/A},~~~~~~u_{1}^{A}=U_{1},~~u_{2}^{A}=U_{2}^{-1}\end{aligned}$$The operators on the noncommutative torus are composed of the Laurant series of $u_{1}$ and $u_{2}$, $$\hat{O}_{T^{2}}=\sum_{m,n}C_{mn}^{\prime }u_{1}^{m}u_{2}^{n} \label{eq:5}$$where $m,n\in Z$ and $C_{00}^{\prime }$is called the trace of the operators, Eq.(\[eq:5\]) includes all operators on the noncommutative torus $T^{2}$, satisfying the relation $U_{i}^{-1}\hat{O}_{T^{2}}U_{i}=\hat{O}_{T^{2}}$. From (\[eq:40\]) we can rewrite the equation(\[eq:5\]) as $$\label{eq:41} \hat{O}_{T^{2}}=\sum_{s,t=0}^{A-1}u_{1}^{s}u_{2}^{t}\Psi _{st}(u_{1}^{A},u_{2}^{A}) \label{eq:6}$$where $\Psi _{st}$ is Laurant series of the operators $u_{1}^{A}$and $% u_{2}^{A}$. we call this formula the standard expression for the operator on the noncommutative torus $T^{2}$. The trace for the operator is the constant term’s coefficient of $\Psi _{00}$. Nextly we introduce rotation $R$ in noncommutative space $R^{2}$ $$R(\theta )=e^{-i\theta \frac{\hat{y_{1}}^{2}+\hat{y_{2}}^{2}}{2}+i\frac{% \theta }{2}}$$with $$R^{-1}\hat{y}_{1}R =\cos \theta \hat{y}_{1}+\sin \theta \hat{y}_{2},~~~~ R^{-1}\hat{y}_{2}R =\cos \theta \hat{y}_{2}-\sin \theta \hat{y}_{1}.$$When $\tau =\tau _{1}+\tau _{2}=e^{2\pi i/N}$, setting $\theta =2\pi /N(N\in Z)$. The noncommutative torus $T^{2}$ keep invariant under rotation $% R_{N}\equiv R(2\pi /N)$ [@16; @11; @9]. Namely $R_{N}^{-1}\hat{O}% _{T^{2}}R_{N} $ is still the operators on the noncommutative Torus $T^{2}$. Now $U_{i}^{\prime }\equiv R_{N}^{-1}U_{i}R_{N}$ can be expressed by monomial of $\{U_{i}\}$ and their inverses [@11].In this case, we call the operators invariant under rotation $R_{N}$ on the noncommutative torus as operators on noncommutative orbifold $T^{2}/Z_{N}$. We can also realize these operators in Fock space. Introduce $$a=\frac{\hat{y}_{2}-i\hat{y}_{1}}{\sqrt{2}},~~~~~~~~~a^{+}=\frac{\hat{y}% _{2}+i\hat{y}_{1}}{\sqrt{2}},$$then $$\begin{aligned} \lbrack a,a^{+}] &=&1, \\ R_{N} &=&e^{-i\theta a^{+}a}.\end{aligned}$$In this paper, we study the projector $P$ on the orbifold $T^{2}/Z_{4}$: $$\begin{aligned} \tau &=&i, \\ P^{2} &=&P, \\ U_{j}^{-1}PU_{j} &=&P,~~~~~~j=1,2 \\ R_{4}^{-1}PR_{4} &=&P.\end{aligned}$$ The $|k,q>$ representation, standard form and eigen value function ================================================================== From the above discussion, we know that the operators $U_{1}$ and $U_{2}$ commute with each other on the integral torus $T^{2}$ when $A$ is an integer. So we can introduce a complete set of their common eigenstates, namely $|k,q>$ representation [@21; @20] $$|k,q>=\sqrt{\frac{l}{2\pi }}e^{-i\tau _{1}\hat{y_{2}}^{2}/2\tau _{2}}\sum_{j}e^{ijkl}|q+jl>,$$where the ket on the right is a $\hat{y_{1}}$ eigenstate. We have $$\begin{aligned} U_{1}|k,q>= &&e^{-ilk}|k,q>,~~~~~~~U_{2}|k,q>=e^{il\tau _{2}q}|k,q>=e^{2\pi iqA/l }|k,q>, \nonumber \label{eq:7} \\ id &=&\int_{0}^{\frac{2\pi }{l}}dk\int_{0}^{l}dq|k,q><k,q|.\end{aligned}$$It also satisfies $$\label{eq:46} |k,q>=|k+\frac{2\pi }{l},q>=e^{ilk}|k,q+l>.$$Consider the equation (\[eq:6\]), namely the standard expansion of operators on $T^{2}$ we have $$\label{eq:15} \Psi _{st}(u_{1}^{A},u_{2}^{A})|k,q>=\Psi _{st}(e^{-ilk},e^{-2\pi iqA/l})|k,q>\equiv \psi _{st}(k,q)|k,q>,$$where $\psi _{st}$ is a function of the independent variables $k$ and $q$, called the eigen value function of $\Psi _{st}(u_{1}^{A},u_{2}^{A})$. From (\[eq:15\]), we see that the function $\psi _{st}$ is invariant when $q\rightarrow q+l/A$, $$\psi _{st}(k,q+\frac{ln}{A})=\psi _{st}(k,q). \label{eq:16}$$As long as the eigen value function is obtained, the operator on the noncommutative torus can be completely determined. Introducing new basis $% |k,q_{0};n>\equiv |k,q_{0}+\frac{ln}{A}>, k\in \lbrack 0,\frac{2\pi }{l}% ), q_{0}\in \lbrack 0,\frac{l}{A})$, we have from (\[eq:7\]) $$\sum_{n=0}^{A-1}\int_{0}^{\frac{2\pi }{l}}dk\int_{0}^{\frac{l}{A}% }dq_{0}|k,q_{0}+\frac{ln}{A}><k,q_{0}+\frac{ln}{A}|=id,$$$$u_{1}|k,q>=|k,q+\frac{l}{A}>, \label{eq:21}$$$$u_{2}|k,q>=e^{-2\pi i\frac{q}{l}}|k,q>. \label{eq:22}$$From the above equation and (\[eq:46\]), we see that when any power of the operators $u_{1}$ and $u_{2}$ act on the $|k,q_{0}+\frac{ln}{A}>$, the result can be expanded in the basis $|k,q_{0}+\frac{ln^{\prime }}{A}>$ with the same $k,q_{0}$. So the operators on the noncommutative torus have the same property, namely don’t change $k$ and $q_{0}$. Thus, for every $k$ and $% q_{0}$ we get a $A\times A$ matrix, called reduced matrix for the operator, as well as the projector: $$\label{eq:23} P_{T^{2}}|k,q_{0}+\frac{ln}{A}>=\sum_{n^{\prime }}M(k,q_{0})_{n^{\prime }n}|k,q_{0}+\frac{ln^{\prime }}{A}>,$$It is easy to find that the sufficient and necessary condition for $P^{2}=P$ is [@9] $$M(k,q_{0})^{2}=M(k,q_{0}).$$When $T^{2}$ satisfies $Z_{N}$ symmetry, since after $R_{N}$ rotation $% U_{i}^{\prime }$ can be expressed by monomial of $\{U_{i}\}$ and their inverses, the state vector $R_{N}|k,q_{0}+\frac{ln}{A}>$ is still the common eigenstate of the operators $U_{1}$ and $U_{2}$. With the completeness of $% \{|k,q+\frac{ls}{A}>\}$ and the A-fold degeneracy eigenvalues of $U_{i}$ in the $kq$ representation, the state can be expanded in the basis $\{|k^{\prime },q^{\prime }+\frac{ls^{\prime }}{A}>\}$ $$\label{eq:17} R_{N}|k,q_{0}+\frac{ln}{A}>=\sum_{n^{\prime }}A(k,q_{0})_{n^{\prime }n}|k,q_{0}+\frac{ln^{\prime }}{A}>$$[^4]where $k^{\prime }\in \lbrack 0,2\pi /l),q^{\prime }\in \lbrack 0,l/A)$ are definite and $$\label{eq:18} R_{N}^{-1}|k^{\prime },q_{0}^{\prime }+\frac{ln^{\prime }}{A}% >=\sum_{n"}A^{-1}(k,q_{0})_{n"n^{\prime }}|k^{\prime },q^{\prime }_{0}+\frac{ln"}{A}>.$$We can get the expression for the relation between $k^{\prime }$,$% q_{0}^{\prime }$ and $k$, $q_{0}$, The mapping $W:(k,q_{0})\longrightarrow (k^{\prime },q_{0}^{\prime }),W^{N}=id$, is essentially a linear relation, and area-preserving. By this fact and since $R_{N}$ is unitary, we conclude that the matrix $A$ is a unitary matrix, that is to say $$A^{\ast }(k,q_{0})_{nn^{\prime }}=A^{-1}(k,q_{0})_{n^{\prime }n}. \label{eq:19}$$The projector on the noncommutative orbifold $T^{2}/Z_{N}$ satisfies $% R_{N}^{-1}PR_{N}=P$, then from (\[eq:23\])(\[eq:17\])(\[eq:18\]) one obtains $$\label{eq:35} R_{N}^{-1}PR_{N}|k,q_{0}+\frac{ln}{A}>=\sum_n^{'}[A^{-1}(k,q_{0})M(k^{\prime },q_{0}^{\prime })A(k,q_{0})]_{n^{'}n}|k,q_{0}+\frac{ln^{'}}{A}>,$$ which should be equal to : $$\label{eq:36} P|k,q_{0}+\frac{ln}{A}>=\sum_{n"}M(k,q_{0})_{n"n}|k,q_{0}+\frac{ln"}{A}>.$$So, we have $$M(k^{\prime },q_{0}^{\prime })=A(k,q_{0})M(k,q_{0})A^{-1}(k,q_{0}) \label{eq:10}$$and the sufficient and necessary condition for the projector on noncommutative orbifold $T^{2}/Z_{N}$ to satisfy is: $$\begin{aligned} M(k,q_{0})^{2} &=&M(k,q_{0}), \\ M(k^{\prime },q_{0}^{\prime }) &=&A(k,q_{0})M(k,q_{0})A^{-1}(k,q_{0}).\label{eq:24}\end{aligned}$$Next we will study the relation between coefficient function $\psi _{st}(k,q) $ and the reduced matrix $M(k,q_{0})$. From (\[eq:16\])(\[eq:21\])(\[eq:22\]) and (\[eq:23\])we have $$\begin{aligned} P|k,q_{0}+\frac{ln}{A}>= &&\sum_{s,t}u_{1}^{s}u_{2}^{t}\Psi _{st}(u_{1}^{A},u_{2}^{A})|k,q_{0}+\frac{ln}{A}> \nonumber \\ &=&\sum_{s,t}e^{-2\pi i(q_{0}/l+n/A)t}\psi _{st}(k,q_{0})|k,q_{0}+\frac{% l(n+s)}{A}> \nonumber \\ &=&\sum_{n^{\prime }}M(k,q_{0})_{n^{\prime }n}|k,q_{0}+\frac{ln^{\prime }% }{A}>.\end{aligned}$$So for $n+s<A$ case, we have $$M(k,q_{0})_{n+s,n}=\sum_{t=0}^{A-1}e^{-2\pi i(q_{0}/l+n/A)t}\psi _{st}(k,q_{0})$$and for $n+s\geq A$ case, we have $$M(k,q_{0})_{n+s-A,n}=\sum_{t=0}^{A-1}e^{-2\pi i(q_{0}/l+n/A)t}\psi _{st}(k,q_{0})e^{-ilk}.$$Setting $$M(k,q_{0})_{n+s,n}=M(k,q_{0})_{n+s-A,n}e^{ilk}, \label{eq:1}$$We can uniformly write as: $$M(k,q_{0})_{n+s,n}=\sum_{t=0}^{A-1}e^{-2\pi i(q_{0}/l+n/A)t}\psi _{st}(k,q_{0}) \label{eq:2}$$and have $$\psi _{st}(k,q_{0})=\frac{1}{A}\sum_{n=0}^{A-1}M(k,q_{0})_{n+s,n}e^{2\pi i(q_{0}/l+n/A)t}. \label{eq:3}$$Eq.(\[eq:2\]) and (\[eq:3\]) is the relation between $\psi _{st}$ and the elements of reduced matrix $M$. Continuous solution for the Projector with Minimal Trace ======================================================== Now one may ask what property the reduced matrix $M$ possess when the coefficient function $\psi _{st}$ is a continuous function. In this section, we mainly answer this question. First we prove the $A\times A$ matrix satisfying the condition $M^{2}=M$ is always diagonalizable. For any vector $% \psi $, $M\psi $ is invariant under $M$, namely $$M(M\psi )=M\psi .$$Assume there are totally $B$ linear independent invariant vectors under transformation $M$, then(1) for $A=B$ case, the matrix $M$ is identity of the space expanded by the vectors, namely $A\times A$ unit matrix. Of course it is diagonal.(2) for $B<A$ case, considering any vector $a$ and setting $b=Ma-a$, we find $Mb=0$. Namely any vector $a$ can be expressed as linear combination of invariant vector $c=Ma$ and null vector $b$ under action of $M$. So the whole linear space is composed of certain invariant vectors and null vectors under action of $M$. $M$ can be diagonalized in the representation with these vectors as basis. So we have: $$M(k,q_{0})=S^{-1}(k,q_{0})\overline{M}(k,q_{0})S(k,q_{0}),$$where $$\overline{M}(k,q_{0})=diag(1,1,\cdots ,1,0,0,\cdots ,0).$$Due to (\[eq:2\]), when $\psi _{st}(k,q_{0})$ is continuous, $M(k,q_{0})$ is also continuous. However $trM(k,q_{0})=tr\overline{M}(k,q_{0})=0,1,2,\cdot \cdot \cdot ,A$, which is discrete, so when $\psi _{st}$ is continuous, the value of $trM(k,q_{0})=A\psi_{00}(k,q_{0})$ is invariant for all $k$ and $q_{0}$. The trace of the projector is the zero order term of $\psi _{00}(k,q_{0})$ in Laurant expression of $e^{-ilk}$ and $e^{-2\pi iqA/l}$, so we have $$\begin{aligned} trP &=&\int_{0}^{\frac{2\pi }{l}}dk\int_{0}^{\frac{l}{A}}dq\frac{A}{2\pi }% \psi _{00}(k,q_{0}) \nonumber\\ &=&\int_{0}^{\frac{2\pi }{l}}dk\int_{0}^{\frac{l}{A}}dq\frac{1}{2\pi }% trM(k,q_{0}) \nonumber\\ &=&\frac{1}{A}trM(k,q_{0}).\end{aligned}$$The projector is trivial for $trM(k,q_{0})=0,A$, indicating $P=0$ and $% identity$. The nontrivial $trP=\frac{1}{A},\frac{2}{A},\cdots ,\frac{A-1}{A}$. In this paper, we only study the nontrivial projector with minimal trace($% trM(k,q_{0})=1$). Thus $$M(k,q_0)=s^{-1}(k,q_0) \left( \begin{array}{cccc} 1 & & & \\ & 0 & & \\ & & 0 & \\ & & & \ddots% \end{array} \right) s(k,q_0), \nonumber$$ $$\label{eq:8} M(k,q_0)_{nn^{\prime}}=s^{-1}(k,q_0)_{n0} s(k,q_0)_{0n^{\prime}}\equiv a(k,q_0)_{n}b(k,q_0)_{n^{\prime}}.$$ Explicit calculation about $R_{N}$ acting on $|k,q;n>$ shows that we can divide the complete area $\Sigma :\{k\in \lbrack 0,2\pi /l),q_{0}\in \lbrack 0,l/A)\}$ into $N$ subarea $\sigma _{0},\cdots ,\sigma _{N-1}$,making $W:\sigma _{i}\rightarrow \sigma _{i+1},(i=0,1,\cdots ,N-2),\sigma _{N-1}\rightarrow \sigma _{0}.$ If we construct a reduced matrix $M(k,q_{0})$ to satisfy ([eq:8]{}) in the area $\sigma _{0}$, then the projector corresponding to continuous $\psi _{st}$ with minimal trace is completely determined. In area $\sigma _{0}$, set $$a_{n}=<k,q_{0}+\frac{ln}{A}|\phi _{1}>,~~~~~~~~b_{n}=<\phi _{2}|k,q_{0}+% \frac{ln}{A}>, \label{eq:9}$$where $$\sum_{n}a_{n}b_{n}=trM(k,q_{0})=1.$$In the other areas $\sigma_{j}$ with $(k,q_{0})\rightarrow (k_{j},q_{0j})$ by mapping $W^{j}$, we demand $$\begin{aligned} a_{n}(k_{j},q_{0j}) &=&<k_{j},q_{0j}+\frac{ln}{A}|\phi _{1}> \nonumber \\ &=&A^{j}(k,q_{0})_{nn^{\prime }}a_{n^{\prime }}(k,q_{0}), \\ b_{n}(k_{j},q_{0j}) &=&<\phi _{2}|k_{j},q_{0j}+\frac{ln}{A}> \nonumber \\ &=&b_{n^{\prime }}(k,q_{0})A^{-j}(k,q_{0})_{n^{\prime }n}.\end{aligned}$$We thus have all coefficients of $|\phi_{1}>, <\phi_{2}|$ in $\sigma_{0},\cdots,\sigma_{N-1}$. Owing to the completeness of $|k,q_{0}+\frac{ln}{A}>$ in the area $% \Sigma $, $|\phi _{1}>$ and$<\phi _{2}|$ can be determined by the coefficient (\[eq:9\]) of $|\phi _{1}>$ and$<\phi _{2}|$. Meanwhile, in the area $\sigma _{j}$, we have $$\begin{aligned} M(k_{j},q_{0j})_{nn^{\prime }} &=&a_{n}(k_{j},q_{0j})b_{n^{\prime}}(k_{j},q_{0j}) \nonumber \\ &=&[A^{j}(k,q_{0})M(k,q_{0})A^{-j}(k,q_{0})]_{nn^{\prime }}.\end{aligned}$$The matrix $M(k,q_{0})$ really satisfies the equation (\[eq:10\]). Consider the state vector $$\begin{aligned} |\phi _{1}>&= &\int dkdq_{0}\sum_{n}|k,q_{0}+\frac{ln}{A}><k,q_{0}+\frac{ln}{% A}|\phi _{1}> \nonumber \\ &=&\sum_{j=0}^{N-1}\int_{\sigma _{j}}dk_{j}dq_{0j}\sum_{n}|k_{j},q_{0j}+\frac{ln}{A}% >a_{n}(k_{j},q_{0j}) \nonumber \\ &=&\sum_{j=0}^{N-1}\int_{\sigma _{j}}dkdq_{0}\sum_{nn_{1}}|k,q_{0}+\frac{ln}{A}% >A_{nn_{1}(k,q_{0})}^{j}a_{n_{1}}(k,q_{0}) \nonumber \\ &=&\sum_{j=0}^{N-1}R_{N}^{j}\int_{\sigma _{0}}dkdq_{0}\sum_{n}|k,q_{0}+\frac{ln}{A}% >a_{n}(k,q_{0}).\end{aligned}$$Thus we have $$\label{eq:25} R_{N}|\phi _{1}>=|\phi _{1>}.$$In the same way, we get $$\label{eq:26} <\phi _{2}|R_{N}=<\phi _{2}|.$$That is to say that the state vectors $|\phi _{1}>$ and $<\phi _{2}|$ are invariant under the rotation $R_{N}$. More generally, we can take any state vectors $|\phi _{1}>$ and $<\phi _{2}|$ satisfying $$\label{eq:50} R_{N}|\phi _{1}>=e^{i\alpha _{1}}|\phi _{1}>,~~~~~~~<\phi _{2}|R_{N}^{-1}=e^{-i\alpha _{2}}<\phi _{2}|$$to construct a projection operator on noncommutative orbifold $T^{2}/Z_{N}$. Let $M(k,q_{0})$ be given by (\[eq:8\]) with $$\begin{aligned} a_{n}(k,q_{0}) &=&\frac{<k,q_{0}+\frac{ln}{A}|\phi _{1}>}{\sqrt{% \sum_{n^{\prime }}<k,q_{0}+\frac{ln^{\prime }}{A}|\phi _{1}><\phi _{2}|k,q_{0}+\frac{ln^{\prime }}{A}>}}, \label{eq:4} \\ b_{n}(k,q_{0}) &=&\frac{<\phi _{2}|k,q_{0}+\frac{ln}{A}>}{\sqrt{% \sum_{n^{\prime }}<k,q_{0}+\frac{ln^{\prime }}{A}|\phi _{1}><\phi _{2}|k,q_{0}+\frac{ln^{\prime }}{A}>}}.\end{aligned}$$The projector of minimal trace and with continuous coefficient functions is surely of this form. It can be verified that $M^{2}=M$. And it is also covariant under $R_{N}$. From (\[eq:18\]) we have $$\begin{aligned} &&<\phi _{2}|k^{^{\prime }},q_{0}^{^{\prime }}+\frac{n^{^{\prime }}l}{A}> \\ &=&<\phi _{2}|R_{N}\sum_{n^{"}}A^{-1}(k,q_{0})_{n"n^{\prime }}|k,q_{0}+\frac{n^{"}l}{A}> \\ &=&e^{i\alpha _{2}}\sum_{n^{"}}A^{-1}(k,q_{0})_{n^{"}n^{^{\prime }}}<\phi _{2}|k,q_{0}+\frac{n^{"}l}{A}>\end{aligned}$$ and similarly $$<k^{^{\prime }},q_{0}^{^{\prime }}+\frac{n^{^{\prime }}l}{A}|\phi _{1}>=e^{-i\alpha _{1}}\sum_{n^{"}}<k,q_{0}+\frac{n^{"}l}{A}|\phi _{1}>A(k,q_{0})_{n^{^{\prime }}n^{"}},$$ giving $$\begin{aligned} &&\sum_{n}<k^{^{\prime }},q_{0}^{^{\prime }}+\frac{ln}{A}|\phi _{1}><\phi _{2}|k^{^{\prime }},q_{0}^{^{\prime }}+\frac{ln}{A}> \nonumber \\ &=&\sum_{n}<k,q_{0}+\frac{ln}{A}|\phi _{1}><\phi _{2}|k,q_{0}+\frac{ln}{A}% >e^{-i(\alpha _{1}-\alpha _{2})}.\end{aligned}$$ Thus $$M(k^{\prime },q_{0}^{\prime })_{nn^{\prime }}=a_{n}(k^{\prime },q_{0}^{\prime })b_{n^{\prime }}(k^{\prime },q_{0}^{\prime })=[AMA^{-1}](k,q_{0})_{nn^{\prime }},$$$P$ is invariant under rotation $R_{N}$ due to (\[eq:24\]) and really gives the projection operator on noncommutative orbifold $T^{2}/Z_{N}$. The form of (\[eq:4\]) is a generalization of $GHS$ construction.[^5]. From the above result, we have $$M(k,q_{0})_{nn^{\prime }}=\frac{<k,q_{0}+\frac{ln}{A}|\phi _{1}><\phi _{2}|k,q_{0}+\frac{ln^{\prime }}{A}>}{\sum_{n^{"}}<k,q_{0}+\frac{ln"}{A}% |\phi _{1}><\phi _{2}|k,q_{0}+\frac{ln"}{A}>}.$$Noticing that this equation satisfies (\[eq:1\]), we have $$\begin{aligned} \label{eq:30} \psi _{st}(k,q_{0}) &=&\frac{1}{A}\sum_{n=0}^{A-1}M(k,q_{0})_{n+s,n}e^{2\pi i(q_{0}/l+n/A)t} \nonumber \\ &=&\frac{\frac{1}{A}\sum_{n=0}^{A-1}<k,q_{0}+\frac{l(n+s)}{A}|\phi _{1}><\phi _{2}|k,q_{0}+\frac{ln}{A}>e^{2\pi i(q_{0}/l+n/A)t}}{% \sum_{n}<k,q_{0}+\frac{ln}{A}|\phi _{1}><\phi _{2}|k,q_{0}+\frac{ln}{A}>} \nonumber \\ &=&\frac{F_{st}(k,q_{0})}{AF_{00}(k,q_{0})},\end{aligned}$$where $$\label{eq:42} F_{st}(k,q_{0})\equiv \sum_{n=0}^{A-1}<k,q_{0}+\frac{l(n+s)}{A}|\phi _{1}><\phi _{2}|k,q_{0}+\frac{ln}{A}>e^{2\pi i(q_{0}/l+n/A)t},$$with $$\begin{aligned} F_{st}(k,q_{0}) &=&F_{st}(k,q_{0}+l/A)=F_{st}(k+2\pi /l,q_{0}), \\ F_{st}(k,q_{0}) &=&F_{s+A,t}(k,q_{0})e^{-ilk} \\ &=&F_{s,t+A}(k,q_{0})e^{-2\pi iq_{0}A/l}.\end{aligned}$$So the function $F_{st}$ is the function of independent variables $% X=e^{-ilk} $ and $Y=e^{-2\pi iq_{0}A/l}$, namely $F_{st}(k,q_{0})=\Phi _{st}(X,Y)$. Similarly $$\psi _{st}(k,q_{0})=\Psi _{st}(X,Y)=\frac{\Phi _{st}(X,Y)}{A\Phi _{00}(X,Y)}. \label{eq:12}$$If we change the variable $X$ and $Y$ into $u_{1}^{A}$ and $u_{2}^{A}$ respectively, the standard form (\[eq:6\]) of the projection operator can be easily obtained. So the key question is to find out $F_{st}(k,q_{0})$. Coherent State Representation ============================== Introduce coherent states $$|z>=e^{-\frac{1}{2}z\bar{z}}e^{a^{+}z}|0>,$$where $z=x+iy,\bar{z}=x-iy$, which satisfies $$\frac{1}{\pi }\int_{-\infty }^{\infty }d^{2}z|z><z|\equiv \frac{1}{\pi }% \int_{-\infty }^{\infty }dxdy|z><z|=identity,$$ $$R_{N}|z>=|\omega _{N}z>.$$ We can show [@9] $$\label{eq:27} <k,q|z>=\frac{1}{\sqrt{l}\pi ^{1/4}}\theta (\frac{q+\frac{\tau }{\tau _{2}}% k-i\sqrt{2}z}{l},\frac{\tau }{A})e^{-\frac{\tau }{2i\tau _{2}}k^{2}+ikq+% \sqrt{2}kz-(z^{2}+z\bar{z})/2},$$where $$\theta (z,\tau )\equiv \theta \left[ \begin{array}{c} 0 \\ 0% \end{array}% \right] (z,\tau )$$ and$$\label{eq:34} \theta \left[ \begin{array}{c} a \\ b% \end{array}% \right] (z,\tau )=\sum_{m}e^{\pi i\tau (m+a)^{2}}e^{2\pi i(m+a)(z+b)}.$$Thus we can expand the state vectors $|\phi _{1}>$ and $<\phi _{2}|$ in terms of coherent state, $$\begin{aligned} |\phi _{1}>&= &\frac{1}{\pi }\int_{-\infty }^{\infty }dxdy|z><z|\phi _{1}> \nonumber \label{eq:28} \\ &\equiv &\frac{1}{\pi }\int_{-\infty }^{\infty }dxdyf_{1}(z)|z>, \\ <\phi _{2}| &=&\frac{1}{\pi }\int_{-\infty }^{\infty }dxdy<\phi _{2}|z><z| \nonumber \\ &=&\frac{1}{\pi }\int_{-\infty }^{\infty }dxdyf_{2}(z)<z|. \label{eq:29}\end{aligned}$$ The condition (\[eq:50\]) is satisfied if and only if $$f_{1}(\omega_{N}^{-1}z)=f_{1}(z)e^{i\alpha _{1}},$$$$f_{2}(\omega_{N}^{-1}z)=f_{2}(z)e^{-i\alpha _{2}}.$$ Here $\omega_{N}=e^{-i\frac{2\pi }{N}}$. We have $$\begin{aligned} \label{eq:31} F_{st}(k,q_{0}) &=&\frac{1}{\pi ^{2}}\sum_{n=0}^{A-1}\int <k,q_{0}+\frac{% l(n+s)}{A}|z_{1}>f_{1}(z_{1})dx_{1}dy_{1} \nonumber \\ &&\times \int <z_{2}|k,q_{0}+\frac{ln}{A}>f_{2}(z_{2})dx_{2}dy_{2}\times e^{2\pi i(q_{0}/l+n/A)t} \nonumber \\ &=&\frac{1}{\pi ^{2}}\int dx_{1}dy_{1}dx_{2}dy_{2}g_{st}(k,q_{0},z_{1},z_{2})f_{1}(z_{1})f_{2}(z_{2}),\end{aligned}$$where $$g_{st}(k,q_{0},z_{1},z_{2})=\sum_{n=0}^{A-1}<k,q_{0}+\frac{l(n+s)}{A}% |z_{1}><z_{2}|k,q_{0}+\frac{ln}{A}>e^{2\pi i(q_{0}/l+n/A)t}.$$We call the kernel $g$ as generating function in coherent state representation. Next, we study the expression of $g$ for $Z_{4}$ case. Through $g$ we can give the integration expression for all the projection operators on the $T^{2}/Z_{4}$ with minimal trace and continuous eigen value function. Consider the equation $$\theta (z,\tau )^{\ast }=\theta (z^{\ast },-\tau ^{\ast }).$$For the $z_{4}$ case, $\tau =i,A=\frac{l^{2}}{2\pi }$, from (\[eq:27\])(\[eq:28\]) and (\[eq:29\]) we get $$\begin{aligned} &<&k,q+\frac{ls}{A}|z_{1}><z_{2}|k,q+\frac{ls^{\prime }}{A}> \nonumber \\ &=&\frac{1}{l\sqrt{\pi }}\theta (\frac{q}{l}+\frac{i}{l}k+\frac{s}{A}-\frac{i% \sqrt{2}z_{1}}{l},\frac{i}{A})\theta (\frac{q}{l}-\frac{i}{l}k+\frac{% s^{\prime }}{A}+\frac{i\sqrt{2}z_{2}^{\ast }}{l},\frac{i}{A}) \nonumber \\ &&\times e^{-k^{2}+\frac{k}{2}(z_{1}+z_{2}^{\ast })+ik\frac{l(s-s^{\prime })% }{A}}e^{-\frac{1}{2}(z_{1}^{2}+(z_{2}^{\ast })^{2}+z_{1}z_{1}^{\ast }+z_{2}z_{2}^{\ast })}\equiv K_{ss^{\prime }}.\end{aligned}$$Let $u=\frac{lk}{2\pi },v=\frac{q}{l},\mu =-i\frac{\sqrt{2}A}{l}z_{1},\nu =i% \frac{\sqrt{2}A}{l}z_{2}^{\ast }$, then $$\begin{aligned} K_{ss^{\prime }} &=&\frac{C_{1}}{l\sqrt{\pi }}\theta (v+\frac{iu}{A}+\frac{% s+\mu }{A},\frac{i}{A})\theta (v-\frac{iu}{A}+\frac{s^{\prime }+\nu }{A},% \frac{i}{A}) \nonumber \\ &&\times e^{\pi i\frac{2i}{A}u^{2}+2\pi i(\frac{s+\mu -s^{\prime }-\nu }{A}% )u}.\end{aligned}$$where $$C_{1}=e^{2\pi i[\frac{-i}{4A}(\mu ^{2}+\nu ^{2})+\frac{i}{4A}(|\mu |^{2}+|\nu |^{2})]}.$$It can be proved that for integer $A$: $$\begin{aligned} &&\sum_{r=0}^{A-1}e^{2\pi irt/A}\theta (x+r/A,\tau /A)\theta (y+r/A,\tau /A) \nonumber \\ &=&A\sum_{d=0,1}\theta (-\frac{\tau }{A}(Ad-t)+x-y,\frac{2\tau }{A})\theta (% \frac{\tau }{A}(-At+A^{2}d)+A(x+y),2\tau A) \nonumber \\ &&\times e^{\pi i\frac{\tau }{A}(Ad-t)^{2}}\times e^{2\pi i(Ad-t)y}\end{aligned}$$and $$\theta (z,\tau )=\sqrt{\frac{i}{\tau }}e^{-\pi iz^{2}/\tau }\theta (\pm \frac{z}{\tau },-\frac{1}{\tau }).$$Thus we have $$\begin{aligned} \label{eq:32} G_{st}(u,v) &\equiv &g_{st}(k,q,z_{1},z_{2}) \nonumber \\ &=&e^{2\pi ivt}\sum_{r}e^{2\pi irt/A}K_{s+r,r}(u,v) \nonumber \\ &=&\frac{AC_{1}}{l\sqrt{\pi }}\sqrt{\frac{A}{2}}\sum_{d=0,1}e^{-\frac{\pi }{% 2A}(Ad-t)^{2}+2\pi i(Ad-t)\frac{\nu }{A}-\frac{\pi }{2A}(s+\mu -\nu )^{2}} \nonumber \\ &&\times e^{\frac{\pi i}{A}(s+\mu -\nu )(Ad-t)+2\pi ivAd}\theta (u-\frac{1}{2% }(Ad-t)+\frac{i}{2}(s+\mu -\nu ),\frac{Ai}{2}) \nonumber \\ &&\times \theta (2Av+s+\mu +\nu -t\tau +iAd,2Ai).\end{aligned}$$Due to $$\begin{aligned} \sum_{a=0,1}\theta (x+\frac{a}{2},\tau ) &=&2\theta (2x,4\tau ), \\ \sum_{a=0,1}(-1)^{a}\theta (x+\frac{a}{2},\tau ) &=&2e^{2\pi i(x+\frac{\tau }{2})}\theta (2x+2\tau ,4\tau ),\end{aligned}$$when $d$ is equal to $0,1$, $$\theta (2x+2\tau d,4\tau )=\frac{1}{2}\sum_{a=0,1}(-1)^{ad}e^{2\pi i(x+\frac{% \tau }{2})d}\theta (x+\frac{a}{2},\tau ).$$The function $G_{st}(u,v)$ can be rewritten as $$\begin{aligned} G_{st}(u,v) &=&\frac{A}{4\pi }e^{2\pi i\phi }\sum_{a,d=0,1}(-1)^{ad}\theta (-u+\frac{1}{2}(Ad+t)+\frac{i}{2}(s+\mu -\nu ),\frac{Ai}{2}) \nonumber \label{eq:13} \\ &&\times \theta (-Av+\frac{1}{2}(s+a-\mu -\nu )+\frac{i}{2}t,\frac{Ai}{2}),\end{aligned}$$where $$\phi =\frac{i}{4A}(s^{2}+t^{2})-\frac{st}{2A}+\frac{i}{2A}s(\mu -\nu )-\frac{% t}{2A}(\mu +\nu )+\frac{i}{4A}(|\mu |^{2}+|\nu |^{2}-2\mu \nu )$$From the above discussion, we know that when $A$ is an even number, only $% a=0 $ contributes, so we have $$G_{st}(u,v)=\frac{A}{2\pi }e^{2\pi i\phi}\theta (-u+\frac{1}{2}t+\frac{i% }{2}(s+\mu -\nu ),\frac{Ai}{2})\theta (-Av+\frac{1}{2}(s-\mu -\nu )+\frac{i}{2}t,\frac{Ai% }{2})$$and when $A$ is an odd number, $$\begin{aligned} G_{st}(u,v) &=&\frac{A}{4\pi }e^{2\pi i\phi }\sum_{a,d=0,1}(-1)^{ad}\theta (-u+\frac{1}{2}(Ad+t)+\frac{i}{2}(s+\mu -\nu ),\frac{Ai}{2}) \nonumber \\ &&\times \theta (-Av+\frac{1}{2}(Aa+s-\mu -\nu )+\frac{i}{2}t,\frac{Ai}{2}).\end{aligned}$$They can be uniformly written as $$\begin{aligned} G_{st}(u,v) &=&\frac{A}{4\pi }e^{2\pi i\phi }\sum_{a,d=0,1}(-1)^{ad}\theta (-u+\frac{1}{2}(Ad+t)+\frac{i}{2}(s+\mu -\nu ),\frac{Ai}{2}) \nonumber \\ &&\times \theta (-Av+\frac{1}{2}(Aa+s-\mu -\nu )+\frac{i}{2}t,\frac{Ai}{2}),\end{aligned}$$ which is $Z_{4}$ covariant. So we have from (\[eq:30\])(\[eq:31\]) and (\[eq:32\]), $$\psi _{st}(k,q)=\frac{1}{A}\{\frac{\int d^{2}\mu d^{2}\nu G_{st}(u,v)f_{1}(\mu )f_{2}(\nu )}{\int d^{2}\mu d^{2}\nu G_{00}(u,v)f_{1}(\mu )f_{2}(\nu )}\} \label{eq:33}$$where functions $f_{i}$ should satisfy$$f_{i}(\omega _{N}\xi )=e^{i\alpha _{i}}f_{i}(\xi ).$$ Let $\hat{u}=\frac{l}{2\pi }\hat{y}_{2}$ and $A\hat{v}=\frac{l}{2\pi }\hat{y}% _{1}$, we may replace $u,v$ by $\hat{u}$ and $\hat{v}$ in (\[eq:33\]) and get $$\Psi _{st}(u_{1}^{A},u_{2}^{A})=\frac{1}{A}\frac{\int d^{2}\mu d^{2}\nu G_{st}(\frac{l}{2\pi }\hat{y}_{2},\frac{l}{2A\pi }\hat{y}_{1})f_{1}(\mu )f_{2}(\nu )}{\int d^{2}\mu d^{2}\nu G_{00}(\frac{l}{2\pi }\hat{y}_{2},\frac{% l}{2A\pi }\hat{y}_{1})f_{1}(\mu )f_{2}(\nu )}.$$The operators $u_{1}$ and $u_{2}$ commute with the operators $u_{1}^{A}$ and $u_{2}^{A}$,and from (\[eq:40\]) $$\begin{aligned} u_{1}^{s} &=&e^{-2\pi i\frac{s}{A}\hat{u}}, \\ u_{2}^{t} &=&e^{-2\pi it\hat{v}}.\end{aligned}$$Further takeing $u_{1}^{s}$ and $u_{2}^{t}$ into account, we can insert them to the corresponding operator form of Eq.(\[eq:13\]). This leads to the function of $\hat{u}$ and $\hat{v}$ $$\begin{aligned} h_{ad} &\equiv &u_{1}^{s}u_{2}^{t}G_{st}(\hat{u},\hat{v})\nonumber\\ &=&\frac{A}{4\pi}\sum_{a,d=0}^{1}(-1)^{ad}e^{2\pi i\phi}e^{-2\pi i\frac{s}{A}\hat{u}}\theta (-% \hat{u}+\frac{1}{2}(Ad+t)+\frac{i}{2}(s+\mu -\nu ),\frac{Ai}{2}) \nonumber \\ &&\times e^{-2\pi i\hat{v}t}\theta (-A\hat{v}+\frac{1}{2}(Aa+s-\mu -\nu )+\frac{i}{% 2}t,\frac{Ai}{2}) \nonumber \\ &=&\frac{A}{4\pi}\sum_{a,d=0}^{1}(-1)^{ad}e^{2\pi i(-\frac{3st}{2A}+\frac{i}{4A}(|\mu |^{2}+|\nu |^{2}-2\mu \nu ))}e^{-2\pi i(\frac{sd}{2}+\frac{ta}{2})} \nonumber \\ &&\times \theta \left[ \begin{array}{l} \frac{s}{A} \\ \frac{t}{2}% \end{array}% \right] (-\hat{u}+\frac{1}{2}Ad+\frac{i}{2}(\mu -\nu ),\frac{Ai}{2}) \nonumber \\ &&\times \theta \left[ \begin{array}{l} \frac{t}{A} \\ \frac{s}{2}% \end{array}% \right] (-A\hat{v}+\frac{1}{2}Aa-\frac{1}{2}(\mu +\nu ),\frac{Ai}{2}).\end{aligned}$$Due to $|\omega _{N}\mu |=|\mu |,|\omega _{N}\nu |=|\nu |$, $e^{2\pi i\frac{i}{4A}(|\mu |^{2}+|\nu |^{2})}$ in the above formula can be attributed to $f_{1}(\mu )$ and $f_{2}(\nu )$. Finally, we have $$\begin{aligned} P &=&\sum_{s,t}u_{1}^{s}u_{2}^{t}\Psi _{st}(u_{1}^{A},u_{2}^{A}) \nonumber \\ &=&\sum_{s,t}e^{2\pi i(-\frac{3st}{2A})}\sum_{a,d=0}^{1}(-1)^{ad}\int d^{2}\mu d^{2}\nu f_{1}(\mu )f_{2}(\nu )e^{\pi i\frac{\mu \nu }{A}}e^{-2\pi i(\frac{sd% }{2}+\frac{ta}{2})} \nonumber \\ &&\times \theta \left[ \begin{array}{l} \frac{s}{A} \\ \frac{t}{2}% \end{array}% \right] (-\frac{l\hat{y}_{2}}{2\pi }+\frac{1}{2}Ad+\frac{i}{2}(\mu -\nu ),% \frac{Ai}{2})\theta \left[ \begin{array}{l} \frac{t}{A} \\ \frac{s}{2}% \end{array}% \right] (-\frac{l\hat{y}_{1}}{2\pi }+\frac{1}{2}Aa-\frac{1}{2}(\mu +\nu ),% \frac{Ai}{2}) \nonumber \\ &&\times \{A\sum_{a,d=0}^{1}(-1)^{ad}\int d^{2}\mu d^{2}\nu f_{1}(\mu )f_{2}(\nu )e^{\pi i\frac{\mu \nu }{A}}\theta (-\frac{l\hat{y}_{2}}{2\pi }+\frac{1}{2}% Ad+\frac{i}{2}(\mu -\nu ),\frac{Ai}{2}) \nonumber \\ &&\times \theta (-\frac{l\hat{y}_{1}}{2\pi }+\frac{1}{2}Aa-\frac{1}{2}(\mu +\nu ),\frac{Ai}{2})\}^{-1}.\end{aligned}$$In the above equation, the two $\theta $ functions can not exchange orders with each other. It holds for any integer number $A$. In the following, we present some discussion.(1) $A$ is an even number, so $\frac{Ad}{2}$ and $\frac{Aa}{2}$ are integers too. Due to (\[eq:34\]) we have$$\begin{aligned} P &=&\sum_{s,t}e^{-3\pi i\frac{st}{A}}\int d^{2}\mu d^{2}\nu f_{1}(\mu )f_{2}(\nu )e^{\pi i\frac{\mu \nu }{A}} \nonumber \\ &&\times \theta \left[ \begin{array}{l} \frac{s}{A} \\ \frac{t}{2}% \end{array}% \right] (-\frac{l\hat{y}_{2}}{2\pi }+\frac{i}{2}(\mu -\nu ),\frac{Ai}{2}% )\theta \left[ \begin{array}{l} \frac{t}{A} \\ \frac{s}{2}% \end{array}% \right] (-\frac{l\hat{y}_{1}}{2\pi }-\frac{1}{2}(\mu +\nu ),\frac{Ai}{2}) \nonumber \\ &&\times \{A\int d^{2}\mu d^{2}\nu f_{1}(\mu )f_{2}(\nu )e^{\pi i\frac{\mu \nu }{A}}\theta (-\frac{l\hat{y}_{2}}{2\pi }+\frac{i}{2}(\mu -\nu ),\frac{Ai% }{2}) \nonumber \\ &&\times \theta (-\frac{l\hat{y}_{1}}{2\pi }-\frac{1}{2}(\mu +\nu ),\frac{Ai% }{2})\}^{-1}.\end{aligned}$$the above equation is the generalization of the Boca’s formula Proposition $% 3.1(i)$ [@16].(2) $A$ is an odd number $$\begin{aligned} P &=&\sum_{s,t=0}^{A-1}e^{-3\pi i\frac{st}{A}}\int d^{2}\mu d^{2}\nu f_{1}(\mu )f_{2}(\nu )e^{\pi i\frac{\mu \nu }{A}} \nonumber \\ &&\times \sum_{a,d=0}^{1}(-1)^{ad}e^{-2\pi i(\frac{sd}{2}+\frac{ta}{2})}\theta % \left[ \begin{array}{l} \frac{s}{A} \\ \frac{t}{2}% \end{array}% \right] (-\frac{l\hat{y}_{2}}{2\pi }+\frac{Ad}{2}+\frac{i}{2}(\mu -\nu ),% \frac{Ai}{2})\nonumber\\ &&\times\theta \left[ \begin{array}{l} \frac{t}{A} \\ \frac{s}{2}% \end{array}% \right] (-\frac{l\hat{y}_{1}}{2\pi }+\frac{Aa}{2}-\frac{1}{2}(\mu +\nu ),% \frac{Ai}{2}) \nonumber \\ &&\times \{A\int d^{2}\mu d^{2}\nu f_{1}(\mu )f_{2}(\nu )e^{\pi i\frac{\mu \nu }{A}}\sum_{a,d=0}^{1}(-1)^{ad}\theta (-\frac{l\hat{y}_{2}}{2\pi }+\frac{Ad}{2}+% \frac{i}{2}(\mu -\nu ),\frac{Ai}{2}) \nonumber \\ &&\times \theta (-\frac{l\hat{y}_{1}}{2\pi }+\frac{Aa}{2}-\frac{1}{2}(\mu +\nu ),\frac{Ai}{2})\}^{-1}.\end{aligned}$$Due to $$\begin{aligned} &&\theta \left[ \begin{array}{l} \frac{s}{A} \\ \frac{t}{2}% \end{array}% \right] (x+\frac{Ad}{2},\tau ) \nonumber \\ &=&\theta \left[ \begin{array}{l} \frac{s}{2A} \\ 0% \end{array}% \right] (2x,4\tau )(-1)^{sd}e^{2\pi i\frac{st}{2A}}+\theta \left[ \begin{array}{l} \frac{s+A}{2A} \\ 0% \end{array}% \right] (2x,4\tau )(-1)^{(s+A)d}e^{2\pi i\frac{(s+A)t}{2A}}\end{aligned}$$the numerator of $P$ can be written as $$\begin{aligned} &&2\int d^{2}\mu d^{2}\nu f_{1}(\mu )f_{2}(\nu )e^{\pi i\frac{\mu \nu }{A}% }\sum_{s,t=0}^{A-1}\{\bar{\theta}_{0}\theta _{0}++\bar{\theta}_{1}\theta _{0}(-1)^{s}+\bar{\theta}_{0}\theta _{1}(-1)^{t}+\bar{\theta}_{1}\theta _{1}(-1)^{s+t-1}\} \nonumber \\ &=&2\int d^{2}\mu d^{2}\nu f_{1}(\mu )f_{2}(\nu )e^{\pi i\frac{\mu \nu }{A} }\sum_{s,t=0}^{2A-1}e^{\frac{\pi i}{A}st}\bar{\theta}_{0}\theta _{0}\end{aligned}$$Where $\bar{\theta}_{\delta }=\theta \left[ \begin{array}{c} \frac{t+A\delta }{2A} \\ 0% \end{array}% \right] (2y,4\tau ),\theta _{\delta }=\theta \left[ \begin{array}{c} \frac{t+A\delta }{2A} \\ 0% \end{array}% \right] (2x,4\tau ),\delta =0,1$ with $x=-\frac{l\hat{y}_{2}}{2\pi }+\frac{i% }{2}(\mu -\nu )$ and $y=-\frac{l\hat{y}_{1}}{2\pi }-\frac{1}{2}(\mu +\nu )$. The denominator of $P$ is $2A\int d^{2}\mu d^{2}\nu f_{1}(\mu )f_{2}(\nu )e^{\pi i\frac{\mu \nu }{A} }\sum_{\delta_1,\delta_2 =0}^{1}e^{\pi i\delta_1\delta_2}\bar{\theta}_{\delta_1}^{'}\theta _{\delta_2}^{'}$, here $\theta^{\prime} _{\delta }=\theta \left[ \begin{array}{c} \frac{\delta }{2} \\ 0% \end{array}% \right] (2x,4\tau )$, $\bar{\theta}^{\prime} _{\delta }=\theta \left[ \begin{array}{c} \frac{\delta }{2} \\ 0% \end{array}% \right] (2y,4\tau )$. This formula gives another result compared with the Boca’s when $f_{1}(z)=f_{2}(z)=\delta ^{2}(z-0).$ Finally, we will give another explicit form of $P$ in terms of the derivative of elliptic functions. Note that the basis $\{|n>\}$ of Fock space produce a phase $\omega ^{n}$ under action of $R_{N}$. It is not difficult to find that $e^{i\alpha _{1}}$ and $e^{i\alpha _{2}}$ in ([eq:50]{}) are both integral powers of $\omega _{N}$ because of $% (R_{N})^{N}=identity.$ Therefore $|\phi _{1}>$ and $<\phi _{2}|$ can respectively be expanded in the basis $\{|n>\}$ and $\{<n|\}$, where$$|n>=\frac{(a^{+})^{n}}{\sqrt{n!}}|0>.$$We have the relation between coherent state and particle number eigenstate as following:$$<z|n>=e^{-\frac{1}{2}z\bar{z}}\frac{z^{n}}{\sqrt{n!}}. \label{eq:43}$$ Obviously, the general forms of $|\phi _{1}>$ and $<\phi _{2}|$ in the expansion in terms of particle number eigenstates are $\sum_{m=0}^{\infty }c_{m}$ $|i+4m>$ and $\sum_{n=0}^{\infty }d_{n}$ $<j+4n|,$ where $i$ and $j$ are nonnegative integers and $c_{m}$ and $d_{n}$ are arbitrary constant coefficients. So$$\begin{aligned} |\phi _{1} >&=&\sum_{m}c_{m}|i+4n> \nonumber \\ &=&\frac{1}{\pi }\sum_{m}\int_{-\infty }^{\infty }dxdyc_{m}|z><z|i+4m>, \label{eq:44}\end{aligned}$$$$\begin{aligned} <\phi _{2}|&=&\sum_{m=0}^{\infty }d_{m}<j+4m| \nonumber \\ &=&\frac{1}{\pi }\sum_{n=0}^{\infty }\int_{-\infty }^{\infty }dxdyd_{n}<j+4n|z><z|. \label{eq:45}\end{aligned}$$We let $R_{N}$ act on $|\phi _{1}>$ and $<\phi _{2}|$ and get $$R_{N}|\phi _{1}>=\omega ^{i}|\phi _{1}>,$$ $$<\phi _{2}|R_{N}^{-1}=<\phi _{2}|\omega ^{-j}.$$ Subsequently, we substitute (\[eq:43\])(\[eq:44\])(\[eq:45\]) into (\[eq:42\]) and make use of the formula $$<k,q|n>=\frac{1}{\sqrt{n!}}\frac{d^{n}}{dz^{n}}\left( e^{\frac{1}{2}z\bar{z}% }<k,q|z>\right) \Vert _{z=0}$$ to obtain$$\begin{aligned} F_{st}(k,q_{0}) &=&\sum_{m,n}\sum_{h=0}^{A-1}c_{m}d_{n}<k,q_{0}+\frac{l(h+s)% }{A}|i+4m>\times <j+4n|k,q_{0}+\frac{lh}{A}>\times e^{2\pi i(q_{0}/l+h/A)t} \nonumber\\ &=&\sum_{m,n}\sum_{h=0}^{A-1}c_{m}d_{n}\frac{1}{\sqrt{(i+4m)!(j+4n)!}}\frac{% d^{n+m}}{dz_{1}^{m}\bar{z_{2}}^{n}}(e^{\frac{1}{2}(z_{1}\bar{z}_{1}+z_{2}\bar{z}% _{2})}\sum_{h=0}^{A-1}<k,q_{0}+\frac{l(h+s)}{A}|z_{1}> \nonumber\\ &&\times<z_{2}|k,q_{0}+\frac{lh}{A}>\times e^{2\pi i(q_{0}/l+h/A)t})\Vert _{z_{1}=\bar{z}_{2}=0} \nonumber\\ &=&\sum_{m,n}\sum_{h=0}^{A-1}c_{m}d_{n}\frac{1}{\sqrt{(i+4m)!(j+4n)!}}\frac{% d^{n+m}}{dz_{1}^{m}\bar{z_{2}}^{n}}\left( e^{\frac{1}{2}(z_{1}\bar{z}_{1}+z_{2}% \bar{z}_{2})}g_{st}(k,q_{0},z_{1},z_{2})\right) \Vert _{z_{1}=\bar{z}% _{2}=0}.\nonumber\\\end{aligned}$$So, we get the projector in the case of $z_{4}$ $$\begin{aligned} P &=&\sum_{m,n}c_{m}d_{n}\frac{1}{\sqrt{(i+4m)!(j+4n)!}}\frac{d^{n+m}}{dz_{1}^{m}% \bar{z}_{2}^{n}}(e^{z_{1}\bar{z}_{1}+z_{2}\bar{z}_{2}}\times e^{2\pi i(-% \frac{3st}{2A})}\times e^{4\pi ^{2}iz_{1}\bar{z}_{2}}\times e^{-2\pi i(% \frac{sd}{2}+\frac{ta}{2})} \nonumber\\ &&\times \sum_{a,d=0}^{1}(-1)^{ad}\theta \left[ \begin{array}{l} \frac{s}{A} \nonumber\\ \frac{t}{2}% \end{array}% \right] (-\frac{l\hat{y}_{2}}{2\pi }+\frac{1}{2}Ad+\frac{\sqrt{2}A}{2l}% (z_{1}+\bar{z}_{2}),\frac{Ai}{2}) \nonumber\\ &&\times \theta \left[ \begin{array}{l} \frac{t}{A} \nonumber\\ \frac{s}{2}% \end{array}% \right] (-\frac{l\hat{y}_{1}}{2\pi }+\frac{1}{2}Aa+\frac{i\sqrt{2}A}{2l}% (z_{1}-\bar{z}_{2}),\frac{Ai}{2}))\Vert _{z_{1}=\bar{z}_{2}=0} \nonumber\\ &&\times \{A\sum_{m,n}\sum_{a,d=0}^{1}(-1)^{ad}c_{m}d_{n}\frac{d^{n+m}}{% dz_{1}^{m}\bar{z}_{2}^{n}}(e^{z_{1}\bar{z}_{1}+z_{2}\bar{z}_{2}}\times e^{4\pi ^{2}iz_{1}\bar{z}_{2}}\times \theta (-\frac{l\hat{y}_{2}}{2\pi }+\frac{1% }{2}Ad+\frac{\sqrt{2}A}{2l}(z_{1}+\bar{z}_{2}),\frac{Ai}{2}) \nonumber\\ &&\times \theta (-\frac{l\hat{y}_{1}}{2\pi }+\frac{1}{2}Aa+\frac{i\sqrt{2}A}{% 2l}(z_{1}-\bar{z}_{2}),\frac{Ai}{2}))\}^{-1}\Vert _{z_{1}=\bar{z}% _{2}=0}.\end{aligned}$$ Thus, We derive two forms of explicit expressions of the projector $P$ in terms of the integration and derivative of the classical theta functions. Discussion ========== In this paper, $P$ is represented by a form of fraction which make sense only when the denominator has inverse. The formula demands: $$D=A\int d^{2}\mu d^{2}\nu f_{1}(\mu )f_{2}(\nu )G_{00}(u,v)$$is unequal to zero for any real variables $u$ and $v$. It is easy to prove that when $f_{1}$ is equal to $f_{2}^{\ast }$, the related denominator $$D=A\sum_{n}<k,q+\frac{ln}{A}|\phi ><\phi |k,q+\frac{ln}{A}>=A\sum_{n}|<k,q+% \frac{ln}{A}|\phi >|^{2}.$$Thus if $D=0$,then $$<k,q+\frac{ln}{A}|\phi >=0~~~~~~~~n=0,1,\cdot \cdot \cdot ,A-1. \label{eq:11}$$The zero points of the state vector $|\phi >$ in $|k,q>$ representation should be points equally spaced along $q$ with interval of $% \frac{l}{A}$. The mapping from $k$ and $q$ to $<k,q+\frac{ln}{A}|\phi >\in C$ is a mapping from plane to plane. In general, $<k,q+\frac{ln}{A}|\phi >=0$ are some discrete points, and thus it is casual that $D$ is equal to zero. So in this sense, for most of $f_{1}=f_{2}^{\ast }$, this still not happen (in some sense, the measure of $D=0$ event is zero.) Specially, when the state $|\phi _{1}>=|\phi _{2}>=|0>$, It can be proved [@16] that $D$ is not equal to zero everywhere. Thus set $$|\phi _{1}>=|0>+\epsilon |\psi _{1}>,~~~~~~~<\phi _{2}|=<0|+\epsilon <\psi _{2}|$$$D$ is also not equal to zero everywhere for small enough $\epsilon $. But we don’t know the situation for general $f_{1}\neq f_{2}^{\ast }.$ [99]{} A. Connes, Non-commutative Geometry, Academic Press, 1994. G. Landi,“ An introduction to non-commutative space and their geometry”, hep-th/9701078; J. Varilly, “An introduction to non-commutative Geometry”, physics/9709045. J. Madore, “An introduction to non-commutative Differential Geometry and its physical Applications”, Cambridge University press 2nd edition, 1999. A. Connes, M. Douglas, A. Schwartz, Matrix theory compactification on Tori, JHEP 9802 (1998) 003, hep-th/9711162; M. dougals, C. 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Phys. 202 (1999) 325. H.Bacry, A.Grassman and J. Zak, Phys.Rev B12(1975) 1112. J. Zak, In solid State Physics, edited by H. Ehrenreich, F. Seitz and D. Turnbull (Academic,new York,1972), Vol. 27. [^1]: Email:byhou@phy.nwu.edu.cn [^2]: Email:yzy@phy.nwu.edu.cn [^3]: Email:yue@phy.nwu.edu.cn [^4]: It is necessary to point out that the matrix $A$ defined here is the transposed matrix of $A$ defined in Formula (93) in paper [@9]. [^5]: The condition $P^{\dag }=P$ isn’t satisfied by $P$ like this, which might represent the solitons in a ”complex” field.

--- abstract: | We present a detailed analysis of present and future Cosmic Microwave Background constraints of the value of the fine-structure constant, $\alpha$. We carry out a more detailed analysis of the WMAP first-year data, deriving state-of-the-art constraints on $\alpha$ and discussing various other issues, such as the possible hints for the running of the spectral index. We find, at $95 \%$ C.L. that $0.95 < \alpha_{\text{dec}} / \alpha_0 < 1.02$. Setting $dn_S /dlnk=0$, yields $0.94< \alpha_{\text{dec}} / \alpha_0 < 1.01$ as previously reported. We find that a lower value of $\alpha / \alpha_0$ makes a value of $d n_S /dlnk = 0$ more compatible with the data. We also perform a thorough Fisher Matrix Analysis (including both temperature and polarization, as well as $\alpha$ and the optical depth $\tau$), in order to estimate how future CMB experiments will be able to constrain $\alpha$ and other cosmological parameters. We find that Planck data alone can constrain $\tau$ with a accuracy of the order 4% and that this constraint can be as small as 1.7% for an ideal cosmic variance limited experiment. Constraints on $\alpha$ are of the order 0.3% for Planck and can in principle be as small as 0.1% using CMB data alone - tighter constraints will require further (non-CMB) priors. author: - 'G. Rocha' - 'R. Trotta' - 'C.J.A.P. Martins' - 'A. Melchiorri' - 'P. P. Avelino' - 'R. Bean' - 'P.T.P. Viana' bibliography: - 'paper.bib' title: | Measuring $\alpha$ in the Early Universe:\ CMB Polarization, Reionization and the Fisher Matrix Analysis --- \#1 Introduction ============ The recent release of the Wilkinson Microwave Anisotropy Probe (WMAP) first-year data [@Bennett; @Hinshaw; @Kogut; @Verde] has pushed cosmology into a new stage. On one hand, it has quantitatively validated the broad features of the ‘standard’ cosmological model—the optimistically called ‘concordance’ model. But at the same time, it has also pushed the borderline of research to new territory. We now know that ‘dark components’ make up the overwhelming majority of the energy budget of the universe. Most of this is almost certainly in some non-baryonic form, for which there is at present no direct evidence or solid theoretical explanation. One must therefore try to understand the nature of this dark energy, or at least (as a first step) look for clues of its origin. It is clear that such an effort must be firmly grounded within fundamental physics, and indeed that recent progress in fundamental physics may shed new light on this issue. On the other hand, this is not a one-way street. Cosmology and astrophysics are playing an increasingly more important role as fundamental physics testbeds, since they provide us with extreme conditions (that one has no hope of reproducing in terrestrial laboratories) in which to carry out a plethora of tests and search for new paradigms. Perhaps the more illuminating example is that of multidimensional cosmology. Currently preferred unification theories [@Polchinski; @Damour1] predict the existence of additional space-time dimensions, which will have a number of possibly observable consequences, including modifications in the gravitational laws on very large (or very small) scales [@Will] and space-time variations of the fundamental constants of nature [@Essay; @Uzan]. There have been a number of recent reports of evidence for a time variation of fundamental constants [@Webb; @Jenam; @Murphy; @Ivanchik], and apart from their obvious direct impact if confirmed they are also crucial in a different, indirect way. They provide us with an important (and possibly even unique) opportunity to test a number of fundamental physics models that might otherwise be untestable. A case in point is that of string theory [@Polchinski]. Indeed here the issue is not *if* such a theory predicts such variations, but *at what level* it does so, and hence if there is any hope of detecting them in the near future (or if we have done it already). Indeed, it has been argued [@Damour1; @Damour2]. that even the results of Webb and collaborators [@Webb; @Jenam; @Murphy] may be hard to explain in the simplest, best motivated models where the variation of alpha is driven by the spacetime variation of a very light scalar field. Playing devil’s advocate, one could certainly conceive that cosmological observations of this kind could one day prove string theory wrong. The most promising case, and the one that has been the subject of most recent work (and speculation), is that of the fine-structure constant $\alpha$, for which some fairly strong statistical evidence of time variation at redshifts $z\sim2-3$ already exists [@Webb; @Jenam; @Murphy], together with weaker (and somewhat more controversial) evidence from geophysical tests using the Oklo natural nuclear reactor [@Fujii]. Interesting and quite tight constraints can also be derived from local laboratory tests [@Marion], and indeed this is a context where improvements of several orders of magnitude can be expected in the coming years. On the other hand, the theoretical expectation in the simplest, best motivated model is that $\alpha$ should be a non-decreasing function of time [@Damour; @Santiago; @Barrow]. This is based on rather general and simple assumptions, in particular that the cosmological dynamics of the fine-structure constant is governed by a scalar filed whose behavior is akin to that of a dilaton. If this is so, then it is particularly important to try to constrain it at earlier epochs, where any variations relative to the present-day value should therefore be larger. In this regard, note that one of the interpretations of the Oklo results [@Fujii] is that $\alpha$ was *larger* at the Oklo epoch (effectively $z\sim0.1$) than today, whereas the quasar results [@Webb; @Jenam; @Murphy] indicate that $\alpha$ was smaller at $z\sim2-3$ than today. Both results are not necessarily incompatible, since they refer to two different cosmological epochs, and hence comparing them necessarily requires specifying not only a *background* cosmological model but also a model for the variation of the fine-structure constant with redshift, $\alpha=\alpha(z)$. However, if both results are validated by future experiments, then the above theoretical expectation must clearly be wrong (with clear implications for both the dilaton hypothesis and on a wider scale), which would be a perfect example of using astrophysics to learn about fundamental physics. Cosmic microwave background (CMB) anisotropies provide an ideal way of measuring the fine-structure constant at high redshift, being mostly sensitive to the epoch of decoupling, $z \sim 1100$ (one could also envisage searching for spatial variations at the last scattering surface [@Sigu]). Here we continue our ongoing work in this area [@Old; @Avelino; @Martins], and particularly extend our most recent analysis [@Martinsw] of the WMAP first-year data, providing updated constraints on the value of $\alpha$ at decoupling, studying some crucial degeneracies with other cosmological parameters and discussing what improvements can be expected with forthcoming datasets. We emphasize that in previous (pre-WMAP) work, CMB-based constraints on $\alpha$ were obtained with the help of additional cosmological datasets and priors. This has raised some eyebrows among skeptics, as different datasets could possibly have different systematic errors that are impossible to control and could conceivably conspire to produce the results we quoted (statistically consistency with the value of $\alpha$ at decoupling being the same as today’s, though with a slight preference towards smaller values). Here, by contrast, we will present results of an analysis of the WMAP dataset alone (we will only briefly discuss what happens when other datasets are added). We also discuss how these constraints can be improved in the future, especially when more precise CMB polarization data is available. In particular, we show that the existence of an early reionization epoch is a significant help in further constraining $\alpha$, and indeed the prospects for measuring $\alpha$ from the CMB are much better than if the optical depth $\tau$ was much smaller. Moreover, now that CMB polarization data is available, there are two approaches one can take. One is to treat CMB temperature and polarization as different datasets, and carry out independent analyses (and, more to the point, cosmological parameter estimations), to check if the results of the two are consistent. The other one is to combine the two datasets, thus getting smaller errors on the parameters. We will show that there are advantages to both approaches, and also that the combination of the two can often by itself break many of the cosmological degeneracies that plague this kind of analysis pipeline. On the other hand, we will also show that in ideal circumstances (*id est*, a cosmic variance limited experiment) CMB polarization is much better than CMB temperature in determining cosmological parameters. This result is not new, and it is of course somewhat obvious, but it has never been quantified in detail as will be done below. On the other hand, because cosmic variance limited experiments are expensive and experimentalists work with limited budgets, it is important to provide detailed forecasts for future experiments. We provide detailed forecalsts for the full (4-year) WMAP dataset, as well as for ESA’s Planck Surveyor (to be launched in 2007). It will be shown that Planck is almost cosmic variance limited (taken into account the range of multipoles covered by this instrument) when it comes to CMB temperature, but far from it for CMB polarization. Again this was previously known, but had not been quantified. This, and the intrinsic superiority of CMB polarization in measuring cosmological parameters, are therefore arguments for a post-Planck, polarization-dedicated experiment. CMB Temperature and Polarization ================================ Following [@zaldarriaga1; @kosowsky; @waynehu1; @waynehu2], one can describe the CMB anisotropy field as a 2x2, $I_{ij}$, intensity tensor which is a function of direction on the sky $\vec{n}$ and 2 other directions perpendicular to $\hat{\bi{n}}$ which define its components ${\hat{\bi{e_1}},\hat{\bi{e_2}}}$. The CMB radiation is expected to be polarised due to Thomson scattering of temperature anisotropies at the time when CMB photons last scattered. Polarised light is traditionally described via the Stokes parameters, $Q,U,V$, where $Q=(I_{11}-I_{22})/4$ and $U=I_{12}/2$, while the temperature anisotropy is given by $T=(I_{11}+I_{22})/4$ and $V$ can be ignored since it describes circular polarization which cannot be generated through Thomson scattering. Both Q and U depend of the choice of coordinate system in that they transform under a right handed rotation in the plane perpendicular to direction $\hat{\bi{n}}$ by an angle $\psi$ as: $$\begin{aligned} Q^{\prime}&=&Q\cos 2\psi + U\sin 2\psi \nonumber \\ U^{\prime}&=&-Q\sin 2\psi + U\cos 2\psi \,, \label{QUtrans} \end{aligned}$$ where ${\bf \hat{e_1}}^{\prime}=\cos \psi{\bf \hat{e_1}}+\sin\psi{\bf \hat{e_2}}$ and ${\hat{\bf e_2}}^{\prime}=-\sin \psi{\bf \hat{e_1}}+\cos\psi{\bf \hat{e_2}}$. In order to compute the rotationally invariant power spectrum a general method to analyse polarization over the whole sky is required. This is so because the calculation of the power spectrum involves the superposition of the different modes contributing to the perturbations. While it is simple to compute $Q$ and $U$ in the coordinate system where the wavevector defining the perturbation is aligned with the z axis, it is more complicated to do so when superimposing the different modes since one needs to rotate $Q$ and $U$ to a common coordinate frame before this superposition is done, and only in the small scale limit does this rotation have a simple expression [@uros]. Most of the literature on the polarization of the CMB uses three alternative representations based on either the Newman-Penrose spin-weight 2 harmonics [@zaldarriaga1], or a coordinate representation of the tensor spherical harmonics [@kamionkowski1; @kamionkowski2], or the coordinate-independent, projected symmetric trace free (PSTF) tensor valued multipoles [@challinor]. Here we follow the first by expanding the polarization in the sky in terms of spin-weighted harmonics which form a basis for tensor functions in the sky. One starts by defining two other quantities $(Q\pm iU)'$: $$(Q\pm iU)'(\hat{\bi{n}})=e^{\mp 2i\psi}(Q\pm iU)(\hat{\bi{n}}).$$ These quantities are then expanded in the appropriate spin-weighted basis: $$\begin{aligned} T(\hat{\bi{n}})&=&\sum_{lm} a_{T,lm} Y_{lm}(\hat{\bi{n}}) \nonumber \\ (Q+iU)(\hat{\bi{n}})&=&\sum_{lm} a_{2,lm}\;_2Y_{lm}(\hat{\bi{n}}) \nonumber \\ (Q-iU)(\hat{\bi{n}})&=&\sum_{lm} a_{-2,lm}\;_{-2}Y_{lm}(\hat{\bi{n}})\,, \label{Pexpansion}\end{aligned}$$ where $Y_{lm}$ are the spherical harmonics and ${}_2Y_{lm}$ are the so-called spin-2 spherical harmonics, which form a complete and orthonormal basis for spin-2 functions. A function $\;_sf(\theta,\phi)$ defined on the sphere has spin-s if under a right-handed rotation of ($\hat{{\bi e}}_1$,$\hat{{\bi e}}_2$) by an angle $\psi$ it transforms as $\;_s f^{\prime}(\theta,\phi)=e^{-is\psi}\;_sf(\theta,\phi)$. Here we are interested in the polarizatin of the CMB which is a quantity of spin $\pm 2$. $Q$ and $U$ are defined at a given direction $\hat{\bi{n}}$ with respect to the spherical coordinate system $(\hat{{\bf e}}_\theta, \hat{{\bf e}}_\phi)$. The expansion coefficients for the polarization variables satisfy $a_{-2,lm}^*=a_{2,l-m}$. For temperature the relation is $a_{T,lm}^*=a_{T,l-m}$, where $$\begin{aligned} a_{T,lm}&=&\int d\Omega\; Y_{lm}^{*}(\hat{\bi{n}}) T(\hat{\bi{n}}) \nonumber \\ a_{2,lm}&=&\int d\Omega \;_2Y_{lm}^{*}(\hat{\bi{n}}) (Q+iU)(\hat{\bi{n}}) \nonumber \\ \nonumber \\ a_{-2,lm}&=&\int d\Omega \;_{-2}Y_{lm}^{*}(\hat{\bi{n}}) (Q-iU)(\hat{\bi{n}}) \nonumber \\ \label{alm}\end{aligned}$$ Usually one considers the following linear combinations: $$\begin{aligned} a_{E,lm}=-(a_{2,lm}+a_{-2,lm})/2 \nonumber \\ a_{B,lm}=i(a_{2,lm}-a_{-2,lm})/2\,. \label{aeb}\end{aligned}$$ The following rotationally invariant quantities then define the power spectra $$\begin{aligned} C_{Tl}&=&{1\over 2l+1}\sum_m \langle a_{T,lm}^{*} a_{T,lm}\rangle \nonumber \\ C_{El}&=&{1\over 2l+1}\sum_m \langle a_{E,lm}^{*} a_{E,lm}\rangle \nonumber \\ C_{Bl}&=&{1\over 2l+1}\sum_m \langle a_{B,lm}^{*} a_{B,lm}\rangle \nonumber \\ C_{Cl}&=&{1\over 2l+1}\sum_m \langle a_{T,lm}^{*}a_{E,lm}\rangle \,, \label{Cls}\end{aligned}$$ in terms of which, $$\begin{aligned} \langle a_{T,l^\prime m^\prime}^{*} a_{T,lm}\rangle&=& C_{Tl} \delta_{l^\prime l} \delta_{m^\prime m} \nonumber \\ \langle a_{E,l^\prime m^\prime}^{*} a_{E,lm}\rangle&=& C_{El} \delta_{l^\prime l} \delta_{m^\prime m} \nonumber \\ \langle a_{B,l^\prime m^\prime}^{*} a_{B,lm}\rangle&=& C_{Bl} \delta_{l^\prime l} \delta_{m^\prime m} \nonumber \\ \langle a_{T,l^\prime m^\prime}^{*} a_{E,lm}\rangle&=& C_{Cl} \delta_{l^\prime l} \delta_{m^\prime m} \nonumber \\ \langle a_{B,l^\prime m^\prime}^{*} a_{E,lm}\rangle&=& \langle a_{B,l^\prime m^\prime}^{*} a_{T,lm}\rangle=0\,. \label{stat}\end{aligned}$$ In real space one describes the polarization field in terms of two quantities that are scalars under rotation, E and B modes, defined as: $$\begin{aligned} \tilde{E}(\hat{{\bi n}})=\sum_{lm}\left[{(l+2)! \over (l-2)!}\right]^{1/2} a_{E,lm}Y_{lm}(\hat{{\bi n}}) \nonumber \\ \tilde{B}(\hat{\bi n})=\sum_{lm}\left[{(l+2)! \over (l-2)!}\right]^{1/2} a_{B,lm}Y_{lm}(\hat{\bi n}) \,. \label{EBexpansions} \end{aligned}$$ These quantities are closely related to the rotationally invariant Laplacian of $Q$ and $U$. In multipole space the relation is as follows $$a_{(\tilde{E},\tilde{B}),lm}=\left[{(l+2)! \over (l-2)!}\right]^{1/2} a_{(E,B),lm}. \label{eblm}$$ While E remains unchanged under parity transformation, B changes its sign (similar to the behaviour of electric and magnetic fields). This decomposition is also useful because the B mode is a direct signature of the presence of a background of gravitational waves, since it cannot be produced by density fluctuations [@zaldarriaga1; @kamionkowski1], Many models of inflation predict a significant gravity wave background. These tensor fluctuations generated during inflation have their largest effects on large angular scales and add in quadrature to the fluctuations generated by scalar modes. Whilst recent WMAP results placed limits on the amplitude of these tensor modes one still lacks an experimental evidence for the presence of a stochastic background of gravitational waves. As mentioned above the detection of the pseudo-scalar field B would provide invaluable information about Inflation in that they reflect the presence of such a background. Therefore to fully characterize the CMB anisotropies only four power spectra are needed–those for T,E,B and the cross-correlation between T and E. (Given that B has the opposite parity of E and T their cross-correlations with B vanishes.) The first detection of polarization of the CMB was due to the DASI experiment [@dasi], and more recently the WMAP experiment [@Kogut] has measured the TE cross-correlation power spectrum. An important result from these is the existence of reionization at larger redshifts then expected from the Gunn-Petterson through, an issue that we will discuss at length below. The CMB, $\alpha$ and $\tau$ ============================ The reason why the CMB is a good probe of variations of the fine-structure constant is that these alter the ionisation history of the universe [@steen; @Kap; @Old; @vsl]. The dominant effect is a change in the redshift of recombination, due to a shift in the energy levels (and, in particular, the binding energy) of Hydrogen. The Thomson scattering cross-section is also changed for all particles, being proportional to $\alpha^2$. A smaller effect (which has so far been neglected) is expected to come from a change in the Helium abundance [@Trotta:2003xg]. Increasing $\alpha$ increases the redshift of last-scattering, which corresponds to a smaller sound horizon. Since the position of the first Doppler peak ($\ell_{peak}$) is inversely proportional to the sound horizon at last scattering, increasing $\alpha$ will produce a larger $\ell_{peak}$ [@Old]. This larger redshift of last scattering also has the additional effect of producing a larger early ISW effect, and hence a larger amplitude of the first Doppler peak [@steen; @Kap]. Finally, an increase in $\alpha$ decreases the high-$\ell$ diffusion damping (which is essentially due to the finite thickness of the last-scattering surface), and thus increases the power on very small scales. These effects have been implemented in a modified CMBFAST algorithm which allows a varying $\alpha$ parameter [@Old; @Avelino]. These follow the extensive description given in [@steen; @Kap], with one important exception that will be discussed below. ![\[figcells\] Contrasting the effects of varying $\alpha$ (right)and reionization (left) on the CMB temperature (top) and polarization (bottom). Here $\zeta={\alpha}_{dec}/{\alpha}_0$. See the text for further details.](figureI.eps){width="3in"} Fig. \[figcells\] illustrates the effect of $\alpha$ and $\tau$ on the CMB temperature and polarization power spectra. The CMB power spectrum is, to a good approximation, insensitive to [*how*]{} $\alpha$ varies from last scattering to today. Given the existing observational constraints, one can therefore calculate the effect of a varying $\alpha$ in both the temperature and polarization power spectra by simply assuming two values for $\alpha$, one at low redshift (effectively today’s value, since any variation of the magnitude of [@Webb] would have no noticeable effect) and one around the epoch of decoupling, which may be different from today’s value. (In earlier works [@steen; @Kap; @Old; @Battye] one assumed a constant value of $\alpha$ throughout, *id est* the values at reionization and the present day were always the same.) For the CMB temperature, reionization simply changes the amplitude of the acoustic peaks, without affecting their position and spacing (top left panel); a different value of $\alpha$ at the last scattering, on the other hand, changes both the amplitude and the position of the peaks (top right panel). ![\[figpeaks\] The separation in $\ell$ between the reionization bump and the first (solid lines), second (dashed) and third (dotted) peaks in the polarization spectrum, as a function of $\alpha$ at decoupling and $\tau$. A (somewhat idealized) description of how $\alpha$ and $\tau$ can be measured using CMB polarization. ](figure3.eps){width="3.5in"} The outstanding effect of reionization is to introduce a bump in the polarization spectrum at large angular scales (lower left panel). This bump is produced well after decoupling (at much lower redshifts), when $\alpha$, if varying, is much closer to the present day’s value. If the value of $\alpha$ at low redshift is different from that at decoupling, the peaks in the polarization power spectrum at small angular scales will be shifted sideways, while the reionization bump on large angular scales won’t (lower right panel). It follows that by measuring the separation between the normal peaks and the bump, one can measure both $\alpha$ and $\tau$, as illustrated in Fig. \[figpeaks\]. Thus we expect that the existence of an early reionization epoch will, when more accurate cosmic microwave background polarization data is available, lead to considerably tighter constraints on $\alpha$. A possible concern with the interpretation of our results is related to the implicit assumption of a sharp transition on the value of $\alpha$ happening sometime between recombination and the epoch of reionization. Hence, it is crucial to understand if this is a valid approximation. Appart from the value of $\alpha$ at the time of recombination the knowledge of its value at two other epochs is relevant as far as the CMB anisotropies are concerned. One such epoch is the period *just before* recombination which is very important for the damping of CMB anisotropies on small angular scales. The other period is the epoch of reionization. In this work we effectively assume that $\alpha$ is equal to $\alpha_{rec}$ before recombination and to $\alpha_0$ at the reionization epoch. A value of $\alpha$ different from $\alpha_0$ at the epoch of reionization will affect the CMB anisotropies through a change in the optical depth $\tau$, *once a single cosmological model is assumed*. However, it is also well known that $\tau$ is itself dependent on the cosmological model through its cosmological parameters ($\Omega_m$ and $\Omega_\Lambda$ for example) as well as on the cosmological density perturbations (in our case through the initial power spectrum) [@pedrolidle] . The exact dependence is difficult to determine since there are several astrophysical uncertainties related to a number of relevant non-linear physical processes which affect the accuracy of reionization models. In general, this problem is solved by treating $\tau$ as a free parameter (independent of the other cosmological parameters and initial power spectrum), which accounts for the relatively poor knowledge of the dependence of $\tau$ on the cosmological model and in our case on the uncertainty about the exact value of $\alpha$ during the reionization epoch. Hence, we find that provided we treat $\tau$ as a free parameter the lack of a precise knowledge of value of $\alpha$ during the epoch of reionization will not affect our results. In the present work, we assume that the universe was completely reionized in a relatively small redshift interval (sudden reionization). A more refined modelling of the reionization history is not yet required by WMAP data, but will be necessary at noise levels appropriate for Planck and beyond [@Bruscoli02; @Hu03; @K03; @Holder03]. On the more practical side, there are of course observational constraints on the value of $\alpha$ at redshifts of a few [@Webb; @Jenam; @Murphy], indicating that at that epoch the possible changes relative to the present day are already very small (and would not be detectable, on their own, through the CMB due to cosmic variance). The knowledge of the value of $\alpha$ before recombination is also crucial for the details of the damping of small scale CMB anisotropies. Let’s assume that the variation of $\alpha$ around the time of recombination is given by some functional, $f$: $$\frac{\alpha}{\alpha_{rec}}=f\left(\frac{1+z}{1+z_{rec}}\right)$$ One can determine the dependence of the Silk damping scale [@kolbturner] $$R_S=\left(\int_{0}^{t_{dec}(\alpha)} dt \frac{\lambda_{\gamma}(\alpha)}{R^{2}(t)} \right)^\frac{1}{2}$$ (where, $\lambda_{\gamma}$, is the photon mean free path) on this functional $f$ and determine $\alpha_{eff}$ (relevant for the damping of CMB anisotropies) as the constant value of $\alpha$ that gives the same Silk damping scale as the variable one. Even though we did not treat $\alpha_{eff}$ as another parameter in the present investigation (this will be done in future work) we expect that our constraints on $\alpha_{rec}$ should also be valid (to a good approximation) for $\alpha_{eff}$. This means that we are already able to constrain a combination of both $\alpha$ and $f$ at the time of recombination. Also, we see that we may be able to rule out particular models for the time variation of $\alpha$ on the basis of the details of such variation, even if the value of $\alpha$ at the time of recombination is not ruled out by our analysis. Finally, we must emphasize that the effects discussed above are direct effects of an $\alpha$ variation, and that indirect effects are usually present as well since any variation of $\alpha$ is necessarily coupled with the dynamics of the Universe [@Mota]. In this paper we take a pragmatic approach and say that, since the CMB is quite insensitive to the details of $\alpha$ variations from decoupling to the present day, *we do not in fact need to specify a redshift dependence for this variation*—although we could have specified one if we so chose. The price to pay would be that, since this coupling is very dependent on the particular model we consider we would end up with very model-dependent constraints. Therefore, at this stage, and given the lack of detailed and well-motivated cosmological models for $\alpha$ variations we prefer to focus on model-independent constraints, and hence do not attempt to include this extra degree of freedom in our analysis. Nevertheless, given some model-independent constraints one can always translate them into constraints on the parameters of one’s favourite model. In fact we expect that some models will be ruled out on the basis of the indirect effect of a variation of $\alpha$ on the dynamics of the Universe rather than the direct effects we described above. This is actually a simpler case in which only the modifications to the background evolution ($a(t)$) would need to be taken into account in order to test the model, with the direct effects of a varying $\alpha$ being negligible. We conclude this section by emphasising that although a more detailed analysis taking into account the expected variation of $\alpha$ with time (and its direct and indirect implications for CMB anisotropies) for specific models is certainly possible, our more general work can easily be used to impose very strong constraints to more complex varying $\alpha$ theories once the relevant variables are computed. Up-to-date CMB constraints on $\alpha$ with WMAP ================================================ We compare the recent WMAP temperature and cross-polarization dataset with a set of flat cosmological models adopting the likelihood estimator method described in [@Verde]. We restrict the analysis to flat universes. The models are computed through a modified version of the CMBFAST code with parameters sampled as follows: physical density in cold dark matter $0.05 < \Omega_ch^2 < 0.20$ (step $0.01$), physical density in baryons $0.010 < \Omega_bh^2 < 0.028$ (step $0.001$), $0.500 < \Omega_{\Lambda} < 0.950$ (step $0.025$), 0$0.900 < \alpha_{\text{dec}} / \alpha_0 <1.050$ (step $0.005$). Here $h$ is the Hubble parameter today, $H_0 \equiv 100h$ km s$^{-1}$ Mpc$^{-1}$ (determined by the flatness condition once the above parameters are fixed), while $\alpha_{\text{dec}}$ ($\alpha_0$) is the value of the fine structure constant at decoupling (today). We also vary the optical depth $\tau$ in the range $0.06-0.30$ (step $0.02$), the scalar spectral index of primordial fluctuations $0.880 < n_s < 1.08$ (step $0.005$) and its running $-0.15 < dn_s/dlnk < 0.05$ (step $0.01$) both evaluated at $k_0=0.002 Mpc^{-1}$ . We don’t consider gravity waves or iso-curvature modes since these further modifications are not required by the WMAP data (see e.g. [@Spergel]). A different model for the dark energy from a cosmological constant could also change our results, but again, is not suggested by the WMAP data (see e.g. [@mmot]). An extra background of relativistic particles is also well constrained by Big Bang Nucleosynthesis (see e.g. [@bhm]) and it will not be considered here. ![\[figalpha\] Likelihood distribution function for variations in the fine structure constant obtained by an analysis of the WMAP data (TT+TE, one-year).](alphanew.eps){width="3in"} ![\[figalphavstau\] $2-$D Likelihood contour plot in the $\alpha / \alpha_0$ vs $\tau$ plane for $2$ analysis: $<TT>$ only and $<TT>$+$<TE>$. As we can see, the inclusion of polarization data, breaks the degeneracy between these $2$ parameters.](alphavstau_p.eps){width="3in"} ![\[figalphavsdn\] $2-$D Likelihood contour plot in the $\alpha / \alpha_0$ vs $dn_S / dlnk$ plane ($<TT>$+$<TE>$ one year). A zero scale dependence, as expected in most of the inflationary models, is more consistent with a value of $\alpha / \alpha_0 < 1$](alphavsdn_p.eps){width="3in"} The likelihood distribution function for $\alpha_{\text{dec}} / \alpha_0$, obtained after marginalization over the remaining parameters, is plotted in Figure \[figalpha\]. We found, at $95 \%$ C.L. that $0.95 < \alpha_{\text{dec}} / \alpha_0 < 1.02$, improving previous bounds, (see [@Martins]) based on CMB and complementary datasets. Setting $dn_S /dlnk=0$, yields $0.94< \alpha_{\text{dec}} / \alpha_0 < 1.01$ as already reported in (see [@Martinsw]). It is interesting to consider the correlations between a $\alpha / \alpha_0$ and the other parameters in order to see how this modification to the standard model can change our conclusions about cosmology. In Figure \[figalphavstau\] we plot the $2-D$ likelihood contours in the $\alpha / \alpha_0$ vs the optical depth $\tau$ for $2$ different analysis: using the temperature only WMAP data and including the $<TE>$ cross spectrum temperature-polarization data. As we can see, there is a clear degeneracy between these $2$ parameters if one consider just the $<TT>$ spectrum: increasing the optical depth, allows for an higher value of the spectral index $n_S$ and a lower value of $\alpha / \alpha_0$ (again, see [@Martins]). As we can see from Figure \[figalphavstau\], the inclusion of the $<TE>$ data, is already able to partially break the degeneracy between $\tau$ and $\alpha / \alpha_0$. However, as we explain below, more detailed measurements of the polarization spectra are needed to fully break this degeneracy. One of the most unexpected results from the WMAP data is the hint for a scale-dependence of the spectral index $n_S$ (see e.g. [@peiris], [@kkmr]). Such dependence is not predicted to be detectable in most of the viable single field inflationary model and, if confirmed, will therefore have strong consequences on the possibilities of reconstructing the inflationary potential. In Figure \[figalphavsdn\] we plot a $2-D$ likelihood contour in the $\alpha / \alpha_0$ vs $d n_S /dlnk$ plane. As we can see, a lower value of $\alpha / \alpha_0$ makes a value of $d n_S /dlnk \sim 0$ more compatible with the data. As already noticed in [@bms], a modification of the recombination scheme can therefore provide a possible explanation for the high value of $dn_S /dlnk$ compatible with the WMAP data. Fisher Matrix Analysis Setup ============================ In our previous work [@Martins], a Fisher Matrix Analysis was carried out, using only the CMB temperature, in order to estimate the precision with which cosmological parameters can be reconstructed in future experiments. Here we extend this analysis by including also E-polarization measurements as well as the TE cross-correlation. We consider the planned Planck satellite (HFI only) and an ideal experiment which would measure both temperature and polarization to the cosmic variance limit (in the following, ’CVL experiment‘) for a range of multipoles, $l$, up to 2000. For illustration purposes, and particularly as a way of checking that our method is producing credible results, we will also present the FMA analysis for WMAP, and compare the corresponding ‘predictions’ with existing results. The Fisher Matrix is a measure of the width and shape of the likelihood around its maximum and as such can also provide useful insight into the degeneracies among different parameters, with minimal computational effort. For a review of this technique, see [@fisher; @tegmark; @jungman1; @jungman2; @knox; @zaldarriaga2; @bond; @efstathiou1; @efstathiou2]. In what follows we will present a brief description of our analysis procedure, emphasizing the aspects that are new. We refer the reader to our previous work [@Martins] for further details. We will assume that cosmological models are characterized by the 8 dimensional parameter set $${\bf \Theta} = (\Omega_b h^2, \Omega_m h^2, \Omega_\Lambda h^2, {{\mathcal R}}, n_s, Q, \tau, {\alpha})\,,$$ where $\Omega_m = \Omega_c +\Omega_b$ is the energy density in matter, $\Omega_\Lambda$ the energy density due to a cosmological constant, and $h$ is a dependent variable which denotes the Hubble parameter today, $H_0 \equiv 100h$ km s$^{-1}$ Mpc$^{-1}$. The quantity ${{\mathcal R}}\equiv \ell_{\rm ref} / \ell$ is the ‘shift’ parameter (see [@melch:01; @Bowen:01] and references therein), which gives the position of the acoustic peaks with respect to a flat, $\Omega_\Lambda = 0$ reference model, The shift parameter ${{\mathcal R}}$ depends on $\Omega_m$, on the curvature $\Omega_{\kappa} \equiv 1 - \Omega_{\Lambda} - \Omega_m - \Omega_{\rm rad}$ through $$\begin{aligned} \label{eq:def_r} {{\mathcal R}}&=& 2 \left( 1 - \frac{1}{\sqrt{1 + z_{\rm dec} }} \right) \nonumber \\ && \times \frac{\sqrt{| \Omega_{\kappa}| }}{ \Omega_m} \frac{1}{\chi(y)} \left[ \sqrt{\Omega_{\rm rad} + \frac{\Omega_m}{1 + z_{\rm dec} } } - \sqrt{\Omega_{\rm rad}} \right] ,\end{aligned}$$ where $z_{\rm dec} $ is the redshift of decoupling, $\Omega_{\rm rad}$ is the energy parameter due to radiation ($\Omega_{\rm rad}=4.13 \cdot 10^{-5}/h^2$ for photons and 3 neutrinos) and $$\begin{aligned} \label{eq:ydef} y &=& \sqrt{|\Omega_{\kappa}|}\int_0^{z_{\rm dec} } \, dz\\ && {[\Omega_{\rm rad} (1+z)^4 + \Omega_m(1+z)^3+\Omega_{\kappa}(1+z)^2+\Omega_{\Lambda}]^{-1/2}}. \nonumber\end{aligned}$$ The function $\chi(y)$ depends on the curvature of the universe and is $y$, $\sin(y)$ or $\sinh(y)$ for flat, closed or open models, respectively. Inclusion of the shift parameter ${{\mathcal R}}$ into our set of parameters takes into account the geometrical degeneracy between $\omega_\Lambda$ and $\omega_m$ [@efstathiou1]. With our choice of the parameter set, ${{\mathcal R}}$ is an independent variable, while the Hubble parameter $h$ becomes a dependent one. $n_s$ is the scalar spectral index and denotes the overall normalization, where the mean is taken over the multipole range $2 \leq \ell \leq 2000$. We assume purely adiabatic initial conditions and we do not allow for a tensor contribution. In the FM approach, the likelihood distribution ${\cal L}$ for the parameters $\bf \Theta$ is expanded to quadratic order around its maximum ${\cal L}_{m}$. We denote this maximum likelihood (ML) point by $\bf \Theta_0$ and call the corresponding model our “ML model”, with parameters $\omega_b = 0.0200$, $\omega_m = 0.1310$, $\omega_\Lambda = 0.2957$ (and $h = 0.65$), ${{\mathcal R}}= 0.9815$, $n_s = 1.00$, $Q = 1.00$, $\tau=0.20$ and ${\alpha}/{\alpha}_0=1.00$. For the value of $z_{\rm dec}$ (which is weakly dependent on $\omega_b$ and $\omega_{\rm tot}$) we have used the fitting formula from [@HuandSugiyama]. For the ML model we have $z_{\rm dec} = 1115.52$. As mentioned above we also present the FMA for the WMAP best fit model as the fiducial model. (ie, $\omega_b = 0.0200$, $\omega_m = 0.1267$, $\omega_\Lambda = 0.2957$, ${{\mathcal R}}= 0.9636$, $n_s = 0.99$, $Q = 1.00$, $\tau=0.17$ and ${\alpha}/{\alpha}_0=1.00$.) Note that we will discuss cases with and without reionization (in the latter case $\tau=0.0$) as well as with and without varying $\alpha$. To compute the derivatives of the power spectrum with respect to a particular cosmological parameter one varies the considered parameter and keeps fixed the value of the others to their ML value. In particular given that we are not constraining our analysis to the case of a flat universe a variation in $\mathcal R$ is considered with all the other parameters fixed and equal to their ML value. Therefore such variation implies a variation of the dependent parameter $h$. --------------------------------------- --------- --------- --------- --------- --------- --------- $\nu$ (GHz) $40$ $60$ $90$ $100$ $143$ $217$ $\theta_c$ (arcmin) $31.8$ $21.0$ $13.8$ $10.7$ $8.0$ $5.5$ $\sigma_cT$ ($\mu$K) $19.8$ $30.0$ $45.6$ $5.4$ $6.0$ $13.1$ $\sigma_{cE}$ ($\mu$K) $28.02$ $42.43$ $64.56$ $n/a$ $11.4$ $26.7$ $w^{-1}_c \cdot 10^{15}$ (K$^2$ ster) $33.6$ $33.6$ $33.6$ $0.215$ $0.158$ $0.350$ $\ell_c$ $254$ $385$ $586$ $757$ $1012$ $1472$ $\ell_{\rm max}$ $f_{\rm sky}$ --------------------------------------- --------- --------- --------- --------- --------- --------- : \[exppar\] Experimental parameters for WMAP and Planck (nominal mission). Note that we express the sensitivities in $\mu$K. In our previous work [@Martins] we assumed a flat fiducial model, and differentiating around it requires computing open and closed models, which are calculated using different numerical techniques. We have found that this can limit the accuracy of the FMA. Here we instead differentiate around a slightly closed model (as preferred by WMAP) with $\Omega_{\rm{tot}} = 1.01$ to avoid extra sources of numerical inaccuracies. We refer to [@Martins] for a detailed description of the numerical technique used. The experimental parameters used for the Planck analysis are in Table \[exppar\]. Note that we use the first 3 channels of the Planck High Frequency Instrument (HFI) only. Adding the 3 channels of Planck’s Low Frequency Instrument leaves the expected errors unchanged: therefore they can be used for other important tasks such as foreground removal and various consistency checks, leaving the HFI channels for direct cosmological use. For the CVL experiment, we set the experimental noise to zero, and we use a total sky coverage $f_{\rm{sky}} = 1.00$. Although this is never to be achieved in practice, the CVL experiment illustrates the precision which can be obtained *in principle* from CMB temperature and E-polarization measurements. If the errors $\Theta - \Theta_{0}$ about the ML model are small, a quadratic expansion around this ML leads to the expression, $${\cal L} \approx {\cal L}_m \exp\left[-{1 \over 2} \label{eq:3} \sum_{ij}{F}_{ij} \delta \Theta_i \delta \Theta_j\right]$$ where $F_{ij}$ is the Fisher matrix, given by derivatives of the CMB power spectrum with respect to the parameters ${\bf \Theta}$ In [@Martins] we computed the [*Fisher information matrix*]{} using temperature information alone. In this case for each $l$ a derivative of the temperature power spectrum with respect to the parameter under consideration is computed and then summed over all $l$, weighted by $Cov^{-1}(\hat{C}^{2}_{Tl})=\Delta C_\ell^2$, that is $$F_{ij} = \sum_{\ell=2}^{\ell_{\rm max}} \frac{1}{\Delta C_\ell^2} \frac{\partial C_\ell}{\partial \Theta_i}\frac{\partial C_\ell}{\partial \Theta_j} \vert_{\bf \Theta_0}\,. \label{eq:fisher}$$ The quantity $\Delta C_\ell$ is the standard deviation on the estimate of $C_{\ell}$: $$\Delta C_\ell^2 = \frac{2}{(2 \ell + 1) f_{\rm sky} } ( C_\ell + {B}_{\ell}^{-2})^2\,;$$ the first term is the cosmic variance, arising from the fact that we exchange an ensemble average with a spatial average. The second term takes into account the expected error of the experimental apparatus [@knox; @efstathiou1], $${B}_{\ell}^2 = \sum_c w_c e^{- \ell (\ell +1)/\ell_c^2}\,.$$ The sum runs over all channels of the experiment, with the inverse weight per solid angle $w_c^{-1} \equiv (\sigma_c \theta_c)^{-2}$ and $\ell_c \equiv \sqrt{8 \ln2}/\theta_c$, where $\sigma_c$ is the sensitivity (expressed in $\mu$K) and $\theta_c$ is the FWHM of the beam (assuming a Gaussian profile) for each channel. Furthermore, we can neglect the issues arising from point sources, foreground removal and galactic plane contamination assuming that once they have been taken into account we are left with a “clean” fraction of the sky given by $f_{\rm sky}$. In the more general case with polarization information included, instead of a single derivative we have a vector of four derivatives with the weighting given by the the inverse of the covariance matrix [@zaldarriaga1], $$F_{ij}=\sum_l \sum_{X,Y}{\partial \hat{C}_{Xl} \over \partial \Theta_i} {\rm Cov}^{-1}(\hat{C}_{Xl}\hat{C}_{Yl}){\partial \hat{C}_{Yl} \over \partial \Theta_j}\,,$$ where $F_{ij}$ is the Fisher information or curvature matrix as above, $Cov^{-1}$ is the inverse of the covariance matrix, $\Theta_i$ are the cosmological parameters we want to estimate and $X,Y$ stands for $T$ (temperature), $E,B$ (polarization modes), or $C$ (cross-correlation of the power spectra for $T$ and $E$). For each $l$ one has to invert the covariance matrix and sum over $X$ and $Y$. The diagonal terms of the covariance matrix between the different estimators are given by $$\begin{aligned} {\rm Cov }(\hat{C}_{Tl}^2)&=&\frac{2}{(2 \ell + 1) f_{\rm sky} }(\hat{C}_{Tl}+ {B}_{T\ell}^{-2})^2 \nonumber \\ {\rm Cov }(\hat{C}_{El}^2)&=&\frac{2}{(2 \ell + 1) f_{\rm sky}}(\hat{C}_{El}+ {B}_{P\ell}^{-2})^2 \nonumber \\ {\rm Cov }(\hat{C}_{Cl}^2)&=&\frac{1}{(2 \ell + 1) f_{\rm sky}}\left[\hat{C}_{Cl}^2+ (\hat{C}_{Tl}+{B}_{T\ell}^{-2}) (\hat{C}_{El}+{B}_{P\ell}^{-2})\right] \nonumber \\ {\rm Cov }(\hat{C}_{Bl}^2)&=&\frac{2}{(2 \ell + 1) f_{\rm sky} }(\hat{C}_{Bl}+ {B}_{P\ell}^{-2})^2. $$ The non-zero off diagonal terms are $$\begin{aligned} {\rm Cov }(\hat{C}_{Tl}\hat{C}_{El})&=&\frac{2}{(2 \ell + 1) f_{\rm sky} }\hat{C}_{Cl}^2 \nonumber \\ {\rm Cov }(\hat{C}_{Tl}\hat{C}_{Cl})&=&\frac{2}{(2 \ell + 1) f_{\rm sky}}\hat{C}_{Cl} (\hat{C}_{Tl}+{B}_{T\ell}^{-2}) \nonumber \\ {\rm Cov }(\hat{C}_{El}\hat{C}_{Cl})&=&\frac{2}{(2 \ell + 1) f_{\rm sky}}\hat{C}_{Cl} (\hat{C}_{El}+{B}_{P\ell}^{-2})\,,\end{aligned}$$ where ${B}_{T\ell}^{-2}={B}_{\ell}^{-2}$ as above and ${B}_{P\ell}^2$ is obtained using a similar expression but with the experimental specifications for the polarized channels. For Gaussian fluctuations, the covariance matrix is then given by the inverse of the Fisher matrix, $C = F^{-1}$ [@bond]. The $1\sigma$ error on the parameter $\Theta_i$ with all other parameters marginalised is then given by $\sqrt{C_{ii}}$. If all other parameters are held fixed to their ML values, the standard deviation on parameter $\Theta_i$ reduces to $\sqrt{1/F_{ii}}$ (conditional value). Other cases, in which some of the parameters are held fixed and others are being marginalized over can easily be worked out. In the case in which all parameters are being estimated jointly, the joint error on parameter $i$ is given by the projection on the $i$-th coordinate axis of the multi-dimensional hyper-ellipse which contains a fraction $\gamma$ of the joint likelihood. The equation of the hyper-ellipse is $$({\bf \Theta - \Theta_0}) {\bf F } ({\bf \Theta - \Theta_0})^t = q_{1-\gamma},$$ where $q_{1-\gamma}$ is the quantile for the probability $1-\gamma$ for a $\chi^2$ distribution with 6,7 and 8 degrees of freedom. For $\gamma = 0.683$ ($1\sigma$ c.l.) we have for 6,7 and 8 degrees of freedom, $q_{1-\gamma} = 7.03$, $q_{1-\gamma} = 8.18$ and $q_{1-\gamma} = 9.30$, respectively. As observed in [@Martins] the accuracy with which parameters can be determined depends on their true value as well as on the number of parameters considered. Note that the FMA *assumes* that the values of the parameters of the true model are in the vicinity of ${\bf \Theta_0}$. The validity of the results therefore depends on this assumption, as well as on the assumption that the $a_{\ell m}$’s are independent Gaussian random variables. If the FMA predicted errors are small enough, the method is self-consistent and we can expect the FMA prediction to reproduce in a correct way the exact behaviour. This is indeed the case for the present analysis, with the notable exception of $\omega_\Lambda$, which as expected suffers from the geometrical degeneracy. Also, special care must be taken when computing the derivatives of the power spectrum with respect to the cosmological parameters. This differentiation strongly amplifies any numerical errors in the spectra, leading to larger derivatives, which would artificially break degeneracies among parameters. In the present work we implement double–sided derivatives, which reduce the truncation error from second order to third order terms. The choice of the step size is a trade-off between truncation error and numerical inaccuracy dominated cases. For an estimated numerical precision of the computed models of order $10^{-4}$, the step size should be approximately 5% of the parameter value [@Numerical:92], though it turns out that for derivatives in direction of $\alpha$ and $n_s$ the step size can be chosen to be as small as 0.1%. After several tests, we have chosen step sizes varying from 1% to 5% for $\omega_b, \omega_m, \omega_\Lambda$ and ${{\mathcal R}}$. This choice gives derivatives with an accuracy of about 0.5%. The derivatives with respect to $Q$ are exact, being the power spectrum itself. Quantity ------------------ --------- -------- --------- ------- ------- -------- ------- ------- -------- marg. fixed joint marg. fixed joint marg. fixed joint $\omega_b$ 1437.41 52.93 4111.09 6.40 0.99 18.31 0.48 0.25 1.38 $\omega_m$ 619.43 31.47 1771.62 3.57 0.33 10.22 0.70 0.03 2.01 $\omega_\Lambda$ 1397.45 980.08 3996.79 38.76 34.40 110.84 11.28 9.94 32.27 $n_s$ 260.43 33.68 744.83 1.47 0.91 4.20 0.30 0.08 0.86 $Q$ 474.57 25.13 1357.31 2.21 0.45 6.32 0.24 0.07 0.68 ${{\mathcal R}}$ 666.04 22.10 1904.92 3.53 0.30 10.09 0.66 0.03 1.88 $\omega_b$ 2.79 1.26 7.97 0.82 0.59 2.36 0.55 0.38 1.59 $\omega_m$ 4.58 0.83 13.11 1.44 0.12 4.12 1.09 0.08 3.11 $\omega_\Lambda$ 115.59 86.53 330.59 91.65 86.37 262.11 80.68 77.25 230.74 $n_s$ 1.50 0.52 4.30 0.48 0.13 1.36 0.33 0.07 0.96 $Q$ 0.80 0.34 2.29 0.19 0.10 0.55 0.17 0.07 0.48 ${{\mathcal R}}$ 4.17 0.73 11.92 1.41 0.11 4.03 1.05 0.07 2.99 $\omega_b$ 2.78 1.26 7.95 0.77 0.51 2.20 0.32 0.21 0.91 $\omega_m$ 4.56 0.83 13.05 1.16 0.12 3.32 0.55 0.03 1.58 $\omega_\Lambda$ 114.34 86.09 327.03 31.79 31.72 90.92 9.87 9.49 28.24 $n_s$ 1.50 0.52 4.28 0.39 0.13 1.12 0.20 0.06 0.57 $Q$ 0.80 0.34 2.28 0.18 0.10 0.52 0.14 0.05 0.40 ${{\mathcal R}}$ 4.15 0.73 11.86 1.14 0.10 3.25 0.52 0.03 1.49 Quantity ------------------ --------- -------- ---------- -------- ------- -------- ------- --------- -------- marg. fixed joint marg. fixed joint marg. fixed joint $\omega_b$ 4109.93 52.93 11754.68 6.42 0.99 18.36 1.10 0.25 3.16 $\omega_m$ 844.65 31.47 2415.75 7.14 0.33 20.43 1.64 0.03 4.69 $\omega_\Lambda$ 1483.80 980.08 4243.77 41.78 34.40 119.50 12.03 9.94 34.41 $n_s$ 365.06 33.68 1044.09 3.90 0.91 11.16 0.79 0.08 2.25 $Q$ 2415.47 25.13 6908.40 3.24 0.45 9.28 0.24 0.07 0.69 ${{\mathcal R}}$ 4847.40 22.10 13863.91 10.13 0.30 28.98 1.19 0.03 3.39 $\alpha$ 887.24 3.51 2537.58 2.62 0.05 7.50 0.40 $<0.01$ 1.15 $\omega_b$ 10.41 1.26 29.78 0.97 0.59 2.78 0.77 0.38 2.21 $\omega_m$ 8.51 0.83 24.34 2.54 0.12 7.27 2.04 0.08 5.85 $\omega_\Lambda$ 125.00 86.53 357.51 107.64 86.37 307.85 93.06 77.25 266.16 $n_s$ 3.05 0.52 8.73 1.32 0.13 3.76 1.04 0.07 2.97 $Q$ 2.11 0.34 6.05 0.20 0.10 0.57 0.17 0.07 0.50 ${{\mathcal R}}$ 21.12 0.73 60.40 1.50 0.11 4.29 1.06 0.07 3.02 $\alpha$ 4.64 0.12 13.27 0.43 0.02 1.22 0.31 0.01 0.88 $\omega_b$ 10.00 1.26 28.60 0.87 0.51 2.49 0.38 0.21 1.09 $\omega_m$ 8.23 0.83 23.54 1.61 0.12 4.60 0.67 0.03 1.90 $\omega_\Lambda$ 123.13 86.09 352.17 31.79 31.72 90.92 9.96 9.49 28.49 $n_s$ 2.97 0.52 8.48 0.85 0.13 2.44 0.32 0.06 0.91 $Q$ 2.04 0.34 5.82 0.18 0.10 0.53 0.14 0.05 0.41 ${{\mathcal R}}$ 20.34 0.73 58.18 1.36 0.10 3.88 0.60 0.03 1.72 $\alpha$ 4.46 0.12 12.75 0.31 0.02 0.88 0.11 $<0.01$ 0.32 FMA without reionization ======================== We will now start to describe the results of our analysis in detail. In order to avoid confusion, we will begin in this chapter by describing the results for the case $\tau=0$ (since most of the crucial degeneracies can be understood in this case), and leave the more relevant case of non-zero $\tau$ for the following chapter. While it may seem pointless after WMAP to discuss the cases without (or with very little) reionization, we shall see that a lot can be learned by comparing the results for the various cases. Analysis results: The FMA forecast ---------------------------------- Tables \[fmast\]–\[fmaevnotauTT\] summarize the results of our FMA for WMAP, Planck and a CVL experiment. We consider the cases of models with and without a varying $\alpha$ being included in the analysis, for $\tau=0$. We also consider the use of temperature information alone (TT), E-polarization alone (EE) and both channels (EE+TT) jointly. Table \[fmast\] shows the $1\sigma$ errors on each of the parameters of our FMA for a ‘standard model’, that is with no reionization or variation of $\alpha$. The inclusion of polarization data does indeed increase the accuracy on each parameter for Planck and for a CVL experiment. For the Planck mission the polarization data helps to better constrain each of the parameters though the increase in accuracy is only of the order 10% in most cases. The error in $\omega_{\Lambda}$ is still large, and larger than those of the other parameters. Indeed, this error is almost insensitive to the experimental details when only temperature is considered in the analysis, which of course is a manifestation of the so-called geometrical degeneracy [@efstathiou1; @efstathiou2]. The existence of this nearly exact degeneracy limits in a fundamental way the accuracy on measurements of the Hubble constant as well as of the curvature of the universe obtained with the CMB observations, and hence limits the accuracy on $\omega_{m}$ and $\omega_{\Lambda}$. This degeneracy can only be removed when constraints on the geometry of the universe from other complementary observations, such as Type Ia supernova or gravitational lensing, are jointly considered [@efstathiou1; @efstathiou2]. Our plots show that actually using polarization data the confidence contours can narrow significantly on the $ \omega_{\Lambda}$ axis. This case is very different from other degeneracies between parameters which actually can be broken with good enough CMB data and by probing a larger set of angular scales ie an enlarged range of multipoles $l$, as well as using the CMB polarised data. The geometrical degeneracy gives rise to almost identical CMB anisotropies in universes with different background geometries but identical matter content, lines of constant $\mathcal R$ are directions of degeneracy. This degeneracy along $\delta(\omega_{m}^{-1/2} \mathcal R)=0$ results in a linear relation between $\delta \omega_{k}$ and $\delta \omega_{\Lambda}$, with coefficients that depend on the fiducial model. This is why we used the $\mathcal R$ parameter to replace $\omega_{k}$ in our fisher analysis instead of the $\omega_{D}$ parameter of [@efstathiou1; @efstathiou2]. The accuracy on the parameter $\mathcal R$ is related to the ability of fixing the positions of the Doppler peaks. Hence Planck is expected to determine $\mathcal R$ with high accuracy given that it samples the Doppler peak region almost entirely. Indeed this is the case with the error reducing from $4\%$ for WMAP to $1\%$ for Planck and to $0.5\%$ for a CVL experiment (see Table II). Table \[fmaal\] shows the $1\sigma$ errors on each of the parameters of our FMA for a model with a time-varying $\alpha$. While the inclusion of a varying $\alpha$ as a parameter (with the nominal value equal to that of the standard model) has no noticeable effect on the accuracy of the other parameters for a CVL experiment, for Planck and most notoriously for WMAP this is not the case (compare Table \[fmast\] with Table \[fmaal\]). For these two satellite missions the accuracy of most of the other parameters is reduced by inclusion of this extra parameter as should be expected (for allowing an extra degree of freedom). The same trend as before is observed with the inclusion of polarization data. From our WMAP predictions one would expect to be able to constrain $\alpha$ to about 5% accuracy at $1 \sigma$ while the actual analysis presented in previous section gives an accuracy of the order of $7\%$ at $2 \sigma$. This is in reasonable agreement with our prediction with the discrepancy being due to the effect of a $\tau \neq 0$ (see next section). On the other hand, the results of our forecast are that Planck and a CVL experiment will be able to constrain variations in $\alpha$ with an accuracy of 0.3% and 0.1% respectively ($1\sigma$ c.l., all other parameters marginalized). If all parameters are being estimated simultaneously, then these limits increase to about 0.9% and 0.3% respectively. This is therefore the best that one can hope to do with the CMB alone—it is somewhat below the $10^{-5}$ level of the claimed detection of a variation using quasar absorption systems [@Webb; @Jenam; @Murphy], but it is also at a much higher redshift, where any variations relative to the present day are expected to be larger than at $z\sim3$. Therefore, for specific models such limits can be at least as constraining as those at low redshift. On the other hand, there *is* a way of doing better than this, which is to combine CMB data with other observables—this is the approach we already took in [@Avelino; @Martins], for example. From these tables we conclude that for WMAP the inclusion of polarization information does not improve significantly the accuracy on each of the parameters, since its accuracy from polarization data alone is expected to be worse than that from temperature alone by a factor of $\simeq 10^{2}- 10^{3}$. With Planck though there is room for improvement, with the accuracy from polarization alone at most only a factor 10 poorer than from temperature. Also for this case a better accuracy on $\omega_{\Lambda}$ is obtained using polarization data alone vs using temperature data alone, for both cases with and without inclusion of a varying $\alpha$. For the CVL experiment the polarization makes a real difference, with the accuracy of polarization alone being *slightly better* than that of the temperature alone. Combining the two typically increases the accuracy on most parameters by a factor of order 2. As expected this is most noticeably so for $\omega_{\Lambda}$. Assuming that the improvement was only owing to the use of independent sets of data we should expect an improvement by at least a factor of $\sqrt{2}$. {width="3.5in"} {width="3.5in"} {width="3.5in"} {width="3.5in"} {width="3.5in"} {width="3.5in"} [|c c c c c c c c c|]{}\ Direction $i$ & $1/\sqrt{\lambda_i}$ & $\omega_b$ & $\omega_m$ & $\omega_\Lambda$ & $n_s$ & $Q$ & ${{\mathcal R}}$ & $\alpha$\ 1& 2.50E-04& 9.9446E-01\* & -9.9203E-02$\dag$ & -2.5224E-05 & -2.7487E-02 & -3.9411E-03 & 1.2295E-02 & 1.6954E-02\ 2& 8.84E-04& 8.1778E-02 & 7.0553E-01\* & -5.6359E-04 & -6.8131E-02 & 2.4777E-02 & -1.1338E-01 & -6.9096E-01$\dag$\ 3& 2.24E-03& 4.8801E-02 & 5.2913E-01 $\dag$& 9.3752E-04 & 2.6766E-01 & -6.3566E-01\* & 4.0924E-02 & 4.9016E-01\ 4& 1.24E-02& 4.2341E-02 & 2.5947E-01 & 1.2292E-02 & 6.5656E-01$\dag$ & 6.6964E-01\* & 4.5581E-02 & 2.2174E-01\ 5& 1.48E-02& 1.0147E-02 & 3.7938E-01 & -3.5290E-02 & -6.9349E-01\* & 3.7432E-01 & 2.0829E-01 & 4.3623E-01$\dag$\ 6& 1.94E-01& -9.0774E-03 & -2.9295E-02 & 2.2193E-01$\dag$ & 8.9661E-02 & -7.8874E-02 & 9.4671E-01\* & -1.9819E-01\ 7& 3.71E-01& 1.9270E-03 & 1.7036E-02 & 9.7435E-01\* & -5.4121E-02 & 2.3700E-02 & -2.0877E-01$\dag$ & 5.7273E-02\ \ Direction $i$ & $1/\sqrt{\lambda_i}$ & $\omega_b$ & $\omega_m$ & $\omega_\Lambda$ & $n_s$ & $Q$ & ${{\mathcal R}}$ & $\alpha$\ 1& 9.02E-05& 7.9666E-01\* & -4.4311E-01$\dag$ & -1.2864E-05 & 4.6149E-03 & -1.4650E-02 & 6.7622E-02 & 4.0518E-01\ 2& 1.38E-04& 6.0235E-01\* & 5.7873E-01$\dag$ & 1.1892E-05 & -3.3913E-02 & -5.8211E-02 & -8.4462E-02 & -5.3905E-01\ 3& 4.80E-04& 2.5914E-02 & 6.1004E-01$\dag$ & -1.5285E-05 & 3.4825E-01 & 3.4725E-01 & 2.8006E-02 & 6.2011E-01\*\ 4& 1.88E-03& 4.0978E-02 & -2.0888E-01 & 2.2619E-04 & -7.9733E-02 & 9.3426E-01\* & 2.5167E-03 & -2.7474E-01$\dag$\ 5& 8.88E-03& 1.2289E-02 & -2.2979E-01 & -3.4989E-03 & 9.1281E-01\* & -4.7146E-02 & -2.2075E-01 & -2.5072E-01$\dag$\ 6& 1.36E-02& -1.1477E-03 & 1.1923E-02 & 7.0929E-03 & 1.9486E-01$\dag$ & -2.7260E-02 & 9.6887E-01\* & -1.4961E-01\ 7& 9.40E-02& 4.5352E-05 & -8.4463E-04 & 9.9997E-01\* & 1.8356E-03 & -1.7712E-04 & -7.6431E-03$\dag$ & 2.6714E-04\ \ Direction $i$ & $1/\sqrt{\lambda_i}$ & $\omega_b$ & $\omega_m$ & $\omega_\Lambda$ & $n_s$ & $Q$ & ${{\mathcal R}}$ & $\alpha$\ 1& 2.67E-05& -1.2198E-01 & 7.5184E-01\* & 1.6953E-05 & -4.5292E-03 & -1.5331E-03 & -1.0829E-01 & -6.3883E-01$\dag$\ 2& 4.30E-05& 9.8787E-01\* & 1.5297E-01$\dag$ & -4.0577E-06 & 2.3058E-02 & -2.0123E-03 & -1.1111E-02 & -6.8706E-03\ 3& 2.26E-04& -8.5658E-02 & 5.3126E-01$\dag$ & -1.6190E-04 & 3.8153E-01 & 4.0338E-01 & 2.5197E-02 & 6.3365E-01\*\ 4& 1.30E-03& 4.2889E-02 & -2.9019E-01 & 4.2863E-03 & 6.5704E-02 & 8.8528E-01\* & 1.3375E-02 & -3.5457E-01$\dag$\ 5& 3.31E-03& 7.1120E-03 & -2.0415E-01 & -3.3636E-02 & 9.1855E-01\* & -2.2722E-01 & -8.6918E-02 & -2.3286E-01$\dag$\ 6& 5.95E-03& -2.8965E-05 & 5.6352E-02 & 1.1741E-02 & 7.0270E-02 & -4.2538E-02 & 9.8973E-01\* & -1.0183E-01$\dag$\ 7& 2.95E-02& 4.7963E-05 & -6.2146E-03 & 9.9936E-01\* & 2.9871E-02$\dag$ & -1.0880E-02 & -1.4605E-02 & -5.0067E-03\ [|c c c c c c c c c|]{}\ Direction $i$ & $1/\sqrt{\lambda_i}$ & $\omega_b$ & $\omega_m$ & $\omega_\Lambda$ & $n_s$ & $Q$ & ${{\mathcal R}}$ & $\alpha$\ 1& 2.50E-04& 9.9447E-01\* & -9.9159E-02$\dag$ & -2.5230E-05 & -2.7515E-02 & -3.9234E-03 & 1.2276E-02 & 1.6802E-02\ 2& 8.84E-04& 8.1646E-02 & 7.0565E-01\* & -5.6626E-04 & -6.8099E-02 & 2.4756E-02 & -1.1338E-01 & -6.9086E-01$\dag$\ 3& 2.24E-03& 4.8844E-02 & 5.2886E-01$\dag$ & 9.4022E-04 & 2.6766E-01 & -6.3596E-01\* & 4.0937E-02 & 4.9006E-01\ 4& 1.24E-02& 4.2256E-02 & 2.5530E-01 & 1.2657E-02 & 6.6444E-01$\dag$ & 6.6515E-01\* & 4.4102E-02 & 2.1685E-01\ 5& 1.49E-02& 1.0648E-02 & 3.8232E-01 & -3.5272E-02 & -6.8593E-01\* & 3.8154E-01 & 2.0973E-01 & 4.3865E-01$\dag$\ 6& 2.00E-01& -9.0958E-03 & -2.9575E-02 & 2.3865E-01$\dag$ & 8.8766E-02 & -7.9309E-02 & 9.4276E-01\* & -1.9779E-01\ 7& 3.78E-01& 2.0990E-03 & 1.7737E-02 & 9.7038E-01\* & -5.5730E-02 & 2.5328E-02 & -2.2492E-01$\dag$ & 6.0883E-02\ \ Direction $i$ & $1/\sqrt{\lambda_i}$ & $\omega_b$ & $\omega_m$ & $\omega_\Lambda$ & $n_s$ & $Q$ & ${{\mathcal R}}$ & $\alpha$\ 1& 1.01E-04& 7.2972E-01\* & -5.0661E-01$\dag$ & 1.3138E-06 & 6.7011E-03 & -6.9279E-03 & 7.5433E-02 & 4.5284E-01\ 2& 1.54E-04& 6.8066E-01\* & 5.0680E-01 & -1.8992E-05 & -4.2986E-02 & -6.6152E-02 & -7.8097E-02 & -5.1724E-01$\dag$\ 3& 4.94E-04& 5.4883E-02 & 6.2059E-01\* & 2.1300E-05 & 3.4975E-01 & 3.5572E-01 & 2.4796E-02 & 6.0198E-01$\dag$\ 4& 1.95E-03& 3.3117E-02 & -2.1287E-01 & -8.9072E-04 & -9.9243E-02 & 9.3131E-01\* & 3.4607E-03 & -2.7638E-01$\dag$\ 5& 1.14E-02& 9.3798E-03 & -2.3387E-01 & 2.6011E-02 & 9.2519E-01\* & -3.5345E-02 & -1.1636E-01 & -2.7161E-01$\dag$\ 6& 1.49E-02& -2.3014E-03 & 3.6400E-02 & 2.0129E-03 & 9.6754E-02 & -2.1077E-02 & 9.8694E-01\* & -1.2173E-01$\dag$\ 7& 3.18E-01& -1.9911E-04 & 5.8193E-03 & 9.9966E-01\* & -2.4365E-02$\dag$ & 1.7831E-03 & 1.0413E-03 & 7.0427E-03\ \ Direction $i$ & $1/\sqrt{\lambda_i}$ & $\omega_b$ & $\omega_m$ & $\omega_\Lambda$ & $n_s$ & $Q$ & ${{\mathcal R}}$ & $\alpha$\ 1& 5.86E-05& 6.7177E-01\* & -4.8930E-01 & 8.7005E-07 & 3.9437E-02 & 2.7795E-02 & 8.1011E-02 & 5.4811E-01$\dag$\ 2& 1.16E-04& 7.3379E-01\* & 5.3949E-01$\dag$& -1.2978E-05 & -2.8027E-05 & -2.4037E-02 & -7.2166E-02 & -4.0585E-01\ 3& 2.91E-04& -9.4755E-02 & 6.0914E-01$\dag$ & 1.7574E-05 & 3.4556E-01 & 3.4843E-01 & 1.1703E-02 & 6.1564E-01\*\ 4& 1.71E-03& 3.4674E-02 & -2.0806E-01 & -8.4424E-04 & -8.2888E-02 & 9.3517E-01\* & 1.4441E-02 & -2.7182E-01$\dag$\ 5& 9.14E-03& 9.1126E-03 & -1.9125E-01 & 2.0128E-02 & 8.9154E-01\* & -5.1488E-02 & 2.8905E-01$\dag$ & -2.8615E-01\ 6& 1.05E-02& -3.6709E-03 & 1.3640E-01 & -9.7678E-03 & -2.7727E-01$\dag$ & -7.0379E-03 & 9.5096E-01\* & 6.0169E-03\ 7& 2.75E-01& -1.7944E-04 & 5.0041E-03 & 9.9975E-01\* & -2.0734E-02$\dag$ & 1.7511E-03 & 3.4827E-03 & 5.5738E-03\ Analysis results: Confidence contours ------------------------------------- In order to provide better intuition for the various effects involved, we show in Figs. \[figellipse1\]-\[figellipse2\] joint 2D confidence contours for all pairs of parameters (all remaining parameters marginalized) for the cases shown in Tables \[fmast\]-\[fmaal\] respectively (that is, the cases $\tau=0$ without and with a varying $\alpha$). For each case we show plots corresponding to our three experiments (WMAP, Planck and CVL), and contours for TT only, EE only and all combined. Note that all contours are $2 \sigma$. To notice that in the WMAP case the errors from E only are very large, hence the contours for T coincide almost exactly with the temperature-polarization combined case. In the CVL case it is the E contours that almost coincide with the combined ones. Again, starting with the standard model in Fig. \[figellipse1\] we can observe the expected degeneracies between parameters, as previously discussed in [@efstathiou1; @efstathiou2]. These degeneracies among parameters limit our ability to disentangle one parameter from another, using CMB observations alone. The search for means to break such degeneracies is therefore of extreme importance. The contour plots for WMAP exhibit the degeneracy directions in the planes ($\omega_{\Lambda}$,$\mathcal R$), ($n_{s}$,$\mathcal R$), for example $\mathcal R$ suffers strong degeneracy with $\omega_{m}$, $\omega_{\Lambda}$. A correlation between $\omega_{\Lambda}$ and both $n_{s}$ and $Q$ is also noticeable. The contour plot in the plane ($\omega_{\Lambda}$,$\mathcal R$) prevents a good constraint of both parameters in agreement with results tabulated in Table \[fmast\]. For both Planck and a CVL experiment the direction of degeneracy for polarization alone is almost orthogonal to this direction while the direction for temperature alone corresponds to $\mathcal R=constant$. The degeneracy direction on the ($\omega_{m}$,$\mathcal R$) plane is defined by $\delta (\omega_{m}^{1/2} \mathcal R$)=0. The contour plots for Planck are perhaps the perfect example of a case where the degeneracy directions between $\mathcal R$ and $\omega_{\Lambda}$ are different and almost orthogonal for Temperature and Polarization alone. This therefore explains why the joint use of T and E data helps to break degeneracies. For example the degeneracy between $\mathcal R$ and $\omega_{b}$ present when polarization is considered alone, disappears when temperature information is included. It is interesting to notice, when comparing WMAP and Planck plots, that the joint use of T and E does not necessarily break degeneracies between the parameters, whilst narrowing down the width of the contour plots without affecting the degeneracy directions. For the CVL experiment the effect of polarization is to better constrain all parameters in particular $\omega_{\Lambda}$, helping to narrow down the range of allowed values in the $\omega_{\Lambda}$ direction as compared with Temperature alone. For instance in the plane ($n_{s}$,$\omega_{\Lambda}$) the direction $n_{s}$ is well constrained but there is no discriminatory power on the $\omega_{\Lambda}$ direction until polarization data is included. For all but the 2D planes containing $\omega_{\Lambda}$, the contours are narrowed to give better constraints to each of the parameters. This is due to the exact degeneracy mentioned above: more accurate CMB measurements simply narrow the likelihood contours around the degeneracy lines on the ($\omega_{\Lambda}$, $\omega_{k}$) plane [@efstathiou1; @efstathiou2]. Fig. \[figellipse1\] also shows that $\omega_{b}$ and $\omega_{m}$ are slightly anticorrelated for the Planck experiment. For the WMAP experiment the plot shows a degeneracy between $\omega_{m}$ and $\omega_{\Lambda}$. If we restrict ourselves to spatially flat models there is a relationship between these two parameters that will result in similar position of the Doppler peaks. The degeneracy direction can be obtained by differentiating $l_{D}$, the location of the maximum of the first Doppler peak [@efstathiou1; @efstathiou2] These degeneracy lines in the $\omega_{c}$ - $\omega_{\Lambda}$ plane are given by (assuming that $\omega_{b}$ is held fixed in the expression of $l_{D}$): $$\begin{aligned} \omega_{c}=(\omega_{c})_{t}+b \omega_{\Lambda}; b=-\frac{(\partial l_{D}/ \partial \omega_{\Lambda})_{t}}{(\partial l_{D}/ \partial \omega_{c})_{t}}\end{aligned}$$ Unlike the geometrical degeneracy, this is not exact. Both the height and the amplitude of the peaks depend upon the parameter $\omega_{m}$, hence an experiment such as Planck which probes high multipoles will be able to break this degeneracy. This is clearly visible in Fig. \[figellipse1\] for both Planck and a CVL experiment (compare with the case for WMAP). Similarly the condition of constant height of the first Doppler peak determines the degeneracies among $\omega_{b}$, $\omega_{c}$ $n_{s}$ and $Q$. Both WMAP and Planck are sensitive to higher multipoles than the first Doppler peak. The other peaks help to pin down the value of $\omega_{b}$ and therefore these degeneracies can actually be broken. The plots for WMAP show a mild degeneracy in the ($n_{s}$,$\omega_{b}$) plane for the EE+TT+ET joint analysis, which seems to be lifted for the Planck experiment. In our previous works [@Old; @Avelino; @Martins] we observed a degeneracy between $\alpha$ and some of the other parameters, most notably $w_{b}$, $n_{s}$ and ${{\mathcal R}}$. Our previous FMA analysis with temperature information alone [@Martins] showed that these degeneracies could be removed by using higher multipole measurements, e.g., from Planck. The question we want to address here is whether the use of polarization data allows further improvements. As previously pointed out, a variation in $\alpha$ affects both the location and height of the Doppler peaks, hence this parameter will be correlated with parameters that determine the peak structure. Therefore, from the previous discussion on degeneracies among parameters for a standard model, one can anticipate the degeneracies exhibited in Fig. \[figellipse2\] in the planes ($\alpha$,$n_{s}$), ($\alpha$,$\mathcal R$), ($\alpha$,$Q$), ($\alpha$,$\omega_{b}$) and ($\alpha$,$\omega_{m}$). In our previous work [@Martins] we showed that using temperature alone the degeneracies of $\alpha$ with $\omega_{b}$ and $\alpha$ with $n_{s}$ are lifted as we move from WMAP to Planck when higher multipoles measurements can break it. All the degeneracy directions for these pairs of parameters for the WMAP joint analysis (which actually is dominated by the temperature data alone) are approximately preserved by using polarization data alone for the Planck experiment. A joint analysis of temperature and polarization helps to narrow down the confidence contours without necessarily breaking the degeneracy. With the inclusion of the new parameter $\alpha$ the WMAP contour plots get wider as compared with Fig. \[figellipse1\], while leaving almost unchanged the degeneracy directions in most planes of pairs of parameters. For Planck the contour plots are still wider whilst the degeneracy directions for polarization alone change for some of the parameters. For example, the direction of degeneracy between the ($\mathcal R$,$n_{s}$) changes when compared with Fig. \[figellipse1\], which is due to the presence of the degeneracy between $\alpha$ and $n_{s}$ which is almost orthogonal to the direction of degeneracy in the plane ($\alpha$,$\omega_{m}$). Another changed direction of degeneracy is that of ($\omega_{b}$,$\mathcal R$), with wider contour plots. The degeneracy present in the WMAP plot for the plane ($\alpha$,$\omega_{b}$) seems to be broken with Planck data. Notice the strong degeneracy between $\alpha$ and $\mathcal R$ which still persists when using jointly temperature and polarization data. Using Temperature and Polarization data jointly seems either to help to break some of the degeneracies or at least to narrow down the contours without lifting the degeneracy, in particular for those cases where the degeneracy directions for each of the temperature and polarization are different (in some cases almost orthogonal see for example the planes containing $\omega_{\Lambda}$ as one of the parameters). For the CVL experiment most of the plots remain unchanged when compared with no inclusion of $\alpha$, with the temperature alone contour plot slightly wider in the ($n_{s}$,$\omega_{\Lambda}$) plane. A large range of possibilities along the $\omega_{\Lambda}$ direction still remains as expected from the exact geometrical degeneracy mentioned above. Analysis results: Principal directions -------------------------------------- The power of an experiment can be roughly quantified by looking at the eigenvalues $\lambda_i$ and eigenvectors ${\bf u^{(i)}}$ of its FM: The error along the direction in parameter space defined by ${\bf u^{(i)}}$ (principal direction) is proportional to $\lambda_i^{-1/2}$. It can be measured by assessing how the principal components mix inflationary variables (such as $n_{s}$) with physical cosmic densities. The accuracy on the former is typically limited by cosmic variance (the derivatives of $C_{l}$ with respect to these variables has large amplitude for low multipoles; the accuracy on the latter is set by the accuracy with which the $C_{l}$ is measured at high multipoles (the derivatives of the angular power spectrum with respect to these variables is larger for $l\sim2000 -3000$). But we are interested in determining the errors on the physical parameters rather then on their linear combinations along the principal directions. Therefore in the ideal case we want the principal directions to be as much aligned as possible to the coordinate system defined by the physical parameters. We display in Table \[fmaevnotau\] eigenvalues and eigenvectors of the FM for WMAP and Planck and a CVL experiment. Planck’s errors, as measured by the inverse square root of the eigenvalues, are smaller by a factor of about $6$ on average that those for WMAP (to be compared with a factor of $4$ using temperature alone obtained in our previous analysis [@Martins]) While a CVL experiment’s errors are smaller by a factor of about $3$ on average than those for Planck. For 5 of the 7 eigenvectors Planck also obtains a better alignment of the principal directions with the axis of the physical parameters. This is established by comparing the ratios between the largest (marked with an asterisk in Tables \[fmaevnotau\]) and the second largest (marked with a dagger) cosmological parameters’ contribution to the principal directions. This is of course in a slightly different form the statement that Planck will measure the cosmological parameters with less correlations among them. It is to be noticed that for Planck direction 7 is mostly aligned with $\omega_{\Lambda}$. While $\alpha$ is the second largest parameter contribution to two of the principal directions for both WMAP and Planck, this is the case for four principal directions for a CVL experiment, and is also the largest parameter contribution to two and one of the principal directions for Planck and a CVL experiment respectively. For comparison we also display in Table \[fmaevnotauTT\] eigenvalues and eigenvectors of the FM for WMAP and Planck and a CVL experiment using Temperature information alone. Comparing Tables \[fmaevnotau\] and \[fmaevnotauTT\] we conclude that for WMAP the largest and second largest parameter contribution to the principal direction are exactly the same. On the other hand for Planck 2 of the principal directions change namely direction 7, whose main contribution is from $\omega_{\Lambda}$ and $n_{s}$ when Temperature information alone is used while when Polarization is included the second largest contribution comes now from $\mathcal R$. For direction 3 the largest and second largest contribution are interchanged (arising from $\omega_{m}$ and $\alpha$) when polarization is included. Finally for a CVL experiment for direction 1 the second largest contribution from $\alpha$ is replaced by $\omega_{m}$ when polarization is included. For direction 2 both largest contribution change from $\omega_{b}$ to $\omega_{m}$ and that from $\omega_{m}$ (second largest) to $\alpha$. The major contributions for the remaining directions remain the same while the second largest contribution changes for all of them. Only for 2 and 3 of the 7 eigenvectors Planck and a CVL experiment respectively obtain a better alignment of the principal directions with the axis of the physical parameters (with the other directions equally aligned), when polarization is included. Therefore we conclude that indeed polarization does not necessarily help to further break degeneracies between parameters when no information on reionization or tensor component of the CMB is included. Quantity ------------------ --------- -------- --------- -------- ------- -------- ------- ------- -------- marg. fixed joint marg. fixed joint marg. fixed joint $\omega_b$ 223.67 22.18 639.70 6.21 1.11 17.75 0.48 0.25 1.38 $\omega_m$ 104.48 22.12 298.81 3.37 0.39 9.64 0.70 0.03 1.99 $\omega_\Lambda$ 1231.56 113.78 3522.35 37.37 22.87 106.89 11.40 9.99 32.61 $n_s$ 107.77 5.31 308.22 1.53 0.96 4.38 0.30 0.08 0.86 $Q$ 139.04 18.38 397.68 2.23 0.51 6.38 0.24 0.07 0.67 ${{\mathcal R}}$ 91.43 20.44 261.50 3.33 0.35 9.52 0.65 0.03 1.86 $\tau$ 156.71 9.64 448.22 5.74 2.78 16.42 1.81 1.52 5.18 $\omega_b$ 10.59 1.35 30.28 0.86 0.60 2.46 0.57 0.38 1.64 $\omega_m$ 13.54 0.88 38.72 1.51 0.13 4.31 1.10 0.08 3.14 $\omega_\Lambda$ 114.06 96.36 326.22 110.15 96.15 315.03 98.15 86.00 280.72 $n_s$ 8.64 0.53 24.72 0.54 0.13 1.56 0.36 0.07 1.04 $Q$ 1.46 0.36 4.19 0.20 0.11 0.56 0.17 0.07 0.50 ${{\mathcal R}}$ 13.98 0.78 39.98 1.47 0.12 4.21 1.05 0.07 3.01 $\tau$ 107.58 13.26 307.68 16.50 8.28 47.20 14.02 5.89 40.09 $\omega_b$ 3.10 1.34 8.86 0.80 0.53 2.30 0.32 0.21 0.92 $\omega_m$ 5.09 0.88 14.56 1.24 0.12 3.55 0.55 0.03 1.58 $\omega_\Lambda$ 89.62 72.75 256.33 30.58 22.04 87.46 10.72 9.85 30.65 $n_s$ 1.66 0.52 4.76 0.43 0.13 1.23 0.20 0.05 0.58 $Q$ 0.96 0.36 2.74 0.19 0.10 0.53 0.14 0.05 0.41 ${{\mathcal R}}$ 4.49 0.78 12.85 1.22 0.11 3.48 0.52 0.03 1.49 $\tau$ 12.38 7.90 35.41 4.04 2.65 11.56 1.73 1.48 4.96 Quantity ------------------ --------- -------- --------- -------- ------- -------- ------- --------- -------- marg. fixed joint marg. fixed joint marg. fixed joint $\omega_b$ 281.91 22.18 806.27 6.46 1.11 18.47 1.09 0.25 3.12 $\omega_m$ 446.89 22.12 1278.15 7.75 0.39 22.17 1.61 0.03 4.60 $\omega_\Lambda$ 1248.94 113.78 3572.04 41.61 22.87 119.01 11.60 9.99 33.17 $n_s$ 126.90 5.31 362.93 4.14 0.96 11.85 0.77 0.08 2.22 $Q$ 200.97 18.38 574.78 2.99 0.51 8.55 0.24 0.07 0.68 ${{\mathcal R}}$ 254.76 20.44 728.63 9.56 0.35 27.33 1.19 0.03 3.40 $\alpha$ 111.52 3.74 318.96 2.66 0.06 7.62 0.40 $<0.01$ 1.14 $\tau$ 275.13 9.64 786.88 8.81 2.78 25.19 2.26 1.52 6.45 $\omega_b$ 13.56 1.35 38.78 1.09 0.60 3.12 0.83 0.38 2.37 $\omega_m$ 17.73 0.88 50.71 3.76 0.13 10.74 2.64 0.08 7.55 $\omega_\Lambda$ 137.68 96.36 393.77 111.61 96.15 319.21 98.97 86.00 283.05 $n_s$ 10.10 0.53 28.88 2.18 0.13 6.24 1.49 0.07 4.26 $Q$ 2.41 0.36 6.89 0.20 0.11 0.57 0.18 0.07 0.50 ${{\mathcal R}}$ 23.86 0.78 68.25 1.58 0.12 4.53 1.06 0.07 3.04 $\alpha$ 5.16 0.13 14.76 0.66 0.02 1.88 0.41 0.01 1.18 $\tau$ 111.97 13.26 320.24 26.93 8.28 77.02 20.32 5.89 58.11 $\omega_b$ 7.37 1.34 21.07 0.91 0.53 2.61 0.38 0.21 1.09 $\omega_m$ 6.94 0.88 19.85 1.81 0.12 5.17 0.67 0.03 1.91 $\omega_\Lambda$ 89.69 72.75 256.51 30.89 22.04 88.36 10.79 9.85 30.85 $n_s$ 2.32 0.52 6.65 0.97 0.13 2.77 0.33 0.05 0.93 $Q$ 1.63 0.36 4.67 0.19 0.10 0.54 0.14 0.05 0.41 ${{\mathcal R}}$ 14.22 0.78 40.68 1.43 0.11 4.08 0.60 0.03 1.72 $\alpha$ 3.03 0.13 8.68 0.34 0.02 0.97 0.11 $<0.01$ 0.32 $\tau$ 12.67 7.90 36.23 4.48 2.65 12.80 1.80 1.48 5.15 {width="3.5in"} {width="3.5in"} {width="3.5in"} {width="3.5in"} {width="3.5in"} {width="3.5in"} [|c c c c c c c c c c|]{}\ Direction $i$ & $1/\sqrt{\lambda_i}$ & $\omega_b$ & $\omega_m$ & $\omega_\Lambda$ & $n_s$ & $Q$ & ${{\mathcal R}}$ & $\alpha$& $\tau$\ 1&2.67E-04& 9.9485E-01\* & -9.5907E-02$\dag$ & 3.4445E-06 & -2.9885E-02 & 1.8838E-03 & 1.0970E-02 & 7.0563E-03 & 1.4101E-03\ 2&9.34E-04& 7.1608E-02 & 7.0264E-01\* & -5.5848E-04 & -7.4249E-02 & 2.9712E-02 & -1.1371E-01 & -6.9403E-01$\dag$ & 1.2867E-02\ 3& 2.37E-03& 5.6495E-02 & 5.3116E-01$\dag$ & 7.4496E-04 & 2.6323E-01 & -6.4191E-01\* & 3.8949E-02 & 4.8141E-01 & -7.5954E-03\ 4& 1.11E-02& 1.5030E-02 & -1.1183E-01 & 2.6785E-02 & 7.5467E-01\* & 8.1770E-02 & -1.0330E-01 & -1.8322E-01 & -6.0506E-01$\dag$\ 5&1.56E-02& 4.0281E-02 & 4.2554E-01 & -1.1637E-02 & 1.8055E-01 & 7.5707E-01\* & 1.4249E-01 & 4.2651E-01$\dag$ & 9.5864E-02\ 6& 2.87E-02& 4.3952E-03 & -1.4321E-01 & -1.9274E-02 & 5.5866E-01$\dag$& -3.4689E-02 & -1.1323E-01 & -1.7258E-01 & 7.8942E-01\*\ 7&1.43E-01& -8.9145E-03 & -2.9306E-02 & -6.9614E-02 & 1.0039E-01 & -7.7565E-02 & 9.6787E-01\* & -2.0293E-01$\dag$ & 1.3044E-02\ 8& 2.66E-01& -4.7667E-04 & 3.1536E-03 & 9.9696E-01\* & -5.9578E-04 & 1.0494E-03 & 6.9739E-02$\dag$ & -8.3540E-03 & 3.3561E-02\ \ Direction $i$ & $1/\sqrt{\lambda_i}$ & $\omega_b$ & $\omega_m$ & $\omega_\Lambda$ & $n_s$ & $Q$ & ${{\mathcal R}}$ & $\alpha$& $\tau$\ 1&9.52E-05& 8.0730E-01\* & -4.3681E-01$\dag$ & 7.2721E-05 & 2.1375E-03 & -1.7179E-02 & 6.5966E-02 & 3.9091E-01 & -2.5317E-04\ 2& 1.44E-04& 5.8762E-01\* & 5.8285E-01$\dag$& 1.1644E-04 & -3.6480E-02 & -5.9816E-02 & -8.5667E-02 & -5.5021E-01 & 2.1552E-03\ 3& 5.11E-04& 3.5099E-02 & 6.0865E-01$\dag$ & -3.5134E-05 & 3.5326E-01 & 3.4997E-01 & 2.8062E-02 & 6.1633E-01\* & -1.9763E-02\ 4& 1.89E-03& 4.0108E-02 & -2.1450E-01 & 2.1539E-03 & -6.8889E-02 & 9.3126E-01\* & -1.9318E-03 & -2.8091E-01$\dag$ & -3.8245E-02\ 5&4.95E-03& -4.2534E-03 & 1.0972E-01 & -4.8443E-02 & -4.2498E-01$\dag$ & 6.7120E-02 & 9.4317E-02 & 1.2132E-01 & 8.8140E-01\*\ 6&1.10E-02& 1.0598E-02 & -2.0227E-01 & -4.5823E-02 & 8.0262E-01\* & -3.4226E-02 & -2.1882E-01 & -2.1656E-01 & 4.6553E-01$\dag$\ 7& 1.43E-02& -1.1429E-03 & 6.9110E-03 & -1.3268E-02 & 2.0966E-01$\dag$ & -2.6773E-02 & 9.6472E-01\* & -1.5502E-01 & 1.9638E-02\ 8& 9.16E-02& 5.2242E-05 & -3.4224E-03 & 9.9768E-01\* & 1.9182E-02 & -6.5898E-04 & 7.3693E-03 & -5.4534E-03 & 6.4522E-02$\dag$\ \ Direction $i$ & $1/\sqrt{\lambda_i}$ & $\omega_b$ & $\omega_m$ & $\omega_\Lambda$ & $n_s$ & $Q$ & ${{\mathcal R}}$ & $\alpha$& $\tau$\ 1& 2.67E-05& -1.2266E-01 & 7.5163E-01\* & 8.1216E-06 & -4.6840E-03 & -1.6753E-03 & -1.0827E-01 & -6.3895E-01$\dag$ & 1.9959E-04\ 2&4.30E-05& 9.8772E-01\* & 1.5389E-01$\dag$ & 4.2633E-05 & 2.3505E-02 & -1.5836E-03 & -1.1188E-02 & -6.8644E-03 & -5.1213E-04\ 3& 2.27E-04& -8.6274E-02 & 5.2862E-01$\dag$ & -2.5551E-04 & 3.8678E-01 & 4.0618E-01 & 2.3505E-02 & 6.3051E-01\* & -2.1039E-02\ 4& 1.28E-03& 4.3348E-02 & -2.9931E-01$\dag$ & 5.8579E-03 & 9.2070E-02 & 8.7287E-01\* & 6.9630E-03 & -3.6458E-01 & -7.1777E-02\ 5&2.70E-03& -2.8246E-03 & 1.2912E-01 & -1.2398E-02 & -6.1385E-01$\dag$& 2.2772E-01 & 8.4123E-02 & 1.4231E-01 & 7.2606E-01\*\ 6& 3.93E-03& 5.6917E-03 & -1.4889E-01 & -4.9020E-02 & 6.7866E-01$\dag$ & -1.4028E-01 & -2.2646E-02 & -1.7680E-01 & 6.8071E-01\*\ 7& 5.96E-03& -1.3903E-04 & 5.9101E-02 & 5.2914E-03 & 5.7747E-02 & -3.8594E-02 & 9.8992E-01\* & -9.8529E-02$\dag$ & -4.4876E-02\ 8& 3.19E-02& -7.2460E-05 & -4.1402E-03 & 9.9869E-01\* & 2.4943E-02 & -8.8701E-03 & -5.3456E-03 & -4.0841E-03 & 4.3079E-02$\dag$\ [|c c c c c c c c c c|]{}\ Direction $i$ & $1/\sqrt{\lambda_i}$ & $\omega_b$ & $\omega_m$ & $\omega_\Lambda$ & $n_s$ & $Q$ & ${{\mathcal R}}$ & $\alpha$& $\tau$\ 1& 2.68E-04& 9.9485E-01\* & -9.5954E-02$\dag$ & -3.8023E-05 & -2.9988E-02 & 1.9066E-03 & 1.1031E-02 & 6.9603E-03 & 1.6935E-03\ 2& 9.35E-04& 7.1581E-02 & 7.0263E-01\* & -5.4546E-04 & -7.3932E-02 & 2.9625E-02 & -1.1370E-01 & -6.9410E-01$\dag$ & 1.1876E-02\ 3& 2.37E-03& 5.6582E-02 & 5.3096E-01$\dag$ & 7.0218E-04 & 2.6304E-01 & -6.4218E-01\* & 3.9069E-02 & 4.8138E-01 & -6.9515E-03\ 4& 1.23E-02& 2.0318E-02 & -8.8391E-02 & 2.0436E-02 & 8.6583E-01\* & 1.5967E-01 & -1.0780E-01 & -1.6237E-01 & -4.2215E-01$\dag$\ 5& 1.58E-02& 3.8004E-02 & 4.4637E-01 & -1.3439E-02 & 5.1628E-02 & 7.4470E-01\* & 1.6739E-01 & 4.5598E-01$\dag$ & 7.7154E-02\ 6& 1.72E-01& -4.5506E-03 & -6.9729E-02 & 3.1859E-01 & 3.0219E-01 & -5.4673E-02 & 6.8576E-01\* & -2.0902E-01 & 5.3419E-01$\dag$\ 7& 2.71E-01& 8.7518E-03 & -4.7970E-02 & -1.4708E-02 & 2.5961E-01 & 5.4289E-02 & -6.4792E-01$\dag$ & 4.5317E-02 & 7.1078E-01\*\ 8& 4.26E-01& 1.8060E-03 & 3.0947E-02 & 9.4746E-01\* & -1.1576E-01 & 2.6839E-02 & -2.3604E-01$\dag$ & 8.0201E-02 & -1.5838E-01\ \ Direction $i$ & $1/\sqrt{\lambda_i}$ & $\omega_b$ & $\omega_m$ & $\omega_\Lambda$ & $n_s$ & $Q$ & ${{\mathcal R}}$ & $\alpha$& $\tau$\ 1& 1.05E-04& 7.4727E-01\* & -4.9695E-01$\dag$ & -5.0448E-07 & 3.3342E-03 & -1.0958E-02 & 7.3355E-02 & 4.3488E-01 & 4.5857E-04\ 2& 1.57E-04& 6.6077E-01\* & 5.1936E-01& -1.9827E-05 & -4.4896E-02 & -6.7337E-02 & -7.9760E-02 & -5.2983E-01$\dag$ & 3.8563E-03\ 3& 5.25E-04& 6.1332E-02 & 6.1587E-01\* & 1.2590E-05 & 3.5518E-01 & 3.5940E-01 & 2.5334E-02 & 6.0046E-01$\dag$ & -2.0469E-02\ 4& 1.96E-03& 3.3384E-02 & -2.1678E-01 & -7.5551E-04 & -9.5447E-02 & 9.2928E-01\* & 1.9606E-03 & -2.8126E-01$\dag$ & -1.0701E-02\ 5& 1.04E-02& 9.3302E-03 & -2.2248E-01 & 1.7817E-02 & 8.6225E-01\* & -4.7210E-02 & -8.6676E-02 & -2.6306E-01 & -3.5733E-01$\dag$\ 6& 1.55E-02& -3.0476E-03 & 4.7288E-02 & 1.5170E-03 & 4.6850E-02 & -1.9767E-02 & 9.8892E-01\* & -1.0832E-01$\dag$ & -7.3922E-02\ 7& 5.73E-02& 1.9693E-03 & -7.2588E-02 & -1.4866E-02 & 3.4196E-01$\dag$ & -8.5608E-04 & 4.6134E-02 & -9.7738E-02 & 9.3053E-01\*\ 8& 3.30E-01& -9.4432E-05 & 2.6522E-03 & 9.9973E-01\* & -1.0430E-02 & 1.5550E-03 & 7.2972E-04 & 3.1688E-03 & 2.0310E-02$\dag$\ \ Direction $i$ & $1/\sqrt{\lambda_i}$ & $\omega_b$ & $\omega_m$ & $\omega_\Lambda$ & $n_s$ & $Q$ & ${{\mathcal R}}$ & $\alpha$& $\tau$\ 1& 5.85E-05& 6.7166E-01\* & -4.8908E-01 & 2.0754E-07 & 4.0071E-02 & 2.8188E-02 & 8.0901E-02 & 5.4839E-01$\dag$ & -1.5833E-03\ 2& 1.16E-04& 7.3358E-01\* & 5.4117E-01$\dag$& -1.3289E-05 & 5.7146E-04 & -2.3329E-02 & -7.2058E-02 & -4.0405E-01 & 1.3803E-03\ 3& 2.93E-04& -9.6933E-02 & 6.0597E-01$\dag$ & 1.5493E-05 & 3.5035E-01 & 3.5105E-01 & 1.0230E-02 & 6.1395E-01\* & -1.9331E-02\ 4& 1.72E-03& 3.5113E-02 & -2.1156E-01 & -7.1277E-04 & -7.6886E-02 & 9.3352E-01\* & 1.4161E-02 & -2.7618E-01$\dag$ & -1.2973E-02\ 5& 8.45E-03& 9.6096E-03 & -1.9881E-01 & 1.4790E-02 & 8.6840E-01\* & -6.2132E-02 & 1.6748E-01 & -2.7495E-01 & -3.1390E-01$\dag$\ 6& 1.05E-02& -2.6411E-03 & 1.1073E-01 & -4.8547E-03 & -1.5346E-01$\dag$ & -1.0584E-02 & 9.7974E-01\* & -3.0626E-02 & 5.6666E-02\ 7& 4.19E-02& 1.9014E-03 & -6.4709E-02 & -2.4494E-02 & 3.0336E-01$\dag$ & 2.7857E-05 & -2.5345E-03 & -7.9100E-02 & 9.4706E-01\*\ 8& 2.93E-01& -7.2266E-05 & 1.7408E-03 & 9.9958E-01\* & -6.2208E-03 & 1.5285E-03 & 2.2272E-03 & 1.7693E-03 & 2.8118E-02$\dag$\ Quantity ------------------ --------- -------- --------- -------- -------- -------- -------- ------- -------- marg. fixed joint marg. fixed joint marg. fixed joint $\omega_b$ 241.44 50.73 690.54 6.36 1.01 18.18 0.48 0.25 1.38 $\omega_m$ 99.44 31.71 284.39 3.55 0.34 10.14 0.70 0.03 2.01 $\omega_\Lambda$ 1201.35 719.21 3435.95 39.02 33.98 111.61 11.55 10.20 33.05 $n_s$ 125.97 19.26 360.29 1.48 0.91 4.22 0.30 0.08 0.86 $Q$ 151.63 25.09 433.68 2.20 0.45 6.30 0.24 0.07 0.68 ${{\mathcal R}}$ 87.25 22.00 249.55 3.50 0.31 10.01 0.66 0.03 1.89 $\tau$ 228.76 63.74 654.28 11.45 10.29 32.75 4.23 4.10 12.10 $\omega_b$ 6.00 1.27 17.16 0.83 0.59 2.37 0.56 0.38 1.59 $\omega_m$ 8.63 0.83 24.69 1.47 0.13 4.20 1.09 0.08 3.12 $\omega_\Lambda$ 173.23 89.11 495.44 94.22 88.94 269.48 83.32 79.55 238.30 $n_s$ 4.42 0.52 12.64 0.50 0.13 1.43 0.34 0.07 0.98 $Q$ 0.90 0.35 2.58 0.19 0.10 0.55 0.17 0.07 0.49 ${{\mathcal R}}$ 8.78 0.74 25.10 1.43 0.11 4.10 1.05 0.07 3.00 $\tau$ 659.96 195.96 1887.52 163.30 126.81 467.05 132.38 96.66 378.61 $\omega_b$ 2.77 1.26 7.93 0.77 0.51 2.21 0.32 0.21 0.91 $\omega_m$ 4.54 0.83 12.99 1.17 0.12 3.34 0.55 0.03 1.58 $\omega_\Lambda$ 109.71 87.68 313.79 32.15 31.29 91.95 10.36 9.88 29.63 $n_s$ 1.47 0.52 4.21 0.39 0.13 1.13 0.20 0.06 0.57 $Q$ 0.81 0.35 2.33 0.18 0.10 0.52 0.14 0.05 0.41 ${{\mathcal R}}$ 4.10 0.74 11.72 1.14 0.11 3.27 0.52 0.03 1.49 $\tau$ 63.32 60.36 181.09 10.38 10.06 29.69 3.87 3.81 11.07 Quantity ------------------ --------- -------- --------- -------- -------- -------- -------- --------- -------- marg. fixed joint marg. fixed joint marg. fixed joint $\omega_b$ 569.33 50.73 1628.32 6.41 1.01 18.32 1.11 0.25 3.17 $\omega_m$ 716.71 31.71 2049.84 7.22 0.34 20.66 1.65 0.03 4.71 $\omega_\Lambda$ 1439.68 719.21 4117.59 42.43 33.98 121.36 12.22 10.20 34.96 $n_s$ 299.32 19.26 856.07 3.91 0.91 11.19 0.79 0.08 2.25 $Q$ 174.27 25.09 498.41 3.15 0.45 9.00 0.24 0.07 0.69 ${{\mathcal R}}$ 419.62 22.00 1200.15 9.87 0.31 28.23 1.19 0.03 3.40 $\alpha$ 192.47 3.57 550.48 2.59 0.05 7.42 0.40 $<0.01$ 1.15 $\tau$ 875.90 63.74 2505.14 15.15 10.29 43.34 4.73 4.10 13.52 $\omega_b$ 14.24 1.27 40.73 1.02 0.59 2.92 0.79 0.38 2.27 $\omega_m$ 9.93 0.83 28.41 2.94 0.13 8.42 2.23 0.08 6.37 $\omega_\Lambda$ 173.24 89.11 495.49 108.85 88.94 311.31 93.56 79.55 267.59 $n_s$ 4.59 0.52 13.12 1.58 0.13 4.51 1.16 0.07 3.32 $Q$ 2.44 0.35 6.99 0.20 0.10 0.56 0.17 0.07 0.50 ${{\mathcal R}}$ 26.80 0.74 76.65 1.51 0.11 4.31 1.06 0.07 3.03 $\alpha$ 5.00 0.12 14.31 0.49 0.02 1.41 0.34 0.01 0.96 $\tau$ 710.55 195.96 2032.22 193.10 126.81 552.27 148.41 96.66 424.46 $\omega_b$ 9.59 1.26 27.43 0.87 0.51 2.50 0.38 0.21 1.10 $\omega_m$ 8.25 0.83 23.59 1.63 0.12 4.65 0.67 0.03 1.91 $\omega_\Lambda$ 120.16 87.68 343.67 32.15 31.29 91.95 10.45 9.88 29.89 $n_s$ 2.97 0.52 8.51 0.86 0.13 2.47 0.32 0.06 0.92 $Q$ 1.99 0.35 5.69 0.19 0.10 0.53 0.14 0.05 0.41 ${{\mathcal R}}$ 19.47 0.74 55.69 1.36 0.11 3.90 0.60 0.03 1.72 $\alpha$ 4.32 0.12 12.34 0.31 0.02 0.89 0.11 $<0.01$ 0.32 $\tau$ 64.65 60.36 184.91 10.52 10.06 30.09 3.91 3.81 11.18 FMA with reionization ===================== The existence of a period when the intergalactic medium was reionized as well as its driving mechanism are still to be understood. One possible way of studying this phase is via the CMB polarization anisotropy. The optical depth to electrons of the CMB photons enhances the polarization signal at large angular scales (see Fig. \[figcells\]) introducing a bump in the polarization spectrum at small multipoles. On the other hand reionization decreases the amplitude of the acoustic peaks on the temperature power spectrum at intermediate and small angular scales This signal has now been detected by WMAP via the temperature polarization cross power-spectrum [@Kogut]. In the absence of polarization observations, the optical depth to Thomson scattering is degenerate with the amplitude of the fluctuations, $Q$ (with $Qe^{\tau} = constant$). From previous Fisher Matrix Analysis for a standard model, e.g. [@zaldarriaga2], one expects that the inclusion of polarization measurements will help to better constrain some of the cosmological parameters, by probing the ionization history of the universe, hence constraining $\tau$ and breaking degeneracies of this with other parameters. We will now repeat the analysis of the previous chapter for the case $\tau\neq0$. Analysis results: The FMA forecast ---------------------------------- Tables \[fmasttau\]–\[fmaaltaulow\] summarize the results of our FMA for WMAP, Planck and a CVL experiment. We consider the cases of models with and without a varying $\alpha$ being included in the analysis, and also two values of the optical depth, $\tau=0.2$ (close to the one preferred by WMAP) and $\tau=0.02$. We also consider the use of temperature information alone (TT), E-polarization alone (EE) and both channels (EE+TT) jointly. To show that our FMA fiducial model is close enough to the WMAP best fit model to produce similar FMA results, we display in Table \[fmaalwmap\] the results of our FMA using as fiducial model the WMAP best fit model. For the sake of completeness we also consider the case (TE) alone as well as (EE+TE) and (EE+TT+TE) for WMAP 4-years. Table \[fmaalwmap4\] display the results of our FMA for WMAP 4-years using the WMAP fiducial model. The FMA predictions for WMAP - 4years are to be compared with the recent WMAP 1-year results. The errors in most of the other cosmological parameters are unaffected by the presence of reionization *if* one has both temperature and polarization data. If one has just one of them then the accuracy is quite different, and also it will depend on whether has high or low $\tau$. This is because different degeneracies may be dominant in each case, while combining temperature and polarization information helps break such degeneracies. The inclusion of the new parameter $\tau$ for a standard model reduces the accuracy in other parameters for all but the CVL experiment (and in this case for all but $\omega_{\Lambda}$) as can be seen from a comparison of Table \[fmasttau\] with Table \[fmast\]. Comparing Tables \[fmasttau\] (for $\tau=0.20$) and \[fmasttaulow\] (for $\tau=0.02$) an immediate effect of considering a large value of $\tau$ is to increase the accuracy on $\tau$ itself. For example the case with temperature and polarization information used jointly, the accuracy on the other parameters is not necessarily reduced by considering a larger value of $\tau$ while its accuracy remains almost the same for a CVL experiment. Whilst comparing Tables \[fmaaltau\] (for $\tau=0.20$) and \[fmaaltaulow\] (for $\tau=0.02$) the effect of a large value of $\tau$, considering the case temperature and polarization used jointly, for WMAP is to increase the accuracy on most of the parameters particularly noticeable for the parameters $\alpha$ and $\tau$; for Planck only the accuracy on $\tau$ is improved while the other parameters have slightly worse accuracy; finally for a CVL experiment the accuracy is the same for all but $\omega_{\Lambda}$ which is slightly worse, and $\tau$ which is much better. It is interesting to note that while for WMAP a large value of $\tau$ does indeed help to improve the accuracy on most parameters, for Planck and a CVL experiment the accuracy is improved using polarization data alone but the inverse is true using temperature data alone. Hence it is not surprising the results obtained when one considers temperature and polarization jointly. As we go from Table \[fmaalwmap\] to Table \[fmaalwmap4\] the accuracy on all parameters increases as should be expected. For the WMAP - 4 years one predicts an accuracy of 3% and 11% on $\alpha$ and $\tau$ respectively as opposed to 4% and 14% respectively, for the 2-year mission. The results of our forecast are that WMAP (2-years mission) is able to constrain $\tau$ with accuracy of the order 13%, which is approximately two times better than the current precision obtained from the WMAP 1-year observations, of the order of 23%. While our FMA predictions for WMAP - 4 years, gives an accuracy of the order 10% using all (TT+EE+TE) temperature, polarization and temperature-polarization cross correlation information. Planck and a CVL experiment can constrain $\alpha$ with accuracies of the order 0.3% and 0.1% respectively and $\tau$ with accuracies of the order 4.5% and 1.8% respectively. For WMAP the accuracy on $\tau$ from polarization data alone is worse by a factor of 2 than from temperature alone. On the other hand, for Planck and the CVL experiment the accuracy from polarization is better by a factor of 3 and 8 respectively, than from temperature alone. While the accuracy on $\alpha$ from polarization alone is worse by a factor of the order 22 and 4 than from temperature alone for WMAP and Planck respectively. For a CVL experiment the accuracies are similar for both polarization and temperature data alone. The accuracy on $\tau$ obtained with Planck using Temperature data alone is roughly the same as a CVL experiment. This suggests that Planck is indeed a cosmic variance limited experiment with respect to Temperature. The inclusion of polarization the accuracy for the CVL experiment is improved by a factor of 4 when compared to Planck satellite. Analysis results: Confidence contours ------------------------------------- As before, we show in Figs. \[figellipse3\]-\[figellipse4\] all joint 2D confidence contours (all remaining parameters marginalized). As previously in the WMAP case the errors from E only are very large, hence the contours for T coincide almost exactly with the temperature-polarization combined case. In the CVL case it is the E contours that almost coincide with the combined ones. From Fig. \[figellipse3\] without $\alpha$, we can infer a good agreement between our predictions and WMAP observations. Particularly striking is the good agreement for the contour plots in the ($n_{s}$,$\tau$) plane which clearly exhibits the observed degeneracy [@Spergel]. For Planck the inclusion of polarization data helps to break degeneracies in particular between $\tau$ and the other parameters for example with $n_{s}$. For a CVL experiment the contours are further narrowed with the joint temperature polarization analysis in agreement with the tabulated accuracies on $\tau$. Again, looking at Fig. \[figellipse4\] with $\alpha$, our predictions for the contour plots in the plane ($\tau$,$n_{s}$) are in close agreement with the observed degeneracy [@Verde]. This same plot shows that the degeneracy direction between $\alpha$ and $n_{s}$ is almost orthogonal to that between $\tau$ and $n_{s}$, The net result of this is a better accuracy on $\alpha$ when the parameter $\tau$ is included (compare Tables \[fmaal\] and \[fmaaltau\]) while the accuracy on $\tau$ itself remains almost unchanged with inclusion of $\alpha$ (compare Tables \[fmasttau\] and \[fmaaltau\]). This is in agreement with our discussion in section III, and quantitatively explains why our $\alpha$ mechanism (summarized in Fig. \[figpeaks\]) works. The accuracy on $n_{s}$ is similar to that obtained without $\tau$ (compare Tables \[fmaal\] and \[fmaaltau\]) but gets worse with inclusion of $\alpha$ (compare Tables \[fmasttau\] and \[fmaaltau\]). In other words the inclusion of reionization helps to lift most of the degeneracies when using information from both the temperature and polarization jointly hence increasing the accuracies for the cases of interest , ie, $\alpha$ and $\tau$. Quantity ------------------ -------- -------- --------- -------- ------- -------- ------- --------- -------- marg. fixed joint marg. fixed joint marg. fixed joint $\omega_b$ 285.33 26.18 816.08 5.84 0.87 16.70 0.96 0.12 2.73 $\omega_m$ 445.06 28.16 1272.90 7.48 0.46 21.41 1.40 0.03 4.00 $\omega_\Lambda$ 184.17 144.61 3386.80 44.12 24.08 126.18 12.83 9.33 36.70 $n_s$ 161.11 6.14 460.78 4.22 1.00 12.08 0.71 0.08 2.04 $Q$ 191.24 21.06 546.95 2.91 0.55 8.32 0.25 0.07 0.73 ${{\mathcal R}}$ 221.83 21.69 634.44 8.81 0.35 25.19 0.79 0.02 2.26 $\alpha$ 113.11 4.52 323.49 2.61 0.07 7.48 0.32 0.00 0.91 $\tau$ 336.62 11.25 962.75 9.25 3.05 26.45 2.32 1.30 6.63 $\omega_b$ 18.50 0.98 52.91 0.98 0.35 2.80 0.73 0.24 2.08 $\omega_m$ 17.89 0.94 51.17 3.30 0.14 9.45 2.31 0.08 6.60 $\omega_\Lambda$ 149.92 83.49 428.77 107.48 83.30 307.39 94.61 74.50 270.59 $n_s$ 9.50 0.54 27.17 2.07 0.14 5.91 1.42 0.07 4.06 $Q$ 3.27 0.37 9.36 0.21 0.11 0.60 0.19 0.07 0.53 ${{\mathcal R}}$ 34.95 0.72 99.97 1.34 0.10 3.84 0.86 0.06 2.45 $\alpha$ 7.95 0.13 22.75 0.59 0.02 1.69 0.37 0.01 1.06 $\tau$ 119.62 17.00 342.11 32.86 9.93 93.98 25.31 6.84 72.38 $\omega_b$ 9.15 0.98 26.18 0.84 0.32 2.39 0.37 0.11 1.07 $\omega_m$ 7.55 0.94 21.58 1.62 0.13 4.65 0.61 0.03 1.75 $\omega_\Lambda$ 95.34 71.51 272.68 32.24 22.94 92.22 11.80 9.21 33.76 $n_s$ 2.58 0.54 7.39 0.93 0.14 2.67 0.33 0.05 0.94 $Q$ 1.77 0.37 5.06 0.19 0.11 0.56 0.15 0.05 0.43 ${{\mathcal R}}$ 17.55 0.71 50.19 1.19 0.10 3.42 0.49 0.02 1.40 $\alpha$ 3.89 0.13 11.12 0.31 0.02 0.88 0.10 $<0.01$ 0.30 $\tau$ 13.57 9.49 38.81 4.71 2.92 13.48 1.81 1.28 5.18 ------------------ -------- ------- --------- -------- ------- -------- Quantity marg. fixed joint marg. fixed joint $\omega_b$ 110.64 16.58 316.44 7.33 0.81 20.96 $\omega_m$ 49.48 17.16 141.52 8.91 0.77 25.49 $\omega_\Lambda$ 622.34 97.58 1779.93 113.30 83.39 324.06 $n_s$ 69.43 4.89 198.58 6.68 0.53 19.11 $Q$ 79.22 13.51 226.58 0.90 0.32 2.58 ${{\mathcal R}}$ 46.52 13.04 133.06 9.25 0.59 26.47 $\tau$ 100.84 8.21 288.40 102.72 16.70 293.79 $\omega_b$ 2.14 0.80 6.11 2.13 0.80 6.08 $\omega_m$ 3.09 0.77 8.85 3.08 0.77 8.81 $\omega_\Lambda$ 90.70 63.84 259.41 86.97 62.69 248.75 $n_s$ 1.46 0.52 4.18 1.45 0.52 4.15 $Q$ 0.52 0.32 1.48 0.52 0.32 1.48 ${{\mathcal R}}$ 2.86 0.59 8.17 2.84 0.59 8.12 $\tau$ 10.52 7.45 30.08 10.41 7.44 29.78 ------------------ -------- ------- --------- -------- ------- -------- ------------------ -------- ------- --------- -------- ------- -------- Quantity marg. fixed joint marg. fixed joint $\omega_b$ 173.74 16.58 496.91 14.09 0.81 40.30 $\omega_m$ 260.62 17.16 745.40 13.76 0.77 39.36 $\omega_\Lambda$ 637.28 97.58 1822.66 133.73 83.39 382.47 $n_s$ 108.18 4.89 309.41 7.86 0.53 22.47 $Q$ 96.60 13.51 276.30 2.33 0.32 6.67 ${{\mathcal R}}$ 133.23 13.04 381.04 26.29 0.59 75.19 $\alpha$ 69.10 2.48 197.62 5.83 0.12 16.66 $\tau$ 228.69 8.21 654.07 103.86 16.70 297.05 $\omega_b$ 7.50 0.80 21.44 7.41 0.80 21.18 $\omega_m$ 5.48 0.77 15.66 5.46 0.77 15.62 $\omega_\Lambda$ 91.57 63.84 261.91 87.48 62.69 250.20 $n_s$ 2.03 0.52 5.82 2.03 0.52 5.81 $Q$ 1.31 0.32 3.73 1.30 0.32 3.71 ${{\mathcal R}}$ 14.34 0.59 41.01 14.17 0.59 40.53 $\alpha$ 3.08 0.11 8.80 3.05 0.11 8.71 $\tau$ 10.65 7.45 30.46 10.52 7.44 30.08 ------------------ -------- ------- --------- -------- ------- -------- {width="3.5in"} {width="3.5in"} Analysis results: Principal directions -------------------------------------- Our previous discussion of principal directions changes completely when reionization is included, as polarization data helps to better constrain the fine structure constant and removes the existing degeneracies between $\alpha$ and $\tau$ see Table \[fmaev\] In Table \[fmaev\] we display the eigenvectors and eigenvalues for WMAP, Planck and a CVL experiment when reionization is included (with $\tau=0.20$) for Temperature and Polarization considered jointly. Planck’s errors, as measured by the inverse square root of the eigenvalues, are smaller by a factor of about $5$ on average that those for WMAP. In the case of a CVL experiment’s errors are smaller by a factor of about $3$ on average than those for Planck. The physical parameter $\tau$ is the largest parameter contribution to the principal direction 6 for both WMAP and Planck, and is the second largest to direction 4 for WMAP and to direction 6 and 8 for Planck. While for a CVL experiment it becomes the main contributor for principal directions 5 and 6 and the second largest for direction 8. For 4 of the 8 eigenvectors Planck obtains a better alignment of the principal directions with the axis of the physical parameters when compared with WMAP. This indicates that the inclusion of the reionization parameter $\tau$ already helps to break degeneracies for WMAP, when we compare the number 4 in 8 against 6 in 7 for the case without reionization. On the other hand, only for 4 of the 8 eigenvectors CVL obtains a better alignment of the principal directions with the axis of the physical parameters when compared with Planck, against 6 in 7 for the case without reionization. The physical parameter $\alpha$ is the second largest contributor for principal directions 2, 5 and 7 for WMAP and for direction 4 for Planck being the main contributor for direction 3. For a CVL experiment it becomes the second largest for directions 1 and 7 and main contributor for direction 3 just like for Planck. In Table \[fmaevTT\] we display the principal directions considering Temperature only. Comparing Tables \[fmaevTT\] and \[fmaev\], we conclude that for 4 of the 8 eigenvectors WMAP obtains a better alignment of the principal directions with the axis of the physical parameters when polarization is included (with similar alignement for the others). While for Planck and a CVL experiment only for 3 of the 8 eigenvectors the alignement is better (with similar alignement for the others). When polarization is included, the largest and second largest physical parameter contributors remain the same for all but for directions 6 and 7 for WMAP, directions 2,3,6 and 7 for Planck, and directions 1,4,6 and 7 for a CVL experiment. For Planck for the case with temperature only, the second largest contributor to direction 2 and 6, the physical parameter $\alpha$ is shifted to $\omega_{m}$ and $n_{s}$ respectively while direction 3 becomes mainly contributed by $\alpha$, when polarization is included. This indicates that when including polarization the degeneracies with $\alpha$ are indeed being broken. The CVL case shows that the changes ocurring with inclusion of Polarization when reionization is considered is not a simple rescaling of contributions from the physical parameters to the principal directions but a rescaling by different factors for each of these physical parameters resulting in changes of the degeneracy directions. To demonstrate that this is indeed the case let us analyse both Tables in detail for the CVL case. For instance direction 1 remains unchanged with inclusion of polarization, while for direction 2 $\alpha$ is the third contributor by an amount similar to $\omega_{m}$ but is much reduced when polarization is included. Also this direction is better aligned with $\omega_{b}$ when temperature and polarization are considered jointly. This indicates that inclusion of polarization helped to break the degeneracy between $\omega_{b}$ and $\alpha$. Direction 3 remains aligned with $\omega_{\Lambda}$ for both cases. Direction 5 exchanges the largest and second largest contributions from $n_{s}$ to $\tau$ when polarization is included in the analysis reducing the contribution from $\alpha$. So the inclusion of polarization helps to better define a direction of degeneracy between $\tau$ and $n_{s}$ by breaking the degeneracy with $\alpha$. The degeneracy between $Q$ and $\alpha$ is also broken by shifting the second largest contributor to direction 4 from $\alpha$ to $\omega_{m}$. The second largest contributor to direction 7 is shifted from $n_{s}$ to $\alpha$ when polarization is included indicating that the degeneracy between $\mathcal R$ and $\alpha$ is now dominating over the other degeneracies with $\alpha$. ---------- ------- ------- ------- ------- --------- ------- marg. fixed joint marg. fixed joint $\alpha$ 2.66 0.06 7.62 0.40 $<0.01$ 1.14 $\tau$ 8.81 2.78 25.19 2.26 1.52 6.45 $\alpha$ 0.66 0.02 1.88 0.41 0.01 1.18 $\tau$ 26.93 8.28 77.02 20.32 5.89 58.11 $\alpha$ 0.34 0.02 0.97 0.11 $<0.01$ 0.32 $\tau$ 4.48 2.65 12.80 1.80 1.48 5.15 ---------- ------- ------- ------- ------- --------- ------- : \[fmaresults\]Fisher matrix analysis results for a model with varying $\alpha$ and reionization: expected $1\sigma$ errors for the Planck satellite and for the CVL experiment (see the text for details). The column [*marg.*]{} gives the error with all other parameters being marginalized over; in the column [*fixed*]{} the other parameters are held fixed at their ML value; in the column [*joint*]{} all parameters are being estimated jointly. ![\[figlike\] Ellipses containing $95.4\%$ ($2\sigma$) of joint confidence in the $\alpha$ vs. $\tau$ plane (all other parameters marginalized), for the Planck and cosmic variance limited (CVL) experiments, using temperature alone (red), E-polarization alone (yellow), and both jointly (white). The dashed contour represents the WMAP - 4years forecast using (TT+EE+TE) jointly. ](alpha_vs_tau.ps){width="3.5in"} The $\alpha$-$\tau$ degeneracy ------------------------------ Our results clearly indicate a crucial degeneracy between $\alpha$ and $\tau$. In order to study it in more detail, we have extracted the relevant results from Table \[fmaaltau\] and Fig. \[figellipse4\] and re-displayed them in Table \[fmaresults\] and Fig. \[figlike\]. Both of these summarize the forecasts for the precision in determining both parameters with Planck and the CVL experiment. It is apparent from Fig. \[figlike\] that TT and EE suffer from degeneracies in different directions, for the reasons explained above. Thus combining high-precision temperature and polarization measurements one can constrain most effectively constrain both variations of $\alpha$ and $\tau$. Planck will be essentially cosmic variance limited for temperature but there will still be considerable room for improvement in polarization. This therefore argues for a post-Planck polarization-dedicated experiment, not least because polarization is, in itself, better at determining cosmological parameters than temperature. We conclude that Planck data alone will be able to constrain variations of $\alpha$ at the epoch of decoupling with 0.34 % accuracy ($1\sigma$, all other parameters marginalized), which corresponds to approximately a factor 5 improvement on the current upper bound. On the other hand, the CMB *alone* can only constrain variations of $\alpha$ up to ${\cal O}(10^{-3})$ at $z \sim 1100$. Going beyond this limit will require additional (non-CMB) priors on some of the other cosmological parameters. This result is to be contrasted with the variation measured in quasar absorption systems by Ref.[@Webb], $\delta \alpha / \alpha_0 = {\cal O}(10^{-5})$ at $z \sim 2$. Nevertheless, there are models where deviations from the present value could be detected using the CMB. Conclusions =========== We have presented a detailed analysis of the current WMAP constraints on the value of the fine-structure constant $\alpha$ at decoupling. We have found that current constraints on $\alpha$, coming from WMAP alone, are as strong as all previously existing cosmological constraints (CMB combined with additional data, e.g. coming from type Ia supernovae or the HST Key project) put together. On the other hand, we have also shown that the CMB *alone* can determine $\alpha$ to a maximum accuracy of $0.1\%$ - one can only improve on this number by again combining CMB data with other observables. Note that such combination of datasets is not without its subtleties—see [@Martins] for a discussion of some specific issues related to this case. Hence this accuracy is well below the $10^{-5}$ detection of Webb *et al.* [@Webb]. However one must keep in mind that one is dealing with much higher redshifts (about one thousand rather than a few). Given that in the simplest, best motivated models for $\alpha$ variation, one expects it to be a non-decreasing function of time, one finds that a constraint of $10^{-3}$ at the epoch of decoupling can be as constraining for these models as the Webb *et al.* results. In addition, there are also constraints on variations of $\alpha$ at the epoch of nucleosynthesis, which are at the level of $10^{-2}$ [@Avelino]. The main difference between them is that while CMB constraints are model independent, the BBN ones are not (they rely on the assumption of the Gasser-Leutwyler phenomenological formula for the dependence of the neutron-proton mass difference on $\alpha$). As discussed in the main text, we focused our analysis on model independent constraints, and in fact explicitly avoided discussing constraints for specific models. Nevertheless it is quite easy, given the constraints (and forecasts) presented here, to translate them into constraints for the specific free parameters of one’s preferred model. We have also presented a thorough analysis of future CMB constraints on $\alpha$ and the other cosmological parameters, specifically for the WMAP and Planck Surveyor satellites, and compared them to those for an ideal (cosmic variance limited) experiment. Comparisons with currently published (1 year) WMAP data indicates that our Fisher Matrix Analysis pipeline is quantitatively robust and accurate. By separately studying the temperature and polarization channels, we have explicitly shown that the degeneracy directions can be quite different in the two cases, and hence that by combining them many such degeneracies can be broken. We have also shown that in the ideal case CMB (EE) polarization is a much more accurate estimator of cosmological parameters than CMB temperature. Nevertheless, polarization measurements are much harder to do in practice. For example, for the case of WMAP the (EE) channel will provide a quite modest contribution for the overall parameter estimation analysis. This situation is quite different for Planck: here the contributions of the temperature and polarization channels are quite similar. In fact we have also shown that Planck’s temperature measurements will be almost cosmic variance limited, while its polarization measurements will be well below this ideal limit. (This fact was previously known, but it had never been quantified as was done in the present paper.) Hence this, together with the fact that polarization is intrinsically superior for the purpose of cosmological parameter estimation, make a strong case for a post-Planck, polarization-dedicated experiment. Our analysis can readily be repeated for other experiments. It should be particularly enlightening to study cases of interferometer experiments and compare them with the WMAP and Planck satellites. On the other hand it would also be possible to extend it to include gravity waves, iso-curvature modes, or a dark energy component different from a cosmological constant. However, none of these is currently required by existing (CMB and other) data, and the latter two are in fact strongly constrained. To conclude, the prospects of further constraining $\alpha$ at high redshift are definitely bright. In addition, further progress is expected at low redshift, where at least three (to our knowledge) independent groups are currently trying to confirm the Webb *et al.* [@Webb; @Jenam; @Murphy] claimed detection of a smaller $\alpha$. All of these are using VLT data, while the original work [@Webb; @Jenam; @Murphy] used Keck data. This alone will provide an important test of the systematics of the pipeline, plus in addition the three groups are using quite different methods. These and other completely new methods that may be devised thus offer the real prospect of an accurate mapping of the cosmological evolution of the fine-structure constant, $\alpha=\alpha(z)$. Finally, a point which we have not discussed at all for reasons of space, but which should be kept in mind in the context of forthcoming experiments, is that any time variation of $\alpha$ will be related (in a model-dependent way) to violations of the Einstein Equivalence principle [@Will]. Thus a strong experimental and/or observational confirmation of either of them will have revolutionary implications not just for cosmology but for physics as a whole. acknowledgments =============== We would like to thank Anthony Challinor and Anthony Lasenby for useful discussions. G.R. acknowledges a Leverhulme Fellowship at the University of Cambridge, R.T. is partially supported by the Swiss National Science Foundation and the Schmidheiny Foundation and C.M. is funded by FCT (Portugal), under grant FMRH/BPD/1600/2000. This work was done in the context of the European network CMBnet, and was performed on COSMOS, the Origin3800 owned by the UK Computational Cosmology Consortium, supported by Silicon Graphics/Cray Research, HEFCE and PPARC.

--- abstract: 'The phonon dispersions of [[$\mathrm{Nd_{1.86}Ce_{0.14}CuO_{4+\delta}}$ ]{}]{}along the $[\xi,0,0]$ direction have been determined by inelastic x-ray scattering. Compared to the undoped parent compound, the two highest longitudinal phonon branches, associated with the Cu-O bond-stretching and out-of-plane oxygen vibration, are shifted to lower energies. Moreover, an anomalous softening of the bond-stretching band is observed around $\mathbf{q}=(0.2,0,0)$. These signatures provide evidence for strong electron-phonon coupling in this electron-doped high-temperature superconductor.' author: - 'M. d’Astuto' - 'P. K. Mang' - 'P. Giura' - 'A. Shukla' - 'P. Ghigna' - 'A. Mirone' - 'M. Braden' - 'M. Greven' - 'M. Krisch' - 'F. Sette' title: 'Anomalous Dispersion of Longitudinal Optical Phonons in $\mathrm{\mathbf{Nd_{1.86}Ce_{0.14}CuO_{4+\bm{\delta}}}}$ Determined by Inelastic X-ray Scattering' --- While the coupling between electrons and phonons is known to be the driving mechanism for Cooper-pair formation in conventional superconductors, its role in the high-critical-temperature superconductors (HTcS) is the subject of intense research efforts. Recently, evidence for electron-phonon coupling has been invoked in the interpretation of inelastic neutron scattering (INS) and angle-resolved photoemission spectroscopy (ARPES) experiments. The INS studies, carried out on [[$\mathrm{La_{1.85}Sr_{0.15}CuO_{4+\delta}}$ ]{}]{}[@pintscB; @pingin; @mcqueeny; @pinbrief], oxygen-doped [[$\mathrm{La_{2}CuO_{4+\delta}}$ ]{}]{}[@lacuod] and [[$\mathrm{YBa_{2}Cu_{3}O_{6+\delta}}$ ]{}]{}[@pingin; @reichardt] reveal an anomalous softening with doping of the highest longitudinal optical (LO) phonon branch, in particular along the $q=[\xi,0,0]$ direction. This branch is assigned to the Cu-O bond-stretching mode [@raman; @pingin]. The observed softening has been interpreted as a signature of a strong electron-phonon coupling [@pingin; @mcqueeny], which has been discussed since the discovery of HTcS [@bednorz; @weber]. Furthermore, in an energy range similar to the LO bond-stretching phonon band, ARPES studies on three different families of hole-doped HTcS reveal a distinct “kink” anomaly in the quasiparticle dispersion [@lanzara]. The scenario emerging from the above INS and ARPES works suggests a strong coupling between the charge carriers and the Cu-O bond-stretching phonon modes to be ubiquitous in HTcS materials, at least for hole-doped compounds. However, its role in the pairing mechanism remains completely unclear [@pinbrief]. At the moment, it may not even be excluded that the electron-phonon interaction is pair-breaking for the d-wave superconducting order parameter. Therefore, it appears very important to analyze the strength of the phonon anomalies in as many cuprate families as possible and to compare them with their superconducting properties. In this context, the electron-doped cuprates are of central importance due to the distinct character of the doped charges in this material: Cu $3d_{x^2-y^2}$ (O $2p$) for n(p)-type cuprates [@ttu], leading to a very different electronic structure [@armitage]. Since the phonon anomalies are related to a coupling between charge fluctuations and phonons, charges with different character may induce quite different electron-phonon interactions. In this Letter, we present an [inelastic x-ray scattering ]{}(IXS) study of the phonon dispersion in the n-type cuprates [[$\mathrm{Nd_{1.86}Ce_{0.14}CuO_{4+\delta}}$ ]{}]{}(NCCO). Inelastic x-ray scattering can overcome the main limitation of inelastic neutron scattering, [*i.e.*]{} the need for sufficiently large single crystals of high chemical and structural quality [@pingin]. Lateral x-ray beam sizes of few tens of $\mu$m are routinely obtained. Moreover, at photon energies around 10-20 keV and $Z >$ 3, the total cross section is dominated by photoelectric absorption, and therefore the typical x-ray penetration depths for high-$Z$ materials is of the order of 10 - 100 $\mu$m. Consequently, very small samples (down to less than $10^{-4}$ mm$^{3}$) can be studied with signal rates comparable to typical INS experiments on cm$^3$-sized samples. Despite these advantages, little work has been done using IXS on the HTcS compounds [@burkel]. We choose $\mathrm{Nd_{2-x}Ce_{x}CuO_{4+\delta}}$ for our IXS study, since its crystallographic structure is one of the simplest among the HTcS, and because extensive INS studies exist for its undoped parent compound [[$\mathrm{Nd_2CuO_{4+\delta}}$ ]{}]{}(NCO) [@pintscB; @pingin]. Our interest is focused on the $[\xi,0,0]$ direction, where the LO branch displays its strongest anomaly for hole-doped HTcS [@mcqueeny; @pinbrief]. The present results reveal that, near the zone center, the two highest longitudinal optical branches, assigned to the Cu-O bond-stretching and O(2) vibration modes, are shifted to lower frequencies with respect to the undoped parent compound. The interpretation of our data is supported by lattice dynamics calculations, taking into account a *Thomas-Fermi* screening mechanism. Furthermore, we observe an anomalous softening of the highest branch around $\mathbf{q}=(0.2,0,0)$. Our results demonstrate that this anomalous behavior of the high-energy LO phonon branch is a universal property of both hole- and electron-doped HTcS compounds, therefore providing further evidence that electron-phonon interactions may play an important role in high-Tc superconductivity. Furthermore, the present results are an important demonstration of IXS as a powerful tool for the study of the lattice dynamics in small, high-quality crystals of complex transition metal oxides. The experiment was carried out at the very-high-energy-resolution IXS beam-line ID16 at the European Synchrotron Radiation Facility (ESRF). X-rays from an undulator source were monochromated using a Si (111) double-crystal monochromator, followed by a high-energy-resolution backscattering monochromator [@ixsmono], operating at 15816 eV (Si (888) reflection order). A toroidal gold-coated mirror refocused the x-ray beam onto the sample, where a beam size of $250 \times 250~ \mu$m$^2$ full-width-half-maximum (FWHM) was obtained. The scattered photons were energy-analyzed by a spherical silicon crystal analyzer 3 m in radius, operating at the same Bragg reflection as the monochromator [@ixsana]. The total energy resolution was 1.6 THz (6.6 meV) FWHM. The momentum transfer $\mathbf{Q}$ was selected by rotating the 3 m spectrometer arm in the scattering plane perpendicular to the linear x-ray polarization vector of the incident beam. The momentum resolution was set to $\approx 0.087~\mathrm{\AA^{-1}}$ in both the horizontal and the vertical direction by an aperture of $20\times20$ mm$^2$ in front of the analyzer. Further experimental details are given elsewhere ([@ixsmono; @ixsana] and references therein). The sample is a single crystal grown by the traveling-solvent floating-zone method in 4 atm of $\mathrm{O_2}$ at Stanford University. It has been reduced under pure Ar atmosphere at $920^{\circ} $C for 20 h, followed by a further 20 h of exposure at $500^{\circ}$C to a pure $\mathrm{O_2}$ atmosphere. Such a procedure is necessary to produce a superconducting phase in $\mathrm{Nd_{1.86}Ce_{0.14}CuO_{4+\delta}}$, although its exact effect is not understood. Following this treatment the sample had a narrow superconducting transition with an onset temperature of $T_c=24$ K. The sample is of very good crystalline quality, with a rocking curve width of 0.02$^{\circ}$ (FWHM) around the $[h,0,0]$ direction. It was mounted on the cold finger of a closed-loop helium cryostat, and cooled to 15 K. The experiment was performed in reflection geometry, and the probed scattering volume corresponded to about $1.5\times10^{-3}$ mm$^3$. IXS scans were performed in the -2$<\nu<$24 THz range, in the $\bm{\tau}=(6,0,0)$ and $\bm{\tau}=(7,1,0)$ Brillouin zones [^1]. The data were collected along the following three lines:\ I) $\mathbf{Q}=(6+\xi,0,0)$, in longitudinal configuration ([*i*.e.]{} with $\mathbf{q}=(\xi,0,0)~ \parallel\mathbf{Q}$),\ II) $\mathbf{Q}=(7\pm\xi,1,0)$ in almost longitudinal configuration ([*i*.e.]{} with $\mathbf{q} = (\xi,0,0) $ and $(\mathbf{Q} \cdot \mathbf{q})/Q \approx q$),\ III) $\mathbf{Q}=(7,1-\xi,0)$ in almost transverse configuration ($\mathbf{q} = (0,\xi,0)$ and $(\mathbf{Q} \cdot \mathbf{q})/Q \approx 0$).\ The low temperature and high momentum transfer were chosen so as to optimize the count rate on the high-frequency optical mode while limiting the loss of contrast due to the contribution from the tails of the intense low frequency acoustic modes. Fig. \[figa\] shows a typical energy scan in almost longitudinal geometry at $\mathbf{Q}=(6.8,1,0)$, corresponding to $\xi=0.2$. The data are shown together with the results of a fit, where the excitations where modeled by harmonic oscillators, convoluted with the instrumental resolution function. Three features can be clearly distinguished: (i) an elastic peak at 0 THz, due to chemical disorder and, possibly, strain due to the different thermal expansion coefficients between the sample and the sample support during cooling, (ii) the longitudinal acoustic phonon, centered near 2.7 THz and (iii) a weaker feature around 13 THz. In the intermediate energy region (from about 4 to 10 THz), no distinct phonon peaks are resolved due to the dominating contribution from the tails of the elastic and acoustic phonon signals. The inset of Fig. 1 emphasizes the high-energy region of the spectrum. One can clearly distinguish two phonons, centered around 12 and 13.5 THz, respectively. The weak shoulder at around 16 THz can be attributed to an admixture of the transverse optical (TO) mode, as indicated by the comparison (after normalization to the same intensity at 20 THz) with the equivalent scan in transverse geometry $\mathbf{Q}=(7,0.8,0)$. Consequently, the two stronger peaks are unambiguously assigned to longitudinal optical (LO) modes. From the absence of other higher frequency modes up to 23 THz, we conclude that the two modes at 12 and 13.5 THz can be identified with the O(2) vibration and Cu-O bond-stretching modes, respectively (see Refs. ). ![\[figb\] IXS spectra in the $\bm{\tau}=(7, 1, 0)$ Brillouin zone with propagation vector $\mathbf{q}$ along $a^*$, as indicated in the figure. The experimental data (circles) are shown together with their corresponding harmonic oscillator model best fits (solid lines), as discussed in the text.](scans1s.eps) In order to determine the dispersion of the two highest LO branches, IXS spectra were recorded for $0<\xi\le1$. In Fig. \[figb\] we show three spectra taken along the $(7-\xi,1,0)$ direction. Close to the zone center, at $\mathbf{q}=(0.1, 0, 0 )$, the highest frequency mode is observed slightly above 15 THz. At $\mathbf{q}=(0.2, 0, 0 )$ the highest mode is found at the much lower energy of 13.5 THz. Finally, for $\mathbf{q}=(0.4, 0, 0)$, we again find a high-frequency mode around 15.5 THz. ![\[figc\] Right hand side: ($\circ$, $\bullet$) experimental longitudinal phonon frequencies determined from IXS spectra in NCCO at $T=15$ K along the $[\xi,0,0]$ direction. Solid circles ($\bullet$) emphasize the highest energy frequencies measured. Solid (dot-dashed) lines indicate lattice dynamics calculation with a screened (unscreened) Coulomb interaction.\ Left hand side: magnification of the high energy portion of the right hand side graph showing the dispersion of the top two phonon branches. The highest frequency branch ($\triangle$), as measured by INS (from Ref. ) in the insulating parent compound [[$\mathrm{Nd_2CuO_{4+\delta}}$ ]{}]{}is shown for comparison. The experimental data are in agreement with calculations, except for the anomalous softening at $q=(0.2,0,0)$ of the two higher energy branches (see text).](nccoLasc7.eps) The peak positions extracted from these and many other scans are summarized in Fig. \[figc\]. The highest branch exhibits a sharp dip around $\mathbf{q}=(0.2, 0, 0)$ and recovers for larger $q$-values. This behavior is most likely due to an anti-crossing with the second highest branch which is mainly associated with vibrations of the O(2) position in the $\xi$-direction. Therefore, within a standard anti-crossing framework, one should interpret the highest longitudinal intensities for $\mathbf{q}=(0.3, 0, 0)$ and above as being mainly due to O(2) vibration. Within that scenario, this second-highest branch increases its frequency in the middle of the zone as in the undoped compound, and, except for the fact that the *Lyddane-Sachs-Teller* (LO-TO) gap closes, seems to be insensitive to doping. The LO bond-stretching mode just above $\mathbf{q}=(0.2, 0, 0)$ is then found at quite low energies, $\sim$ 12 THz, but can not be unambiguously followed to larger q-values. Nevertheless, our data document that the LO bond-stretching branch in NCCO is strongly renormalized compared to undoped NCO, in particular it bends down anomalously from the zone center to $\mathbf{q}=(0.2, 0, 0)$. In order to further validate the correctness of our assignments, we performed a lattice dynamical calculation [@mirone] based on a shell model. We used a common potential model for cuprates, in which the interatomic potentials have been derived from a comparison of INS results for different HTcS compounds by Chaplot [*et al.*]{} [@chaplot], using a screened Coulomb potential, in order to simulate the effect of the free carriers introduced by doping. Following Ref. [@chaplot] for metallic [[$\mathrm{La_{1.85}Sr_{0.15}CuO_{4+\delta}}$ ]{}]{}and $\mathrm{YBa_2Cu_3O_7}$, we replaced the long-range Coulomb potential $V_c(q)$ by $V_c(q)/\epsilon(q)$, and for the dielectric function we take the semi-classical *Thomas-Fermi* limit of $\epsilon(q)=1+\kappa_s^2/q^2$, where $\kappa_s^2$ indicates the screening vector. The results of the calculation (without screening: dot-dashed lines; with screening: solid lines) are shown as well in Fig. \[figc\]. The lattice dynamics calculations without screening have been included, since they reproduce very well the experimental dispersion of the undoped parent compound [@pintscB]. The shift at the zone center of the high-energy phonon branches of NCCO with respect to NCO is due to the closing of a large LO-TO splitting. Indeed, the corresponding $\Delta_1$ and $\Delta_3$ branches in [[$\mathrm{Nd_2CuO_{4+\delta}}$ ]{}]{} are separated by almost 3 THz at the zone center, as observed by INS [@pintscB]. We point out that in our case a strong softening due to *Thomas-Fermi* screening does not imply a higher *Thomas-Fermi* parameter $\kappa_s$: actually, we find a screening vector $\kappa_s$ of about $0.39~\mathrm{\AA}^{-1}$, which is comparable to that for [[$\mathrm{La_{1.85}Sr_{0.15}CuO_{4+\delta}}$ ]{}]{}[@chaplot]. Though our modified calculation reproduces the closing of the LO-TO splitting, we still observe an anomalous additional softening of the highest bond-stretching LO branch near $\mathbf{q}=(0.2,0,0)$, which is not reproduced by our calculations (see Fig. \[figc\]). This branch softens in frequency from $\mathbf{q}=(0.1, 0, 0)$ to $\mathbf{q}=(0.2, 0, 0)$ by about $\Delta\nu\approx$ 1.5 THz, which is a shift comparable to the anomalous shift observed in [[$\mathrm{La_{1.85}Sr_{0.15}CuO_{4+\delta}}$ ]{}]{}at slightly larger $\mathbf{q}$ [@pintscB; @mcqueeny; @pinbrief]. Therefore, we believe that this anomalous softening is of the same nature as the one observed in p-type $\mathrm{La_{1.85}Sr_{0.15}CuO_{4+\delta}}$. A comparison between the experimental and calculated intensities for the two highest phonon branches is shown in Fig. \[figd\]. The good agreement of the observed integrated intensities with the calculated ones for the upper branches validates the correctness of our phonon branch assignment, at least for $\xi \le 0.2$. For $\xi > 0.2$ we would have expected an intensity exchange between the two highest branches, which seems to be not observed. ![\[figd\] Comparison between experimental (diamonds) and calculated (line) phonon intensities for the two highest optical phonon branches along $(7-\xi,1,0)$. a) bond-stretching LO mode starting around 15 THz and b) O(2) vibration LO branch starting around 10.5 THz.](compint6.eps) The main difference between [[$\mathrm{La_{1.85}Sr_{0.15}CuO_{4+\delta}}$ ]{}]{}and [[$\mathrm{Nd_{1.86}Ce_{0.14}CuO_{4+\delta}}$ ]{}]{}is, besides the screening effect, that in NCCO the Cu-O bond-stretching branch is closer in energy to the out-of-plane oxygen vibration one, having almost the same energy at $\xi = 0.2$. These two branches belong to the same symmetry and therefore cannot cross, so that for $\xi>0.2$ softening implies interaction with the out-of-plane oxygen vibration mode with the same symmetry. In the region between $\mathbf{q}=(0.25,0,0)$ and $(0.3,0,0)$, the two modes are poorly defined in energy, which is consistent with what is observed in [[$\mathrm{La_{1.85}Sr_{0.15}CuO_{4+\delta}}$ ]{}]{} by Pintschovius and Braden [@pinbrief] and McQueeney *et al.* [@mcqueeny] for $\xi\sim 0.25-0.3$. The corresponding real space periodicity of 3 to 4 unit cells may therefore be linked to the proximity to some charge instability. We remark that a reduced vector $\xi\sim 0.25-0.3$ approximately corresponds to the nesting vector along $[\xi 0 0]$ direction, as can be inferred from the ARPES data of Ref. . In conclusion, the present results reveal that the anomalous softening previously observed in hole-doped compounds [@pintscB; @mcqueeny; @pinbrief; @lacuod; @pingin; @reichardt], is also present in the electron-doped cuprates. This is evidenced by the comparison of the present results on doped NCCO with the previously reported ones on pure NCO [@pintscB]. This implies that: (i) the anomaly also exists in n-type cuprates, giving strength to the hypothesis [@pintscB; @mcqueeny; @pinbrief; @lacuod; @pingin; @reichardt] of an electron-phonon coupling origin of this feature; (ii) this is a generic feature of the high-temperature superconductors, as expected, if phonons are relevant to high temperature superconductivity. Moreover, this Letter demonstrates that high-energy resolution inelastic x-ray scattering has developed into an invaluable tool for the study of the lattice dynamics of complex transition metal oxides, allowing measurements on small high-quality single crystals which are inaccessible to the traditional method of inelastic neutron scattering. We acknowledge L. Paolasini and G. Monaco for useful discussions and H. Casalta for precious help during preliminary tests. The authors are grateful to D. Gambetti, C. Henriquet and R. Verbeni for technical help, to J.-L. Hodeau for help in the crystal orientation and J. -P. Vassalli for crystal cutting. P.K.M. and M.G. are supported by the U.S. Department of Energy under Contracts No. DE-FG03-99ER45773 and No. DE-AC03-76SF00515, by NSF CAREER Award No. DMR-9985067, and by the A.P. Sloan Foundation. [17]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ****, (). , in **, edited by (, , ), p. . , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ** (, ), . , ****, (). , ****, (). , ** (), . , ****, (). [^1]: All reciprocal lattice vectors are expressed in r.l.u.

--- abstract: 'For given $r \geq 1$ elliptic curves $E_1, \ldots, E_r$, there exists a closed Riemann surface of the minimal genus $e(E_{1},\ldots,E_{r})$ whose jacobian variety is isogenous to a product $E_1 \times \cdots \times E_r \times A$, where $A$ is also a product of jacobian varietes. Let $e(r)$ be the maximum of the values $e(E_1,\ldots,E_r)$, where $E_{1}, \ldots, E_{r}$ runs over the space of elliptic curves. If $r \leq 3$, then it is known that $e(E_1,\ldots,E_r)=r=e(r)$; we provide explicit equations of such a minimal genus Riemann surface. If $r \geq 4$, then we obtain that $e(r) \leq \widehat{e}(r)$, where (i) $\widehat{e}(r)=1+2^{(r-2)/2}r$ (for $r$ even) and (ii) $\widehat{e}(r)=1+2^{(r-3)/2}(r-1)$ (for $r$ odd). Our constructions also permits to obtain a $2$-dimensional family of Riemann surfaces of genus $g \in \{5,9\}$ and also of a $1$-dimensional family of genus $g=13$ whose jacobian varieties are isogenous to the product of elliptic curves.' address: 'Departamento de Matemática y Estadística, Universidad de La Frontera. Casilla 54-D, Temuco, Chile' author: - 'Ruben A. Hidalgo' title: Elliptic factors in the jacobian variety of Riemann Surfaces --- [^1] Introduction ============ An abelian variety is called simple if it is not isogenous to a product of two abelian varieties of smaller dimensions. A non-simple abelian variety is then isogenous to a non-trivial product of abelian varieties of smallest positive dimension. Such a decomposition is not unique, but if we, moreover, require all the factors to be simple, then Poincaré reducibility theorem asserts that the decomposition is unique up to permutation of factors. Examples of (principally polarized) abelian varieties are given by the jacobian variety of a closed Riemann surface (the polarization is induced by the intersection form of the first integral homology group of the surface). In this paper we should restrict to these types of principally polarized abelian varieties and isogenous decompositions of them into a product of jacobian varieties. Let $S$ be a closed Riemann surface and $JS$ be its jacobian variety. If $JS$ is a non-simple abelian variety, then it is isogenous to a product $A_{1}^{n_{1}} \times \cdots \times A_{s}^{n_{s}}$, where $A_{j}$ is an abelian variety of positive dimension, $n_{j} \geq 1$ is an integer and $s \geq 2$. If $g \leq 4$, then it possible to assume all the factors $A_{j}$ to be jacobian varieties. For $g \geq 5$, this seems to be a difficult problem to see if one may choose such a decomposition so that all its factors are jacobian varieties. Somehow related to the above is the following converse. Assume we are given $r \geq 1$ closed Riemann surfaces $S_1, \ldots, S_r$, where $S_j$ has genus $g_j \geq 1$. \(a) Is there a closed Riemann surface whose jacobian variety is isogenous to the product $JS_1 \times \cdots \times JS_r \times A$, for $A$ being also a suitable product of jacobian varieties? \(b) If the answer to (a) is affirmative, then which is minimal genus $e(S_1,\ldots,S_r)$ of such a closed Riemann surfaces? Clearly, $e(S_1,\ldots,S_r) \geq g_1 + \cdots g_r$. In this paper, we consider the previous problems in the particular case that each $S_{j}$ is a genus one (i.e., an elliptic curve) Riemann surface $E_{j}$. In this case, we also set $e(r)$ as the maximum of the values $e(E_{1},\ldots,E_{r})$, where $E_1,\ldots,E_r$ run over the space of elliptic curves. Since $e(E_1)=e(1)=1$, the interesting case is $r \geq 2$. In [@E-S] Ekedahl and Serre constructed examples of elliptic curves $E_1, \ldots, E_r$ (for almost every $r \leq 1297$) so that $e(E_1, \ldots, E_r)=r$. It is an open problem to determine if there are similar examples for infinitely many values of $r$ . Another similar constructions can be found, for instance, in [@Earle; @E-S; @HN; @Nakajima; @Paulhus1; @Paulhus2; @RR]. The space of isomorphism classes of principally polarized abelian varieties of dimension $r$ is the upper half Siegel space ${\mathfrak H}_{r}$; of complex dimension $r(r+1)/2$. The rational symplectic group ${\rm Sp}_{2r}({\mathbb Q})$ acts on ${\mathfrak H}_{r}$ as a group of holomorphic automorphism and, moreover, if $z \in {\mathfrak H}_{r}$ and $T \in {\rm Sp}_{2r}({\mathbb Q})$, then $T(z)$ and $z$ represent isogenous principally polarized abelian varieties. If $E_{1},\ldots, E_{r}$ are elliptic curves and $z_{0}$ represents the abelian variety $E_{1} \times \cdots \times E_{r}$, then the ${\rm Sp}_{2r}({\mathbb Q})$-orbit of $z_{0}$ is dense in ${\mathfrak H}_{r}$. As for $r \in \{2,3\}$, the jacobian locus ${\mathfrak J}_{r}$ in ${\mathfrak H}_{r}$ (i.e. the principally polarized abelian varieties obtained as jacobian of Riemann surfaces) is an open dense set, the density of the previous orbit asserts that there are infinitely many closed Riemann surfaces of genus $r$ whose jacobian variety is isogenous to $E_{1}\times \cdots \times E_{r}$; in particular, $e(r)=r$. In Theorems \[r=2\] and \[coro1\] we construct explicit equations for such Riemann surfaces (for $r=2$ this was already done by Gaudry and Schost in [@Gaudry]). If $r \geq 4$, then the dimension of ${\mathfrak J}_{r}$ is $3(r-1)$; which is strictly smaller that that of ${\mathfrak H}_{r}$. So the density property of the ${\rm Sp}_{2r}({\mathbb Q})$-orbit of $z_{0}$ does not ensure intersection of it with ${\mathfrak J}_{r}$ (note that the density asserts that the jacobian variety of any closed Riemann surface of genus $r$ is very near, in some sense, to be isogenous to $E_{1} \times \cdots \times E_{r}$). In Theorem \[coro2\] we observe that, for $r \geq 4$, $$e(r) \leq \left\{ \begin{array}{ll} 1+2^{(r-2)/2}r, & \mbox{for $r$ even,} \\ 1+2^{(r-3)/2}(r-1), & \mbox{for $r$ odd.} \end{array} \right.$$ We believe these upper bounds are sharp. The proof of Theorem \[coro2\] is based in an explicit construction. Given $(2s-3)$ elliptic curves, where $s \geq 3$, we explicitly construct a Riemann surface of genus $1+2^{s-2}(s-2)$ whose jacobian variety is isogenous to the product of $s(s-1)/2$ elliptic curves and some other jacobian varieties of elliptic/hyperelliptic curves (Theorem \[construccion\]). Such a Riemann surface happens to be one of the two irreducible component of a certain fiber product of $s$ elliptic curves (these two components being isomorphic). The construction also permits to obtain a $2$-dimensional family of Riemann surfaces of genus $g \in \{5,9\}$ whose jacobian varieties are isogenous to the product of $g$ elliptic curves (Theorem \[coro3\]). We also provide a similar construction, a fiber product of $r$ given elliptic curves $E_{1}, \ldots, E_{r}$, which turns out to be irreducible. Such a fiber product is a closed Riemann surface of genus $g=1+2^{r-2}(r-1)$ whose jacobian variety is isogenous to the product $E_{1} \times \cdots \times E_{r} \times A_{g-r}$, where $A_{g-r}$ is again the product of jacobian varieties of elliptic/hyperelliptic Riemann surfaces. This construction also permits to obtain a $2$-dimensional family of Riemann surfaces of genus $g=5$ and a $1$-dimensional family of Riemann surfaces of genus $g=13$ whose jacobian varieties are isogenous to the product of elliptic curves (see Corollaries \[g=5\] and \[g=13\]). Main results ============ Before to state the main results, we need to recall some facts on elliptic curves. Set $\Delta_{1}={\mathbb C}-\{0,1\}$ and, for $s \geq 2$, set $\Delta_{s}=\{(\lambda_{1},\ldots,\lambda_{s}) \in {\mathbb C}^{s}: \lambda_{j} \in \Delta_{1};\; \lambda_{i} \neq \lambda_{j},\; i \neq j\}$. If $\lambda \in \Delta_{1}$, then we set the elliptic curve $$E_{\lambda}: y^{2}=x(x-1)(x-\lambda).$$ It is known that $E_{\lambda}$ and $E_{\mu}$ are isomorphic if and only if there is some $T \in {\mathbb G}=\langle u(\lambda)=1/\lambda, V(\lambda)=1-\lambda\rangle \cong {\mathfrak S}_{3}$ so that $\mu=T(\lambda)$. It is also well known that for every $\lambda \in \Delta_{1}$ there exist infinitely many values $\mu \in \Delta_{1}$ (in fact a dense subset) so that $E_{\mu}$ and $E_{\lambda}$ are isogenous. In particular, given $s \geq 2$ elliptic curves $E_{1},\ldots, E_{s}$, there are infinitely many tuples $(\lambda_{1},\ldots,\lambda_{s}) \in \Delta_{s}$ so that $E_{\lambda_{j}}$ and $E_{j}$ are isogenous for each $j=1,\ldots,s$. For $s=2$, we may even replace “isogenous" by “isomorphic". Genus two case: the known situation ----------------------------------- As already noted in the introduction, given any pair $(E_{1},E_{2})$ of elliptic curves, there is a closed Riemann surface $S$ of genus two whose jacobian variety $JS$ is isogenous to $E_{1} \times E_{2}$. The following describes explicit equations for one of them (an argument is provided in Section \[Sec:r=2\]). This is not new and it can be tracked back to, for instance, [@Earle; @Gaudry; @HN; @Hermite]. \[r=2\] Let $E_{1}$ and $E_{2}$ two elliptic curves. Choose $(\lambda_{1},\lambda_{2}) \in \Delta_{2}$ so that $E_{j}$ is isomorphic (or isogenous) to $E_{\lambda_{j}}$, for $j=1,2$, and set $$\eta_{1}=\frac{\lambda_{1}-1}{\lambda_{2}-1}, \quad \eta_{2}=\frac{\lambda_{2}(\lambda_{1}-1)}{\lambda_{1}(\lambda_{2}-1)}.$$ If $S$ is the genus two Riemann surface defined by the hyperelliptic curve $$y^{2}=(x^{2}-1)(x^{2}-\eta_{1})(x^{2}-\eta_{2}),$$ then $JS$ is isogenous to $E_{1} \times E_{2}$. The constructed genus two Riemann surface $S$ in Theorem \[r=2\] admits a non-hyperelliptic involution $\alpha$ so that $S/\langle \alpha \rangle$ is isomorphic to $E_1$ and $S/\langle \alpha \iota \rangle$ is isomorphic to $E_2$, where $\iota$ is the hyperelliptic involution. But, there are also Riemann surfaces of genus two with no extra automorphisms (with the exception of the hyperelliptic involution) whose jacobian variety is also isogenous to $E_1 \times E_2$; these can be obtained by considering non-constant holomorphic maps $h:S \to E$, where $S$ is a closed Riemann surface of genus two and $E$ being some elliptic curve. A general construction ---------------------- Next, given $2s-3$ elliptic curves, for $s \geq 3$, we make an explicit construction of a closed Riemann surface of genus $g=1+2^{s-2}(s-2)$, whose jacobian variety is isogenous to the product of at least $s(s-1)/2$ elliptic curves and jacobian varieties of some elliptic/hyperelliptic Riemann surfaces. Let us consider the set of cardinality $2^{s-1}-1$ defined as $$V_{s}=\{\alpha=(\alpha_{1},\ldots,\alpha_{s}) \in \{0,1\}^{s}-\{(0,\ldots,0)\}: \alpha_{1}+\cdots+\alpha_{s} \; \mbox{is even}\}.$$ As previously noted, given $2s-3$ elliptic curves, $E_{1},\ldots, E_{2s-3}$, we may find a tuple $(\lambda_{1},\ldots,\lambda_{2s-3}) \in \Delta_{2s-3}$ so that $E_{j}$ is isogenous to the elliptic curve $y^{2}=x(x-1)(x-\lambda_{j})$. \[construccion\] Let $s \geq 3$ and $(\lambda,\mu_{1,1},\mu_{1,2},\mu_{2,1},\mu_{2,2},\ldots,\mu_{s-2,1},\mu_{s-2,2}) \in \Delta_{2s-3}$. Set $$\eta_{0}=\frac{-1}{\mu_{s-2,2}}, \; \eta_{1}=\frac{1}{1-\mu_{s-2,2}}, \; \eta_{2}=\frac{1}{\lambda-\mu_{s-2,2}}, \; \eta_{3}=\frac{1}{\mu_{s-2,1}-\mu_{s-2,2}},$$ $$\eta_{j,t}=\frac{1}{\mu_{j,t}-\mu_{s-2,2}}, \quad t=1,2, \quad j=1,\ldots,s-3.$$ For each $\alpha=(\alpha_{1},\ldots,\alpha_{s}) \in V_{s}$, set $K_{\alpha} \in {\mathbb C}^{*}={\mathbb C}-\{0\}$ given by $$K_{\alpha}=(-\mu_{s-2,2})^{\alpha_{1}} (\mu_{s-2,2}-1)^{\alpha_{2}}(\mu_{s-2,2}-\lambda)^{\alpha_{2}} (\mu_{s-2,2}-\mu_{s-2,1})^{\alpha_{s}} \prod_{k=1}^{s-3} (\mu_{s-2,2}-\mu_{k,1})^{\alpha_{k+2}}(\mu_{s-2,\mu_{k,2}}-\lambda)^{\alpha_{k+2}}.$$ If $X \subset {\mathbb C}^{2^{s-1}}$ is the affine curve defined by the following $2^{s-1}-1$ equations $$\left\{ \begin{array}{c} w_{\alpha}^{2}=K_{\alpha} z^{\alpha_{1}}(z-\eta_{0})^{\alpha_{1}} (z-\eta_{1})^{\alpha_{2}}(z-\eta_{2})^{\alpha_{2}}(z-\eta_{3})^{\alpha_{s}} \prod_{k=1}^{s-3}(z-\eta_{k,1})^{\alpha_{k+2}} (z-\eta_{k,2})^{\alpha_{k+2}},\\ \alpha=(\alpha_{1},\ldots,\alpha_{s}) \in V_{s}.\\ \end{array} \right\},$$ then $X$ defines a closed Riemann surface $S$ of genus $g=1+2^{s-2}(s-2)$ whose jacobian variety $JS$ is isogenous to the product of the jacobian varieties of the following $\sum_{j=1}^{[s/2]}\binom{s}{2j}$ elliptic/hyperelliptic curves $$C_{i_{1},\ldots,i_{k}}: \nu^{2}=(\upsilon-\rho_{i_{1},1})(\upsilon-\rho_{i_{1},2})\cdots(\upsilon-\rho_{i_{k},1})(\upsilon-\rho_{i_{k},2}),$$ where $2 \leq k \leq s$ is even, the tuples $(i_{1},\ldots,i_{k})$ satisfy $$1\leq i_{1} < i_{2} < \cdots < i_{k} \leq s,$$ and $$\rho_{i_{j},1}=\left\{\begin{array}{cl} \infty, & i_{j}=1\\ 1, & i_{j}=2\\ \mu_{r-2,1}, & i_{j}=r \geq 3 \end{array} \right. \quad \rho_{i_{j},2}=\left\{\begin{array}{cl} 0, & i_{j}=1\\ \lambda, & i_{j}=2\\ \mu_{r-2,2}, & i_{j}=r\geq 3 \end{array} \right.$$ In the case that $\rho_{i_{j},1}=\infty$, then the factor $(u-\rho_{i_{j},1})$ is deleted from the above expression. 1. The above provides a $(2s-3)$-dimensional family of closed Riemann surfaces of genus $g=1+2^{s-2}(s-2)$. 2. The Riemann surface $S$ constructed in Theorem \[construccion\] has the following properties. 1. $JS$ contains at least $s(s-1)/2$ elliptic curves in its isogenous decomposition. 2. Some of the elliptic curves factors of $JS$ are $$\begin{array}{ll} E_{1}: & y^{2}=x(x-1)(x-\lambda)\\ E_{2}: & y^{2}=(x-1)(x-\lambda)(x-\mu_{1,1})(x-\mu_{1,2})\\ E_{3}: & y^{2}=(x-\mu_{1,1})(x-\mu_{1,2})(x-\mu_{2,1})(x-\mu_{2,2})\\ E_{4}: & y^{2}=(x-\mu_{2,1})(x-\mu_{2,2})(x-\mu_{3,1})(x-\mu_{3,2})\\ &\vdots\\ E_{s-1}: & y^{2}=(x-\mu_{s-3,1})(x-\mu_{s-3,2})(x-\mu_{s-2,1})(x-\mu_{s-2,2})\\ E_{s}: & y^{2}=x(x-\mu_{s-2,1})(x-\mu_{s-2,2}) \end{array}$$ 3. It happens that $S$ is an irreducible connected component of the fiber product of the pairs $(E_{1},\pi_{1}),\ldots,(E_{s},\pi_{s})$, where $\pi_{j}(x,y)=x$. An example of genus nine ------------------------ The construction provided in Theorem \[construccion\] permits to obtain Riemann surfaces whose jacobian variety is isogenous to the product of elliptic curves. Next, as an example, we describe a $2$-dimensional family of genus nine Riemann surfaces whose jacobian is isogenous to the product of nine elliptic curves. \[coro3\] Let $(\lambda, \mu) \in \Delta_{2}$ and set $$\mu_{1,1}=\mu, \; \mu_{1,2}=\frac{\lambda}{\mu}, \; \mu_{2,1}=\frac{\lambda(\mu-1)}{\mu-\lambda}, \; \mu_{2,2}=\frac{\mu-\lambda}{\mu-1},$$ $$K_{1}=(\mu_{2,2}-\mu_{1,1})(\mu_{2,2}-\mu_{1,2})(\mu_{2,2}-\mu_{2,1}), \; K_{2}=(\mu_{2,2}-1)(\mu_{2,2}-\lambda)(\mu_{2,2}-\mu_{2,1}),$$ $$K_{3}=(\mu_{2,2}-1)(\mu_{2,2}-\lambda)(\mu_{2,2}-\mu_{1,1})(\mu_{2,2}-\mu_{1,2}), \; K_{4}=-\mu_{2,2}(\mu_{2,2}-\mu_{2,1}),$$ $$K_{5}=-\mu_{2,2}(\mu_{2,2}-\mu_{1,1})(\mu_{2,2}-\mu_{1,2}), \; K_{6}=-\mu_{2,2}(\mu_{2,2}-1)(\mu_{2,2}-\lambda).$$ If $S$ is the genus nine Riemann surface defined by the curve $$\left\{\begin{array}{lcl} w_{1}^{2}&=&K_{1} \left(z-\dfrac{1}{\mu_{1,1}-\mu_{2,2}} \right) \left(z-\dfrac{1}{\mu_{1,2}-\mu_{2,2}} \right) \left(z-\dfrac{1}{\mu_{2,1}-\mu_{2,2}} \right) \\ w_{2}^{2}&=&K_{2} \left(z-\dfrac{1}{1-\mu_{2,2}} \right) \left(z-\dfrac{1}{\lambda-\mu_{2,2}} \right) \left(z-\dfrac{1}{\mu_{2,1}-\mu_{2,2}} \right) \\ w_{3}^{2}&=&K_{3} \left(z-\dfrac{1}{1-\mu_{2,2}} \right) \left(z-\dfrac{1}{\lambda-\mu_{2,2}} \right) \left(z-\dfrac{1}{\mu_{1,1}-\mu_{2,2}} \right) \left(z-\dfrac{1}{\mu_{1,2}-\mu_{2,2}} \right) \\ w_{4}^{2}&=&K_{4} z\left(z+\dfrac{1}{\mu_{2,2}} \right) \left(z-\dfrac{1}{\mu_{2,1}-\mu_{2,2}} \right) \\ w_{5}^{2}&=&K_{5} z\left(z+\dfrac{1}{\mu_{2,2}} \right) \left(z-\dfrac{1}{\mu_{1,1}-\mu_{2,2}} \right) \left(z-\dfrac{1}{\mu_{1,2}-\mu_{2,2}} \right) \\ w_{6}^{2}&=&K_{6} z\left(z+\dfrac{1}{\mu_{2,2}} \right) \left(z-\dfrac{1}{1-(\mu_{2,2}} \right) \left(z-\dfrac{1}{\lambda-\mu_{2,2}} \right) \\ w_{7}^{2}&=&w_{3}^{2}w_{4}^{2} \end{array} \right\}$$ then $JS$ is isogenous to the product of nine elliptic curves. Genus three case ---------------- As previously noted in the introduction, for every triple $(E_{1},E_{2},E_{3})$ of elliptic curves there is some genus three closed Riemann surface $S$ whose jacobian variety is isogenous to the product $E_{1} \times E_{2} \times E_{3}$. We may use the constructed Riemann surface in Theorem \[construccion\], for $s=3$, to construct explicitly equations for such surface $S$. \[coro1\] Let $E_{1}$, $E_{2}$ and $E_{3}$ be three elliptic curves. Choose $(\lambda_{1},\lambda_{2},\lambda_{3}) \in \Delta_{3}$ so that $E_{j}$ is isogenous to $E_{\lambda_{j}}$, for $j=1,2,3$, and let $\mu$ be a root of $$\lambda_{2}\lambda_{3} \mu^{2}-(\lambda_{1}\lambda_{2}+\lambda_{2}\lambda_{3}+\lambda_{1}\lambda_{3}-\lambda_{1}-\lambda_{3}+1) \mu +\lambda_{1}\lambda_{2}=0.$$ If $S$ is the genus three Riemann surface defined by the curve $$\left\{ \begin{array}{lcl} w_{1}^{2}&=&\mu(\lambda_{3} \mu -1)(\lambda_{3} \mu -\lambda_{1})(\lambda_{3} - 1) z \left( z-\dfrac{1}{\lambda_{1}-\lambda_{3}\mu} \right) \left(z-\dfrac{1}{\mu (1-\lambda_{3})} \right)\\ w_{2}^{2}&=&-\lambda_{3}\mu^{2}(\lambda_{3}-1)z\left(z+\dfrac{1}{\lambda_{3} \mu}\right) \left(z-\dfrac{1}{1-\lambda_{3}\mu} \right)\\ w_{3}^{2}&=&-\lambda_{3}\mu^{2}(\lambda_{3}\mu-1)(\lambda_{3}-1)z^{2}\left(z+\dfrac{1}{\lambda_{3} \mu}\right) \left(z-\dfrac{1}{\mu(1-\lambda_{3})}\right) \\ \end{array} \right\},$$ then $JS$ is isogenous to the product $E_{1} \times E_{2} \times E_{3}$. Upper bounds for $e(r)$, $r \geq 4$ ----------------------------------- Another direct consequence of the construction provided by Theorem \[construccion\] is the following upper bound for $e(r)$. \[coro2\] $$e(r) \leq \left\{ \begin{array}{ll} 1+2^{(r-2)/2} r, & r \geq 4 \quad \mbox{even}\\ 1+2^{(r-3)/2}(r-1), & r \geq 5 \quad \mbox{odd} \end{array} \right.$$ We conjecture that the above inequalities are in fact equalities. Another fiber product construction ---------------------------------- In Section \[Sec:otraconstruccion\] we describe another similar construction which permits to obtain the following construction. \[construccion2\] If $(\lambda_{1},\ldots,\lambda_{r}) \in \Delta_{r}$, then $$X=\left\{(x,y_{1},\ldots,y_{r}) \in {\mathbb C}^{r+1}: y_{j}=x(x-1)(x-\lambda_{j}); \; j=1,\ldots,r \right\}$$ defines a closed Riemann surface of genus $g=1+2^{r-2}(r-1)$ whose jacobian variety is isogenous to a product $E_{\lambda_{1}} \times \cdots \times E_{\lambda_{r}} \times A_{g-r}$, where $A_{g-r}$ is the product of certain explicit elliptic/hyperelliptic Riemann surfaces. Preliminaries: The jacobian variety of a closed Riemann surface =============================================================== Abelian varieties ----------------- A [*polarized abelian variety*]{} of dimension $g$ is a pair $A=(T,Q)$, where $T={\mathbb C}^{g}/L$ is a complex torus of dimension $g$ and $Q$ (called a [*principal polarization*]{} of $A$) is a positive-definite Hermitian product in ${\mathbb C}^{g}$ with ${\rm Im}(Q)$ having integral values over elements of the lattice $L$. There is basis of $L$ for which ${\rm Im}(Q)$ can be represented by the matrix $$\left( \begin{array}{cc} 0 & D\\ -D & 0 \end{array} \right)$$ where $D$ is a diagonal matrix, whose diagonal entries are $d_{1},\ldots,d_{g}$, where $d_{j} \geq 1$ divides $d_{j+1}$. The tuple $(d_{1},\ldots,d_{g})$ is called the [*polarization type*]{}. When $d_{1}=\cdots=d_{g}=1$, we say that the polarization is [*principal*]{} and that the abelian variety is [*principally polarized*]{}. A non-constant surjective morphism $h:A_{1} \to A_{2}$ between abelian varieties is called an [*isogeny*]{} if it has a finite kernel. In this case we say that $A_{1}$ and $A_{2}$ are [*isogenous*]{}. An abelian variety $A$ is called [*decomposable*]{} if it is isogenous to the product of abelian varieties of smaller dimensions. It is called [*completely decompossable*]{} if it is the product of elliptic curves (varieties of dimension $1$). If the abelian variety is not isogenous to a product of lowest dimensional abelian varieties, then we say that it is [*simple*]{}. If $A$ is an abelian variety, then there exist simple polarized abelian varieties $A_{1}, \ldots, A_{s}$ and positive integers $n_{1},\ldots,n_{s}$ such that its jacobian variety $JS$ is isogenous to the product $A_{1}^{n_{1}} \times \cdots A_{s}^{n_{s}}$. Moreover, the factors $A_{j}$ and the integers $n_{j}$ are unique up to isogeny and permutation of the factors. In general, to describe these simple factors of an abelian variety seems to be a very difficult problem. When the abelian variety $A$ admits a non-trivial group $G$ of automorphisms, then there is a method to compute factors (non-necessarilly simple ones) by using the rational representations of $G$ (the isotipical decomposition) [@CR; @LR; @Anita]. The Jacobian variety -------------------- Let $S$ be a closed Riemann surface of genus $g \geq 1$. The first homology group $H_{1}(S,{\mathbb Z})$ is isomorphic to ${\mathbb Z}^{2g}$ and its complex vector space $H^{1,0}(S)$ of its holomorphic $1$-forms is isomorphic to ${\mathbb C}^{g}$. There is a natural injective map $$\iota:H_{1}(S,{\mathbb Z}) \hookrightarrow \left( H^{1,0}(S) \right)^{*} \quad \mbox{(the dual space of $H^{1,0}(S)$)}$$ $$\alpha \mapsto \int_{\alpha}.$$ The image $\iota(H_{1}(S,{\mathbb Z}))$ is a lattice in $\left( H^{1,0}(S) \right)^{*}$ and the quotient $g$-dimensional torus $$JS=\left( H^{1,0}(S) \right)^{*}/\iota(H_{1}(S,{\mathbb Z}))$$ is called the [*jacobian variety*]{} of $S$. The intersection product in $H_{1}(S,{\mathbb Z})$ induces a principal polarization on $JS$; that is, $JS$ is a principally polarized abelian variety. If we fix a point $p_{0} \in S$, then there is a natural holomorphic embedding $$\rho_{p_{0}}:S \to J(S)$$ defined by $\rho(p)=\int_{\alpha}$, where $\alpha \subset S$ is an arc connecting $p_{0}$ with $p$. If we choose a symplectic homology basis for $S$, say $\{\alpha_{1},\ldots,\alpha_{g},\beta_{1},\ldots,\beta_{g}\}$ (i.e. a basis for $H_{1}(S,{\mathbb Z})$ such that the intersection products $\alpha_{i} \cdot \alpha_{j}=\beta_{i} \cdot \beta_{j}=0$ and $\alpha_{i} \cdot \beta_{j}=\delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta function), we may find a dual basis $\{\omega_{1},\ldots,\omega_{g}\}$ (i.e. a basis of $H^{1,0}(S)$ such that $\int_{\alpha_{i}} \omega_{j}=\delta_{ij}$). We may consider the Riemann matrix $$Z=\left( \int_{\beta_{j}} \omega_{i} \right)_{g \times g}.$$ If we now consider the Riemann period matrix $\Omega=(I \; Z)_{g \times 2g}$, then its $2g$ columns define a lattice in ${\mathbb C}^{g}$. The quotient torus ${\mathbb C}^{g}/\Omega$ is isomorphic to $JS$. A decomposition result of the jacobian variety ---------------------------------------------- Next, we recall the following decomposition result due to Kani-Rosen [@K-R] which will be used in our constructions. \[coroKR\] Let $S$ be a closed Riemann surface of genus $g \geq 1$ and let $H_{1},\ldots,H_{s}<{\rm Aut}(S)$ such that: 1. $H_{i} H_{j}=H_{j} H_{i}$, for all $i,j =1,\ldots,s$; 2. $g_{H_{i}H_{j}}=0$, for $1 \leq i < j \leq s$ 3. $g=\sum_{j=1}^{s} g_{H_{j}}$. Then $$JS \cong_{isog.} \prod_{j=1}^{s} J(S_{H_{j}}).$$ Proof of Theorem \[r=2\] {#Sec:r=2} ======================== Let us consider the fiber product $C$ of $(E_{\lambda_{1}},\pi_{1})$ and $(E_{\lambda_{2}},\pi_{2})$, where $\pi_{j}(x,y)=x$, that is, $$C: \left\{ (x,y_{1},y_{2}): y_{1}=x(x-1)(x-\lambda_{1}), \quad y_{2}=x(x-1)(x-\lambda_{2}) \right\}.$$ We consider the projection $\pi:C \to \widehat{\mathbb C}$, where $\pi(x,y_{1},y_{2})=x$. On $C$ we have the automorphisms $$A_{1}(x,y_{1},y_{2})=(x,-y_{1},y_{2}), \quad A_{2}(x,y_{1},y_{2})=(x,y_{1},-y_{2}),$$ with $H=\langle A_{1}, A_{2} \rangle \cong {\mathbb Z}_{2}^{2}$, $H$ being the deck group of $\pi$, $C/H$ being of signature $(0;2,2,2,2,2)$ and branch values being $\infty$, $0$, $1$, $\lambda_{1}$ and $\lambda_{2}$. The Riemann surface $S$ defined by $C$ has genus two If $H_{j}=\langle A_{j} \rangle$, it can be seen that the orbifold $C/H_{j}$ has underlying Riemann surface $E_{\lambda_{3-j}}$. We may apply Proposition \[coroKR\], using $H_{1}$ and $H_{2}$, to obtain that $JS$ is isogenous to $E_{1} \times E_{2}$. The automorphism $A_{2}\circ A_{1}(x,y_{1},y_{2})=(x,-y_{1},-y_{2})$ is the hyperelliptic one. Let us consider a two-fold branched cover $P:C \to \widehat{\mathbb C}$ with deck group $\langle A_{2}\circ A_{1} \rangle$. We may assume that the involution $A_{1}$ descent to the involution $T(x)=-x$ (this because all involutions are conjugated in the Möbius group). So the branch values of $P$ may be assume to be $\pm 1$, $\pm \rho_{1}$ and $\pm \rho_{2}$. In this way, $C$ can be described by the hyperelliptic curve $$y^{2}=(x^{2}-1)(x^{2}-\rho_{1}^{2})(x^{2}-\rho_{2}^{2}).$$ Let us consider the two-fold branched cover $Q(x)=x^{2}$ whose deck group is $\langle T \rangle$. The branch values of $Q$ are $1$, $\rho_{1}^{2}$ and $\rho_{2}^{2}$. The images of $\infty$ and $\pm 1$ are $\infty$ and $1$. Let us consider the Möbius transformation $$L(x)=\frac{(1-\rho_{1}^{2})(x-\rho_{2}^{2})}{(1-\rho_{2}^{2})(x-\rho_{1}^{2})}.$$ Then $L(1)=1$, $L(\rho_{1}^{2})=\infty$, $L(\rho_{2}^{2})=0$, $L(\infty)=(1-\rho_{1}^{2})/(1-\rho_{2}^{2})$ and $L(0)=\rho_{2}^{2}L(\infty)/\rho_{1}^{2}$. By making $L(\infty)=\lambda_{1}$ and $L(0)=\lambda_{2}$, we obtain that $$\rho_{1}^{2}=\frac{\lambda_{1}-1}{\lambda_{2}-1}, \quad \rho_{2}^{2}=\frac{\lambda_{2}}{\lambda_{1}} \rho_{1}^{2}$$ Proof of Theorem \[construccion\] ================================= Let us consider the following $s$ elliptic curves $$\begin{array}{ll} E_{1}: & y^{2}=x(x-1)(x-\lambda)\\ E_{2}: & y^{2}=(x-1)(x-\lambda)(x-\mu_{1,1})(x-\mu_{1,2})\\ E_{3}: & y^{2}=(x-\mu_{1,1})(x-\mu_{1,2})(x-\mu_{2,1})(x-\mu_{2,2})\\ E_{4}: & y^{2}=(x-\mu_{2,1})(x-\mu_{2,2})(x-\mu_{3,1})(x-\mu_{3,2})\\ &\vdots\\ E_{s-1}: & y^{2}=(x-\mu_{r-3,1})(x-\mu_{s-3,2})(x-\mu_{s-2,1})(x-\mu_{s-2,2})\\ E_{s}: & y^{2}=x(x-\mu_{s-2,1})(x-\mu_{s-2,2}) \end{array}$$ If we consider the degree two maps $\pi_{j}:E_{j} \to \widehat{\mathbb C}$ defined as $\pi_{j}(x,y)=x$, then we may perform the fiber product of the $s$ pairs $(E_{1},\pi_{1}),\ldots,(E_{s},\pi_{s})$. Such a fiber product is given by the curve $\widehat{C}$ formed of the tuples $(x,y_{1},\ldots,y_{s})$ so that $(x,y_{j}) \in E_{j}$, for $j=1,\ldots,s$. This curve is reducible and contains two irreducible components, both of them being isomorphic. The curve $\widehat{C}$ admits the group of automorphisms $N=\langle f_{1},\ldots,f_{s}\rangle \cong {\mathbb Z}_{2}^{s}$, where $$f_{j}(x,y_{1},\ldots,y_{s})=(x,y_{1},\ldots,y_{j-1},-y_{j},y_{j+1},\ldots,y_{s}).$$ The two irreducible factors are permuted by some elements of $N$ and each one is invariant under a subgroup isomorphic to ${\mathbb Z}_{2}^{s-1}$. In what follows we will construct a Riemann surface $S$ which is isomorphic to the irreducible components of $\widehat{C}$. Let us now consider the (affine) generalized Humbert curve (see [@C-G-H-R] for details) $$D: \left\{ \begin{array}{ccc} z_{1}^{2}+z_{2}^{2}+z_{3}^{2}&=&0\\ \lambda z_{1}^{2}+z_{2}^{2}+z_{4}^{2}&=&0\\ \mu_{1,1}z_{1}^{2}+z_{2}^{2}+z_{5}^{2}&=&0\\ \mu_{1,2}z_{1}^{2}+z_{2}^{2}+z_{6}^{2}&=&0\\ \vdots & \vdots& \vdots\\ \mu_{k,1}z_{1}^{2}+z_{2}^{2}+z_{2k+3}^{2}&=&0\\ \mu_{k,2}z_{1}^{2}+z_{2}^{2}+z_{2k+4}^{2}&=&0\\ \vdots & \vdots& \vdots\\ \mu_{s-2,1}z_{1}^{2}+z_{2}^{2}+z_{2s-1}^{2}&=&0\\ \mu_{s-2,2}z_{1}^{2}+z_{2}^{2}+1&=&0 \end{array} \right\}.$$ The conditions on the parameters ensure that $D$ is a non-singular algebraic curve, that is, a closed Riemann surface. On $D$ we have the abelian group $\langle b_{1},\ldots,b_{2s-1}\rangle=H_{0} \cong {\mathbb Z}_{2}^{2s-1}$ of conformal automorphisms, where $$b_{j}(z_{1},\ldots,z_{2s-1})=(z_{1},\ldots,z_{j-1}, - z_{j},z_{j+1},\ldots,z_{2s-1}), \; j=1,...,2s-1.$$ Inside the group $H_{0}$, the only non-trivial elements acting with fixed points are $b_{1},\ldots, b_{2s-1}$ and $b_{2s}=b_{1}b_{2}\cdots b_{2s-1}$. The degree $2^{2s-1}$ holomorphic map $$P:D \to \widehat{\mathbb C}: (z_{1},\ldots,z_{2s-1}) \mapsto-\left( \dfrac{z_{2}}{z_{1}} \right)^{2}$$ is a branched regular cover with deck group being $H_{0}$. The projection under $P$ of the set of fixed points are as follows: $$P({\rm Fix}(b_{1}))=\infty, \quad P({\rm Fix}(b_{2}))=0, \quad P({\rm Fix}(b_{3}))=1,\quad P({\rm Fix}(b_{4}))=\lambda_{1},$$ $$P({\rm Fix}(b_{2k+3}))=\mu_{k,1}, \quad P({\rm Fix}(b_{2k+4}))=\mu_{k,2}, \quad k=1,\ldots, s.$$ In particular, the branch locus of $P$ is the set $$\{\infty,0,1,\lambda, \mu_{1,1}, \mu_{1,2},\ldots,\mu_{s-2,1}, \mu_{s-2,2}\}.$$ By the Riemann-Hurwitz formula, $D$ has genus $g_{D}=1+2^{2s-2}(s-2)$. Let us consider the surjective homomorphism $$\theta:H_{0} \to H=\langle a_{1},\ldots,a_{s-1}\rangle \cong {\mathbb Z}_{2}^{s-1}$$ $$\begin{array}{c} b_{1},b_{2} \mapsto a_{1}\\ b_{3},b_{4} \mapsto a_{2}\\ b_{5},b_{6} \mapsto a_{3}\\ \vdots\\ b_{2k-1}, b_{2k} \mapsto a_{k}\\ \vdots\\ b_{2s-3},b_{2s-2} \mapsto a_{s-1}\\ b_{2s-1},b_{s} \mapsto a_{s}=a_{1}a_{2}\cdots a_{s-1} \end{array}$$ If $K$ is the kernel of $\theta$, then $$K=\langle b_{1}b_{2}, b_{3}b_{4},\ldots,b_{2k-1}b_{2k}, \ldots, b_{2s-3}b_{2s-2},b_{1}b_{3}b_{5}\cdots b_{2s-1}\rangle \cong {\mathbb Z}_{2}^{s}.$$ Since $K$ acts freely on $D$, it follows that $S=D/K$ is a Riemann surface. Again, by the Riemann-Hurwitz formula, it can be checked that the genus of $S$ is $g_{S}=1+2^{s-2}(s-2)$. In order to write equations for $S$, we need to compute a set of generators of ${\mathbb C}[z_{1},\ldots,z_{2s-1}]^{K}$, the algebra of $K$-invariant polynomials. Since the linear action of $K$ is given by diagonal matrices, a set of generators can be found to be $$t_{1}=z_{1}^{2}, t_{2}=z_{2}^{2},\ldots, t_{2s-1}=z_{2s-1}^{2},$$ together the monomials of the form $$t_{\alpha}=(z_{1}z_{2})^{\alpha_{1}}(z_{3}z_{4})^{\alpha_{2}}\cdots(z_{2s-3}z_{2s-2})^{\alpha_{s-1}} z_{2s-1}^{\alpha_{s}},$$ where $\alpha=(\alpha_{1},\ldots,\alpha_{s}) \in V_{s}$. As $V_{s}$ has cardinality $2^{s-1}-1$, the number of the above set of generators is $N=2^{s-1}+2s-2$. Using the map $\Phi:D \to {\mathbb C}^{N}$, whose coordinates are $t_{1},\ldots,t_{2s-1}$ and the monomials $t_{\alpha}$, $\alpha \in V_{s}$, one obtain that the Riemann surface induced by $\Phi(D)$ is isomorphic to $S$ and that its equations are given by $$\Phi(D)=\left\{ \begin{array}{ccl} t_{1}+t_{2}+t_{3}&=&0\\ \lambda t_{1}+t_{2}+t_{4}&=&0\\ \mu_{1,1}t_{1}+t_{2}+t_{5}&=&0\\ \mu_{1,2}t_{1}+t_{2}+t_{6}&=&0\\ \vdots & \vdots& \vdots\\ \mu_{k,1}t_{1}+t_{2}+t_{2k+3}&=&0\\ \mu_{k,2}t_{1}+t_{2}+t_{2k+4}&=&0\\ \vdots & \vdots& \vdots\\ \mu_{s-2,1}t_{1}+t_{2}+t_{2s-1}&=&0\\ \mu_{s-2,2}t_{1}+t_{2}+1&=&0\\ t_{\alpha}^{2}&=&(t_{3}t_{4})^{\alpha_{2}}\cdots(t_{2s-3}t_{2s-2})^{\alpha_{s-1}} t_{2s-1}^{\alpha_{s}}, \quad \alpha=(\alpha_{1},\ldots,\alpha_{s}) \in V_{s} \end{array} \right\}.$$ The first linear equations permit to write $t_{2},\ldots,t_{2s-1}$ in terms of $t_{1}$ as follows: $$t_{2}=-1-\mu_{s-2,2}t_{1}$$ $$t_{3}=1+(\mu_{s-2,2}-1)t_{1}$$ $$t_{4}=1+(\mu_{s-2,2}-\lambda)t_{1}$$ $$t_{2k+3}=1+(\mu_{s-2,2}-\mu_{k,1})t_{1}, \quad t_{2k+4}=1+(\mu_{s-2,2}-\mu_{s,2})t_{1}, \quad k=1,\ldots, s-3,$$ $$t_{2s-1}=1+(\mu_{s-2,2}-\mu_{s-2,1})t_{1}.$$ We may then eliminate the variables $t_{2},\ldots,t_{2s-1}$ and just keep the variables $t_{1}$ and $t_{\alpha_{1},\ldots,\alpha_{s}}$. Let us set $t_{1}=z$ and $t_{\alpha}=w_{\alpha}$, for $\alpha \in V_{s}$. In these new $2^{s-1}$ coordinates, the above curve is isomorphic to the one given by $$C=\left\{ \begin{array}{c} w_{\alpha}^{2}=z^{\alpha_{1}}(-1-\mu_{s-2,2}z)^{\alpha_{1}} (1+(\mu_{s-2,2}-1)z)^{\alpha_{2}}(1+(\mu_{s-2,2}-\lambda)z )^{\alpha_{2}}\cdots\\ \cdots (1+(\mu_{s-2,2}-\mu_{k,1})z)^{\alpha_{k+2}}(1+(\mu_{s-2,2}-\mu_{k,2})z)^{\alpha_{k+2}} \cdots \\ \cdots (1+(\mu_{s-2,2}-\mu_{s-3,1})z)^{\alpha_{s-1}}(1+(\mu_{s-2,2}-\mu_{s-3,2})z)^{\alpha_{s-1}} (1+(\mu_{s-2,2}-\mu_{s-2,1})z)^{\alpha_{s}},\\ \alpha=(\alpha_{1},\ldots,\alpha_{s}) \in V_{s}.\\ \end{array} \right\}$$ By making the choices as described in the hypothesis of the theorem for $K_{\alpha}$ and the values of $\eta_{0}$, $\eta_{1}$, $\eta_{2}$, $\eta_{3}$ and $\eta_{k,j}$, then the above curve can be written in the desired algebraic form. If $\Phi_{1}:D \to {\mathbb C}^{2^{s-1}}$ is the map whose coordinates are $z$ and $w_{\alpha}$, where $\alpha \in V_{s}$, then $\Phi_{1}(D)=C$. If $\alpha=(\alpha_{1},\ldots,\alpha_{s})$ and $j=1,\ldots,s-1$, then the induced automorphisms $a_{j}$ acts by multiplication by $-1$ at coordinates $w_{\alpha}$ if $\alpha_{j}=1$ and acts by the identity on the rest of coordinates. The map $$\pi:C \to \widehat{\mathbb C}: (z,\{w_{\alpha \in V_{s}}\}) \mapsto \frac{1+\mu_{s-2,2}z}{z}$$ is a regular branched cover with $H$ as its deck group and its satisfies that $P=\pi \circ \Phi_{1}$. The branch locus of $\pi$ is the set $$\{\infty,0,1,\lambda, \mu_{1,1}, \mu_{1,2},\ldots,\mu_{s-2,1}, \mu_{s-2,2}\}.$$ \[lema01\] The only non-trivial elements of $H$ acting with fixed points are $a_{1},\ldots,a_{s-1}$ and $a_{s}=a_{1}a_{2} \cdots a_{s-1}$. Moreover, $$\pi({\rm Fix}(a_{1}))=\{\infty,0\}, \quad \pi({\rm Fix}(a_{2}))=\{1,\lambda\},$$ $$\pi({\rm Fix}(a_{k}))=\{\mu_{k-2,1}, \mu_{k-2,2}\}, \quad k=3,\ldots, s-1,$$ $$\pi({\rm Fix}(a_{s}))=\{\mu_{s-2,1}, \mu_{s-2,2}\}.$$ It can be seen that, for $j=1,\ldots,s$, ${\rm Fix}(a_{j})$ has cardinality $2^{s-1}$. This follows directly from all the above. If $2 \leq k \leq s$ is even and $1 \leq i_{1}<i_{2}< \cdots < i_{k} \leq s$, then we consider the subgroup $$H_{i_{1},i_{2},\ldots,i_{k}}=\langle a_{i_{1}}a_{i_{2}}, a_{i_{1}}a_{i_{3}},\ldots,a_{i_{1}}a_{i_{k}},a_{j}; j\in\{1,\ldots,s\}-\{i_{1},\ldots,i_{k}\}\rangle \cong{\mathbb Z}_{2}^{s-2}.$$ In the case $k=2$ we have $s(s-1)/2$ such subgroups. Between them are the following ones $$H_{1,2}=\langle a_{1}a_{2},a_{3},\ldots,a_{s}\rangle$$ $$H_{2,3}=\langle a_{2}a_{3},a_{4},\ldots,a_{1}\rangle$$ $$H_{3,4}=\langle a_{3}a_{4},a_{5},\ldots,a_{2}\rangle$$ $$\vdots$$ $$H_{s-1,s}=\langle a_{s-1}a_{s},a_{1},\ldots,a_{s-2}\rangle$$ $$H_{s,1}=\langle a_{s}a_{1},a_{2},\ldots,a_{s-1}\rangle$$ The quotient orbifold $C/H_{j,j+1}$ has underlying Riemann surface structure $E_{j} \cong E_{\lambda_{j}}$, for $j=1,\ldots,s-1$, and $C/H_{s,1}$ has underlying Riemann surface structure $E_{s} \cong E_{\lambda_{s}}$. \[lema02\] With the above notations, the following hold for the above defined subgroups. 1. Any two such subgroups $H_{i_{1},i_{2},\ldots,i_{k}}$ and $H_{j_{1},j_{2},\ldots,j_{l}}$ commute. 2. The quotient $C/H_{i_{1},i_{2},\ldots,i_{k}}$ has genus $k-1$ and its underlying Riemann surface is given by the (elliptic) hyperelliptic curve $$\nu^{2}=(\upsilon-\rho_{i_{1},1})(\upsilon-\rho_{i_{1},2})\cdots(\upsilon-\rho_{i_{k},1})(\upsilon-\rho_{i_{k},2}),$$ where $$\rho_{i_{j},1}=\left\{\begin{array}{cl} \infty, & i_{j}=1\\ 1, & i_{j}=2\\ \mu_{r-2,1}, & i_{j}=r \geq 3 \end{array} \right. \quad \rho_{i_{j},2}=\left\{\begin{array}{cl} 0, & i_{j}=1\\ \lambda, & i_{j}=2\\ \mu_{r-2,2}, & i_{j}=r\geq 3 \end{array} \right.$$ In the case that $\rho_{i_{j},1}=\infty$, then the factor $(u-\rho_{i_{j},1})$ is deleted from the above expression. 3. The group generated by any two different such subgroups is $H$. Property (1) holds trivially as $H$ is an abelian group. Property (2) follows from Riemann-Hurwitz formula and Lemma \[lema01\]. Property (3) is clear as in the product we obtain all the generators. The next result states that the sum of the genera appearing in all quotients of the form $C/H_{i_{1},i_{2},\ldots,i_{k}}$ is equal to the genus of $C$ (that is, the genus of $S$). Recall that we are considering $k$ even and $2 \leq k \leq s$. \[lema03\] $$\sum_{k=2}^{s} \binom{r}{k} (k-1) \left(\frac{1+(-1)^{k}}{2}\right) =1+2^{s-2}(s-2).$$ Consider the function $$f(x)=\frac{(1+x)^{s}}{2x}=\frac{1}{2}\sum_{k=0}^{s} \binom{s}{k} x^{k-1}$$ and its derivative $$f'(x)=\frac{(1+x)^{s-1}((s-1)x-1)}{2x^{2}}=\frac{1}{2}\sum_{k=0}^{s} \binom{s}{k} (k-1) x^{k-2}.$$ Next, we evaluate at $x=1$ and $x=-1$ and then we add the results to obtain $$2^{s-2}(s-2)=f'(1)+f'(-1)=\frac{1}{2}\sum_{k=0}^{s} \binom{s}{k} (k-1) (1+(-1)^{k}) =-1+\sum_{k=2}^{s} \binom{s}{k} \frac{(1+(-1)^{k})}{2} (k-1).$$ We may apply Proposition \[coroKR\] for $C$ using all the subgroups $H_{i_{1},\ldots,i_{k}}$ in order to obtain that $JC$ (so $JS$) is isogenous to the product of the jacobian varieties of all Riemann surfaces $C/H_{i_{1},\ldots,i_{k}}$ as desired. The equations of these curves are provided in Lemma \[lema02\]. Proof of Theorem \[coro3\] ========================== As a consequence of Theorem \[construccion\] (and Lemma \[lema02\]), the jacobian variety of $S$ is isogenous to the product of the following six elliptic curves $$\begin{array}{ll} E_{1}: & y^{2}=x(x-1)(x-\lambda)\\ E_{2}: & y^{2}=(x-1)(x-\lambda)(x-\mu_{1,1})(x-\mu_{1,2})\\ E_{3}: & y^{2}=(x-\mu_{1,1})(x-\mu_{1,2})(x-\mu_{2,1})(x-\mu_{2,2})\\ E_{4}: & y^{2}=x(x-\mu_{2,1})(x-\mu_{2,2})\\ E_{5}: & y^{2}=x(x-\mu_{1,1})(x-\mu_{1,2})\\ E_{6}: & y^{2}=(x-1)(x-\lambda)(x-\mu_{2,1})(x-\mu_{2,2}) \end{array}$$ and the jacobian variety of the following genus three hyperelliptic Riemann surface $$R: y^{2}=x(x-1)(x-\lambda)(x-\mu_{1,1})(x-\mu_{1,2})(x-\mu_{2,1})(x-\mu_{2,2}).$$ Let us observe that the group $J=\langle f_{1}(x)=\lambda/x, f_{2}(x)=\lambda (x-1)/(x-\lambda) \rangle \cong {\mathbb Z}_{2}^{2}$ keeps invariant the collection $$\infty, 0, 1, \lambda, \mu_{1,1}, \mu_{1,2}, \mu_{2,1}, \mu_{2,2}.$$ In this way, $R$ admits the following automorphisms $$F_{1}(x,y)=\left( \frac{\lambda}{x}, \frac{\lambda^{2} y}{x^{4}}\right)$$ $$F_{2}(x,y)=\left( \frac{\lambda(x-1)}{x-\lambda}, \frac{\lambda^{2}(\lambda-1)^{2} y}{(x-\lambda)^{4}}\right)$$ We may see that $\langle F_{1}, F_{2}\rangle \cong {\mathbb Z}_{2}^{2}$ and that each of the invoilutions $F_{1}$, $F_{2}$ and $F_{1}\circ F_{2}$ acts with exactly $4$ fixed points on $R$. The quotients $R/\langle F_{1} \rangle$, $R/\langle F_{2} \rangle$ and $R/\langle F_{1} \circ F_{2}\rangle$ have genus one. We may apply Proposition \[coroKR\] to $R$ using the three cyclic groups of order two in order to see that $JR$ is isogenous to the product of three elliptic curves. Proof of Theorem \[coro1\] ========================== Given the triple $(\lambda_{1},\lambda_{2},\lambda_{3}) \in \Delta_{3}$ and $\mu$ a root of $$\lambda_{2}\lambda_{3} \mu^{2}-(\lambda_{1}\lambda_{2}+\lambda_{2}\lambda_{3}+\lambda_{1}\lambda_{3}-\lambda_{1}-\lambda_{3}+1) \mu +\lambda_{1}\lambda_{2}=0,$$ then we set $\lambda=\lambda_{1}$, $\mu_{1,1}=\mu$ and $\mu_{1,2}=\lambda_{3}\mu$. It can be seen that $(\lambda,\mu_{1,1},\mu_{1,2}) \in \Delta_{3}$ and that $$F_{1}: y^{2}=x(x-1)(x-\lambda_{1}) := E_{\lambda_{1}},$$ $$F_{2}: y^{2}=(x-1)(x-\lambda_{1})(x-\mu)(x-\lambda_{3}\mu):\cong E_{\lambda_{2}},$$ $$F_{3}: y^{2}=x(x-\mu)(x-\lambda_{3}\mu) :\cong E_{\lambda_{3}}.$$ Theorem \[construccion\], applied to the triple $(\lambda,\mu_{1,1},\mu_{1,2})$, asserts that $JS$ is isogenous to the product $F_{1} \times F_{2} \times F_{3}$, so isogenous to the product $E_{1} \times E_{1} \times E_{3}$. In this case the Riemann surface $S$ is described by the curve $$\left\{ \begin{array}{lcl} w_{1}^{2}&=&\mu(\lambda_{3} \mu -1)(\lambda_{3} \mu -\lambda_{1})(\lambda_{3} - 1) z \left( z-\dfrac{1}{\lambda_{1}-\lambda_{3}\mu} \right) \left(z-\dfrac{1}{\mu (1-\lambda_{3})} \right)\\ w_{2}^{2}&=&-\lambda_{3}\mu^{2}(\lambda_{3}-1)z\left(z+\dfrac{1}{\lambda_{3} \mu}\right) \left(z-\dfrac{1}{1-\lambda_{3}\mu} \right)\\ w_{3}^{2}&=&-\lambda_{3}\mu^{2}(\lambda_{3}\mu-1)(\lambda_{3}-1)z^{2}\left(z+\dfrac{1}{\lambda_{3} \mu}\right) \left(z-\dfrac{1}{\mu(1-\lambda_{3})}\right) \\ \end{array} \right\},$$ the group $H=\langle a_{1},a_{2}\rangle \cong {\mathbb Z}_{2}^{2}$ is generated by $$a_{1}(z,w_{1},w_{2},w_{3})=(z,w_{1},-w_{2},-w_{3}),$$ $$a_{2}(z,w_{1},w_{2},w_{3})=(z,-w_{1},w_{2},-w_{3}),$$ the three automorphisms $a_{1}$, $a_{2}$ and $a_{3}=a_{1}a_{2}$ acts with exactly four fixed points each one, and the corresponding regular branched cover with $H$ as deck group is $$\pi:S \to \widehat{\mathbb C}: (z,w_{1},w_{2},w_{3}) \mapsto \frac{\lambda_{3} \mu z +1}{z}.$$ The subgroups of $H$, in this case ($k=2$ is the only option) are $$H_{1,2}=\langle a_{1}a_{2}\rangle$$ $$H_{1,3}=\langle a_{1}a_{3}\rangle=\langle a_{2} \rangle$$ $$H_{2,3}=\langle a_{2}a_{3}\rangle=\langle a_{1}\rangle$$ The quotients $S/\langle a_{1} \rangle$, $S/\langle a_{2} \rangle$ and $S/\langle a_{1}a_{2} \rangle$ are of genus one and they correspond, respectively, to the elliptic curves $E_{\lambda_{2}}$, $E_{\lambda_{1}}$ and $E_{\lambda_{3}}$. Proof of Theorem \[coro2\] ========================== Let us first consider the case $r \geq 3$ odd. Write $r=2s-3$, where $s \geq 3$, and let us fix $(\lambda_{1},\ldots,\lambda_{r}) \in \Delta_{r}$. Set $\lambda=\lambda_{1}$ and, for $j=1,\ldots,s-2$, we set $\mu_{j,2}=\lambda_{j+1} \mu_{j,1}$ and $\mu_{j,1}$ a root of the polynomial $$\lambda_{j+1}(1-\lambda_{s-2+j})\mu_{j,1}^{2}+(\lambda_{s-2+j}-\lambda_{j+1}-\lambda_{1}+\lambda_{1}\lambda_{j+1}\lambda_{s-2+j})\mu_{j,1}+(1-\lambda_{1}\lambda_{s-2+j})=0.$$ Note that a value $\lambda_{j}$ may be changed to some other value $\lambda_{j}'$ (inside an infinite set of values) so that $E_{\lambda_{j}}$ and $E_{\lambda_{j}'}$ are isogenous. This observation permits to ensure that the constructed tuple $(\lambda,\mu_{1,1},\mu_{1,2},\mu_{2,1},\mu_{2,2},\ldots,\mu_{s-2,1},\mu_{s-2,2}) \in \Delta_{2s-3}$. Let us consider the Riemann surface $S$ constructed in Theorem \[construccion\]. The jacobian variety of $S$ is isogenous to a product of certain explicit jacobian varieties. It can be seen, from the proof of that theorem, that some of these factors are (isogenous to) the elliptic curves $$y^{2}=x(x-1)(x-\lambda)$$ $$y^{2}=x(x-\mu_{j,1})(x-\mu_{j,2}), \quad j=1,\ldots,s-2,$$ $$y^{2}=(x-1)(x-\lambda)(x-\mu_{j,2}), \quad j=1,\ldots,s-2.$$ The choice we have done for the tuple $(\lambda,\mu_{1,1},\mu_{1,2},\mu_{2,1},\mu_{2,2},\ldots,\mu_{s-2,1},\mu_{s-2,2}) \in \Delta_{2s-3}$ ensures that they are isomorphic to the elliptic curves $$y^{2}=x(x-1)(x-\rho)$$ where $\rho \in \{\lambda_{1},\ldots,\lambda_{r}\}$. The case $r \geq 4$ even can be worked similarly, but in this case we add an extra elliptic curve to the $r$ given ones in order to obtain the result as a consequence of the odd situation. An irreducible fiber product of elliptic curves: Theorem \[construccion2\] {#Sec:otraconstruccion} ========================================================================== In the above main construction, the fiber product turned-out to be reducible; it has two isomorphic irreducible components. In this section we describe a similar construction, but the fiber product we obtain is irreducible. Unfortunately, the genus is bigger than the previous one. The fiber product construction ------------------------------ Let $(\lambda_{1},\ldots,\lambda_{r}) \in \Delta_{r}$, $E_{\lambda_{j}}: y_{j}^{2}=x_{j}(x_{j}-1)(x_{j}-\lambda_{j})$ and $\pi_{j}:E_{\lambda_{j}} \to \widehat{\mathbb C}$ defined by $\pi_{j}(x_{j},y_{j})=x_{j}$. The locus of branch values of $\pi_{j}$ is the set $\{\infty,0,1,\lambda_{j}\}$. Let $X$ be the fiber product of these $r$ pairs $(E_{\lambda_{1}},\pi_{1}),\ldots,(E_{\lambda_{r}},\pi_{r})$, that is, $$X=\left\{(x,y_{1},\ldots,y_{r}): y_{j}^{2}=x(x-1)(x-\lambda_{j}), \; j=1,\ldots,r \right\}.$$ The curve $X$ is irreducible but it has singular points at those points with first coordinate $x\in \{\infty,0,1\}$. The Riemann surface defined by $X$ (after desingularization) admits a group of conformal automorphisms $H \cong {\mathbb Z}_{2}^{r}$ so that $X/H$ is the Riemann sphere with conical points (each one of order two) at $\infty$, $0$, $1$, $\lambda_{1},\ldots, \lambda_{r}$. In particular, it has genus $g=1+2^{r-2}(r-1)$. We should to proceed to see that $JX$ is isogenous to a product of the form $E_{\lambda_{1}} \times \cdots \times E_{\lambda_{r}} \times A_{g-r}$, where $A_{g-r}$ is a suitable product of jacobian varieties of elliptix/hyperelliptic Riemann surfaces (Theorem \[descompone\]). Another way to describe $X$ --------------------------- Let us consider the generalized Humbert curve (see [@C-G-H-R]) $$F: \left\{ \begin{array}{ccc} z_{1}^{2}+z_{2}^{2}+z_{3}^{2}&=&0\\ \lambda_{1}z_{1}^{2}+z_{2}^{2}+z_{4}^{2}&=&0\\ \lambda_{2}z_{1}^{2}+z_{2}^{2}+z_{5}^{2}&=&0\\ \vdots & \vdots& \vdots\\ \lambda_{r}z_{1}^{2}+z_{2}^{2}+z_{r+3}^{2}&=&0 \end{array} \right\} \subset {\mathbb P}^{n}.$$ The conditions on the parameters $\lambda_{j}$ ensure that $F$ is a non-singular projective algebraic curve, that is, a closed Riemann surface. On $F$ we have the Abelian group $\langle b_{1},\ldots,b_{r+2}\rangle=H_{0} \cong {\mathbb Z}_{2}^{r+2}$ of conformal automorphisms, where $$b_{j}([z_{1}:\cdots:z_{r+3}])=[z_{1}:\cdots:z_{j-1}: - z_{j}:z_{j+1}:\cdots:z_{r+3}], \; j=1,...,r+2.$$ Inside the group $H_{0}$, the only non-trivial elements acting with fixed points are $b_{1},\ldots, b_{r+2}$ and $b_{r+3}=b_{1}b_{2}\cdots b_{r+2}$. The degree $2^{r+2}$ holomorphic map $$P:F \to \widehat{\mathbb C}: [z_{1}:\cdots:z_{r+3}]) \mapsto-\left( \dfrac{z_{2}}{z_{1}} \right)^{2},$$ is a branched regular cover with deck group being $H_{0}$. The projection under $P$ of the set of fixed points are as follows: $$P({\rm Fix}(b_{1}))=\infty, \quad P({\rm Fix}(b_{2}))=0, \quad P({\rm Fix}(b_{3}))=1,$$ $$P({\rm Fix}(b_{3+j}))=\lambda_{j}, \quad j=1,\ldots, r.$$ In partricular, the branch locus of $P$ is the set $$\{\infty,0,1,\lambda_{1},\ldots,\lambda_{r}\}.$$ By the Riemann-Hurwitz formula, $F$ has genus $1+2^{r}(r-1)$. Let us consider the subgroup $K^{*}=\langle b_{1}b_{2}, b_{2}b_{3}\rangle \cong {\mathbb Z}_{2}^{2}$. It can be seen that $X=F/K^{*}$ and, in particular, that $X$ has genus $g_{X}=1+2^{r-2}(r-1)$. The quotient group $H_{0}/\langle b_{1}b_{2}, b_{2}b_{3}\rangle$ induces the group of automorphisms of $X$ given by $$L=\langle c_{1},\ldots,c_{r}\rangle \cong {\mathbb Z}_{2}^{r},$$ where $$c_{j}(x,y_{1},\ldots,y_{r})=(x,y_{1},\ldots,y_{j-1}, - y_{j},y_{j+1},\ldots,y_{r}), \; j=1,...,r.$$ The map $$\pi:X \to \widehat{\mathbb C}: (x,y_{1},\ldots,y_{r}) \mapsto x$$ is a regular branched cover with $H$ as its deck group. The branch locus of $\pi$ is the set $$\{\infty,0,1,\lambda_{1},\ldots,\lambda_{r}\}.$$ With the above description, we obtain another set of equations for $X$ as $$\left\{\begin{array}{c} w_{1}^{2}=(\lambda_{r}-\lambda_{1})u+1\\ w_{2}^{2}=(\lambda_{r}-\lambda_{2})u+1\\ \vdots \\ w_{r-1}^{2}=(\lambda_{r}-\lambda_{r-1})u+1\\ w_{r}^{2}=-u(\lambda_{r}u+1)((\lambda_{r}-1)u+1) \end{array} \right.$$ The equations for $E_{\lambda_{j}}$ can also be written as (for $j=1,\ldots,r-1$) $$\left\{\begin{array}{c} w_{1}^{2}=(\lambda_{r}-\lambda_{1})u+1\\ \vdots\\ w_{j-1}^{2}=(\lambda_{r}-\lambda_{j-1})u+1\\ w_{j}^{2}=(\lambda_{r}-\lambda_{j})u+1\\ \vdots \\ w_{r-1}^{2}=(\lambda_{r}-\lambda_{r-1})u+1\\ w_{r}^{2}=-u(\lambda_{r}u+1)((\lambda_{r}-1)u+1)((\lambda_{r}-\lambda_{j})u+1) \end{array} \right.$$ and for $E_{\lambda_{r}}$ as $$\left\{\begin{array}{c} w_{1}^{2}=(\lambda_{r}-\lambda_{1})u+1\\ \vdots\\ w_{r-1}^{2}=(\lambda_{r}-\lambda_{r-1})u+1\\ w_{r}^{2}=-u(\lambda_{r}u+1)((\lambda_{r}-1)u+1) \end{array} \right.$$ \[lema1\] The only non-trivial elements of $L$ acting with fixed points are $c_{1},\ldots,c_{r}$ and $c_{r+1}=c_{1}c_{2} \cdots c_{r}$. Moreover, $$\pi({\rm Fix}(c_{j}))=\lambda_{j}, \quad j=1,\ldots,r,$$ $$\pi({\rm Fix}(c_{r+1}))=\{\infty,0,1\}.$$ A non-trivial element of $L$ has the form $$c(x,y_{1},\ldots,y_{r})=(x,(-1)^{\alpha_{1}}y_{1},\ldots,(-1)^{\alpha_{r}}y_{r}),$$ where $\alpha_{1},\ldots,\alpha_{r} \in \{0,1\}$ and $\alpha_{1}+\cdots+\alpha_{r}>0$. A point $(x,y_{1},\ldots,y_{r}) \in C$ is a fixed point of $c$ if and only if $y_{j}=0$ for $\alpha_{j}=1$. The equality $y_{j}=0$ is equivalent to have $x \in \{\infty,0,1,\lambda_{j}\}$. The values $x \in \{\infty,0,1\}$ produce fixed points for $c_{r+1}$. Also, as we are assume the values $\lambda_{j}$ to be different, it follows that the only possibility is to have only one $j$ with $\alpha_{j}=1$. It can be seen that, for $j=1,\ldots,r$, ${\rm Fix}(c_{j})$ has cardinality $2^{r-1}$ and that ${\rm Fix}(c_{1}\cdots c_{r})$ has cardinality $3 \times 2^{r-1}$. In particular, for $r=3$, the surface $X$ is hyperelliptic with $c_{1}c_{2}c_{3}$ as its hyperelliptic involution. Some subgroups of $L$ --------------------- If either $1 \leq k \leq r$ is odd or $4 \leq k \leq r$ is even, and $\{i_{1},\ldots,i_{k}\} \subset \{1,\ldots,r\}$ with $1 \leq i_{1}<i_{2}< \cdots < i_{k} \leq r$, then we consider the subgroup $$L_{i_{1},i_{2},\ldots,i_{k}}=\langle c_{i_{1}}c_{i_{2}}, c_{i_{1}}c_{i_{3}},\ldots,c_{i_{1}}c_{i_{k}},c_{j}; j\in\{1,\ldots,r\}-\{i_{1},\ldots,i_{k}\}\rangle \cong{\mathbb Z}_{2}^{r-1}.$$ Note that for $k=1$ we have the subgroups $$L_{j}=\langle c_{1},\ldots,c_{j-1},c_{j+1},\ldots,c_{r}\rangle \cong {\mathbb Z}_{2}^{r-1}.$$ If $$Q_{j}:X \to E_{\lambda_{j}}: (x,y_{1},\ldots,y_{r}) \mapsto (x,y_{j}),$$ then $Q_{j}$ is a regular branched cover with deck group being $L_{j}$. The branch locus of $Q_{j}$ is the set $$\{(\lambda_{j},y_{i}): y_{i}^{2}=\lambda_{i}(\lambda_{i}-1)(\lambda_{i}-\lambda_{j}), \; i=1,\ldots,r, i \neq j\}.$$ \[lema2\] With the above notations, the following hold for the above defined subgroups. 1. Any two such subgroups commute. 2. The quotient $X/L_{i_{1},i_{2},\ldots,i_{k}}$ is an orbifold of genus $(k+1)/2$ if $k$ is odd and genus $(k-2)/2$ if $k$ is even. Moreover, the underlying Riemann surface is given by the hyperelliptic curve $$w^{2}=z(z-1)(z-\lambda_{i_{1}})(z-\lambda_{i_{2}})\cdots(z-\lambda_{i_{k}}), \quad \mbox{ if $k$ is odd},$$ $$w^{2}=(z-\lambda_{i_{1}})(z-\lambda_{i_{2}})\cdots(z-\lambda_{i_{k}}), \quad \mbox{ if $k$ is even}.$$ 3. The group generated by any two different such subgroups is $L$. Property (1) holds trivially as $L$ is an abelian group. Property (2) follows from Riemann-Hurwitz formula and Lemma \[lema1\]. Property (3) is clear. The next result states that the sum of the genera appearing in all quotients of the form $X/L_{i_{1},i_{2},\ldots,i_{k}}$ is equal to the genus of $X$. \[lema3\] $$\sum_{k=1}^{r} \binom{r}{k} \frac{(1-(-1)^k)}{2} \frac{(k+1)}{2} + \sum_{k=4}^{r} \binom{r}{k} \frac{(1+(-1)^k)}{2} \frac{(k-2)}{2} =1+2^{r-2}(r-1).$$ Consider the functions $$f_{1}(x)=x\frac{(1+x)^{r}}{4}=\frac{1}{4}\sum_{k=0}^{r} \binom{r}{k} x^{k+1},$$ $$f_{2}(x)=\frac{(1+x)^{r}}{4x^{2}}= \frac{1}{4}\sum_{k=0}^{r} \binom{r}{k} x^{k-2},$$ and their derivetaives $$f'_{1}(x)=\frac{(1+x)^{r}(1+(1+r)x)}{4}=\frac{1}{4}\sum_{k=0}^{r} \binom{r}{k} (k+1) x^{k},$$ $$f'_{2}(x)=\frac{(1+x)^{r-1}(r-2x(1+x))}{4x^{2}}=\frac{1}{4}\sum_{k=0}^{r} \binom{r}{k} (k-2)x^{k-3}.$$ Next, we evaluate at $x=\pm 1$ to obtain $$2^{r-3}(r+2)=f'_{1}(1)-f'_{1}(-1)=\frac{1}{4}\sum_{k=0}^{r} \binom{r}{k} (k+1)(1-(-1)^{k})=\sum_{k=1}^{r} \binom{r}{k} \frac{(1-(-1)^{k})}{2} \frac{(k+1)}{2},$$ $$2^{r-3}(r-4)=f'_{2}(1)+f'_{2}(-1)=\frac{1}{4}\sum_{k=0}^{r} \binom{r}{k} (k-2)(1+(-1)^{k})=-1+\sum_{k=4}^{r} \binom{r}{k} \frac{(1+(-1)^k)}{2} \frac{(k-2)}{2}.$$ By adding the above equalities we obtain the desired result. Decomposition of $JX$ --------------------- We may apply Proposition \[coroKR\] for $X$ using the subgroups $L_{i_{1},\ldots,i_{k}}$ in order to obtain the following. \[descompone\] $JX$ is isogenous to a product of the form $E_{\lambda_{1}} \times \cdots \times E_{\lambda_{r}} \times A_{g-r}$, where $A_{g-r}$ is the product of the jacobian varieties of all elliptic/hyperelliptic Riemann surfaces $X/L_{i_{1},\ldots,i_{k}}$, for $k \geq 2$. Case $r=3$: A construction of a $2$-dimensional family of curves $X$ of genus five with $JX$ isogenous to the product of five elliptic curves {#casog=5} --------------------------------------------------------------------------------------------------------------------------------------------- In this case, the jacobian variety of the fiber product $X$ (being of genus $5$) of the three elliptic curves $$E_{\lambda_{1}}: y^{2}=x(x-1)(x-\lambda_{1}),\; E_{\lambda_{2}}: y^{2}=x(x-1)(x-\lambda_{2}),\; E_{\lambda_{3}}: y^{2}=x(x-1)(x-\lambda_{3}),$$ is isogenous to the product $$E_{\lambda_{1}} \times E_{\lambda_{2}} \times E_{\lambda_{3}} \times JS_{0},$$ where $$S_{0}: y^{2}=x(x-1)(x-\lambda_{1})(x-\lambda_{2})(x-\lambda_{3}).$$ The corresponding subgroups of $H$ are in this case $$L_{1}=\langle c_{2},c_{3}\rangle, \quad L_{2}=\langle c_{1},c_{3}\rangle, \quad L_{3}=\langle c_{1},c_{2}\rangle, \quad L_{1,2,3}=\langle c_{1}c_{2}, c_{1}c_{3}\rangle.$$ \[g=5\] If $\lambda_{3}=\lambda_{1}/\lambda_{2}$, then $JX$ isogenous to the product of $5$ elliptic curves. If we assume that $\lambda_{3}=\lambda_{1}/\lambda_{2}$, then $S$ admits the involution $(x,y) \mapsto (\lambda_{1}/x, \lambda_{1}^{3/2} y/x^{3})$. Such an involution has exactly two fixed points. It follows that, under this restriction, $JS_{0}$ is isogenous to the product of two elliptic curves. Corollary \[g=5\] provide a $2$-dimensional family of curves $X$ of genus five with $JX$ isogenous to the product of five elliptic curves. Case $r=4$: A construction of a $1$-dimensional family of curves $X$ of genus $13$ with $JX$ isogenous to the product of $13$ elliptic curves {#casog=3} --------------------------------------------------------------------------------------------------------------------------------------------- In this case, the jacobian variety of the fiber product $X$ (being of genus $13$) of the four elliptic curves $$E_{\lambda_{1}}: y^{2}=x(x-1)(x-\lambda_{1}),\; E_{\lambda_{2}}: y^{2}=x(x-1)(x-\lambda_{2}),$$ $$E_{\lambda_{3}}: y^{2}=x(x-1)(x-\lambda_{3}),\; E_{\lambda_{4}}: y^{2}=x(x-1)(x-\lambda_{4}),$$ is isogenous to the product $$E_{\lambda_{1}} \times E_{\lambda_{2}} \times E_{\lambda_{3}} \times E_{\lambda_{4}} \times JS_{1} \times JS_{2} \times JS_{3} \times JS_{4} \times E_{5},$$ where $$S_{1}: y^{2}=x(x-1)(x-\lambda_{1})(x-\lambda_{2})(x-\lambda_{3}),$$ $$S_{2}: y^{2}=x(x-1)(x-\lambda_{1})(x-\lambda_{2})(x-\lambda_{4}),$$ $$S_{3}: y^{2}=x(x-1)(x-\lambda_{1})(x-\lambda_{3})(x-\lambda_{4}),$$ $$S_{4}: y^{2}=x(x-1)(x-\lambda_{2})(x-\lambda_{3})(x-\lambda_{4}),$$ $$E_{5}: y^{2}=(x-\lambda_{1})(x-\lambda_{2})(x-\lambda_{3})(x-\lambda_{4}).$$ The corresponding subgroups of $H$ are in this case $$L_{1}=\langle c_{2},c_{3},c_{4}\rangle, \quad L_{2}=\langle c_{1},c_{3},c_{4}\rangle, \quad L_{3}=\langle c_{1},c_{2},c_{4}\rangle, \quad L_{4}=\langle c_{1},c_{2},c_{3}\rangle,$$ $$L_{1,2,3}=\langle c_{1}c_{2}, c_{1}c_{3}, c_{4}\rangle, \quad L_{1,2,4}=\langle c_{1}c_{2}, c_{1}c_{4}, c_{3}\rangle, \quad L_{1,3,4}=\langle c_{1}c_{3}, c_{1}c_{4}, c_{2}\rangle,$$ $$L_{2,3,4}=\langle c_{2}c_{3}, c_{2}c_{4}, c_{1}\rangle, \quad L_{1,2,3,4}=\langle c_{1}c_{2},c_{1}c_{3},c_{1}c_{4}\rangle.$$ \[g=13\] If $\lambda_{3}=\lambda_{1}/\lambda_{2}$, $\lambda_{4}=\lambda_{1}(\lambda_{2}-1)/(\lambda_{2}-\lambda_{1})$ and $\lambda_{2}^{2}(1+\lambda_{1})-4\lambda_{1}\lambda_{2}+\lambda_{1}(1+\lambda_{1})=0$, then $JX$ isogenous to the product of $13$ elliptic curves. If $a_{1}(z)=\lambda_{1}/z$ and $a_{2}(z)=\lambda_{1}(z-1)/(z-\lambda_{1})$, then the group generated by them is isomorphic to ${\mathbb Z}_{2}^{2}$. Since $a_{1}$ permutes in pairs the elements in $\{\infty,0,1,\lambda_{1},\lambda_{2},\lambda_{3}\}$, it follows that $JS_{1}$ is isogenous to the product of two elliptic curves. Similarly, as $a_{2}$ permutes in pairs the elements in $\{\infty,0,1,\lambda_{1},\lambda_{2},\lambda_{4}\}$, it follows that $JS_{2}$ is isogenous to the product of two elliptic curves and as $a_{2}a_{1}$ permutes in pairs the elements in $\{\infty,0,1,\lambda_{1},\lambda_{3},\lambda_{4}\}$, it follows that $JS_{3}$ is isogenous to the product of two elliptic curves. In this way, under the above assumptions, $JX$ is isogenous to the product of $11$ elliptic curves and $JS_{4}$. If we also assume that $\lambda_{2}^{2}(1+\lambda_{1})-4\lambda_{1}\lambda_{2}+\lambda_{1}(1+\lambda_{1})=0$, then $a_{3}(z)=\lambda_{2}(z-\lambda_{3})/(z-\lambda_{2})$ permutes in pairs the elements of the set $\{\infty,0,1,\lambda_{2},\lambda_{3},\lambda_{4}\}$. In this case, $JS_{4}$ is also isogenous to the product of two elliptic curves Corollary \[g=13\] provides a $1$-dimensional family of curves $X$ of genus $13$ with $JX$ isogenous to the product of $13$ elliptic curves. Examples of values as in the above proposition are $\lambda_{1}=2$ and $\lambda_{2}=(4+i\sqrt{2})/3$; so $\lambda_{3}=(4-i\sqrt{2})/3$ and $\lambda_{4}=-i\sqrt{2}$. Case $r=5$ ---------- In this case, the jacobian variety of the fiber product $X$ (being of genus $33$) of the four elliptic curves $$E_{\lambda_{1}}: y^{2}=x(x-1)(x-\lambda_{1}),\; E_{\lambda_{2}}: y^{2}=x(x-1)(x-\lambda_{2}),\; E_{\lambda_{3}}: y^{2}=x(x-1)(x-\lambda_{3}),$$ $$E_{\lambda_{4}}: y^{2}=x(x-1)(x-\lambda_{4}),\; E_{\lambda_{5}}: y^{2}=x(x-1)(x-\lambda_{5}),$$ is isogenous to the product $$E_{\lambda_{1}} \times E_{\lambda_{2}} \times E_{\lambda_{3}} \times E_{\lambda_{4}} \times E_{\lambda_{5}}\times JS_{1} \times \cdots \times JS_{10} \times JS_{11} \times E_{6} \times \cdots \times E_{10},$$ where $$S_{1}: y^{2}=x(x-1)(x-\lambda_{1})(x-\lambda_{2})(x-\lambda_{3}),$$ $$S_{2}: y^{2}=x(x-1)(x-\lambda_{1})(x-\lambda_{2})(x-\lambda_{4}),$$ $$S_{3}: y^{2}=x(x-1)(x-\lambda_{1})(x-\lambda_{2})(x-\lambda_{5}),$$ $$S_{4}: y^{2}=x(x-1)(x-\lambda_{1})(x-\lambda_{3})(x-\lambda_{4}),$$ $$S_{5}: y^{2}=x(x-1)(x-\lambda_{1})(x-\lambda_{3})(x-\lambda_{5}),$$ $$S_{6}: y^{2}=x(x-1)(x-\lambda_{1})(x-\lambda_{4})(x-\lambda_{5}),$$ $$S_{7}: y^{2}=x(x-1)(x-\lambda_{2})(x-\lambda_{3})(x-\lambda_{4}),$$ $$S_{8}: y^{2}=x(x-1)(x-\lambda_{2})(x-\lambda_{3})(x-\lambda_{5}),$$ $$S_{9}: y^{2}=x(x-1)(x-\lambda_{2})(x-\lambda_{4})(x-\lambda_{5}),$$ $$S_{10}: y^{2}=x(x-1)(x-\lambda_{3})(x-\lambda_{4})(x-\lambda_{5}),$$ $$S_{11}: y^{2}=x(x-1)(x-\lambda_{1})(x-\lambda_{2})(x-\lambda_{3})(x-\lambda_{4})(x-\lambda_{5}),$$ $$E_{6}: y^{2}=(x-\lambda_{1})(x-\lambda_{2})(x-\lambda_{3})(x-\lambda_{4}),$$ $$E_{7}: y^{2}=(x-\lambda_{1})(x-\lambda_{2})(x-\lambda_{3})(x-\lambda_{5}),$$ $$E_{8}: y^{2}=(x-\lambda_{1})(x-\lambda_{2})(x-\lambda_{4})(x-\lambda_{5}),$$ $$E_{9}: y^{2}=(x-\lambda_{1})(x-\lambda_{3})(x-\lambda_{4})(x-\lambda_{5}),$$ $$E_{10}: y^{2}=(x-\lambda_{2})(x-\lambda_{3})(x-\lambda_{4})(x-\lambda_{5}).$$ The corresponding subgroups of $H$ are in this case $$L_{1}=\langle c_{2},c_{3},c_{4},c_{5}\rangle, \quad L_{2}=\langle c_{1},c_{3},c_{4},c_{5}\rangle, \quad L_{3}=\langle c_{1},c_{2},c_{4},c_{5}\rangle,$$ $$L_{4}=\langle c_{1},c_{2},c_{3},c_{5}\rangle, \quad L_{5}=\langle c_{1},c_{2},c_{3},c_{4}\rangle,$$ $$L_{1,2,3}=\langle c_{1}c_{2}, c_{1}c_{3}, c_{4}, c_{5}\rangle, \quad L_{1,2,4}=\langle c_{1}c_{2}, c_{1}c_{4}, c_{3}, c_{5}\rangle, \quad L_{1,2,5}=\langle c_{1}c_{2}, c_{1}c_{5}, c_{3}, c_{4}\rangle,$$ $$L_{1,3,4}=\langle c_{1}c_{3}, c_{1}c_{4}, c_{2}, c_{5}\rangle, \quad L_{1,3,5}=\langle c_{1}c_{3},c_{1}c_{5}, c_{2}, c_{4}\rangle, \quad L_{1,4,5}=\langle c_{1}c_{4}, c_{1}c_{5}, c_{2}, c_{3}\rangle,$$ $$L_{2,3,4}=\langle c_{2}c_{3}, c_{2}c_{4}, c_{1}, c_{5}\rangle, \quad L_{2,3,5}=\langle c_{2}c_{3}, c_{2}c_{5}, c_{1}, c_{4}\rangle, \quad L_{2,4,5}=\langle c_{2}c_{4}, c_{2}c_{5}, c_{1}, c_{3}\rangle,$$ $$L_{3,4,5}=\langle c_{3}c_{4},c_{3}c_{5}, c_{1}, c_{2}\rangle, \quad L_{1,2,3,4}=\langle c_{1}c_{2},c_{1}c_{3},c_{1}c_{4}\rangle,\quad L_{1,2,3,5}=\langle c_{1}c_{2},c_{1}c_{3},c_{1}c_{5}\rangle,$$ $$L_{1,2,4,5}=\langle c_{1}c_{2},c_{1}c_{4},c_{1}c_{5}\rangle,\quad L_{1,3,4,5}=\langle c_{1}c_{3},c_{1}c_{4},c_{1}c_{5}\rangle, \quad L_{2,3,4,5}=\langle c_{2}c_{3},c_{2}c_{4},c_{2}c_{5}\rangle.$$ [99]{} A. 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--- author: - Clare Burrage - 'Andrew Kuribayashi-Coleman' - James Stevenson - and Ben Thrussell bibliography: - 'draft.bib' title: Constraining symmetron fields with atom interferometry --- Introduction ============ Theories of dark energy that introduce new, light scalar fields coupled to matter have inspired the study of screening mechanisms to explain why the associated fifth forces have not yet been detected [@Joyce:2014kja; @Clifton:2011jh]. Screening mechanisms allow the scalar field theory to have non-trivial self-interactions, and so the properties of the scalar, and the resulting fifth force, can vary with the environment. Whilst screening mechanisms were introduced in order to explain the absence of an observation of a fifth force to date, that does not mean that such fifth forces are intrinsically unobservable. Experimental searches need only to be carefully designed to take advantage of the non-linear screening behaviour. Given a background field profile, the self interactions of the screened scalar field can have three possible consequences on the properties of scalar fluctuations on top of that background [@Joyce:2014kja]: 1. The mass of the fluctuations becomes dependent on the background. If the field becomes heavy in dense environments and light in diffuse ones, this can explain why the scalar force would not be detected around the macroscopic dense sources used in current fifth force experiments. This is known as the chameleon mechanism after the archetypal chameleon model [@Khoury:2003rn; @Khoury:2003aq]. 2. The strength of the coupling to matter becomes dependent on the background. If the field becomes weakly coupled in experimental environments it is clear that it will be harder to detect. Examples of models that employ this mechanism include the symmetron [@Hinterbichler:2010es; @Hinterbichler:2011ca] and density dependent dilaton [@Damour:1994zq; @Brax:2010gi]. 3. The coefficient of the scalar kinetic term becomes dependent on the background. If the coefficient becomes large in experimental searches it becomes difficult for the scalar to propagate, and so the force is suppressed. This effect occurs in any model which has gradient self interactions, including Galileon [@Nicolis:2008in] and k-essence models [@Babichev:2009ee; @Brax:2012jr; @Burrage:2014uwa], and is called the Vainshtein mechanism [@Vainshtein:1972sx]. It has recently been demonstrated that atomic nuclei inside a high quality vacuum chamber are very sensitive probes of chameleon screening, this is because the nucleus is so small that the screening cannot work efficiently. Forces on individual atoms can now be measured to a very high precision using atom interferometry, and as a result new constraints on chameleon models have been derived. Further improvements to these experiments are cureently underway. It remains to be determined whether the power of atom interferometry can be extended to constrain theories which screen through other means. Vainshtein screening will not be accessible, because the gradient self-interactions mean that the screening takes place over much longer distance scales than are achievable in a terrestrial laboratory. In contrast, however, theories which screen by varying their coupling constant with the environment are phenomenologically similar to chameleon models, and so it is expected that atom interferometry could also provide useful constraints. In this work we will focus on the symmetron model, as an example of a theory which screens by varying its coupling constant. This model is chosen because it has been shown that the model can be constructed in such a way that it is radiatively stable and quantum corrections remain under control [@Burrage:2016xzz]. Earlier work studied a similar model but with a different motivation [@Pietroni:2005pv; @Olive:2007aj], and string-inspired models with similar phenomenology have also been proposed [@Damour:1994zq; @Brax:2011ja]. In Section \[sec:symm\] we will review the symmetron model, and how the force between two extended objects can be screened. In Section \[sec:atom\] we apply the results of existing atom interferometry experiments to find new constraints on the symmetron model which are presented in Figure \[fig:constraints\]. In Section \[sec:domain\] we discuss the possibility that domain walls could form inside the vacuum chamber, leading to the possibility that atoms could experience a symmetron force, even in the absence of a source inside the vacuum chamber. We conclude in Section \[sec:conclusions\]. The Symmetron {#sec:symm} ============= The simplest version of the symmetron model is as a canonical scalar field with potential $$V(\phi) = \frac{\lambda}{4} \phi^4 -\frac{\mu^2}{2}\phi^2\;,$$ where $\lambda$ (which is dimensionless) and $\mu$ (which has mass dimensions) are the parameters of the theory which must be determined by experiment. The scalar field couples to matter through dimension six terms in the Lagrangian of the form $$\mathcal{L} \supset \frac{\phi^2}{2 M^2}T^{\mu}_{\mu}\;,$$ where $T_{\mu\nu}$ is the energy-momentum tensor of all of the matter fields and $M$ is an energy scale which controls the strength of the coupling to matter. The interactions with matter mean that in the presence of a non-relativistic, static background matter density $\rho$ the symmetron field moves in an effective potential $$V_{\rm eff}(\phi) = \frac{1}{2}\left(\frac{\rho}{M^2}-\mu^2\right)\phi^2 +\frac{\lambda}{4}\phi^4\;,$$ from which it can be seen that when the density is sufficiently high, $\rho>M^2 \mu^2$, the effective potential has only one minimum and the field is trapped at $\phi=0$. As the density is decreased the potential undergoes a symmetry breaking transition, and the field can roll into one of two minima with $\phi^2 = (\mu^2 -\rho/M^2)/\lambda$. In Ref. [@Hinterbichler:2011ca], the symmetry-breaking scale is chosen close to the cosmological density today, i.e. $\mu^2 M^2\sim H_0^2M_{\rm Pl}^2$, where $H_0$ is the present-day Hubble scale. In addition, the symmetron force in vacuum is required to have approximately gravitational strength, i.e. $\phi/M^2 \sim 1/M_{\rm Pl}$, such that there may be observable consequences without fine-tuning of the coupling scale. However, other choices of parameters are possible. In particular it has been shown that both Eöt-Wash experiments [@Upadhye:2012rc], and measurements of exo-planets [@Santos:2016rdg] constrain a very different region of parameter space with coupling constants $M\lesssim 10 \mbox{ TeV}$ and the mass scale $\mu$ around the electronvolt scale. The self coupling parameter $\lambda$ is very poorly constrained currently. The radiatively stable model derived in [@Burrage:2016xzz] has a slightly more complicated potential $$V(\phi) =\left(\frac{\lambda}{16\pi}\right)^2 \phi^4 \left(\ln \frac{\phi^2}{m^2}-Y\right)\;,$$ for constant $m$, $\lambda$ and $Y$. However, the structure of the symmetry breaking transition, and the resulting phenomenology remains essentially the same as the original symmetron model, and so we will focus our attention on the simpler model in what follows. Around a static, spherically symmetric object of density $\rho_{\rm in}$ and radius $R$, embedded in a background density $\rho_{\rm out}$, the symmetron field profile is $$\phi=\phi_{\rm out} -\frac{(\phi_{\rm out}-\phi_{\rm in})R e^{m_{\rm out}(R-r)}}{r}\left(\frac{m_{\rm in}R-\tanh m_{\rm in} R}{m_{\rm in}R+Rm_{\rm out}\tanh m_{\rm in}R}\right)\;,$$ where $m_{\rm in}$ and $m_{\rm out}$ are respectively the mass of the field inside the source object, and outside, and $\phi_{\rm in}$ and $\phi_{\rm out}$ are the values of the scalar field that minimize the effective potential inside and outside the source. The force on a test particle moving on top of this field profile is then given by $F= \phi\nabla \phi/M^2$. If everywhere in the experiment the density is higher than that required for the symmetry breaking transition, $(\rho_{\rm out},\rho_{\rm in})>M^2 \mu^2$, then the field will be constrained to be $\phi=0$ everywhere and it will never be possible to see the associated fifth forces. If $\rho_{\rm out}< \mu^2 M^2$ then the field has a non-trivial profile inside the vacuum chamber, and the value of the field at the center of the vacuum chamber (assumed to be spherical for simplicity) depends on the relative sizes of the radius of the vacuum chamber $L$ and the Compton wavelength of the field. If $m_{\rm out} L\gtrsim1$ then $$\phi_{\rm out} =\frac{1}{\sqrt{\lambda}}\left(\mu^2 -\frac{\rho_{\rm out}}{M^2}\right)^{1/2}\;,$$ otherwise the field does not have room to evolve away from $\phi=0$ in the walls of the chamber and we have $$\phi_{\rm out}=0\;.$$ If $\rho_{\rm in}< \mu^2 M^2$, then the source object causes only a small perturbation of the background field. In this case there is no screening, and the force has the usual Yukawa form. On the other hand, if $\rho_{\rm in}> \mu^2 M^2$ and $m_{\rm in}R \gg 1$, then the symmetry is restored inside the source and the resulting force on a test particle is suppressed [@Hinterbichler:2010es]. We are not interested, however, in the force on an infinitesimal test particle but instead in the force between two extended objects either or both of which could be screened. Following the arguments of [@Hui:2009kc] if we assume a hierarchy between mass A and mass B, so that the field profile due to B can be considered a small perturbation on the field profile of A [*at the surface of mass B*]{}, then the force can be found by considering the change in momentum of ball B and using the Bianchi identity. The symmetron force between two objects A and B is therefore $$F_{\rm symm}=4 \pi \lambda_A\lambda_B (1+m_{\rm out}R_B)(1+m_{\rm out}r)\frac{e^{m_{\rm out}(R_A-r)}}{r^2}\;,$$ where $$\lambda_i = \left.(\phi_{\rm out}-\phi_{\rm in})R\left(\frac{m_{\rm in}R-\tanh m_{\rm in} R}{m_{\rm in}R+Rm_{\rm out}\tanh m_{\rm in}R}\right)\right|_{i}\;, \label{eq:lambda}$$ where all of the quantities on the right hand side of Equation (\[eq:lambda\]) are evaluated for the object in question. $\lambda_i$ can therefore be considered the symmetron ‘charge’ for the object. The fact that we are treating the field profiles due to A and B hierarchically explains the slight asymmetry in the dependencies on $R_A$ and $R_B$. If $\lambda_i \ll 1$ then we say that the object is screened, and the force is suppressed. Atom Interferometry {#sec:atom} =================== Atom interferometry has been shown to be a powerful technique for constraining chameleon models with screening [@Burrage:2014oza]. These experiments work by putting an atom into a superposition of states which travel on two different paths. If the wavefunction accumulates a phase difference between the two paths this can be detected as an interference pattern when the two path are merged [@Storey:1994oka; @feynmanhibbs]. If the atoms experience a constant acceleration in the same direction as the separation between the paths then this results in exactly such a phase difference, allowing the force the atoms have experienced to be measured very precisely. Atom interferometry measurements looking for chameleons have reached a sensitivity of $10^{-6} g$ (where $g$ is the acceleration due to free fall at the surface of the earth) and are forecast to reach $10^{-9} g$ [@Hamilton:2015zga; @Elder:2016yxm]. For chameleon screening two properties make atom interferometry particularly powerful. Firstly, because the atoms are so small they are unscreened and so the chameleon force is less suppressed than it would be in a comparable macroscopic fifth force experiment. Secondly the walls of a vacuum chamber are sufficiently thick that they screen the interior of the vacuum chamber from any chameleon gradients or fluctuations in the exterior. This simplifies the computation of the chameleon forces in the experiment, but does have the consequence that the source mass must be placed inside the vacuum chamber, unlike many other tests of gravity performed with atom interferometry. Do these same advantages also apply to symmetron screening? Assuming the walls of the vacuum chamber have a density of $\rho_{\rm wall}\sim \mbox{g/cm}^3$ then the field can reach $\phi=0$ and restore the symmetry inside the walls if the thickness of the walls is greater than $\sim 1/m_{\rm wall}$. We will see shortly that atom interferometry experiments constrain a fairly narrow region in $\mu$ around $\mu \sim 10^{-4}\mbox{ eV}$ and coupling strengths in the range $10^{-4} \mbox{ GeV}< M < 10^{4} \mbox{ GeV}$. In this region of the parameter space the Compton wavelength of the field in the symmetry restored vacuum in the wall has a maximum value of $1/m_{\rm wall} \sim 1\mbox{ mm}$. Therefore we should expect the symmetry to be restored in the wall in the region of parameter space we consider, and so the interior is effectively decoupled from the behaviour of the symmetron in the exterior. For a compact object to be screened from the symmetron force we need both $\rho_{\rm in}/M^2 >\mu^2$ and $m_{\rm in}R\gg1$. The first condition is actually [*easier* ]{} to satisfy for an atomic nucleus than for a macroscopic test mass, because the nuclear density is much higher than the density of, for example, silicon. The second condition is harder to satisfy for atoms than macroscopic masses because of the small size of the atomic nuclei. It is therefore not always the case that atoms make better probes of the symmetron field than macroscopic objects do, however they will be sensitive in some region of the parameter space which we will now determine. We apply the results of reference [@Hamilton:2015zga], which measures the acceleration of cold caesium 133 atoms. The atoms were held $8.8\mbox{ mm}$ away from an aluminium sphere of radius $9.5\mbox{ mm}$. The experiment was performed in a vacuum chamber of radius $5 \mbox{ cm}$ and pressure $6 \times 10^{-10}\mbox{ Torr}$. No anomalous acceleration of the atoms is measured, restricting the acceleration due to the symmetron field to satisfy $a< 6.8 \times 10^{-6}\mbox{ m/s}^2$. The constraints that this places on the symmetron parameter space can be seen in Figure \[fig:constraints\]. ![\[fig:constraints\] Constraints on the symmetron parameters from the atom interferometry experiment of [@Hamilton:2015zga]. The excluded regions are shaded blue. The different regions are for different values of $\mu$; from left to right $\mu= 10^{-4}\mbox{ eV}$, $\mu= 10^{-4.5}\mbox{ eV}$, $\mu= 10^{-5}\mbox{ eV}$, $\mu= 10^{-5.5}\mbox{ eV}$. The black dashed line on the left shows constraints from observations of exo-planets (points to the left of the line are excluded), with $\mu \rightarrow 0$ the most constraining choice [@Santos:2016rdg]. The black dotted line on the right shows the constraints from torsion pendulum experiments (points to the right of the line are excluded) with $\mu=10^{-4} \mbox{ eV}$ chosen for reference [@Upadhye:2012rc].](figure.pdf) Constraints are restricted to a narrow range of the mass parameter $\mu$, around $\sim 10^{-4}, 10^{-5}\mbox{ eV}$. For smaller $\mu$ the Compton wavelength of the symmetron in the vacuum is larger than the size of the vacuum chamber and so the field cannot vary its value over the scale of the experiment. For larger $\mu$ the Compton wavelength of the symmetron in vacuum is so small that the Yukawa term exponentially suppresses the force. The peak in the $\mu= 10^{-4}\mbox{ eV}$ plot occurs because there is a value of $M$ for which the Compton wavelength of the field in vacuum exactly matches the distance between the aluminium sphere and the atoms. We see that, whilst the range of $\mu$ values that are accessible is relatively constrained, where atom interferometry experiments do give constraints they explore a region of parameter space that is inaccessible to other experiments and observations. Other Experiments with Unscreened Test Particles ------------------------------------------------ Atoms are not the only objects that can be unscreened in a laboratory vacuum. Experiments that measure forces on neutrons [@Brax:2011hb; @Ivanov:2012cb; @Brax:2013cfa; @Jenke:2014yel; @Lemmel:2015kwa; @Li:2016tux] and on silicon microspheres [@Rider:2016xaq] have also been shown to be sensitive to chameleon forces, precisely because the test particles are sufficiently small that they are not screened. However, these have not yet reached the sensitivity of the atom interferometry experiments and so do not provide better constraints on symmetron models than those presented in Figure \[fig:constraints\]. We note, however, that silicone microspheres have a lower average density than neutrons and atomic nuclei, and so, if the sensitivity can be improved, they may provide the best prospect for searching for symmetron forces. Domain Walls {#sec:domain} ============ Symmetron fields open another possibility for laboratory searches that is not present for the chameleon model. Since the symmetron effective potential has two minima, as gas is pumped from the vacuum chamber the field could settle in either minimum with equal probability, and there is no reason for different regions of the chamber to all settle into the same one; if the Compton wavelength is comparable to or smaller than the size of the vacuum chamber then it is possible for a domain wall or a network of domain walls to form. If we approximate the wall as being straight and static, then its field profile is $$\phi(z)=\phi_0\tanh\left(\frac{\tilde{\mu}z}{\sqrt{2}}\right)\;, \label{eq:soliton}$$ where $$\label{eq_muTilde} \tilde{\mu}^2:=\mu^2-\frac{\rho}{M^2}\;,$$ and $\phi_0 = \frac{\tilde{\mu}}{\sqrt{\lambda}}$, meaning that the thickness of the wall is $\sim 1/\tilde{\mu}$, and its tension is $= 4\tilde{mu}^3/(3\lambda)$ [@Vilenkin:2000jqa]. Taking into account that the atom may be screened, the acceleration experienced by atom moving in the neighbourhood of a domain wall is $$\vec{a} = 4 \pi \lambda_{\rm atom} (1+m_{\rm out}R_{\rm atom}) \nabla \phi\;,$$ where we should remember that $\lambda_{\rm atom}$ depends on the background field value $\phi_{\rm out}$ which in this case should be replaced with the domain wall field profile $\phi$ evaluated at the position of the atom. The maximum acceleration that an atom may experience in such a situation is roughly proportional to $|\phi\nabla\phi|$. We can find an approximation for this by assuming the domain walls are small compared to the radius of the chamber such that we can use the planar solution for a domain wall given above in equation (\[eq:soliton\]). We find that this maximum acceleration $a_\phi\approx\frac{\phi\nabla\phi}{M^2}\approx 0.27\frac{\tilde{\mu}^3}{\lambda M^2}$, which is always much less than $10^{-10}\;g$ within the parameter space we have examined. This means that any domain walls that form will have a negligible effect on searches for symmetron fifth forces between atoms and source masses in the vacuum chamber, and that sensitivity must be improved if we are to detect the forces due to the domain wall directly. Of course, depending on the correlation length more than one domain wall can form creating a network. The symmetry is restored in the walls of the vacuum chamber, and in the core of the domain wall, so from the point of view of the field the walls can be viewed as a fixed sphere of $\phi=0$ surrounding the domain wall network. We know from cosmological studies [@Vilenkin:2000jqa] that networks of domain walls want to evolve towards the configuration with the minimum wall length. Therefore we can assume that the network inside the vacuum chamber is not stable, and the domain walls will straighten, and merge with one another and with the walls of the vacuum chamber. It is therefore reasonable to expect that the end point of this evolution will be the vacuum chamber entirely filled with one domain and no domain walls are present. An example of such an evolution is shown in Figure \[fig:network\]. This figure was constructed using two-dimensional numerical simulation with unphysical values for the symmetron parameters, and so should be considered only as an example of what kind of evolution is possible. We leave a full numerical simulation for future work. ![\[fig:network\] A two dimensional domain wall network inside a circular cavity. The colours indicate the value of the scalar field. Blue and grey regions represent the positive and negative symmetry broken vacua respectively, and in the black regions the symmetry is restored at $\phi=0$. The system evolves in time from left to right. This simulation was performed with unphysical values for the symmetron parameters due to numerical limitations. ](symmetronConfigs.pdf) It remains to be determined how long such an evolution takes. We can use some results from cosmological studies of domain walls as a guideline of what to expect. In a true vacuum it is expected that the domain walls move with relativistic velocities. If this were the case in our vacuum chamber the walls would exist for an undetectably short period of time. However, this motion can be slowed down by friction if the walls interact with a surrounding particle bath. The force per unit area on the wall can be approximated by [@Kibble:1976sj] $$F_{\rm friction} \sim N n T v\;,$$ where $N$ is the number of light particles interacting with the domain wall, $n$ is the number density of these particles, $T$ is their temperature and $v$ the velocity of the wall compared to the background. Taking the number density corresponding to the hydrogen gas pressure in the atom interferometry experiment described above, which is performed at room temperature, we find $$F_{\rm friction} \sim v \times 1.3 \times 10^{-45} \mbox{ GeV}^4\;.$$ This frictional force is comparable to the domain wall tension if $F_{\rm friction} \sim 4 \mu^3/(3 \lambda R)$, where $R$ is the mean curvature radius. From this we can deduce that the strings will move non-relativistically if $$0.02 \left(\frac{\mbox{cm}}{R}\right) \ll \lambda\;.$$ This suggests that at least for some values of $\lambda$, the domain walls could be long lived inside the vacuum chamber. As mentioned above, a full numerical study of the evolution of the domain walls in a vacuum chamber remains a topic for future work. Searches for the forces due to domain walls are not the most sensitive way to search for symmetron fields, although they do have the technical advantage that they do not rely on the presence of a source mass that can be moved inside the vacuum chamber. However, they only occur in theories of screening, such as the symmetron, which undergo a symmetry breaking transition. If a fifth force with screening is detected in an upcoming experiment, the presence or absence of a network of domain walls in the experiment could be used to discriminate between models. Conclusions {#sec:conclusions} =========== We have shown that symmetron fifth forces, inspired by theories of dark energy, can be constrained by terrestrial experiments using cold atoms. The constraints we have found in Figure \[fig:constraints\], are particularly interesting as they fill a previously empty region of parameter space between the constraints coming from Eöt-Wash experiments, and those coming from observations of exo-planets. We have also discussed the possibility that symmetron domain walls may form in the vacuum chamber, leading to the atoms experiencing a fifth force without the need to place a source mass inside the vacuum chamber. Whilst we find that the accelerations experienced by the atoms are smaller than the sensitivity of current experiments, they are not so small that it would be impossible to detect them in the future. Additionally, as the domain walls only form for symmetron models, if a screened fifth force is ever detected in a terrestrial experiment the presence or absence of these domain walls would provide a way to discriminate between different models of screening. {#section .unnumbered} In the final stages of writing this article it has come to our attention that Brax and Davis have derived the same constraints on the symmetron model using the tomographic model of screening . Acknowledgements {#acknowledgements .unnumbered} ================ We would like to thank Ed Copeland for useful discussions during the completion of this work. CB is supported by a Royal Society University Research Fellowship. JS is supported by the Royal Society

--- abstract: 'Using density functional theory based calculations, we have carried out in-depth studies of effect of Co substitution on the magnetic properties of Ni and Pt-based shape memory alloys. We show the systematic variation of the total magnetic moment, as a function of Co doping. A detailed analysis of evolution of Heisenberg exchange coupling parameters as a function of Co doping has been presented here. The strength of RKKY type of exchange interaction is found to decay with the increase of Co doping.' author: - 'Tufan Roy$^{1,2}$[^1], Aparna Chakrabarti$^{1,2}$' title: 'Ab initio Study of Effect of Co Substitution on the Magnetic Properties of Ni and Pt-based Heusler Alloys' --- Introduction ============ Full Heusler alloys (with typical formula A$_{2}$BC) have drawn considerable attention of the researchers over the last decades because of their possible technological applications. Upon cooling, some of the Heusler alloys undergo a structural transition from a high temperature cubic phase, namely austenite phase to a lower symmetry phase, called martensite phase below a certain temperature. This type of structural transition is referred as martensite transition, and the particular temperature at which the transition takes place is called martensite transition temperature. Ni$_{2}$MnGa belongs to this category of Heusler alloys.[@phil-web-1984; @apl-ullakko-1996; @apl-sozinov-2002] The Heusler alloys of this category may find their application as various devices, such as actuators, antenna, sensors etc. For the application purpose it is always desired that martensitic transition temperature is above the room temperature. In case of conventional shape memory effect, which is governed by the temperature, the actuation process is much more slow compared to a magnetically controlled actuation. So it is desirable to have a magnetic shape memory alloy with the Curie temperature (T$_{C}$) higher than the room temperature. It has been observed that both T$_{M}$ and T$_{C}$ values are very much dependent on the composition of a particular Heusler alloy.[@prb-sroy-2009; @prb-kataoka-2010; @prb-achakrabarti-2005; @prb-barman-2008; @apl-achakrabarti-2009; @jpcm-khan-2004; @apl-mario-2011; @apl-stadler-2006; @prb-achakrabarti-2013; @jalcom-troy-2015; @prb-troy-2016; @jmmm-troy-2016; @jmmm-troy-2017] There is also another category of full Heusler alloys, which are known to be metallic for one kind of spin channel and insulator for the other kind of spin channel because of their very high spin polarization (HSP) at the Fermi level. They are often called as half metallic Heusler alloys.[@prl-Groot-1983] Most of the Co-based Heusler alloys, like Co$_{2}$MnSn, Co$_{2}$MnGa belong to this category.[@JPD40HCK; @PRB-76-024414-2007] These Heusler alloys may have potential application in spintronic devices. Apart from the technological application, these Heusler alloys are very interesting because of their wide diversity in terms of magnetic property. These alloys may be ferromagnetic, ferrimagnetic, anti-ferromagnetic and also non-magnetic depending on the chemical composition. So it is of immense interest to have an in depth study on the magnetic interactions present in these systems. In most of the full Heusler alloys, A$_{2}$BC, $B$ is the primary moment carrying atom, in many of the Heusler alloys, A$_{2}$BC, there is presence of a delocalized-like common d-band formed by the d-electrons of the $A$ and $B$ atoms, which are both typically first-row transition metal atoms.[@PRB28JK] Additionally, there is also an indirect RKKY-type exchange mechanism[@RKKY] between the $B$ atoms, primarily mediated by the electrons of the $C$ atoms, which also plays an important role in defining the magnetic properties of these materials.[@PRB28JK; @prb-sasioglu-2008] Staunton et al[@staunton-jpcssp-1988] reported the role of RKKY interaction behind the origin of magnetic anisotropy of a system. For the magnetic shape memory alloys, magnetic anisotropy energy plays an important role. In this regard also, it will be interesting to study of RKKY interaction in detail in these systems. In a very recent paper, we have shown in detail the similarities and differences between the Heusler alloys which are likely to show shape memory alloy (SMA) property and which are not, in terms of the electronic, magnetic as well as mechanical properties.[@prb-troy-2016] In this paper, we focus our interest as to how the magnetic exchange interactions, mainly the RKKY type of interaction between the $B$ atoms of the A$_{2}$BC systems, are evolving in going from the materials which are prone to martensite transition (which are generally metallic in nature) to the other class of Heusler alloys (which are typically half-metallic in nature) i.e. which do not show SMA property. Here we study about the nature of RKKY types of interaction for four sets of materials Ni$_{2-x}$Co$_{x}$MnGa, Ni$_{2-x}$Co$_{x}$FeGa, Pt$_{2-x}$Co$_{x}$MnGa, Pt$_{2-x}$Co$_{x}$MnSn as a function of x (x=0.00, 0.25, 0.50, 0.75, 1.25, 1.50, 1.75, 2.00). In all the cases the material is likely to show SMA property for x=0.00 and is predicted to be half-metallic for x=2.00. In the section following the methodology, the results of the work and the relevant discussion are presented. Finally, we summarize and conclude in the last section. Method ====== The Heusler alloys (A$_{2}$BC) studied here possess L2$_{1}$ structure that consists of four interpenetrating face-centered-cubic (fcc) sub-lattices with origin at fractional positions, (0.25, 0.25, 0.25), (0.75, 0.75, 0.75), (0.5, 0.5, 0.5), and (0.0, 0.0, 0.0). For the conventional Heusler alloy structure, the first two sub-lattices are occupied by $A$ atom and the third by $B$ and fourth by $C$ atom. In total, there are 16 atoms in the cell. While we study the Co substitution in A$_{2}$BC systems, the Co atom substitutes the $A$ atom only. First we carry out full geometry optimization of the materials, of all the materials corresponding to x=0.00, 0.25, 1.75, 2.00, using the 16 atom cell. For the geometry optimization, we employ the Vienna Ab Initio Simulation Package (VASP)[@prb-kreese-1996] in combination with the projector augmented wave method.[@prb-blochl-1994] We use an energy cut-off of minimum 500 eV for the planewave basis set. The calculations have been performed with a $k$ mesh of 15$\times$15$\times$15. The energy and force tolerance used were 10 $\mu$eV and 10 meV/Å, respectively. After obtaining the equilibrium lattice constants of the four above-mentioned materials by using the VASP package we plot the same. A linear variation of the lattice constant is observed. We deduce the lattice constants of the other materials, corresponding to x=0.50, 0.75, 1.25, 1.50 by the method of interpolation. To gain insight into the magnetic interactions of these materials, we calculate and discuss their Heisenberg exchange coupling parameters. We use the Spin-polarized-relativistic Korringa-Kohn-Rostoker method (SPR-KKR) to calculate the Heisenberg exchange coupling parameters, Jij as implemented in the SPR-KKR programme package.[@rep-ebert-2011] The mesh of $k$ points for the SCF cycles has been taken as 21$\times$21$\times$21 in the BZ. The angular momentum expansion for each atom is taken such that lmax=3. The partial and total moments have also been calculated for all the materials studied. We use local density approximation (LDA) for exchange correlation functional.[@LDA] ![x dependence of magnetic moments for Ni$_{2-x}$Co$_{x}$MnGa, Ni$_{2-x}$Co$_{x}$FeGa, Pt$_{2-x}$Co$_{x}$MnSn, Pt$_{2-x}$Co$_{x}$MnGa.[]{data-label="fig:1"}](Figure-1.eps){width="8cm" height="8cm"} ![Variation of the total energy of (a) Co$_{2}$MnGa, Co$_{2}$FeGa, Co$_{2}$MnSn (b) Ni$_{2}$MnGa, Ni$_{2}$FeGa, Pt$_{2}$MnGa, Pt$_{2}$MnSn in their respective ground state magnetic configurations as a function of $c$/$a$. Energy $E$ in the Y-axis signifies the energy difference between the cubic and tetragonal phase. Some results of these figures are part of published literature.[@prb-troy-2016; @jmmm-troy-2016][]{data-label="fig:1"}](Figure-2.eps){width="8cm" height="8cm"} Results and Discussion ====================== ***Total and partial moments*** As mentioned above, we studied here four sets of materials, Ni$_{2-x}$Co$_x$MnGa, Ni$_{2-x}$Co$_x$FeGa, Pt$_{2-x}$Co$_x$MnGa, Pt$_{2-x}$Co$_x$MnSn with x=0.00, 0.25, 0.50, 0.75, 1.25, 1.50, 1.75, 2.00. At the two ends of the composition, i.e. for x=0.00 and x=2.00, all the materials except Pt$_{2}$MnSn already exist in the literature. In this study, all the materials corresponding to x=0.00 are likely to exhibit martensite transition. We predict here that Pt$_{2}$MnSn also possesses conventional Heusler alloy structure in its ground state and exhibits the martensite transition. All the studied materials here are ferromagnetic in nature. We observe from Figure-1 that for the three sets of materials namely, Ni$_{2-x}$Co$_{x}$MnGa, Pt$_{2-x}$Co$_{x}$MnGa, Pt$_{2-x}$Co$_{x}$MnSn the variation of total moment($\mu_{T}$) follows the same trend, which is for lower value of x, $\mu_{T}$ increases and then starts to fall at a higher x value, attaining a maximum value in between the range of x=0.00 to x=2.00. For Ni$_{2-x}$Co$_x$MnGa, the nature of variation of the moment as a function of x matches with the existing literature.[@prb-kanomata-2009] The variation of the total moment as a function of x, can be well understood from the variation of the partial moments for the respective systems. We find that for Ni$_{2-x}$Co$_{x}$MnGa, Pt$_{2-x}$Co$_{x}$MnGa, Pt$_{2-x}$Co$_{x}$MnSn, the partial moment of Co and Mn-atom decreases linearly as a function of x. This may be because, as we move towards the higher value of x, the lattice parameter of the systems decreases which leads to decrease of the Mn and Co partial moment. But as the absolute value of moment of Co-atom is much larger compared than that of Ni or Pt, the total moment increases initially with increasing value of x. However, this increasing factor has to compete with the continuous reduction of the partial moments of Co and Mn-atom, which dominates at higher value of x. This results in a fall of the total value of the moment. Because of these two competing factors, initially we get a maximum value of $\mu_{T}$ and then it falls, finally reaches a value, very close to an integer following the Slater Pauling rule.[@JPD40HCK] However, the total magnetic moment of Ni$_{2-x}$Co$_{x}$FeGa increases linearly as a function of x. This type of variation may be because of the almost constant partial moment of Fe and Co atom over the entire range of x. This is probably due to the fact that the lattice parameters for the two end materials Ni$_{2}$FeGa (a= 5.76 Å) and Co$_{2}$FeGa (a=5.73 Å) are very close. Here the only controlling factor is the change of moment due to Ni substitution by Co-atom, which is always positive and proportional to the substitution and effectively results in a linear increase of the total moments of this system. ***Energy vs $c/a$ curve*** Heusler alloys may be used as shape memory device if they undergo a structural transition from high temperature cubic phase to low temperature non-cubic phase upon cooling. The alloys, which are likely to undergo this structural transition, they must have the non-cubic phase with much lower energy compared to its cubic phase. We have applied a tetragonal distortion on the cubic phase of the stoichiometric material to probe whether they are favourable to undergo tetragonal distortion or not. In the upper panel of the Figure-2, we find that there is no lowering of energy under tetragonal distortion. For this set of materials, namely Co$_{2}$MnGa, Co$_{2}$MnSn, Co$_{2}$FeGa, the cubic phase is the lowest energy state ($c/a=1$) and they are not likely to undergo martensite transition.[@jalcom-troy-2015; @prb-troy-2016; @thesis-Antje] In the lower panel of the Figure-2 we observe that for all the materials shown here (i.e. Ni$_{2}$MnGa, Ni$_{2}$FeGa, Pt$_{2}$MnGa, Pt$_{2}$MnSn), energy of the systems is lowered under tetragonal distortion which indicates to a possibility of martensite transition for these materials. Except Pt$_{2}$MnSn, the other three materials, namely Ni$_{2}$MnGa, Ni$_{2}$FeGa, Pt$_{2}$MnGa, of the lower panel are already reported to undergo martensite transition.[@jpcm-brown-1999; @apl-liu-2003; @apl-mario-2011] ![J$_{ij}$ of Mn atom with its neighbours as a function of normalized distance $d/a$ for Ni$_{2-x}$Co$_x$MnGa system. $a$ is the lattice parameter for x=0.00, 0.25, 0.50, 0.75, 1.25, 1.50, 1.75, 2.00.[]{data-label="fig:1"}](Figure-3.eps){width="8cm" height="8cm"} ![J$_{ij}$ of Fe atom with its neighbours as a function of normalized distance $d/a$ for Ni$_{2-x}$Co$_x$FeGa system. $a$ is the lattice parameter for x=0.00, 0.25, 0.50, 0.75, 1.25, 1.50, 2.00.[]{data-label="fig:1"}](Figure-4.eps){width="8cm" height="8cm"} ![J$_{ij}$ of Mn atom with its neighbours as a function of normalized distance $d/a$ for Pt$_{2-x}$Co$_x$MnSn system. $a$ is the lattice parameter for x=0.00, 0.25, 0.50, 0.75, 1.25, 1.50, 1.75, 2.00.[]{data-label="fig:1"}](Figure-5.eps){width="8cm" height="8cm"} ![J$_{ij}$ of Mn atom with its neighbours as a function of normalized distance $d/a$ for Pt$_{2-x}$Co$_x$MnGa system. $a$ is the lattice parameter for x=0.00, 0.25, 0.50, 0.75, 1.25, 1.50, 1.75, 2.00.[]{data-label="fig:1"}](Figure-6.eps){width="8cm" height="8cm"} Now we plot (Figure-3 to Figure-6) the Heisenberg exchange coupling parameters (J$_{ij}$), between Mn or Fe with other magnetic atoms of Ni$_{2-x}$Co$_x$MnGa, Ni$_{2-x}$Co$_x$FeGa, Pt$_{2-x}$Co$_x$MnGa, Pt$_{2-x}$Co$_x$MnSn (x=0.00, 0.25, 0.50, 0.75, 1.25, 1.50, 1.75, 2.00), as a function of interatomic spacing in the units of lattice parameter(a). We study the evolution of the magnetic interaction in going from a material which shows the SMA property (x=0.00) to one which does not show the SMA property (x=2.00). From Figure-3, we observe that the strength of the direct exchange interaction between Mn and Co is maximum at x=0.25, i.e. for Ni$_{1.75}$Co$_{0.25}$MnGa and it is minimum for x=2.00 i.e. for Co$_{2}$MnGa. We observe that for the value of x=0.25, the partial magnetic moments of Mn and Co atom attain their maximum value among all the intermediate compounds (x=0.00 cannot be considered because there is no Co atom) which leads to the strongest direct exchange interaction between Mn and Co atom. The scenario is exactly opposite for x=2.00. For the other three sets of materials also, i.e. Ni$_{2-x}$Co$_x$FeGa, Pt$_{2-x}$Co$_x$MnGa, Pt$_{2-x}$Co$_x$MnSn, we find similar types variation of direct exchange interaction between Co and the $B$ atom ($B$=Mn for Pt$_{2-x}$Co$_x$MnGa, Pt$_{2-x}$Co$_x$MnSn and Fe for Ni$_{2-x}$Co$_x$FeGa). which has been shown in Figure-4 to Figure-6. For all the materials studied here, we find that the exchange interaction energy between the $B$ and Co atom is much more stronger compared to $B$ and Ni (for Ni$_{2-x}$Co$_x$MnGa, Ni$_{2-x}$Co$_x$FeGa) or Pt atom (for Pt$_{2-x}$Co$_x$MnGa, Pt$_{2-x}$Co$_x$MnSn). This is because the partial moment of Co-atom is much higher compared to that of Ni or Pt-atom. ![J$_{B-B}$ (B=Mn or Fe depending on the systems) as a function of normalized distance $d/a$ for (a)Ni$_{2-x}$Co$_x$MnGa (b) Ni$_{2-x}$Co$_x$FeGa (c)Pt$_{2-x}$Co$_x$MnSn (d)Pt$_{2-x}$Co$_x$MnGa . $a$ is the lattice parameter for different values of x.[]{data-label="fig:1"}](Figure-7.eps){width="8cm" height="8cm"} RKKY type of interaction plays a very important role in the systems where the localized moments are far apart to have any direct exchange interaction. There are an extensive studies on the RKKY interactions in various dilute magnetic systems, where the magnetic atoms like Mn or Fe are present in a very low concentration in the nonmagnetic metallic host material.[@prb-smith-1976] The presence of RKKY interaction between localized-like moments (Mn or Fe) was reported. This interaction was via the conduction electrons of the host material, which may be Au, Ag, Mo, Zn etc. Not only in metallic system, RKKY interaction plays a crucial role in determining the magnetic property of dilute magnetic semiconductor also.[@Priour-prl-2004] In this study all the systems contain Mn or Fe atom, and the magnetic moments are mainly confined to them. As all the systems studied here possess conventional Heusler alloy structure (A$_{2}$BC), the separation between the B atoms (Mn or Fe depending on the systems) are large enough to have a direct exchange interaction between them. For this kind of Heusler alloy structure B atom is surrounded by eight A atoms, which makes A atoms to play a very important role in determining the magnetic exchange interactions between B atoms themselves. The strong local nature of the magnetic moment of the B atoms spin-polarizes the free like electrons present in the system and the spin-polarized conduction electrons effectively couple the B atom.[@stearns-jap-1979] Previously in literature [@PRB28JK] it was mentioned for A$_{2}$MnC systems (A=Cu, Pd; C=Al, In, Sb), that the conduction electrons of the C atom take role in the coupling between Mn atoms. But in a recent study[@prb-sasioglu-2008] the role of conduction electron of A atom has also been confirmed for a number of Mn-based Heusler alloys. In our studied systems here, for a given series of materials, C-atom is fixed which is Ga for Ni$_{2-x}$Co$_x$MnGa, Ni$_{2-x}$Co$_x$FeGa, Pt$_{2-x}$Co$_x$MnGa and Sn for Pt$_{2-x}$Co$_x$MnSn. However with substitution, nature of A atom changes. Here we will discuss only about the role of A-atom in the RKKY interaction between B atoms themselves. It is to be noted that the spin polarization of the conduction electron will depend on the local magnetic moment of the B atom and the number of the conduction electrons present in the system. Now as we move from Ni$_{2}$MnGa to Co$_{2}$MnGa we are effectively reducing the number of conduction electrons of the system, as Ni has one more d-electron compared to Co-atom. This may cause a weaker coupling between the Mn atoms themselves. From Figure-7(a) we find that for Ni$_{2}$MnGa (x=0.00) the Mn-Mn interaction is the most oscillatory in nature (Heisenberg exchange coupling constant varies between 1.39 eV to -0.25 eV at $d/a=1$ and $d/a=1.73$) whereas for Co$_{2}$MnGa the oscillation is minimal (varies between 0.2 eV to -0.02 eV at $d/a=0.71$ and $d/a=1.58$). The strength of the oscillation reduces gradually as we move from Ni$_{2}$MnGa to Co$_{2}$MnGa. For Ni$_{2-x}$Co$_{x}$FeGa system also we observe same kind of variation for Mn-Mn interaction as we move from x=0.00 to x=2.00. For x=0.00 i.e. for Ni$_{2}$FeGa the amplitude of Fe-Fe RKKY interaction varies between 1.96 eV ($d/a=0.71$) and -0.76 eV ($d/a=1.41$) which is the strongest among the Ni$_{2-x}$Co$_{x}$FeGa series. In going from Pt-based systems to Co-based systems (Figure-7(c) and Figure-7(d)) also, we are reducing the number of conduction electrons. One more factor which we must consider when we discus about Pt$_{2-x}$Co$_{x}$MnSn and Pt$_{2-x}$Co$_{x}$MnGa, is the change in lattice parameter between the compounds corresponding to x=0.00 and 2.00. On the other hand, for Ni$_{2-x}$Co$_{x}$MnGa and Ni$_{2-x}$Co$_{x}$FeGa this change is very nominal as both Ni and Co has very close atomic radius. But as we move from Pt$_{2}$MnSn (a=6.46 Å) to Co$_{2}$MnSn (5.98 Å) there is a contraction of lattice parameter of about 0.48 Å. For Pt$_{2}$MnGa (a=6.23 Å) to Co$_{2}$MnGa (a=5.72 Å), a contraction of about 0.51 Åtakes place. This larger lattice parameter for Pt-based systems causes more localization of Mn partial magnetic moment (3.97 $\mu_{B}$ and 3.82 $\mu_{B}$ in Pt$_{2}$MnSn and Pt$_{2}$MnGa respectively) compared to the values in Co-based system (3.19 $\mu_{B}$ and 2.73 $\mu_{B}$ in case of Co$_{2}$MnSn and Co$_{2}$MnGa respectively). In Ref[@prb-bose-2011] Bose etal have mentioned that the strength of exchange interaction between two interacting magnetic moments also depends on value of the respective magnetic moments. Therefore, if we focus on Figure-7(c) we observe that the Mn-Mn exchange interaction energy for x=0.00 (Pt$_{2}$MnSn) oscillates between a maximum value of 1.21 eV ($d/a=1.00$) and minimum value of -1.29 eV ($d/a=1.73$) but oscillation becomes weaker gradually as we increase x and for Co$_{2}$MnSn it varies between 1.61 eV ($d/a=0.71$) and 0.03 eV ($d/a=0.58$). It means RKKY type of interaction is much more strong in Pt$_{2}$MnSn compared to Co$_{2}$MnSn, which may be because of more localized-like Mn-moments in Pt$_{2}$MnSn. For Pt$_{2-x}$Co$_{x}$MnGa system also we find that the for x=0.00, RKKY type of interaction between Mn-Mn is the most oscillatory (for Pt$_{2}$MnGa it varies between -1.33 eV and 1.94 eV at $d/a =0.71, 1.00$ respectively) and gradually with increasing x, the interaction becomes less oscillating nature. Conclusion ========== From density functional theory based calculations we study the effects of Co substitution in Ni and Pt-based Heusler alloys which are likely to show SMA. Our results suggest that there is a decrease in strength of the RKKY interaction as we increase the Co doping at Ni or Pt site. It indicates about the dominant role played by A atom’s d-electron in the formation of coupling between localized moments of B atom in the A$_{2}$BC system studied here. 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--- abstract: 'Features of the energy landscape sampled by supercooled liquids are numerically analyzed for several Lennard-Jones like model systems. The properties of quasisaddles (minima of the square gradient of potential energy $W\!=\!|\nabla V|^2$), are shown to have a direct relationship with the dynamical behavior, confirming that the quasisaddle order extrapolates to zero at the mode-coupling temperature $T_{MCT}$. The same result is obtained either analyzing all the minima of $W$ or the saddles (absolute minima of $W$), supporting the conjectured similarity between quasisaddles and saddles, as far as the temperature dependence of the properties influencing the slow dynamics is concerned. We find evidence of universality in the shape of the landscape: plots for different systems superimpose into master curves, once energies and temperatures are scaled by $T_{MCT}$. This allows to establish a quantitative relationship between $T_{MCT}$ and potential energy barriers for LJ-like systems, and suggests a possible generalization to different model liquids.' author: - 'L. Angelani$^{1,2}$, G. Ruocco$^{1}$, M. Sampoli$^{3,4}$, and F. Sciortino$^{1,2}$' title: 'General features of the energy landscape in Lennard-Jones like model liquids' --- Introduction ============ The investigation of the topological and metric properties of potential energy surface (PES), often referred to as “energy landscape”, is a useful and powerful tool for studying slow dynamics in condensed matter, especially in those cases where the lack of order (as for example in supercooled liquids) inhibits the use of the analytical tools pertaining to the crystalline state [@deb_nature; @land_angell; @land_sastry; @land_buc; @land_sastry2; @land_keyes]. The PES approach has been successfully applied to the study of many different interacting systems (glasses, proteins, sheared materials, and so on). The PES approach started with the introduction of the fruitful concept of [*inherent structures*]{} [@still1]. In the last years, several steps toward a more detailed description of the statistical properties of the PES have been performed, most of them pointing toward a better understanding of the relationship between the landscape properties and the emergent dynamical behavior of the analyzed systems. Among others, two landscape-based approaches have proven to be particularly stimulating. The first one concerns the detailed analysis of the [*inherent structures*]{} (i.e. the configurations at the minima of potential energy) visited by the system at different temperatures. This method has allowed to clarify many interesting phenomena, as, for example, the thermodynamic picture of the supercooled liquid regime based on the configurational entropy [@fs_entropy], the relationship between fragility and properties of inherent structures [@land_sastry2], the analysis of diffusion processes in terms of visited inherent structures [@land_keyes; @fabr; @la_98], or the interpretation of the effective fluctuation-dissipation temperature in the out-of-equilibrium regime in terms of inherent structures visited during aging [@fs_aging], only to cite a few. The second approach is based on the analysis of the eigenvalues (normal modes) of the Hessian at the instantaneous configurations during the dynamic evolution of the system, from here the name [ *instantaneous normal mode approach*]{} (for an introduction and an extended application of this method see the works of Keyes and coworkers [@keyes_inm; @keyes_vari]). This approach allowed to relate the emergent diffusive processes to the features of the landscape, opening the way to the interpretation of diffusion in terms of accessible paths in the multidimensional energy surface. Promising steps was obtained [*i)*]{} using simultaneously both the instantaneous normal mode approach and the inherent structure one, in order to identify the relevant slow diffusive directions [@donati; @lanave], and [*ii)*]{} by analyzing the reaction paths in order to eliminate the non-diffusive unstable modes [@bembe]. Recently, a further approach has been introduced [@noi_sad; @cav_sad] and applied to the study of supercooled liquids[@sad_1; @doye; @sad_3; @sad_4; @sad_5; @grig; @sadBLJ]. This approach is based on the analysis of the [*saddles*]{} of the potential energy surface and has provided new insight in the analysis of the dynamic crossover taking place on lowering the temperature in supercooled liquids. Indeed it allows to characterize the dynamic transition temperature $T_{MCT}$ (mode-coupling temperature [@mct]) as the temperature where the order (fractional number of negative eigenvalues of the Hessian matrix) of the saddles vanishes. This finding suggested the following scenario for the dynamics: above $T_{MCT}$ the representative point in the configuration space lies close to the saddles and the relevant dynamic process is the diffusion among multidimensional saddle points, i.e. the diffusion takes place along paths at almost constant potential energy, and the limiting factors to particles diffusion are “entropic”, rather than “energetic”, barriers. On the contrary, below $T_{MCT}$ the minimum-to-minimum diffusion processes dominate and the “true” barrier jump controls the diffusive dynamics. A clear landscape-based interpretation of the dynamic behavior of the system is then provided. It is important to mention here that the term “saddles” is not mathematically correct, as the way the saddles have been defined in Ref.s [@noi_sad; @cav_sad] is based on the partition of the configuration space in basins of attraction of the minima of the “pseudopotential”[@still_W] $W\!=\!|\nabla V|^2$. It is clear that the absolute minima of $W$, located at $W\!=\!0$, are true saddles of the energy surface (for simplicity of notation we call “saddles” also the minima and maxima of $V$) while the local minima of $W$ (those with $W>0$) correspond to points with (at least) one inflection direction, and are not saddles in mathematical sense, rather they are “shoulders” along the inflection direction. As pointed out by Doye and Wales [@doye], the local minima of $W$, and not the absolute ones, are very often encountered during the minimization procedure. However, as it will be clear soon, the properties of the local and absolute minima of $W$ which are actually important in determining the diffusive behavior are exactly the same. For this reason we call the local minima of $W$ [*quasisaddles*]{}, to emphasize the fact that they carry the same information as saddles, even if they are geometrically different in nature (for a more detailed discussion see [@noi_jcp; @comment; @rensp]). Besides the landscape picture of the dynamic processes, the study of saddles has also permitted a quantitative characterization of the main features of the PES of liquid systems. Indeed, important PES properties, as the mean energy elevation of saddles from underling minima or the Euclidean distances among saddles, can be inferred from the analysis of saddle properties. It emerges an high regularity of the PES, with few parameters describing the spatial and energetic location of saddles. In this work we apply the saddle-approach to different model liquids (Lennard-Jones like pair potentials), in order to better understand the relationship between landscape properties and slow dynamics, and in order to evidencing the existence of general features of the PES. The main result of this work is the existence of master curves both for temperature-dependent properties (saddle order vs. $T$) and for landscape properties (saddle energy vs. order), once energies and temperatures are normalized to $T_{MCT}$. This is a very strict relationship between dynamics and landscape features: differences in the PES for different systems simply define different $T_{MCT}$ values, and once scaled by these values, one obtains exactly the same behavior. In other words, it appears that the PESs are very similar, the only differences being the values of few parameters describing them (like the mean elevation barriers $\Delta E$ - mean elevation of saddles of order one from underlying minima) that lead to different values of dynamical quantities ($T_{MCT}$). The last point is of particular importance: for all the systems investigated we obtain that the value of $T_{MCT}$ is about $1/10$ of the energy barrier $\Delta E$, suggesting a kind of universality in the rearrangement processes governing the diffusion. Models ====== We numerically investigated four different Lennard-Jones like model systems, all composed of $N\!=\!256$ particles inside a cubic box with periodic boundary conditions. These are: 1. the modified monatomic Lennard-Jones (MLJ) [@mlj], at $\rho$=$1.0$ (hereafter all the quantities will be expressed in LJ reduced units) $$V_{MLJ}(r) = 4 \epsilon \left[ (\sigma/r)^{12} - (\sigma/r)^6 \right] + \delta V\ ,$$ where $\delta V$ is a (small) many-body term that inhibits crystallization $$\delta V = \alpha \Sigma_{\vec q} \; \; \theta ( S({\vec q}) - S_0 ) \ \left [ S({\vec q}) - S_0 \right ]^2 \ . \label{castra}$$ $S({\vec q})$ is the static structure factor, the sum is made over all $\vec q$ with $q_{max}$$-$$\Delta$$<$$|{\vec q}|$$<$$q_{max}$+$\Delta$, where $q_{max}$=7.12$(\rho)^{1/3}$ and $\Delta$=0.34, and the values of the parameters are $\alpha$=0.8 and $S_0$=10. 2. the modified monatomic soft spheres (MSS), at $\rho$=$1.0$, $$V_{MSS}(r) = 4 \epsilon (\sigma/r)^{12} + \delta V\ ,$$ where $\delta V$ is defined in Eq. \[castra\]. 3. the binary mixture Lennard-Jones $80$-$20$ (BMLJ) [@bmlj], at density $\rho$=$1.2$, $$V_{BMLJ}(r) = 4 \epsilon_{\alpha \beta} \left[ (\sigma_{\alpha \beta}/r)^{12} - (\sigma_{\alpha \beta}/r)^6 \right] \ , \label{vbmlj}$$ where the values of the parameters are those of the Kob-Andersen mixture ($\sigma_{AA}\!=\!1$, $\sigma_{AB}\!=\!0.8$, $\sigma_{BB}\!=\!0.88$, $\epsilon_{AA}\!=\!1$, $\epsilon_{AB}\!=\!1.5$, $\epsilon_{BB}\!=\!0.5$); 4. a variant of the binary mixture Lennard-Jones (BMLJ$_2$), at $\rho $=$1.2$, in which the values of $\sigma _{AA}$ and $\sigma _{BB}$ were exchanged. In the case of BMLJ and BMLJ$_{2}$, the interaction potential is tapered at long distances between $r_1\!=\!2.43\sigma _{AA} \leq r\leq 2.56\sigma _{AA}\!=\!r_2$ with the following fifth-order smoothing function $\mathcal{T}(r)=1+(r_{1}-r)^{3}(6r^{2}+ (3r+r_{1})(r_{1}-5r_{2})+10r_{2}^{2}) \diagup (r_{2}-r_{1})^{5}$. In this way the potential, the forces and their derivatives are continous, the energy can be kept constant to better than $1/10^{5}$ over $100$ millions of time steps. The MLJ and MSS potential have been simply cut and shifted at 2.5$\sigma$. We performed standard molecular dynamics simulations at equilibrium ($NVE$ ensemble), in a temperature range from $T\!=\!2$ down to the lowest temperature that can be equilibrated in the MD run (this temperature is strongly model dependent). Along the equilibrium molecular dynamics trajectories at a given temperature we analyzed the properties of i) the instantaneous configurations; ii) the inherent structures (minima); and iii) the saddle configurations. About $1000$ configurations have been analyzed for each temperature and for each system. The inherent structures associated to instantaneous configurations are obtained by a conjugate-gradient minimization procedure on the total potential energy. For saddles, a similar minimization procedure has been applied to the pseudo-potential $W\!=\!|\nabla V|^{2}$. The tapering of the BMLJ and BMLJ$_{2}$ potentials allows the minimization procedures of both $V$ and $W$ to work correctly as they are not affected by small discontinuities in the derivative of $V$ and $W$. The importance of avoiding discontinuities in order to obtain good $W$ minimization has been recently underlined in Ref. [@sad_5] where the LBFGS algorithm [@LBFGS] was used. However to obtain good minimizations of $W$, even for a “small” system of $256$ particles (i.e. 768 dimensions), is a stiff problem. We tested different minimization algorithms (steepest-descent, Gauss-Newton, preconditioned conjugate gradient, Levenberg-Marquardt [@numrec]) but they eventually stick in some points of the configuration space, where the algorithm decrease more and more the step size, and the search becomes inefficient and possibly stops. Different algorithms usually stick in different points. Sometimes the same algorithm who stuck in a given point can be effective in overcoming the critical situation if a larger step is used. Therefore, in the present work, a complex flow chart with various algorithms was used to obtain good minima (the details of the numerical algorithms will be presented elsewhere [@MS1]). We want to remark that in this way the calls to the function $W$ are always less than $3500$ (average $\approx 1500$) and less than $1000$ (average $\approx 200$) to the derivative of $W$. For all the analyzed configuration points (instantaneous, minima and saddles) we store the energies per particle ($e$, $e_{_{IS}}$ and $e_s$ respectively), and for instantaneous and saddles we also determine their order $n$ and $n_s$, defined as the fractional number of negative eigenvalues of the Hessian, i.e. the absolute number of negative curvatures over $3N$ (for inherent structures one obviously has $n_{_{IS}}$=$0$). Models $\rho$ $T_{MCT}$ $\Delta E$ $\Delta E^*$ ---------- -------- ----------- ------------ -------------- MLJ 1.0 0.475 4.43 9.3 MSS 1.0 0.210 2.06 9.8 BMLJ 1.2 0.435 4.16 9.6 BMLJ$_2$ 1.2 0.605 5.93 9.8 : \[table\]For the different Lennard-Jones like models we report the investigated density $\rho$, the mode-coupling temperature $T_{MCT}$ (estimated from the apparent power-law vanishing of the diffusion coefficient), the mean barrier values $\Delta E$ (mean elevation of order-one saddles from underlying minima) and the reduced barrier height $\Delta E^* \!=\! \Delta E / T_{MCT}$. All the quantities are in LJ reduced units. Saddles and quasisaddles ======================== First of all we focus our attention on the differences between saddles (absolute minima of $W$) and quasisaddles (local minima of $W$). As an example, Fig. \[fig\_1\] shows, for the case of BMLJ$_2$ model, the histogram of the value of the pseudopotential $W$ at the minima ($6000$ configurations analyzed at $T\!=\!2$). The values of $W$ at the minima are normalized to the values at the corresponding instantaneous configurations, i.e. to the value of $W$ before starting the $W$-minimization procedure. We observe two very well distinct regions: the one with higher $W$ values corresponds to local minima of $W$ (quasisaddles), the lower one corresponds to absolute minima (true saddles). The non-zero values of $W$ on the low-$W$ peak is due to the finite precision and/or threshold employed in the minimization procedure. A closer inspection of the eigenvalues of the Hessian shows that the quasisaddles are points with only one extra zero eigenvalue [@comment] (besides the three connected to the global translations), corresponding to an inflection one-dimensional profile along the corresponding eigenvector. The fact that in the plot the two regions are well separated, allows to discriminate true and false saddles in a clear way. On the contrary, no clear separation has been found between saddles and quasisaddles from the analysis of the eigenvalues: due to the finite precision the found eigenvalues relative to the inflection points are different from zero of the same amount of the lowest frequency eigenvalues of real vibrational (or diffusive) modes. As it is evident from Fig. \[fig\_1\], true-saddles are very few and their number are found to decrease on lowering the temperature (e.g. for BMLJ$_2$ in Fig. \[fig\_1\] about $5\%$ at $T\!=\!2$, and for BMLJ about $2\%$ at $T\!=\!2$ and less than $1\%$ at $T\!=\!0.48$). ![Histogram of the ratio $W/W_{inst,}$, i.e. the value at the minima of $W$ with respect to the value at instantaneous configurations, at $T\!=\!2$ for BMLJ$_2$ ($6000$ configurations analyzed). The higher region corresponds to quasisaddles (local minima of $W$), while the lower one to true saddles (absolute minima).[]{data-label="fig_1"}](fig1.eps){width=".5\textwidth"} An interesting observation arises from the analysis of the behavior of the $T$-dependence of the number of negative curvatures in the “true” saddles and in the quasisaddles separately. In Fig. \[fig\_2\] the saddle order is shown as a function of the temperature using only the true- (full symbols) and the quasi- (open symbols) saddles, for the cases of BMLJ (triangles) and BMLJ$_2$ (squares) models (we note that in the BMLJ$_2$ case, due to the appearance of crystallization, the data are available only for $T\gtrsim 1$). The coincidence between the two set of data indicates that, as far as the temperature dependence of their characteristics (order and energy) is concerned, quasisaddles and true saddles share the same properties. Also other properties, as for example the spectral features (i.e. the density of vibrational states), of quasi-saddles and true-saddles are found indistinguishable [@sadBLJ]. This finding suggests that, no matter if saddles or quasisaddles, the minimization of $W$ leads to points of the PES that are relevant for a landscape-based interpretation of the slow dynamics of the system: the order extrapolates to zero at the mode-coupling temperature $T_{MCT}$ (see Table \[table\] for the values of $T_{MCT}$, estimated from diffusivity data, for the different models), indicating that at this temperature the properties of the landscape probed by the system manifest a kind of discontinuity (the number of open directions, related to the saddle order, goes to zero and the dynamical processes change their characteristics). In other words, the minimization of $W$ seems to be a good method to get ride of the fast degrees of freedom and to keep information only on the slow degrees relevant for the slowing down of the dynamics taking place in supercooled regime. General features of the PES =========================== We now turn our attention to the existence of common features among the different model systems analyzed. ![Temperature dependence of the fractional order of true saddles (full symbols)and quasisaddles (open symbols), for BMLJ (triangles) and BMLJ$_2$ (squares). Dashed lines are power law fits.[]{data-label="fig_2"}](fig2.eps){width=".5\textwidth"} $T$-dependent properties ------------------------ As already pointed out in Ref. [@noi_sad; @noi_jcp], the (quasi-) saddle order, $n_s$, vanishes as $T$ approaches $T_{MCT}$ from above. At a first sight, it seems that the specific behavior of $n_s(T)$ is a model-dependent property (see Fig. \[fig\_2\]). However, we observe that after the scaling the temperature scale by a specific sample dependent quantity, i. e. by $T_{MCT}$, all the models behave similarly. In Fig. \[fig\_3\] the saddle order $n_s$ is reported as a function of reduced temperature $T/T_{MCT}$. All the curves for the different systems collapse into a single master curve. The latter can be fitted by a power law $$n_s = \overline{n} \left( \frac{T}{T_{MCT}} - 1 \right) ^{\gamma} \ ,$$ with $\gamma \!=\!0.85$ and $\overline{n}\!=\!0.025$ (in the fitting procedure, the values of $T_{MCT}$, reported in Table\[table\], are kept fixed to the ones derived by the fit of the power-law behavior of the diffusion coefficient). A similar master plot is obtained also for the relation between the saddle energy and the temperature. These results suggest a universal behavior (at least for the LJ-like model systems analyzed here): at a given reduced temperature $T^*=T/T_{MCT}$ all the systems visit saddles with the same properties (hereafter we will indicate with “$^*$” the temperature and the energy scaled by $T_{MCT}$). One could conjecture that this universality is due to the repulsive part of the pair potential $r^{-12}$ (common to all the systems), that dominates over the attractive one at the studied densities. However, the facts that the curves superimpose each other quite well in the whole temperature range and that non-LJ systems (as, for example, the Morse potential - see the next section) show a similar behavior, seems to indicate that the observed universality is not trivially related to the repulsive part of the interaction potential. Finally, we want to remark that the small value of $\overline{n}$ indicates that even at temperature twice that of the MCT critical point, the system is visiting saddles of low order ($n_s \simeq 0.025$) , so indicating that at $T\!=\!2 T_{MCT}$ the closest saddle, according to the partitioning defined by the minimization of $W$, is far below the top of the landscape.\ ![Saddle order $n_s$ as a function of reduced temperature $T/T_{MCT}$, for all the analyzed systems. The dashed line is a power law with exponent $\gamma\!=\!0.85$. For MLJ and MSS $\rho\!=\!1.0$, while for BMLJ and BMLJ$_2$ $\rho\!=\!1.2$.[]{data-label="fig_3"}](fig3.eps){width=".5\textwidth"} Energy barriers and $T_{MCT}$ ----------------------------- The existence of common and general features of the PES emerges in a clear way from the comparative analysis of the energy and of the order of the saddles. In Fig. \[fig\_4\] part A the energy elevation $\Delta e_s$=$e_s - e_{_{IS}}$ of the saddles from the underling minima is plotted as a function of the saddle order $n_s$ for the different investigated models. As already observed [@noi_sad], there exists a proportionality between these two quantities, indicating a simple organization of the PES: saddles are equally spaced in energy over the minima. The slopes of the different straight lines in Fig. \[fig\_4\] determine the elementary energy elevation $\Delta E$ of saddles of order $n$ from saddles of order $n-1$: $$\Delta E = \frac{1}{3} \frac{d (e_s - e_{_{IS}})}{d n_s} \ , \label{delta}$$ where the factor $3$ is due to the fact that energies are per particles ($N$) and the fractional order per degrees of freedom ($3N$).The values of $\Delta E$ obtained for the various systems are reported in Table \[table\]. A possible explanation of the linear relationship observed in Eq. \[delta\] is that there exist in the system several spatially uncorrelated rearranging regions, each experiencing a mean barrier energy $\Delta E$. In other words, if the system as a whole lies on a saddle of order $m$, this is due to the fact that there are $m$ uncorrelated subsystems each one visiting a saddle of order 1. The analysis of the specific atomic motion associated to these saddles, needed to assess or disprove the validity of this hypothesis, is beyond the aim of the present work. ![ A) Energy elevation of saddles from underling minima $e_s - e_{_{IS}}$ against saddle order $n_s$; B) Energy elevation rescaled by mode-coupling temperature $T_{MCT}$ against saddle order $n_s$. Dashed straight line is a guide to the eyes.[]{data-label="fig_4"}](fig4.eps){width=".5\textwidth"} A very interesting and surprising result is obtained by scaling the energy values reported in Fig. \[fig\_4\] part A to the mode-coupling temperature $T_{MCT}$ ($K_B=1$), obtaining again a single master curve (see Fig. \[fig\_4\] part B). The landscapes of different systems seem to share common features, with only one parameter describing the organization of saddles, i. e. the mean elevation $\Delta E$, that becomes an universal parameter ($\Delta E^* \!=\! \Delta E / T_{MCT} \simeq 9 \div 10$) once normalized to the mode-coupling temperature (see the last column of Table \[table\]). In other words, all the models have the common property that the elementary barrier height is about $10$ times the critical temperature $T_{MCT}$: $$\Delta E \simeq 10\ T_{MCT} \ . \label{univ}$$ This relation have been numerically proved for the four potential models investigated here. The same relation also holds for another LJ-like model, the binary mixture soft-sphere model (BMSS) investigated in Ref. [@grig] (at $\rho=1$). This observation gives further support to the universality of Eq. \[univ\]. If this is a particular characteristic of Lennard-Jones like models or a general feature of a more wide class of simple liquids is a open and interesting question which remains to be answered. We can try to give a first answer to this question analyzing the available data in the literature for other systems. To our knowledge, besides the Lennard-Jones like systems, a saddle-based analysis has been performed only for the Morse potential [@sad_4]. The Morse potential, used in Ref. [@sad_4], is defined as $V_\alpha(r)= \epsilon [ 1 - \exp(\alpha(1-r/r_e))]^2 - \epsilon$, where $\epsilon$ is the well depth, $r_e$ is the interparticle distance and the parameter $\alpha$ is inversely correlated to the range of the potential. Differently from soft-spheres and LJ, the Morse potential is finite as $r \rightarrow 0$. Unfortunately, equilibrium simulation based on the Morse potential are difficult, since the undercooled system crystallizes easily. Therefore simulations reported in Ref. [@sad_4] have been performed only well above $T_{MCT}$. It was found that the larger the value of $\alpha$ is , the further the distance between the temperature of the lowest non-crystalline simulation and $T_{MCT}$ is. In this study, a linear dependence between $n_s$ and $e_s$ have been observed and the values of $de_s/dn_s$ normalized to $T_{MCT}$ are in agreement with Eq. \[univ\] for the three smaller $\alpha$ values $\alpha=4,5$ and $6$ ($\Delta E^*$ are in the range $9.3 \div 10.5$ [@nota_morse]). For the two highest $\alpha$ values, $\alpha=9$ and $12$, the reported values for $\Delta E^*$ are quite different ($4.6$ and $3.5$ respectively). Further studies, for example focussing on binary mixture systems, are requested to find out if such discrepancy is due to an approximate determination of $T_{MCT}$ for $\alpha=9$ and $12$ (which was obtained by extrapolating $n_s(T)$ from a temperature region where $n_s$ is far away from zero, $n_s(T) \gtrsim 0.2$). Uncertanties in the estimates of $T_{MCT}$ at large $\alpha$ are also consistent with the unexpected non-monotonic dependence of $T_{MCT}$ with $\alpha$ reported in Ref. [@sad_4]. We conclude that, for all $\alpha$ values for which the reliability of the data is unquestionable, the Morse potential landscape shares the same characteristic of those of the LJ-like potentials. In all the other model systems studied in the literature we do not have a direct information on the saddle energy elevation. However, the existence of a well defined barrier energy scale $\Delta E$ in the PES is expected to control the activation processes at low temperature, giving rise to an Arrhenius behavior of the transport properties at temperatures below $T_{MCT}$. The existence of Arrhenius law in LJ-like systems - that are basically “fragile”, in the Angell classification scheme [@land_angell] - would be, per se, surprising (however, the degree of fragility of LJ systems is a matter of debate [@angell_pisa]). ![Diffusivity $D$ as a function of inverse temperature $1/T$ for the MLJ model. Straight line represents the mode coupling like power law fit. Dashed line is the Arrhenius law with energy barrier $\Delta E_{Arr} \simeq 1.9 \Delta E = 8.4$ ($\Delta E$ is the energy barrier from saddles - see Table \[table\]), following the corresponding relation obtained for the BMLJ case (Ref. [@sastry_pisa]).[]{data-label="fig_5"}](fig5.eps){width=".5\textwidth"} The simulations below $T_{MCT}$ are very difficult to perform, due to the extremely long relaxation times in this regime and a direct inspection of the expected Arrhenius behavior is not easy to pursuit. Only very recently such a kind of analysis has been performed for the BMLJ model at $\rho=1.2$ [@sastry_pisa]. In that work an Arrhenius behavior was actually found in the temperature dependence of the diffusion coefficient below $T_{MCT}$ : $D \propto \exp(-\Delta E_{Arr} /T)$ (we use the symbol $\Delta E_{Arr}$ for the activation energy in the Arrhenius law of the diffusivity, to distinguish it from the energy barrier $\Delta E$ determined from the saddles analysis of the PES), with a value of $\Delta E_{Arr} \simeq 8.1$. The observed Arrhenius behavior is somewhat surprising in this “fragile” liquid models, and seems to indicate that close to $T_{MCT}$ activated processes start to be relevant and dominate the dynamics. However, the value of the activation energy $\Delta E_{Arr}$ found in Ref. [@sastry_pisa] is not equal to the elementary barrier energy $\Delta E$ estimated from the saddles analysis (see Table \[table\]), but it is about twice that value: $\Delta E_{Arr}/\Delta E \simeq 1.9$. Re-analyzing our data for the MLJ model (for which we have few thermodynamic points equilibrated close to but below $T_{MCT}$), we find that the above reported ratio is compatible with MLJ data (see Fig. \[fig\_5\]), even if statistic is poorer than that of BMLJ case and the equilibrium condition is not fully satisfied by the lowest two temperature points. If such observation has general validity, then Arrhenius behavior should be observed below $T_{MCT}$, with an activation energy value about $2$ times the value of the elementary saddle energy barrier \[so obtaining a value of reduced barrier energy (normalized to $T_{MCT}$) $\Delta E_{Arr}^* \simeq 18 \div 20$\]. The origin of this factor two needs to be further clarified. To this aim, it is important to underline that the “effective” energy barriers for activated processes as seen by the dynamics (i. e. those entering in the Arrhenius law for the diffusivity) can be higher than the minimum-to-saddle energy difference (as measured directly by analyzing the PES). This can be due to the fact that the true diffusive path in the landscape [@demic] could pass higher in energy with respect to the saddle point, in order, for example, to minimize the minimum-to-minimum path length (i. e. for entropic reasons). In this respect, it is worth to mention that a non-coincidence between the relaxation times determined either through MD simulations or through the direct inspection of the PES has been observed in the simulation of a model protein during the folding process [@torcini]. In particular, the results in Ref. [@torcini] indicates that the effective saddle height is larger than the actual one. Models $\Delta E^*$ $\Delta E_{Arr}^*$ ------------------- --------------- -------------------- MLJ 9.3 17.7 MSS 9.8 ... BMLJ 9.6 18.6[^1] BMLJ$_2$ 9.8 ... BMSS[^2] 9.1 ... Morse[^3] $9.3\div10.5$ ... Silica (BKS)[^4] ... 16$\div$18 OTP[^5] ... $20\div28$ Water (SPC/E)[^6] ... 40 : \[table2\] Reduced energy barrier heights estimated from saddles ($\Delta E^*\!=\!\Delta E / T_{MCT}$) and from low-temperature Arrhenius law of diffusivity ($\Delta E_{Arr}^*\!=\!\Delta E_{Arr} / T_{MCT}$) for different model systems. The data of MLJ, MSS, BMLJ and BMLJ$_2$ are from this work (except the $\Delta E_{Arr}^*$ for BMLJ, that is from Ref. [@sastry_pisa]). Having in mind that $\Delta E_{Arr}^* \! \simeq \! 2 \Delta E^*$ and that $\Delta E^*\! \simeq\!10$ (i.e. $\Delta E_{Arr}^* \!\simeq\! 20$), we can try to analyze what is observed for other model potentials existing in the literature where the $D(T)$ has been determined. We found three different models for which a low temperature analysis of $D(T)$ has been performed [*via*]{} molecular dynamics: [*i)*]{} The BKS-silica model [@horbac], for which the values of $\Delta E_{Arr}^*$ are $16.2$ and $18.0$, for the self diffusion of $O$ and $Si$ respectively; [*ii)*]{} the Lewis and Wahnström ortho-terphenyl model [@mossa], for which the temperature dependence of the molecular center of mass diffusion coefficient at five different densities give values of reduced barrier energy $\Delta E_{Arr}^* \simeq 20 \div 28$ (except the lowest density that gives a value of about $10$); [*iii)*]{} The SPC/E-water model [@starr], for which one finds $\Delta E_{Arr}^* \simeq 40$. Table \[table2\] summarizes the known results on energy barrier heights estimated from saddles and from Arrhenius low-temperature dependence of diffusivity. The values for MLJ, MSS, BMLJ and BMLJ$_2$ are from the present work, except the $\Delta E^*_{Arr}$ for BMLJ that is obtained from Ref. [@sastry_pisa]. In future works we will try to determine the saddle-barriers $\Delta E^*$ for non-LJ systems (the last three systems in the Table), in order to have a better understanding of the diversity of the different landscapes. In conclusion, besides the case of water, the other systems seem to be in agreement with the findings of this work (the values of the reduced barrier energies are of the same order), evidencing a quite general universality of the observed relations. A deeper understanding of the differences among various model liquids deserves further investigations. Conclusions =========== In conclusion, despite complex and disordered in nature, the simple liquid PES seems to exhibit few general and regular features, useful both to bring important insight for the understanding of the relevant diffusion processes taking place in supercooled liquids and to construct simplified PES models. The main findings of the present work can be summarized as: - the coincidence between the temperature dependence of the quasisaddles and of the true saddles properties;\ - the existence of master curves for saddle properties, once energies and temperatures are rescaled by the mode coupling critical temperature $T_{MCT}$;\ - the existence of a universal relationship between the mode-coupling temperature and the mean energy barrier height $\Delta E \simeq 10\ T_{MCT}$, that seems to extend beyond the class of the Lennard-Jones like models analyzed here.\ Finally, we would like to point out that it already exists in the literature an hint on the existence of a linear relationship between $\Delta E_{Arr}$ and the mode coupling critical temperature. Indeed, in a large class of glassy system one experimentally observes a linear relationship between the glass transition temperature $T_g$ and the infinite-frequency shear modulus $G_{\infty}$ [@nemilov]: $T_g \propto G_{\infty}$. 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--- abstract: 'We have identified the existence of globally clustered chimera states in delay coupled oscillator populations and find that these states can breathe periodically, aperiodically and become unstable depending upon the value of coupling delay. We also find that the coupling delay induces frequency suppression in the desynchronized group. We provide numerical evidence and theoretical explanations for the above results and discuss possible applications of the observed phenomena.' author: - 'Jane H. Sheeba' - 'V. K. Chandrasekar' - 'M. Lakshmanan' title: 'Globally clustered chimera states in delay–coupled populations' --- The existence of chimera states (states characterized by the separation of identical oscillator groups into synchronized and desynchronized subgroups) in coupled oscillator populations came as a surprise in the study of synchronization phenomenon in complex systems. Since its discovery [@Kuramoto:02; @Abrams:04], various theoretical and numerical developments have been reported on the stability of chimera states and their existence in systems with varied structures [@Abrams:04; @Abrams:08], including time delay [@Sethia:08]. By and large, synchronization in coupled oscillator systems has been analytically and numerically investigated in a rigorous manner over the past years [@Winfree:67; @Strogatz:01]. Possible routes to global synchronization and methods to control synhronization have also been proposed [@Sherman:92; @Rosenblum:04]. However, complete understanding of the effects induced by coupling delay in synchronization of coupled oscillator systems is still an open problem. The consideration of delayed coupling is vital for modeling real life systems. For example, in a network of neuronal populations, there is certainly a significant delay in propagation of signals. In addition there can also be synaptic and dendritic delays. Other examples include, finite reaction time of chemicals, finite transfer time associated with the basic mechanisms that regulate gene transcription and mRNA translation. In this paper, we demonstrate that coupling delay can induce globally clustered chimera (GCC) states in systems having more than one coupled identical oscillator (sub) populations. By a GCC state, here we mean a state where the system, which has more than one (sub) population, splits into two different groups, one synchronized and the other desynchronized, each group comprising of oscillators from both the populations (note that this is in contrast to the chimera state where one of the populations is synchronized while the other is desynchronized [@Abrams:04]). The system under study is a system of two populations of identical oscillators coupled through a finite delay, represented by the following equation of motion $$\begin{aligned} \dot{\theta_i}^{(1,2)}&=& \omega - \frac{A}{N} \sum_{j=1}^{N} f(\theta_i^{(1,2)}(t)-\theta_j^{(1,2)}(t-\tau_1)) \nonumber \\ &&\mp \frac{B}{N}\sum_{j=1}^{N} h(\theta_i^{(1,2)}(t)-\theta_j^{(2,1)}(t-\tau_2)).\label{chim_del01}\end{aligned}$$ ![(Color online) Occurrence of (stable) GCC in system (\[chim\_del01\]) as explained in the text. Top panel: Global clustering phenomenon - synchronized and desynchronized (frequency suppressed) groups have oscillators from both the populations. Bottom panel: One of the populations is synchronized and the other is desynchronized (frequency suppressed). Green (light gray) and red (dark gray) lines represent oscillators in the first and the second populations, respectively. Here $\{f,h\}=\{\sin(\theta),\cos(\theta)\}$, $\tau_1=n\tau_2=n\tau$ with $n=1$ (top panel) $A=1.2$, $B=1$ and $\tau=2$, (bottom panel) $A=1.6$, $B=1$ and $\tau=1$.[]{data-label="Motiv"}](Chim_Del_Fig1.eps){width="8.5cm"} A typical example of such a system is the two groups of interacting neurons in the brain such as those in the cortex and the thalamus [@Jane:08b]. Here $\omega$ is the natural frequency of the oscillators in the populations and it is the same for all oscillators in both the populations making all of them identical. However, the two populations are distinguished by the initial distribution of their phases; the phases are uniformly distributed between $0$ and $\pi$ for the first population and between $\pi$ and $2\pi$ for the second population. $A$ and $B$ refer to coupling strengths within and between populations, respectively. The functions $f$ and $h$ are $2\pi$–periodic that describe the coupling. $N$ refers to the size of the populations. The complex mean field parameters $X^{(1,2)}+iY^{(1,2)}=r^{(1,2)}e^{i\psi^{(1,2)}}= \frac{1}{N}\sum_{j=1}^Ne^{i\theta_j^{(1,2)}}$, characterize synchronization within a population but not global clustering. $\tau_1$ and $\tau_2$ quantify coupling delay within and between populations, respectively. The investigation is motivated by the numerical discovery of the existence of GCC states in a system of two identical populations that are delay–coupled and are given by Eq. (\[chim\_del01\]) (see Fig. \[Motiv\]). We found that coupling delay can induce splitting of identical delay–coupled populations into desynchronized frequency suppressed (vanishing oscillating frequencies) clusters and synchronized clusters. This splitting can occur either within the populations or between the populations. The former represents the chimera and the latter is the GCC, as noted earlier. Further, the GCC state need not be stable but it can either breathe or can be unstable as will be discussed later. ![Illustration of a breathing GCC state with $A=0.3$, $B=0.2$, $n=1$, $\{f,h\}=\{\sin(\theta),\sin(\theta)\}$ and initial condition close to the GCC state. Grey and black lines represent the long and short–periodic breather with $\tau=3.6$ and $\tau=4$, respectively. Order parameter $r^{\mbox{\tiny{DS}}}$ and the corresponding phases $\theta_i^{\mbox{\tiny{DS}}}$ (see text) are plotted against time in the top and bottom panels, respectively.[]{data-label="Rplot"}](breather3.eps){width="6cm"} For illustrative purpose, we simulate system (\[chim\_del01\]) using Runge–Kutta fourth order routine with a time step of 0.01 (the results are not affected by decreasing the time step below 0.01). For all the numerical plots shown, we allow a transient time of 2000 units and take $N=32$ (the results have been verified to be independent of the size of the system) and $\tau_1=n\tau_2=n\tau$, where $n$ is an arbitrary constant. We further found that the GCC need not be stable but can breathe depending upon the value of the coupling delay. Since the coherence parameter $r$ quantifies synchronization within a population, it can also be used to quantify a breathing or unstable chimera. However, as mentioned earlier, global clustering cannot be quantified using this order parameter. Therefore, in order to quantify a breathing GCC numerically, after allowing the transients, we identify those oscillators whose $\theta_i$s are equal for all times and neglect them so as to end up with the desynchronized group (that comprises oscillators from both the populations, whose size is $N^{\mbox{\tiny{DS}}}$) and calculate its order parameter as $$\begin{aligned} \label{dsr} r^{\mbox{\tiny{DS}}}e^{i\psi^{\mbox{\tiny{DS}}}}=\frac{1}{N^{\mbox{\tiny{DS}}}}\sum_{j=1}^{N^{\mbox{\tiny{DS}}}} e^{i\theta_j^{\mbox{\tiny{DS}}}},\end{aligned}$$ where $N^{\mbox{\tiny{DS}}}=2N-N^{\mbox{\tiny{S}}}$. This order parameter $r^{\mbox{\tiny{DS}}}$ can be used to quantify both the chimera and GCC states and also valid for cases where there exists more than one synchronized cluster. Such multi-cluster states also occur for model (\[chim\_del01\]), the details of which will be published elsewhere. While a GCC is breathing, one of the groups is completely synchronized while the desynchronized group continuously fluctuates. Fig. \[Rplot\] illustrates breathing GCC where we plot the order parameters $r^{\mbox{\tiny{DS}}}$ for two different values of $\tau$ in the top panel. The grey line represents a long period breather for $\tau=3.6$ where switching occurs between frequency suppressed synchronized state and the desynchronized state. Increasing $\tau$ further to 4 results in a short period breather (the black line) where the desynchronized state oscillates similar to the previous case but in a faster manner. ![Top panel: Aperiodic breathing GCC with $\tau=5$ and $N^{\mbox{\tiny{DS}}}=17$; Bottom panel: Unstable breathing GCC with $\tau=4$ and $N^{\mbox{\tiny{DS}}}=12$, (left) order parameter $r^{\mbox{\tiny{DS}}}$ and (right) the corresponding phases $\theta_i^{\mbox{\tiny{DS}}}$ of the desynchronized group. Here $A=0.6$, $B=0.3$, $n=1$, $N=32$ and $\{f,h\}=\{\sin(\theta),\cos(\theta)\}$.[]{data-label="LC1"}](lc2.eps){width="8cm"} The GCC can also be unstable where the oscillators in the desynchronized group remain desynchronized for a while after which this state loses its stability and all the oscillators lock to one phase. Thus at this stage the GCC loses stability and a two clustered synchronized state becomes stable. Therefore, finally the system goes from a GCC to a state with two separate synchronized clusters. This phenomenon is depicted in Fig. \[LC1\], where for a sufficiently large value of $\tau$ the GCC breathes in an aperiodic manner (top panel). On decreasing $\tau$, this breather loses stability and the desynchronized group entrains itself to a synchronized state (bottom panel). The regions of occurrence of these phenomena in the phase plane, obtained numerically corresponding to Fig. \[LC1\], is shown in Fig. \[XY\_LC2\]. The black line is the stable limit cycle attractor of the synchronized group (which is always the same whatever the value of the entrainment frequency of the synchronized group is). The grey region represents the aperiodic breather. The GCC is unstable while in the white region between these two; a GCC in this white region is attracted to the limit cycle and a stable synchronized state is established (as shown in Fig. \[LC1\] (bottom panel)). A GCC in the innermost white region is always stable. The sizes of all these regions change with respect to system parameters. Thus we find that, for a given set of system parameters, increasing/decreasing (depending on the values of the parameters $A$, $B$ and $\tau$, since the behaviour repeats itself periodically as will be discussed later under Fig. \[Btau1\]) the coupling delay parameter $\tau$ results in the following sequence of GCC dynamics: stable GCC, long–period breather, short–period breather, aperiodic breather and unstable GCC leading to global synchronization. Further increase in $\tau$ from the global synchronization state leads to a stable GCC by following the above mentioned route in the reverse order. If we are able to discriminate the regions of stability of the synchronized and the desynchronized states, we will be able to expect the occurrence of GCC near these boundaries with reference to the numerical observations. This expectation also depends on the fact that the stability of the GCC state changes periodically with respect to $\tau$ incorporating regions of synchronization and desynchronization. In order to gain a better understanding of the numerically observed phenomena, we analyze system (\[chim\_del01\]) in the continuum limit $N\rightarrow \infty$. We write down the continuity equation [@Winfree:67] for the density of phases $\rho$ and then express $\rho$ and $\{f,h\}$ as Fourier expansions, $\rho=\sum_{k=-\infty}^{\infty}\rho_ke^{ik\theta}$ and $\{f,h\}=\sum_{k=-\infty}^{\infty}\{f,h\}_ke^{ik\theta}$. Now by considering only the non-trivial $k$th Fourier mode we arrive at the eigen value of that mode $\lambda_k=\bar{A}e^{-\lambda_k\tau_1} \pm \bar{B} e^{-\lambda_k\tau_2} -i\omega_0$ which characterizes the stability of the desynchronized state. Here $\bar{A}=ikf_kA$, $\bar{B}=ikh_kB$ and $\omega_0=k(\omega-Af_0\mp Bh_0)$. Assuming $\lambda_k=-i\beta/\tau$, one can find the $kth$ stability region in a parametric form as $$\begin{aligned} \label{chim_del02} B=\pm kA\frac{|f_k|\cos(n\beta-\alpha_f)}{|h_k|\cos(\beta-\alpha_h)}; \; \; \tau=\beta/[k(\omega_0 \nonumber \\ +A|f_k|\sin(n\beta-\alpha_f) \pm B|h_k|\sin(\beta-\alpha_h)]^{-1},\end{aligned}$$ where $\{f,h\}_k=-i|\{f,h\}_k|e^{i\alpha_{\{f,h\}}}$ and $\tau_1=n\tau_2=n\tau$. The overall stability of the desynchronized state is determined by the overlap of these domains for all the modes. ![Phase portraits showing the limit cycle of the synchronized state (the black line) and a breathing GCC (grey region). The white region between these two represents unstable GCC. The innermost white region represents a stable GCC. Parameter values correspond to Fig. \[LC1\]. Here $X=r\cos\psi$ and $Y=r\sin\psi$, where $r$ and $\psi$ are the mean-field parameters.[]{data-label="XY_LC2"}](xy_lc7.eps){width="6cm"} Now it is also of importance to investigate the stability of the synchronized state for which we consider the solution to the synchronization state $\theta_i^{(1,2)}=\Omega t$. With this solution, system (\[chim\_del01\]) becomes $\Omega=\omega-Af(n\Omega \tau)\mp Bh(\Omega \tau)$. Along with this relation, the condition $Af'(n\Omega \tau)\pm Bh'(\Omega \tau)>0$ should also be satisfied in order that the synchronized state is stable. This provides the stability regime $$\begin{aligned} \label{chim_del04} B=\frac{\mp Af'(n\beta)}{h'(\beta)}; \; \; \tau=\frac{\beta }{\omega-Af(n\beta)\mp Bh(\beta)},\end{aligned}$$ where $\beta=\Omega \tau$. The parametric forms (\[chim\_del02\]) and (\[chim\_del04\]) separate the desynchronization and synchronization regimes. A homogeneous perturbation $\theta_i^{(1,2)}= \Omega t+\Delta\theta$ pertaining to the case when all the phases remain equal while their rotation becomes nonuniform in time to the synchronization regimes leads to the following equation for stability $\Delta\dot{\theta}=-(Af'(n\beta)\pm Bh'(\beta))\Delta\theta +Af'(n\beta)\Delta\theta_{n\tau}\mp Bh'(\beta)\Delta\theta_{\tau}$. The stability condition for $n=1$ is [@DVS:07] $$\begin{aligned} \label{chim_del05} \int_{t_0}^{\infty}[Af'(\beta)\pm Bh'(\beta)-|Af'(\beta)\mp Bh'(\beta)|]dt=\infty.\end{aligned}$$ The stability of the global synchronization state is determined by the integrand in this condition. From equations (\[chim\_del02\])-(\[chim\_del05\]) it becomes evident that the stability of the synchronized/desynchronized state switches periodically between stable and unstable states depending on the signs of $A$ and $B$, since $h$ and $f$ are $2\pi$ periodic. This is obvious from Fig. \[Btau1\], where on increasing $\tau$, regions of synchronization and desynchronization alternate each other. This is in agreement with the numerical analysis as pointed out earlier and hence forms a theoretical basis. The GCC state can be expected near the stability boundaries shown in Fig. \[Btau1\]. This is evident from the numerical results depicted in Figs. \[Motiv\] and \[LC1\]. ![(Color online) Stability regions as obtained from theory with $\{f,h\}=\{\sin(\theta),\cos(\theta)\}$. I. Desynchronization, II. Synchronization of the populations individually and III. Global synchronization regions. $\circ$: according to (\[chim\_del02\]), $+$: according to (\[chim\_del04\]) and dotted line: according to (\[chim\_del05\]). The black and grey symbols correspond respectively to the $+$ and $–$ signs in equations (\[chim\_del02\])-(\[chim\_del05\]). Note that the boundaries obtained by the stability analysis on the incoherent ($\circ$) and the synchronization ($+$) regimes are exactly one and the same. The red symbols are the locations of the GCC as from the numerical examples in Figs. \[Motiv\] and \[LC1\]. $\bullet$, $\times$ and $\top$ represent the stable, unstable and the breathing GCC respectively.[]{data-label="Btau1"}](anal_pre_test3.eps){width="6cm"} The knowledge about synchrony control methods is very important because synchronization is desirable somtimes as in neuronal networks while they support cognition via temporal coding [@Singer:99; @Jane:08b] and in the case of lasers and Josephson junction arrays [@Trees:05]. However, synchronization can also be dangerous in cases like epileptic seizures [@Timmermann:03], Parkinson’s tremor [@Percha:05], or pedestrians on the Millennium Bridge [@Strogatz:01]. For example, in [@Jane:08b] a thalamocortical model of asymmetrically interacting neuronal populations has been proposed to simulate the state of emergence from deep to light an[æ]{}sthesia. The model results revealed the fact that successful coding of information and consciousness is achieved by the occuurrence of global synchronization between the thalamus and the cortex. Further, it was eludicated that consciousness and cognition are kept away during deep an[æ]{}sthesia because of the lack of phase locking between the cortex and the thalamus. This is one example of a situation where global clustering/synchronization and hence controlling the same prove to be very crucial. There are various methods to control synchronization (even its rate and velocity). However if we could handle it all with one parameter, it makes life much easier. In summary, a new type of globally clustered chimera states have been identified in delay coupled populations – a system of two identical, delay–coupled populations split into two groups, one synchronized and the other desynchronized, each group having a fraction of oscillators from both the populations. We have found that this state need not be stable always but can breathe periodically, aperiodically or become unstable, depending upon the value of coupling delay. A modified version of the order parameter is introduced in order to capture these phenomena. In the presence of coupling delay, frequency suppression is induced in the desynchronized group. We have also provided analytical explanations of the observed effects on the basis of linear stability theory. The illustrative model presented here can be considered as a phenomenological model of oscillatory neural networks. 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--- abstract: | Planning a path for a nonholonomic robot is a challenging topic in motion planning and it becomes more difficult when the desired path contains narrow passages. This kind of scenario can arise, for instance, when quadcopters search a collapsed building after a natural disaster. Choosing the quadcopter as the target platform, this paper proposes the Kinodynamic Aggressive Trajectory (KAT) motion planning algorithm, which aims to compute aggressive trajectories for narrow passages under nonholonomic constraints. This type of maneuvers is necessary because the dynamics of quadcopters entail that some narrow passages can only be traversed at high speed. To find the best path, the KAT uses RRT to determine a holonomic path first and then adjusts it to satisfy the nonholonomic constraints. The innovations in this process are: 1) The states of the robot are divided into near-holonomic set and non-holonomic set, which makes the constraints local rather than global; 2) For each of the most confined waypoints in the path, KAT plans forward and backward simultaneously around the waypoint to find a feasible local trajectory traversing the narrow passage. Our approach thus transforms a globally-constrained planning problem into a problem with local constraints, and as a result, the computation becomes tractable. We evaluate KAT by applying it to a quadcopter flying through two inclined holes that require aggressive maneuvers in a simulated environment. The average computation time to successfully find a solution for passing two 50$^{\circ}$ inclined holes is around 32 seconds. author: - 'Yaohui Guo$^{1}$, Zhaolun Su$^{2}$, Dmitry Berenson$^{3}$, Ding Zhao$^{4}$[^1] [^2][^3][^4][^5]' bibliography: - './bibtex/bib/598.bib' title: A Kinodynamic Aggressive Trajectory Planner For Narrow Passages --- Introduction ============ A robot’s ability to guide itself is the basis for accomplishing higher level tasks, making motion planning a popular and practical problem in robotics. But due to its high computational complexity [@schwartz1987planning], it is still a challenging problem. The difficulties in planning a path for an informed robot in a complex environment arise from two principal concerns. First, the existence of narrow passages makes sampling method less efficient. Second, some robots have the nonholonomic property, which makes their attainable region a local submanifold of the workspace and thus, similarly, the probability of reaching the goal is low if the robot samples in the whole workspace. Therefore, planning for a nonholonomic robot in an environment with narrow passages could be difficult. Quadcopters are typical examples of such a problem. Because of its outstanding mobility, the quadcopter has been used widely in complex and confined environments for applications such as exploration, inspection, and mapping. For instance, a quadcopter is useful for exploring the inside of a collapsed building in a search-and-rescue scenario, where the environment is usually narrow and complex [@michael2012collaborative]. For example, the robot may need to pass through a tilted window without colliding. Unfortunately, a quadcopter can not maintain a tilted attitude at low speed because all the forces applied by the propellers can not entirely offset the gravitational force. Instead, a quadcopter can achieve an instantaneous tilted state by exploiting its dynamics. Thus, an aggressive maneuver is required. In this paper, we present the Kinodynamic Aggressive Trajectory (KAT) planner for computing a trajectory for a nonholonomic robot in an environment with narrow passages. The main idea of KAT is to eliminate the nonholonomic constraints at the near-holonomic states, and plan the trajectory around narrow points by forward and backward control. This is intended to let the algorithm focus on the bottlenecks of the path, thus the global planning can be broken into several local planning problems. The near-holonomic state assumption assumes that when the quadcopter is flying steadily at very low speed, it is able to change its direction of movement abruptly by a small amount. This assumption allows the quadcopter to move freely in the near-holonomic state and adjust its trajectory to fit the aggressive flying path for the narrow passage. The paper is organized as follows: section II presents a general overview of related work mainly in the aspect of sampling-based planners; section III describes the math model of the problem; section IV are the details on KAT algorithm; section V presents the experiments we designed for KAT and its results; section VII concludes the work and provides an overview of the future work. Related Work ============ The Rapidly-exploring Random Trees (RRT) [@lavalle2006planning], as a sampling-based motion planning algorithm, has been widely used for a broad range of robotic systems. For instance, in [@lavalle2001randomized], LaValle and Kuffner presented the first randomized approach to systems with kinodynamic constraints. Rather than planning in the configuration space, their approach plans the kinodynamic path in the state space considering the kinodynamic planning as a generalization of holonomic planning. This approach solves the path planning for nonholonomic systems like spacecrafts and hovercrafts, but cannot guarantee optimality and is not efficient for more complex systems due to the high dimensionality of the state space. To improve RRT for finding the optimal path, Karaman and Frazzoli proposed RRT\* [@Karaman-RSS-10] for holonomic systems, which grows the same way as RRT except that the tree will locally replan to ensure optimality. In [@karaman2013sampling], Karaman and Frazzoli proposed an extension of RRT\*, which could handle nonholonomic dynamics systems. This algorithm leverages the ball-box theorem to find an optimized extending range for each step while guaranteeing the optimal path. This algorithm works well for nonholonomic planning but is difficult to implement when the system is complex. In [@webb2013kinodynamic], Webb and Berg introduced the kinodynamic RRT\*. Like RRT\* algorithm, it is an asymptotically optimal motion planning algorithm, using a fixed-final-state-free-final-time controller to connect any pair of states optimally for systems with controllable linear dynamics to achieve optimality. However, this algorithm still needs to sample the state space, thus it can be time-consuming on systems with high dimensionality. Moon and Chung [@6872545] presented the kinodynamic planner Dual-Tree RRT (DR-RRT) for high-speed navigation of differential drive robot which is composed of a workspace tree and a state tree. The DT-RRT does not reduce the degree of freedom directly. Instead, it searches in the workspace to reduced the search complexity and tries to validate the path in the state space. However, this algorithm is mainly suitable for the low DOF kinodynamic system like Dubin’s car and hard to implement on high DOF systems. Other approaches like using motion primitives by building a path set [@knepper2009path], constructing the state lattice [@pivtoraiko2009differentially], are to discretize the state space in order to reduce the search complexity. These methods can be used for real-time path planning due to the high efficiency, but compromise the optimality. Based on the previous work, searching in the state space with the high dimensionality is the bottleneck of most sampling based planners, when dealing with nonholonomic systems. The KAT algorithm we proposed here aims to reduce this complexity by planning a path in the configuration space first and then validating it in the state space. Problem Statement ================= We define our object of interest as a time-invariant dynamic system: $$\dot{s}(t) = g(s(t),u(t)),\quad s(0) = s_0$$ where $\mathbb{S} \subset \mathbb{R}^{n_s}$ is the state space of the robot; $s(t) = s_t\in \mathbb{S}$ is the state of the robot at time $t$; $\mathbb{U} \subset \mathbb{R}^{n_u}$ is the input space of the system; $u(t)=u_t\in\mathbb{U} $ is the input of the system at time $t$; $g$ is the nonholonomic constraint of the system, which will also be referred as the update function of the system. For convenience, the following notations are used in this work. The configuration of a robot is the robot’s location and attitude; the state of a robot consists of the configuration and the change rate of the configuration. We define the configuration space of the object as $\mathbb{C} \subset \mathbb{R}^{n_c}$. If the robot is free when it is at configuration $c$, we define $c$ as a free configuration; otherwise $c$ is a collision configuration. Define the free configuration space as $\mathbb{C}_{free} \subset \mathbb{C} $ and the collision space as $\mathbb{C}_{col} \subset \mathbb{C} $. Define function $\alpha:\mathbb{S} \rightarrow \mathbb{C}$ maps a state $s$ to its configuration part $c$. Define the free state set as $\mathbb{S}_{free} = \{s \mid \alpha(s) \in \mathbb{C}_{free}\} \subset \mathbb{S}$, and the collision state set as $\mathbb{S}_{col} = \{s \mid \alpha(s) \in \mathbb{C}_{col}\} \subset \mathbb{S}$. The narrow configuration set $\mathbb{C}_{nar} \subset \mathbb{C}_{free}$ is the set of all the configurations in the narrow passages. The narrow state set is defined as $\mathbb{S}_{nar} = \{s \mid \alpha(s) \in \mathbb{C}_{nar} \} \subset \mathbb{S}_{free}$. Define the start configuration set as $C_s \subset \mathbb{C}_{free}$ and the goal configuration set as $C_g \subset \mathbb{C}_{free}$. Our goal is to find a dynamically feasible path $l:[0,T]\rightarrow S_{free}$ connecting $C_s$ and $C_g$ while passing through some narrow configuration $c_{nar,i},i = 1,2,3,...,n_{nar}$. This is equivalent to $\alpha(l(0)) \in C_s$ and $\alpha(l(T)) \in C_g$, and there exists $t_i\in [0,T], i = 1,2,3,...,n$ such that $\alpha(l(t_i)) = c_{nar,i} \in \{l(t)|t \in [0,T]\},i = 1,2,3,...,n_{nar}$. Also, to satisfy the nonholonomic constraints, there exists valid input $u(t) \in \mathbb{U}$, $t\in [0,T]$ such that $$l(T)=\int_{0}^{T}g(l(t),u(t))dt + l(0)$$ For numerical computation, we replace the integration with summation, and get $$l(T)=\sum_{0}^{T}g(s(t),u(t)) \Delta t + l(0)$$ For the quadcopter, we denote its configuration as $c = [p,r]^\intercal $, where $p = [x, y, z]^\intercal\in \mathbb{R}^3$ is the translation of the robot and $r \in SO(3)$ is the rotation. For computational convenience, here we use the quaternion $r= [q_r, q_i, q_j, q_k]^\intercal$ instead of the Euler angles to represent the rotation. Therefore, we have $c= [x, y, z, q_r, q_i, q_j, q_k]^\intercal$. The velocity of a configuration is represented as: $v_c = [ \dot{x}, \dot{y}, \dot{z}, \omega_x, \omega_y, \omega_z]^ \intercal$ . Then a state can be expressed as $s =[p,r,v,\omega]^ \intercal$, where $v = [ \dot{x}, \dot{y}, \dot{z}]^ \intercal$ is the translational velocity and $\omega=[\omega_x, \omega_y, \omega_z]^ \intercal$ is the angular velocity. Because a quadcopter can respond to small changes in its velocity and pose almost instantly by a linear controller when it is still [@brescianini2013quadrocopter], we can assume the quadcopter is not restricted by the nonholonomic constraints when it is nearly still. Thus here we define the near-holonomic state set as $\mathbb{S}_{holo^*}=\{s|w_{\omega}||\omega||+w_v ||v||+w_r||r-r_0||< \epsilon,s=[p,r,v,\omega]^\intercal \} $ where $w_v$, $w_{\omega}$ and $w_r$ are the weights and $r_0=[1,0,0,0]^\intercal$ is the unit quaternion parallel with *z* axis. The quadcopter with a near-holonomic state means the quadcopter could move freely in any direction within the $ \mathbb{S}_{holo^*}$. In this case, we can constrain the initial and goal states in $\mathbb{S}_{holo^*}$, so the quadcopter does not have to obey nonholonomic constraint when leaving the start configuration and reaching the goal. The nonholonomic constraint is only effective at the states where the quadcopter needs to conduct aggressive maneuver. Method ====== The overview of our algorithm appears in Algorithm \[alg:KAT\]. The method that we employ consists of four principal parts: 1. RRT planning in holonomic space 2. Sampling narrow configurations with maximum margin in narrow passage 3. Identifying escape velocity for each narrow configuration 4. Controller based dual-direction planning with nonholonomic constraints The goal of the first planning in holonomic space is to efficiently gather information regarding the narrow passage. By testing the robot along the smoothed holonomic path, we could identify the exact location of the narrow passage and collect possible poses that would allow the quadcopter to move through them. Since the smoothed trajectory will typically hug the obstacle, it is almost impossible for such a trajectory to be used. We propose implementing the maximum margin sampling inside the narrow passage to avoid such scenarios. From the smoothed holonomic path, we will be able to infer the general configurations where the robot is in a narrow passage. The algorithm will uniformly sample around the cluster centers of the narrow points and replace each cluster center by the configuration with the maximum margin to the surrounding passage. It is obvious that using such a pose is more likely to plan a successful path under nonholonomic constraints. Then the algorithm will search a velocity, defined as the escape velocity, to complete the above configuration as a candidate narrow state on the path. To reduce the risk of collision and make it easier for the system to recover to a near-holonomic state, the escape velocity will be the minimum velocity required to pass through the narrow passage. Next, starting from the narrow state, a dual-direction controller is employed to find a trajectory through the narrow passage. The dual-direction controller plans both forward and backward the dynamics function. If the planner can reach a near-holonomic point $s_{f} \in \mathbb{S}_{holo^*}$ by forward planning and a near-holonomic point $s_{b} \in \mathbb{S}_{holo^*}$ by backward planning, it will return a local path connecting these two states for the corresponding narrow passage. Finally, with the local trajectories through each narrow passage, we can use RRT again to find the paths connecting the start point and end point of all these trajectories sequentially within $\mathbb{S}_{holo^*}$. Thus we have a global path satisfying the nonholonomic constraints. $C_g,C_s, Env,robot$ $ l{global} $\ *Initialization*: KAT $\leftarrow C_g,C_s, Env,robot$ $l_{holo}$ $\leftarrow$ RRT$(C_g,C_s, Env,robot)$ $C_{nar}$ $\leftarrow$ NarrowPoints$(l_{holo},Env,robot)$ $c_{nar,i}$ $\leftarrow$ MaxMargin$(c_{nar,i},Env,robot)$ $s_{nar,i}$ $\leftarrow$ $c_{nar,i}$, EscapeVelocity$(c_{nar,i},Env)$ $l_{local,i} \leftarrow$ planFB$(s_{nar,i},Env) $ $ l_{global} $ $\leftarrow$ RRTConnectLocalPath$($ all $l_{local,i}) $ $ l_{global} $ Planning in Holonomic Space with White-listed RRT ------------------------------------------------- The planner begins by sampling in the holonomic space using the RRT algorithm [@lavalle2006planning]. The purpose of planning in holonomic space is to gain information about the direction and possible poses for crossing the narrow passage. In conventional RRT, if the new sample is biased as the goal configuration, the nearest neighbor will be found from the entire explored tree structure. However, we found this algorithm could be inefficient, particularly when a narrow passage presents. For example, in Figure \[fig:wl\], the samples in this 2D environment would not be able to form a direct connection from the goal configuration to the nearest neighbor A. However, it should be able to connect with node B. We eliminate this kind of scenario by adding an additional feature called whitelisting on top of the RRT algorithm. It will keep a list of newly added nodes in the tree structure and make sure every node will only be tested once. Every time RRT samples the goal bias, instead of the entire explored tree structure, the white list will be used to find the nearest neighbor node for testing the goal connectivity and the tested node will be deleted from the it. This means node A in Figure \[fig:wl\] will only be tested its connectivity with the goal once and yield for other nodes after it fails. Following this pattern nodes like B will succeed much earlier during planning. $path$, $Env$ ,$robot$ $C_{nar}$\ *Initialization*: $C_{nar} \leftarrow \emptyset$, $ C'_{nar} \leftarrow \emptyset$ $C_{nar}$ Maximum Margin Sampling in Narrow Passage ------------------------------------------- Since the algorithm plans under delicate conditions, it is preferable to find a way to go through the narrow passage while staying as far from the obstacles as possible. In order to achieve this, each waypoint should be optimized to have the margin to the nearest obstacle maximized. Unfortunately, a smoothed holonomic path would tightly pass through obstacles and leave very little room to work with. The KAT algorithm resolves this problem with a maximum margin sampling scheme. After the RRT planning, KAT has found a collision-free holonomic trajectory. The next step is to identify the narrow points on this trajectory. This process is shown in Algorithm \[alg:narrowpoints\]. KAT first checks every point on the path and records those that have more than four 4-connected neighbors in $\mathbb{C}_{col}$. Since there may be many points around one narrow passage, KAT uses K-centroids clustering to adaptively select the cluster centers $c_{nar,i}^*$ identified for each narrow passage. The set $C_{nar}^*=\{c_{nar,i}^*\mid i=1,2,3,...,n_{nar} \}$ constitutes the hardest part of the trajectory. For each cluster center $c_{nar,i}^*$ generated from Algorithm \[alg:narrowpoints\], KAT will sample configurations uniformly in the plane perpendicular to the planned holonomic path $l_{holo}$, which is denoted as $\mathbb{C}_i^\perp=\{c|c - c_{nar,i}^* \perp l_{holo} \}$. For the sampled poses that are not in collision with an obstacle, we will find the node in them with the lowest objective function value. This objective function calculates the sum of squared distances between this collision-free pose and all of the in-collision poses. The objective function is formulated as: $$\begin{aligned} c_{nar,i} = \mathop{\arg\min}_{c \in \mathbb{C}_{free} \cap \mathbb{C}_i^\perp} \sum_{c_{col} \in \mathbb{C}_{col}}{(c - c_{col})(c - c_{col})^T } \end{aligned}$$ Since the value of the objective function is not sensitive to the points far away from the narrow passage, we can simplify the equation above by only considering the collision points near each narrow point. For instance, replace the constraint on $c_{col}$ from $c_{col} \in \mathbb{C}_{col}$ to $c_{col} \in \mathbb{C}_{col} \cap \mathbb{B}_i^\delta$, where $\mathbb{B}_i^\delta =\{c\mid ||c-c_{nar,i}^*||<=\delta\}$ and $\delta$ is a parameter related to the scale of the environment and robot. Fig.\[fig:grouped\](b) shows the location of this maximum margin sample derived from the narrow configurations in a passage. [.48]{} ![Four different stages of planning.[]{data-label="fig:grouped"}](holonomicratio.png "fig:"){width="\columnwidth"} [.48]{} ![Four different stages of planning.[]{data-label="fig:grouped"}](narrowP.png "fig:"){width="\columnwidth"} [.48]{} ![Four different stages of planning.[]{data-label="fig:grouped"}](atCenterRatio.png "fig:"){width="\columnwidth"} [.48]{} ![Four different stages of planning.[]{data-label="fig:grouped"}](globalratio.png "fig:"){width="\columnwidth"} Escape Velocity ---------------- In this step, we will complete the narrow point $c_{nar,i}$ in configuration space to a narrow state $s_{nar,i}$ by appending a translation velocity $v_t$ to it while setting the angular velocity to zero. If $v_t$ is the velocity with the minimum norm that can lead the robot through the narrow passage using the forward and backward planning algorithm described in the next part, it is called the escape velocity, denoted by $v_{escape}$. KAT will deduce the direction of $v_{escape}$ by taking the weighted mean of a set of direction vectors, shown in Algorithm \[alg:escapevelocity\]. Then the $v_{escape}$ will be fully determined by the forward and backward planning in the next part. The algorithm first samples a direction set $D = \{d_i|d_i =(x_i,y_i,z_i), |d_i| = 1, i =1,2,3...n\}$ uniformly distributed on the unit sphere $\mathbb{S}^2$. By reusing the cluster center $c_{nar,i}^*$, we can find a heuristic direction $d_{nar,i}$ for each narrow passage. This is to first find the point $c_{nar,i}^{n}$ which is the nearest to $c_{nar,i}^*$ on $l_{holo}$ by $$c_{nar,i}^{n} = \underset{c\in l_{holo}}{\text{arg\;min}} |c-c_{nar,i}^*|$$ and then identify the tangent direction at $c_{nar,i}^{n}$ by $$d_{nar,i}=\frac{\text{d}l_{holo}}{\text{d}t_{nar,i}}, \quad l_{holo}(t_{nar,i}) =c_{nar,i}^{n}$$ Next, for each $d_i$ in $D$, if $\langle d_i,d_{nar,i} \rangle<=0$, it will be removed from $D$. After constructing $D$, a length $t_i$ is generated for each $d_i$ following a normal distribution $N(\mu_{nar},\sigma_{nar})$, where $\sigma_{nar}$ and $\mu_{nar}$ should be selected according to the property of the environment, and translate the robot from the narrow point $c_{nar,i}$ by each pair of $(t_i, d_i)$ to $c_{nar,i}^t$. If $c_{nar,i}^t \in C_{col}$, delete $(d_i)$ from $D$. Finally, if $D$ is not empty, the direction of the escape velocity will be calculated as $$v_{escape,i}^* = \frac{\sum_{d_i \in D}d_it_i}{ \left| {\sum_{d_i \in D}d_it_i} \right| }$$ $c_{nar}$, $Env$ ,$robot$ $v_{escape}^*$\ *Initialization*: $D= \{d_i|d_i\in \mathbb{S}^2, i =1,2,3...n\}$ $v_{escape}^* $ Forward and Backward Planning With Controller --------------------------------------------- This step finds local dynamically feasible paths through each narrow passage by exploiting $c_{nar,i}$ and $v_{escape,i}^* $ deduced above. If such paths can be found, the bottlenecks of the planning are solved since the remaining task is to connect the starts and ends of each local path to form a global path. The algorithm here will generate $s_{nar,i}$ by adding $v_{escape,i}$ to $c_{nar,i}$. The direction of $v_{escape,i} $ is determined by $v_{escape,i}^* $ and its norm increases each time. For each generated $s_{nar,i}$ the algorithm will use a forward controller $C_f:(\mathbb{S},\mathbb{S})\rightarrow \mathbb{U}$ and a time-inverse controller $C_b:(\mathbb{S},\mathbb{S})\rightarrow \mathbb{U}$ to stabilize $s_{nar,i}$ respectively. This process should satisfy $$\begin{aligned} &s_{t+1} = g(s_t,u_t)\Delta t + s_t\\ &u_t=\begin{cases} C_f(s_t,s_{still}),\quad t>t_{nar,i}\\ C_b(s_t,s_{still}),\quad t<=t_{nar,i} \end{cases} \end{aligned}$$ for $\forall t \in [0,T_{nar,i}]$, and $\exists t_{nar,i}\in [0,T_{nar,i}]$ such that $s_{t_{nar,i}} = s_{nar,i}$. If $s_0,s_{T_{nar,i}} \in \mathbb{S}_{holo^*}$ and $s_t\in \mathbb{S}_{free},\forall t \in [0,T_{nar,i}]$, the path $l_{nar,i}(t)=s_t,t\in [0,T_{nar,i}]$ is a local dynamically feasible path through the corresponding narrow passage with both start and end points in $\mathbb{S}_{holo^*}$. If the algorithm can find a local passage $l_{local,i}$ for every $c_{nar,i}$ on $l_{holo}$, the global path $l_{global}$ will be constructed from connecting the start points and end points of these local paths sequentially. Since the start and end of each $l_{local,i}$ are in $\mathbb{S}_{holo^*}$, the algorithm will use RRT to plan the path by sampling in $\mathbb{S}_{holo^*}$ without restricted by the nonholonomic constraints. Experiments and Results ======================= This section describes the dynamics model that is used for our experiment, results from different sections of the KAT algorithm as well as the final path generated for different environment settings. Our algorithms were implemented in Python with Openrave. All experiments were executed on a laptop with an Intel(R) Core(TM) i7-5600U at 2.6GHz, 8GB of RAM. Each experiment ran until a trajectory was found, or 1 minutes had elapsed. We performed 30 trials for each experiment and removed the fastest and slowest. The video of the whole computational processing is shown in [@algovideo1] Dynamic model ------------- We exploit the quadcopter dynamics model and controller developed in [@loianno2017estimation]. The configuration of the drone model is shown in Figure \[fig:quad\]. The inertia frame with axes $e_1$,$e_2$,$e_3$ is a reference frame attached to the ground; the body frame with axes $b_1$,$b_2$,$b_3$ is a frame attached to the drone. The dynamics of the drone is described by the following equations: $$\begin{aligned} &\dot{x} = v,\quad \dot{v} = mge_3-fRe_3 \\ &\dot{R} = R\hat{\Omega}, \quad J\dot{\Omega}+\Omega\times J \Omega=M \\ &\begin{bmatrix} f\\M_1\\M_2\\M_3 \end{bmatrix} = \begin{bmatrix} 1&1&1&1\\ 0 & -d & 0 & d\\ d&0&-d&0\\ -c&c&-c&c \end{bmatrix} \begin{bmatrix} f_1\\ f_2\\ f_3\\ f_4 \end{bmatrix} \end{aligned}$$ where $m\in \mathbb{R}$ is the total mass of the quadcopter; $J\in\mathbb{R}^{3\times3}$ is the inertia matrix with respect to the drone frame; $R\in SO(3)$ is the rotation matrix from the body-fixed frame to the inertial frame; $\Omega\in\mathbb{R}^3$ is the angular velocity in the drone frame; $x\in\mathbb{R}^3$ is the position of the center of mass in the inertial frame; $v\in\mathbb{R}^3$ is the velocity of the center of mass in the inertial frame; $d\in\mathbb{R}$ is the distance from the each axis of the rotor to the drone center; $f_i\in\mathbb{R}$ is the thrust of the $i$th propeller; $f\in\mathbb{R}$ is the total thrust; $\tau_i\in\mathbb{R}$ is the torque applied to the drone by $i$th rotor along $i$th axis; $M\in\mathbb{R}^3$ is the moment vector in the body frame. The controller is modified from [@loianno2017estimation]. The forward controller is $$\begin{aligned} M_f = &-k_\Omega e_\Omega+\Omega \times J \Omega \\ &+Y(k_{zv}R^\top\hat{\omega}^\top-k_zR)e_3, \\ f_f = & (-k_ve_v+mge_3) \cdot Re_3 \end{aligned}$$ and the backward controller is $$\begin{aligned} M_b = &k_\Omega e_\Omega-\Omega \times J \Omega \\ &-Y(k_{zv}R^\top\hat{\omega}^\top-k_zR)e_3, \\ f_b = & (k_ve_v+mge_3) \cdot Re_3 \end{aligned}$$ where $$Y = \begin{bmatrix} 0&1&0\\ 1&0&0\\ 0&0&0 \end{bmatrix}$$ The desired translational velocity and angular velocity are set to zero; $e_\Omega$, $e_v$ are the angular velocity error and translational velocity error; $k_\Omega$, $k_{zv}$, $k_z$, $k_v$ are control parameters;$M_f$, $f_f$, $M_b$, $M_b$ are the total moment and force of forward control and backward control respectively. Experiment Setup ---------------- The experiment setting is shown in Figure \[fig:rrt\]. The start configuration and goal configuration are at the different sides of the wall; all paths connecting the two sides contain a narrow passage on the wall, which is an inclined hole and the quadcopter cannot reach the goal without passing it. Experiment Result ----------------- The holonomic path found by the RRT algorithm is shown in Figure \[fig:rrt\]. Table \[compholo\] shows the comparison of conventional RRT with white-list RRT on the same environment setting. By limiting the computation time to 1 minute, the conventional RRT has a slightly lower success rate than modified RRT. In addition, the white-listed RRT finishes in less time and with fewer sampled nodes. [c||c|c|c|c]{} & **Time&** ------------- **Sampled\ **Nodes**** ------------- : Computation time of holonomic sampling[]{data-label="compholo"} & ------------ **Success\ **Rate**** ------------ : Computation time of holonomic sampling[]{data-label="compholo"} & **Path Length\ **RRT & 23.3s & 12197 & 100% & 9.05\ **Modified RRT & 17.7s & 9648 & 100% & 9.03\ ****** This holonomic path identifies a collision-free trajectory that can connect the goal and start, but violates the dynamics of the quadcopter. One can easily find that the quadcopter should fly at a low speed in order to be able to make a sharp turn near the hole, which conflicts with the need of a high speed pass for the inclined hole. By sampling and clustering adjacent points of each node on the path, the algorithm identifies the narrow passage and refines the configuration for passing, as illustrated in Figure \[fig:max\]. From Figure \[fig:max\], we can see the refined point allows the quadcopter to leave a safe margin from the wall. The next step is to sample forward and backward to generate a feasible path passing the narrow passage while making it possible to connect the start and end nodes with a holonomic path. The local path built here is shown in Figure \[fig:local\]. Figure \[fig:analysis\] provides an analysis of this process. $F$ is the thrust generated by each propeller; $v_t$ is the translational velocity; $Z$ is the angle between $b_3$ and $e_3$ defined in Figure \[fig:quad\]; $t_{nar}$ is the time when the quadcopter passes the narrow passage. The control input saturates when $t = t_{nar}$, because the feedback error reaches its maximum, which is the difference between the instantaneous state and the near-holonomic state. This is very different from planning the path from one side to the other. At the last step, KAT connects the end and start points of the local path with the corresponding nearest nodes in the holonomic path and returns the result, as shown in Figure \[fig:final\]. For testing the effectiveness of KAT, we derived a more complex environment setup. By setting up two obstacle walls, each with a different window opening angle, we proved that KAT could connect multiple aggressive trajectories. Figure \[fig:wholetra\] shows the finished trajectory for passing these two obstacles. In [@algovideo1], the video demonstrates the entire planning process and simulated execution of the planned trajectory. We also changed the opening angle on both walls and analyzed how it would affect the results. Table \[comptable\] shows that as the opening gets steeper, the maximum speed that KAT has to sample rises. As the opening becomes near-vertical, the maximum speed of the trajectory will be over 10 meters per second. ------------------------------------------------------------------ -- -- -- **Opening & **Computation Time& **Max Velocity & **Success Rate\ **Rotation ($^{\circ}$) & (second) & (m/s) & (%)\ 0 & 17.1 & 0.3 & 100.0\ 30 & 23.2 & 2.1 & 100.0\ 50 & 32.5 & 6.4 & 89.47\ 85 & 52.4 & 15.2 & 70.0\ ********** ------------------------------------------------------------------ -- -- -- : Average result of 30 trials for dual-obstacle setup[]{data-label="comptable"} Discussion ========== Compared to other sampling-based planners, the advantage of KAT is using the dual-direction control scheme to generate local paths around narrow passages, saving a large amount of computation time from sampling in the high dimensional state space. This innovation reduces the time complexity dramatically and can be extended to other similar motion planning problems, where a particular subset of the problem poses a much higher challenge than others. Although the KAT algorithm has proven to be able to successfully generate aggressive flying patterns in a simulated environment, it could conceivably encounter some difficulties during implementation in real world environments because the execution of a trajectory will always subject to drift in practice. A robust controller might be required to resolve this issue and make implementation more feasible. Future work on KAT may include generalizing the concept of near-holonomic set on other robotic systems, and designing an evaluation method of the success rate of the local aggressive paths. Conclusion ========== In this work, we have proposed the KAT path planning algorithm for systems with nonholonomic constraints. The KAT is aimed to solve planning problem where aggressive maneuver is required to pass the narrow passages. The algorithm reduces the computation cost significantly by first identifying the states allow the robot to pass the narrow passages and then planning the local path exploiting the dual-direction control scheme. In the simulation, KAT can efficiently plan a quadcopter through two walls with tilted holes, showing it is a effective planner for aggressive trajectories. [^1]: Manuscript created September 14, 2017.(Yaohui Guo and Zhaolun Su are co-first authors.) [^2]: $^{1}$Y. Guo is with the Robotic Institute at the University of Michigan, Ann Arbor [yaohuig@umich.edu]{} [^3]: $^{2}$Z. Su is with the Department of Electrical and Computer Engineering at the University of Michigan, Ann Arbor [zhsu@umich.edu]{} [^4]: $^{3}$D. Berenson is with the Department of Electrical and Computer Engineering and the Robotic Institute at the University of Michigan, Ann Arbor [berenson@eecs.umich.edu]{} [^5]: $^{4}$D. Zhao is with the Department of Mechanical Engineering and Robotic Institute at the University of Michigan, Ann Arbor ([corresponding author: zhaoding@umich.edu]{})

--- abstract: | Magnetic spinels (with chemical formula $AX_{2}$O$_{4}$, with $X$ a 3$d$ transition metal ion) that also have an orbital degeneracy are Jahn-Teller active and hence possess a coupling between spin and lattice degrees of freedom. At high temperatures, MgV$_{2}$O$_{4}$ is a cubic spinel based on V$^{3+}$ ions with a spin $S$=1 and a triply degenerate orbital ground state. A structural transition occurs at T$_{OO}$=63 K to an orbitally ordered phase with a tetragonal unit cell followed by an antiferromagnetic transition of T$_{N}$=42 K on cooling. We apply neutron spectroscopy in single crystals of MgV$_{2}$O$_{4}$ to show an anomaly for intermediate wavevectors at T$_{OO}$ associated with the acoustic phonon sensitive to the shear elastic modulus $\left(C_{11}-C_{12}\right)/2$. On warming, the shear mode softens for momentum transfers near close to half the Brillouin zone boundary, but recovers near the zone centre. High resolution spin-echo measurements further illustrate a temporal broadening with increased temperature over this intermediate range of wavevectors, indicative of a reduction in phonon lifetime. A subtle shift in phonon frequencies over the same range of momentum transfers is observed with magnetic fields. We discuss this acoustic anomaly in context of coupling to orbital and charge fluctuations. This is a pre-print of our paper at <https://link.aps.org/doi/10.1103/PhysRevB.96.184301>, © 2017 American Physical Society. author: - 'T. Weber' - 'B. Roessli' - 'C. Stock' - 'T. Keller' - 'K. Schmalzl' - 'F. Bourdarot' - 'R. Georgii' - 'R. A. Ewings' - 'R. S. Perry' - 'P. Böni' bibliography: - 'mvo.bib' title: 'Transverse acoustic phonon anomalies at intermediate wavevectors in MgV$_{2}$O$_{4}$' --- \[sec:intro\] Introduction ============================ Magnetically frustrated and orbitally degenerate materials are of high interest to study the coupling between the lattice, spin, and orbital degrees of freedom. [@Tokura03; @Millis98; @Rudolf07] One important class are the spinels, i.e. minerals possessing the chemical formula [$\textrm{A}^{2+}\textrm{B}^{3+}_2\textrm{X}^{2-}_4$]{} where A and B are divalent and trivalent metallic cations, respectively, and in most cases X are oxygen ions. The anions form a face-centred cubic lattice with 32 ions in the unit cell. The interstices of the close-packed structure consist of 8 tetrahedral and 16 octahedral sites which – in a normal spinel – are occupied by the smaller A and B cations, respectively (Fig. \[fig:spinel\]). The ions occupying the tetrahedral sites constitute a diamond lattice and the ions in the octahedral sites form a corner-sharing pyrochlore lattice, which is an archetype for geometrically frustrated magnetic systems. [@Anderson56] ![\[fig:spinel\] Conventional unit cell of a spinel. In an [$\textrm{A}\textrm{B}_2\textrm{O}_4$]{} spinel, the A metal ions are in a tetrahedral $\textrm{AO}_4$ environment that constitutes a diamond lattice, the B metal ions form edge-sharing $\textrm{BO}_6$ octahedra arranged in a pyrochlore lattice. ](spinel){width="1\columnwidth"} In spinel vanadates – [$\textrm{A}^{2+}\textrm{V}^{3+}_2\textrm{O}^{2-}_4$]{} – the vanadium ion is situated in a octahedral environment surrounded by oxygen ions. The crystal field of the oxygen ions splits the fivefold degenerate $d$ orbitals into $e_g$ and the $t_{2g}$ states with the $e_g$ orbitals lying higher than the $t_{2g}$ orbitals. The vanadium ions possess two $3d$ electrons that are distributed over the three degenerate $t_{2g}$ orbitals, namely $d_{xy}$, $d_{xz}$, and $d_{yz}$. This distribution of electrons results in one hole which can occupy any of the three degenerate $t_{2g}$ orbitals resulting in a three-fold orbital degeneracy. This orbital occupancy introduces an additional spin-orbit coupling term to the Hamiltonian. [@McClure59:9; @Abragam:book] Because of the cooperative Jahn-Teller effect [@Kaplan:book; @Gehring1975] for materials with an orbital degeneracy, a structural phase transition accompanied by orbital ordering [@Radaelli2005; @Pandy11] sets in for [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} below a temperature $T_{OO}$. For $T < T_{OO}$ one of the crystallographic axes – which is conventionally taken along $c$ – compresses with $c/a=0.9941 <1$, lowering the point group symmetry at the V sites from cubic to tetragonal. The compression along a single axis corresponds to the $C_{11}-C_{12}$ stability condition of the elastic constants $C_{ij}$ for cubic crystals [@Cowley76]. Consequently, the triplet degeneracy of the $t_{2g}$ orbital is partially lifted. One of the vanadium electrons occupies the energetically lower-lying $d_{xy}$ orbital, while the other electron is shared among the degenerate $d_{xz}$ and $d_{yz}$ orbitals. The $d_{xy}$ orbitals on the pyrochlore lattice order in a way that their lobes point to neighbouring vanadium ions in the $ab$ plane, along $\left[110\right]$ and $\left[1\overline{1}0\right]$ directions [@Tcherny04]. The occupation of the $d_{xz}$ and $d_{yz}$ orbitals is an unresolved problem arising from the competition of three different interaction mechanisms [@Khomskii2014]. Namely the coupling of the orbital to the elastic strain via the cooperative Jahn-Teller effect, the Kugel-Khomskii interaction between neighbouring vanadium ions, and the spin-orbit coupling. Depending on which interaction is assumed to dominate, different orbital ordering patterns are obtained. A prominent theoretical model, given by Tsunetsugu and Motome [@Tsunetsugu03], is based on the Kugel-Khomskii interaction between the vanadium ions, and results in a ferro-orbital (antiferro-orbital) ordering of the $d_{xy}$ ($d_{xz}$, $d_{yz}$) orbitals along the \[110\] (\[101\], \[011\]) directions in the pyrochlore lattice. Because this theory has been found to predict a symmetry for the structural lattice which is incompatible with measurements [@Wheeler10], a different model was devised by O. Tchernyshyov [@Tcherny04], treating the spin-orbit interaction to be dominating, followed by the Jahn-Teller, and – on the weakest scale – the Kugel-Khomskii interaction. It predicts an orbital ordering where one of the electrons on each vanadium ion occupies a complex superposition of $d_{xz}$ and $d_{yz}$. For both models [@Tsunetsugu03; @Tcherny04], the spin of the vanadium electron in the $d_{xy}$ orbital orders antiferromagnetically along chains in the $\left[110\right]$ and $\left[1\overline{1}0\right]$ directions on the pyrochlore lattice. In the $bc$ and $ac$ planes, an up-up-down-down spin pattern forms below the Néel temperature $T_N < T_{OO}$. The magnetic moments point along the $c$ axis of the crystal and a strong (weak) magnetic coupling is obtained for perpendicular (parallel) chains. In the model by Tchernyshyov [@Tcherny04], geometric frustration of the spins in the perpendicular chains on the pyrochlore lattice gives rise to two (plus two time-reversed) degenerate ground states. Neutron inelastic scattering and diffraction experiments by Wheeler *et al.* [@Wheeler10] found a mixture of both of these models, a real and a complex superposition of the $d_{xz}$ and $d_{yz}$ orbitals, to best reproduce their experimental data for [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{}. The interplay between lattice and spin/orbital ordering was studied in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} by Watanabe *et al.* [@Watanabe14] using ultrasound measurements. They found that the $C_{11}$ and $C_{44}$ elastic constants show a softening region for cooling towards $T_{OO}$ and a large discontinuity at $T_{OO}$, which indeed suggests that there is a strong coupling between the orbitals of the Jahn-Teller ions and the lattice strain. In addition, Watanabe *et al.* [@Watanabe14] found a sensitivity of $C_{11}$ and $C_{44}$ on external magnetic fields up to 7 T in the \[110\] direction for the softening region near $T > T_{OO}$ in contrast to $\left(C_{11}-C_{12}\right)/2$, which does not depend on field. In this work we present results of inelastic neutron scattering experiments above and below the orbital-ordering temperature $T_{OO}$ that yield the dispersion of the acoustic phonon branches in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} thus complementing the $q \rightarrow 0$ results of Watanabe *et al.* [@Watanabe14] Unlike ultrasound measurements which probe acoustic fluctuations in the $lim_{q \rightarrow 0}$ and Raman spectroscopy which is a strictly $q=0$ probe, neutron spectroscopy can investigate excitations associated with a uniform lattice deformation at all wavevectors, and hence wavelengths, over the entire Brillouin zone. We focus our measurements on the acoustic phonons and not the optical phonons previously reported to exist for energies above $\sim$ 10 meV using optical techniques. [@Popovic03; @Jung08] We will show that there exists a range of acoustic phonon wavelengths associated with the $(C_{11}-C_{12})/2$ elastic constant where the TA mode both softens in energy and also increases in linewidth indicative of decreased lifetime. This paper is divided into three main sections including this introduction. We first describe the neutron scattering experiments studying the low energy acoustic phonons where the softening in energy over intermediate wavevectors and then the linewidth broadening indicative of a shortening of phonon lifetimes is described. We finally conclude with a discussion comparing our results to theories for soft modes in Jahn-Teller systems and also a comparison between the anomalies observed here and in metallic systems with acoustic instabilities. \[sec:exp\] Experiments ========================= \[sec:char\] Sample characterisation -------------------------------------- Three cylindrical single-crystals of [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{}, each of about 2 cm height and 0.75 cm diameter were grown using a mirror image furnace. Characterisation was performed using bulk susceptibility measurements and neutron diffraction confirming the presence of both a magnetic and structural transition. The bulk measurements of the heat capacity and the magnetic susceptibility show well defined structural and antiferromagnetic transitions at temperatures of $T_{OO} \approx 63 \pm 1 \,\textrm{K}$ and $T_N \approx 42\,\textrm{K}$, respectively, demonstrating the high quality of the crystals. Using neutron diffraction on the single-crystals, the space group was confirmed to be face-centred cubic $\mathrm{Fd\bar{3}m}$ above $T_{OO}$ and tetragonal $\mathrm{I4_{1}/a}$ below $T_{OO}$ [^1]. At the neutron diffractometer MIRA [@MIRA; @MIRAnew], the temperature ($T$) dependence of the lattice constants was measured using $\theta$-$2\theta$ scans around the (220) Bragg reflections of the single-crystals. The volume of the unit cell was determined to be $V \approx 593\,{\mbox{\normalfont\AA}}^3$. The very weak $T$-dependence of $V$ (Fig. \[fig:lattice\]) is compatible with the results of Refs. . ![ \[fig:lattice\]$(a)$ Temperature-dependent change of the unit cell volume $V$ as determined using neutron diffraction. Note that the error bars include the instrument resolution as systematic error. $(b)$ The intensities of the $\vec{Q}$=(110) and the $\vec{Q}$=(004) Bragg reflections as a function of temperature illustrating the structural transition at T$_{OO}$=63 K and magnetic ordering temperature of T$_{N}$=42 K. The data in panel $(b)$ was taken using the RITA spectrometer in two-axis mode with E$_{i}$=5 meV. The (004) Bragg peak was measured at the (002) position using $\lambda/2$ with the scattered Be filter removed. Note that the intensity of (110) has been magnified by a factor of 10 relative to the (004) data.](lattice){width="1\columnwidth"} \[sec:energies\] Phonon dispersion ----------------------------------- We first describe the softening in energy of the acoustic fluctuations in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{}. Inelastic neutron scattering experiments were performed using the triple-axis spectrometer (TAS) EIGER [@Eiger2017] at the spallation source SINQ [@Fischer97] at the Paul Scherrer Institute (PSI) in Villigen, Switzerland and the TAS IN22 [@Regnault1999] at the Institute Laue-Langevin (ILL) in Grenoble, France. Both spectrometers were operated in the constant final energy mode with $E_f = 14.68\,\mathrm{meV}$. At EIGER and IN22, the dispersion of the acoustic phonon branches was measured using a vertically focusing monochromator and horizontally focusing analyser without using any collimation. For the EIGER measurements, crystals oriented in the $hk0$ and the $hhl$ planes were used, collecting data around the $(400)$ and $(440)$ Bragg peaks. For IN22 we used the $hk0$ scattering plane. In addition, experiments at small reduced momentum transfer $q$ were conducted using the cold TAS option of MIRA [@MIRA; @MIRAnew] at the MLZ in Garching, Germany, using neutrons with fixed incident energies $3.78\,\mathrm{meV} < E_i < 4.97\,\mathrm{meV}$ yielding an energy resolution (full-width at half-maximum, FWHM) of $0.11\,\mathrm{meV} < \Delta E < 0.17\,\mathrm{meV}$. At MIRA, a vertically focusing monochromator was combined with a flat analyser. No collimation was installed in the incident and scattered beams. Higher order neutrons were removed by means of a cooled beryllium filter. At MIRA, we used the $hk0$ scattering plane and collected data in the $(220)$ Brillouin zone. Furthermore, we employed the time-of-flight spectrometer (TOF) MERLIN [@MERLIN] at the Rutherford Appleton Laboratory in Harwell, UK. MERLIN uses a multi repetition-rate chopper, where we selected incident energies of $E_i = 24\,\mathrm{meV}$ and $E_i = 49\,\mathrm{meV}$ and used a Fermi chopper frequency of 350 Hz. Data was collected with the sample rotation covering the full range of $90^{\circ}$ around the $(440)$ Bragg reflection and the crystal oriented in the $hhl$ plane. Analysis was performed using the software HORACE [@Horace]. The primary goal of our experiments was to establish the dispersion of the TA1, TA2, and LA phonon modes. Note that in our notation, the elastic constant $C_{44}$ is related to the sound velocity of the twofold degenerate transverse mode TA1 propagating along a $[001]$-direction with a polarisation along $[100]$ or $[010]$. $\left(C_{11}-C_{12}\right)/2$ corresponds to the TA2-mode propagating along $[110]$ with a polarisation along $[1\overline{1}0]$. $\left(C_{11}+C_{12}+2C_{44}\right)/2$ is the sound velocity of the LA-phonon propagating along $[110]$. Owing to the neutron cross section and selection rules associated with phonon eigenvectors [@Harada70], sensitivity to both the TA1 and TA2 phonon modes is obtained with triple-axis measurements when the crystal is aligned in the $hk0$ scattering plane while alignment in the $hhl$ plane only affords measurements of the TA1 phonon. Time of flight measurements using chopper spectrometers allow momentum transfers out of the horizontal scattering plane to be measured and therefore afford sensitivity to both TA1 and TA2. Typical data of TA1 and TA2 phonons are shown in Fig. \[fig:scans\] for temperatures $T = 10$ K (tetragonal phase) and $T = 80$ K (cubic phase). In panels (a) and (b) example $(4q0)$ (TA1) and $(4-q, 4+q, 0)$ (TA2) phonon data from IN22 are shown. Panels (c) and (d) depict typical $(44q)$ (TA1) and $(4-q, 4+q, 0)$ (TA2) slices from MERLIN. The TA1 mode does not show any $T$-dependence while the TA2 mode becomes softer in energy in the high temperature cubic phase. We selected the example $q$ values shown in Fig. \[fig:scans\] as they are in the region where the effect is strongest and where the peaks can be clearly separated from other contributions, e.g. incoherent scattering. Fig. \[fig:energies\] summarises the results of all phonon measurements that were conducted at various temperatures in the range $10$ K $\le T \le 200$ K. A transverse optic phonon branch is visible at $E \approx 20\,\mathrm{meV}$ near the zone boundary. Attempts using both triple-axis and time of flight measurements failed to track this mode closer to the zone centre. ![\[fig:scans\] Typical data for transverse phonons TA1 and TA2 are shown for $T = 10$ K and $T = 80$ K. There is a clear softening of the TA2 mode in the cubic phase when compared with the low temperature tetragonal phase. ](samplescans){width="1\columnwidth"} The dispersion of the LA and TA1 phonon branches do not show appreciable changes between the cubic and the tetragonal phases. In contrast, the TA2 branch exhibits an observable $T$-dependence. At intermediate $q$-values, the dispersion shows a “spoon-like” anomalous behaviour, i.e. the phonons soften in energy when entering the cubic phase for $T > T_{OO}$. While the data is suggestive of a softening in energy of the TA2 phonon near the zone centre, the effect is not as large as at intermediate wavevectors. The effect at small $q$ close to the zone centre $\Gamma$ is also less pronounced than the $q \rightarrow 0$ behaviour reported by Watanabe *et al.* [@Watanabe14] using ultrasound sensitive to acoustic fluctuations on the MHz timescale. {width="100.00000%"} The detailed temperature dependence of two TA2 phonons at $q = \left(\overline{0.075}\ 0.075\ 0\right)$, $q = \left(\overline{0.3}\ 0.3\ 0\right)$, and $q=\left( \overline{\frac{1}{2}} \frac{1}{2} 0 \right)$ are shown in Fig. \[fig:detailscan\]. Both the modes at small and large $q$ soften by ca. 0.1 meV and 0.4 meV, respectively, when entering the cubic phase. For $T_{OO} < T < 200 $ K, the low-$q$ phonons harden by ca 0.2 meV, while the large-$q$ modes further soften by ca. 0.2 meV. ![\[fig:detailscan\] The dispersion of the TA2 phonons softens and hardens at $q = (\overline{0.075}\ 0.075\ 0)$ (top) and $q = (\overline{0.5}\ 0.5\ 0)$ (bottom), respectively, when approaching the cubic-tetragonal phase transition near $T_{OO} = 63$ K from high temperatures. For $q = (\overline{0.3}\ 0.3\ 0)$ (middle) there is no change in phonon energies in the $T > T_{OO}$ region. Please note that at MIRA we measured using neutron energy gain ($-E$) for reasons of analyser efficiency and angular constraints at the instrument. The physics of the system is unaffected by this choice. ](q075 "fig:"){width="1\columnwidth"} ![\[fig:detailscan\] The dispersion of the TA2 phonons softens and hardens at $q = (\overline{0.075}\ 0.075\ 0)$ (top) and $q = (\overline{0.5}\ 0.5\ 0)$ (bottom), respectively, when approaching the cubic-tetragonal phase transition near $T_{OO} = 63$ K from high temperatures. For $q = (\overline{0.3}\ 0.3\ 0)$ (middle) there is no change in phonon energies in the $T > T_{OO}$ region. Please note that at MIRA we measured using neutron energy gain ($-E$) for reasons of analyser efficiency and angular constraints at the instrument. The physics of the system is unaffected by this choice. ](q3 "fig:"){width="1\columnwidth"} ![\[fig:detailscan\] The dispersion of the TA2 phonons softens and hardens at $q = (\overline{0.075}\ 0.075\ 0)$ (top) and $q = (\overline{0.5}\ 0.5\ 0)$ (bottom), respectively, when approaching the cubic-tetragonal phase transition near $T_{OO} = 63$ K from high temperatures. For $q = (\overline{0.3}\ 0.3\ 0)$ (middle) there is no change in phonon energies in the $T > T_{OO}$ region. Please note that at MIRA we measured using neutron energy gain ($-E$) for reasons of analyser efficiency and angular constraints at the instrument. The physics of the system is unaffected by this choice. ](35_45_0 "fig:"){width="1\columnwidth"} \[sec:linewidths\] Phonon linewidths -------------------------------------- Having established the presence the softening over intermediate wavevectors of the TA2 acoustic phonon, the linewidth indicative of the phonon lifetime is now discussed. We have determined the linewidth of phonons by analysing the inelastic scattering data from EIGER and IN22 using the dynamical structure factor given by Ref. which obeys detailed balance required for neutron cross sections $$\begin{gathered} \label{eq:Sqw} S\left(q,E\right) = \frac{S_0}{\left|A\cdot\sin\left(x\cdot q\right)\right| \cdot\pi} \cdot \left[ \frac{1}{\exp \left(E / k_BT\right) - 1} + 1 \right] \cdot \\ \left(\frac{\pm\Gamma_p} {\left(E\mp \left|A\cdot\sin\left(x\cdot q\right)\right| \right)^{2}+\Gamma_p^{2}}\right).\end{gathered}$$ Here, the dispersion of the TA2 branch is modeled with a simple sine function that is appropriate to describe acoustic phonons. The lineshape of the phonons is approximated with a Lorentzian function. The reduced momentum transfer is given by $q = 2\pi\sqrt{(h^2 + k^2)/a^2 + l^2/c^2}$, where $a$ and $c$ are the lattice constant defining the tetragonal unit cell. The term in square brackets is the Bose occupation factor for temperature $T$ and energy $E$. $\Gamma_p$ denotes the half-width at half-maximum of the phonon, $S_0$ is an overall scaling factor. For $\Gamma < q < \mathrm{K}$ the TA2 dispersion at low temperature (Fig. \[fig:energies\]) can be approximated by $A = \left( 22.5 \pm 0.1 \right) \, \mathrm{meV}$ and $x = \left(0.825 \pm 0.004 \right)$. Here, $A$ and $x$ serve as scaling factors for the energy and reduced momentum of the dispersion, respectively. $S(q,E)$ as given by Eq. (\[eq:Sqw\]) was convoluted with the Eckold-Sobolev resolution function [@Eckold2014] of a TAS spectrometer using Monte-Carlo integration. This novel algorithm was used because it reproduces the resolution function of focusing TAS better than the algorithm of Ref. . Fitting the data was performed using Minuit’s simplex minimiser [@Root2011]. A full description of the software tool *Takin* that was developed by some of the authors and used for the convolution fits can be found in Refs. . The $q$-dependence of $\Gamma_p$ of the TA2 branch is shown in Fig. \[fig:linewidths\_q\]. While $\Gamma_p$ is essentially $p$ independent at 10 K, it attains a maximum about half-way from the $\Gamma$-point to the K-point at 80 K. The maximum occurs at the $q$-position where we also observe the strongest softening of the TA2 phonon branch (Fig. \[fig:energies\]). No appreciable changes were observed for the other acoustic phonon branches or for the linewidth of the TA2 phonons in the tetragonal phase. ![\[fig:linewidths\_q\] Linewidth $\Gamma_p$ of the TA2 phonon branch. In the tetragonal phase at 10 K (blue triangles), $\Gamma_p$ is independent of momentum transfer. In the cubic phase at $T = 80$ K (red points), $\Gamma_p$ attains a maximum halfway from the $\Gamma$ to the $\textrm{K}$ point.](lw_80_10_Gamma){width="1\columnwidth"} The temperature dependence of the TA2 phonons at $q = (\overline{\frac{1}{2}} \frac{1}{2} 0)$ and $q = (\overline{0.4}\ 0.4\ 0)$ are shown in Fig. \[fig:linewidths\_T\]. $\Gamma_p$ increases when warming the sample from $T = 10\,\mathrm{K}$ towards the Néel-transition at $T_{N} \approx 40\,\mathrm{K}$. Here $\Gamma_p$ seems to saturate at $\approx 200\, \mathrm{\upmu eV}$ before increasing again to a maximum value $\Gamma_p \approx 450\, \mathrm{\upmu eV}$ in the vicinity of the structural phase transition $T_{OO} \approx 63\,\mathrm{K}$. ![\[fig:linewidths\_T\] Temperature dependence of the TA2 phonons at $Q = (3.5\ 4.5\ 0)$ using conventional TAS-spectroscopy (red dots) and $Q = (3.6\ 4.4\ 0)$ using TRISP (blue triangles). A subtle plateau can be identified between $T_N$ and $T_{OO}$. $\Gamma_p$ increases to approximately 0.45 meV in the cubic phase where it remains constant up to $T = 200$ K.](lw_3_5_Gamma){width="1\columnwidth"} Note that the linewidth at $q = (\overline{0.4}\ 0.4\ 0)$ was determined with high accuracy by means of the neutron-resonance spin-echo (NRSE) technique at the triple axis spectrometer TRISP at MLZ [@TRISP]. TRISP was set up in a negative-negative-positive scattering configuration. The NRSE coils were aligned such that the focusing condition for the TA2 branch was fulfilled. The data was corrected for resolution effects reducing polarisation [@Habicht03; @Habicht04]. These are caused by the sample mosaic and the slope of the phonon dispersion. \[sec:fields\] Field dependence of phonon energy -------------------------------------------------- We finally present the magnetic field dependence of the transverse acoustic phonon lifetimes and energies in the same wavevector region where temperature dependent anomalies are observed. While a magnetic field response has been reported for polar phonons in spinels based on optical data [@RudolfA07], magnetic field effects on the acoustic phonon response have been reported in superconductors and also in metals where the Fermi surface topology is relatively flat in momentum space [@Pynn74]. Attempts on semiconducting materials have failed to observe any effect. [@Comes81] The field dependence of $\Gamma_p$ of the TA2 phonon at $q=\left(\overline{0.4}\ 0.4\ 0 \right)$ was determined for [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} by application of a magnetic field $B_{[001]}$ along the $[001]$-direction (Fig. \[fig:field\_001\]). The phonon energy $E$ is independent of $B_{[001]}$ in the cubic phase while $E$ decreases subtly at $T = 40$ K in the tetragonal phase above 7 T by $\approx 0.2$ meV up to the highest magnetic field achievable in this experiment. The results are suggestive of a slight decrease in energy of the acoustic phonon at high magnetic fields at intermediate wavevectors. Further measurements to higher fields and also in other orbitally degenerate spinels would be helpful for establishing this effect. ![\[fig:field\_001\] Panel $(a)$ shows the field-dependence of the TA2 phonon at the $q$-position $\left(\overline{0.4}\ 0.4\ 0\right)$. Panels $(b)$–$(d)$ depict selected phonon data. A small softening of $\approx 0.2$ meV is observed at high fields in the tetragonal phase for $T = 40$ K as shown by blue triangles in panel $(a)$ and in panel $(d)$.](fieldscans){width="1\columnwidth"} \[sec:disc\] Discussion ========================= By measuring the phonon dispersion along the high-symmetry directions $[0\xi0]$ and $[\overline\xi\xi0]$ using neutron scattering, we have demonstrated an anomaly in the TA2 branch along the $\langle\xi\xi0\rangle$-directions associated with the structural and orbital phase transition in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} at $T_{OO}$ (Fig. \[fig:energies\]). An energy softening at intermediate wavevectors mid-way between the $\Gamma$- and the K-point is observed with increasing temperature. This softening leads to a spoon-like dispersion of the TA2 branch. We first consider the possibility of thermal expansion [@Maradudin62] of the lattice as an explanation for the acoustic phonon anomalies in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{}. A change in the unit cell volume would result in a change in the size of the Brillouin zone and, assuming a fixed zone boundary energy, would give a corresponding shift in phonon velocity. This cannot explain our data for two reasons. First, the phonon anomaly is only observable in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} over a range of intermediate wavevectors and no change in the $lim_{q\rightarrow 0}$ slope is observed using neutron scattering on the THz timescale. Second, the change in phonon energy is larger then would be expected from such an effect and is inconsistent with the thermal expansion data presented in Fig. \[fig:lattice\] and also from the higher resolution x-ray data for the unit cell volume presented in Ref. . Therefore, the anomaly observed here is not due to thermal changes in the lattice constants. The phonon anomaly in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} measured with neutron scattering differs from the response observed with ultrasound [@Ishikawa03]. At $T_{OO}$, Watanabe *et al.* [@Watanabe14] reported a temperature dependent change of the sound velocity of the TA2 mode $\Delta v \approx 25\,\%$. We find no observable change for momentum transfers near the zone centre, however measure a spoon-like anomaly for intermediate wavevectors. We speculate that the difference between the neutron low-$q$ data and the ultrasound data in the $lim_{q \rightarrow 0}$ originates from the differing energy and lengthscales of the two measurements. Owing to temporal resolution, neutron scattering measure fluctuations on the THz timescale while ultrasound measures the dynamic response for long wavelengths typically on the MHz timescale. Similar differences between neutron scattering and ultrasound have previously been reported for spinels in Ref. . The neutron scattering results do not reflect an observable softening of the TA2 phonon on the THz timescale, rather an anomaly over a range of intermediate wavevectors of $\Delta v \approx \left(12 \pm 5\right) \,\%$. It is interesting to compare our results with theoretical predictions of soft acoustic phonons near a Jahn-Teller distortion. Considering the Jahn-Teller effect and coupling to acoustic phonons, a theoretical model for the phase transition from the tetragonal to the cubic phase is given in papers by Pytte [@Pytte71; @Pytte73]. In the first paper, Pytte presents a theory which models the temperature dependence of the elastic constants as a result of the coupling between the elastic strain tensor and the twofold degenerate $d$ orbitals. Including higher-order anharmonic coupling, the $T$-dependence of the transverse acoustic phonon mode propagating along $[110]$ with polarisation along $[1\overline{1}0]$ – the tetragonal shear elastic mode – shows a softening with decreasing temperature, a step discontinuity at the transition temperature $T_{OO}$, followed by an increase of the velocity of sound for further decreasing temperature. The second paper [@Pytte73] includes the coupling of the degenerate orbitals to the optical phonon modes and finds a “central peak” in the dynamical structure factor $S\left(q,\omega\right)$. Central peaks have been reported to often co-occur with soft modes and had first been identified in $\mathrm{SrTiO_3}$ [@Riste71] and $\mathrm{Nb_3Sb}$ [@Shirane71; @Axe73]. They are also discussed theoretically in Refs. . The theory developed by Pytte predicts temperature dependent changes to the TA2 phonons (with velocity proportional to $C_{11}-C_{12}$) and this is the same mode which displays anomalies in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} over intermediate wavevectors. However, there are several important differences which make the relevance of this theory to the case here questionable. The theoretical prediction is in the context of a two-fold orbital degeneracy associated with the partial occupancy of the $e_{g}$ orbitals. As outlined above, this is not the case for S=1 V$^{3+}$ where two electrons occupy the $t_{2g}$ orbitals resulting in a three-fold orbital degeneracy. Pytte [@Pytte71] also predict a true soft mode whose energy reaches zero at the phase transition for momenta $q \rightarrow 0$. A general theory for transitions driven by orbital degeneracy by Elliott [@Elliott1977] and Young [@Young75] also reach the same conclusion with regards to the prediction of a soft mode close the zone centre. Experimentally, this would be reflected in the neutron response by a softening of the TA2 acoustic phonon branch and is more compatible with the results from ultrasound and also in agreement with neutron scattering experiments studying the low energy acoustic phonons in Jahn-Teller active PrAl$_{3}$ [@Birgeneau74]. In contrast to these predictions and previous experimental examples for Jahn-Teller driven structural distortions, we observe an anomaly at finite-$q$ over a limited intermediate range in wavevector. Another discrepancy concerns the central peak at $E=0$ [@Pytte73], which we do not observe at finite $q$. The strong increase in zone centre Bragg scattering upon cooling into the tetragonal phase, which had previously been reported by Wheeler *et al.* [@Wheeler10], may be instead related to the central peak, however it appears at a different $q$ than the acoustic phonon anomaly. Based on these comparisons with theory and experiment, it is difficult to associate the TA2 phonon anomaly in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} with a soft mode due to a proximate Jahn Teller distortion resulting in breaking of the ground state orbital degeneracy and a soft acoustic mode in the $lim_{q\rightarrow 0}$. The simultaneous broadening and softening of an acoustic phonon over a range in wavevector transfers is indicative of coupling to some other degree of freedom (see for example discussion of coupling in Ref. ). Coupling of acoustic phonons to crystal field [@Bruhl78] driven distortions have been experimentally studied with examples being the TA2 phonon anomalies reported in DyVO$_{4}$ and TbVO$_{4}$. [@Melcher72; @Sandercock72] Similar coupling effects have also been discussed in UO$_{2}$. [@Allen68] These examples involve the coupling of an acoustic phonon to a crystal field level with a similar energy scale to that of the phonon and this is also predicted to be required in the case of magnetic pyrochlores. [@Yamashita00] While the $dd$ transitions in $3d$ transition metal ions are large ($\sim$ 1-2 eV) [@Haverkort07; @Kant08; @Schooneveld12; @Kim11; @Cowley13] making them unlikely to be involved with this process, the lower energy spin-orbit split levels may have a more appropriate energy scale. As discussed by Tchernyshyov [@Tcherny04], the ground state of V$^{3+}$ derived by diagonalising the spin-orbit Hamiltonian is a $j_{eff}$=2 quintuplet which is separated by an energy scale of $\sim$10 meV [@Abragam:book] to an excited $j_{eff}$=1 triplet state. The splitting of such degenerate ground state due to a local molecular field could provide the spin-orbit crystal field level with the correct energy scale to couple to the acoustic phonon. However, we emphasise that such a coupling in the framework of the theories described above would involve a true softening of the fluctuations near the Brillouin zone centre inconsistent with the phonon anomalies in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{}. Another possibility is coupling between the acoustic phonons and low energy magnon excitations from the V$^{3+}$ S=1 sites. No magnon excitations were observed to cross the phonon branch in our experiments, however, it should be noted that the comparatively large momentum transfers optimised for phonon measurements are where magnetic form factors ensure reduced neutron scattering cross sections from magnons. But, such magnetic excitations have been reported to be highly dispersive [@Wheeler10], gapped [@Gleason2014], and located in reciprocal space away from the low-energy acoustic lattice fluctuations, making coupling in the context of the theories discussed above likely weak. While full magnon dispersion curves from neutron scattering are limited in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{}, neutron measurements in ZnV$_{2}$O$_{4}$ [@Lee04] observe a significant gap of $\sim$ 10 meV at the magnetic zone centre while we observe anomalies at a displaced momentum and at lower energies. Based on the energy scale in comparison to analogous systems and the different momentum transfer away from either the zone boundary or centre, we conclude that coupling to magnons is unlikely. It is interesting to compare the results in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} to metallic systems with acoustic instabilities. Our results show similarities to the measurements by F. Weber *et al.* [@Weber2011] who determined the dispersion of the TA-phonons in a metallic manganite. In this material, a charge and orbital ordering (COO) sets in at low temperatures. Above $T_{COO}$, F. Weber *et al.* [@Weber2011] report a softening of the TA phonon at $q = \left( \frac{1}{4} \overline{\frac{1}{4}} 0 \right)$ by $\Delta E \approx 0.25 \ \mathrm{meV}$ accompanied by an increase in linewidth $\Delta \Gamma_{HWHM} \approx 0.25 \,\mathrm{meV}$. Similar anomalies have been reported in other metallic manganites on decreasing temperature. [@Hoesch2013] The anomaly in these example manganites is associated with electron-phonon coupling for wave vectors associated with Fermi surface nesting, given the metallic electronic response. Indeed, similar intermediate acoustic phonon anomalies have been reported in a number of materials where there is a coupling between lattice and charge degrees of freedom. One example is the one-dimensional conductor TTF-TCNQ [@Shirane76] which displays a softening of a longitudinal acoustic phonon over intermediate wavevectors near a Peierls transition and also in superconductors where the change in linewidth has been related to the onset of a gap in the electronic quasiparticle response [@Shapiro75; @Weber2008; @Weber2014; @Keller06; @Bullock57]. The wave vector at which the acoustic mode is unstable in these examples is determined by the electronic Fermi surface and hence the metallic properties. While the analogy between soft acoustic modes in metallic systems is not obvious given that [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} is an insulator, there are several properties that indicate that charge fluctuations may be playing a role in this spinel. There is evidence that [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{}, ZnV$_{2}$O$_{4}$, and CoV$_{2}$O$_{4}$ are proximate to an insulator-metallic transition under pressure evidenced through both calculations [@Canosa07] and also thermodynamic and transport measurements [@Kism11]. This has led to theoretical studies indicating that both [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} and ZnV$_{2}$O$_{4}$ contain large charge fluctuations and should be considered in a partially delocalised regime. [@Kato12] Therefore, both theory and experiment are suggestive of electronic or charge fluctuations in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} and the possible coupling between the acoustic phonons and the charge channel is further corroborated by our magnetic field measurements which would alter the chemical potential. [@Maitra07] While [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} is an insulator, the similarity between the acoustic phonon anomaly presented above and the studies in metallic systems discussed here support the notion of charge fluctuations in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{}. The similarity between the [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} phonon anomalies and metallic systems discussed above is reminiscent of the Peierls transition in one dimensional systems, with TTF-TCNQ discussed above being an example. This analogy may be appropriate for [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} [@Tcherny02] given the one dimensional magnetic V$^{3+}$ chain interactions which have been measured using neutron spectroscopy [@Wheeler10]. Similar one dimensional magnetic correlations have been reported in ZnV$_{2}$O$_{4}$ [@Lee04] and also in MgTi$_{2}$O$_{4}$ [@Schmidt04], both of which have a triply orbital degenerate ground state. Supporting this possibility further, spin dimerisation has been reported in the spinel CuIr$_{2}$S$_{4}$ [@Radaelli2002] where both Ir$^{3+}$ and Ir$^{4+}$ are present. Given this similarity to other spinels and the one dimensional magnetic correlations, an instability towards dimerisation resulting from an “anti-Jahn Teller" distortions has been predicted to apply to [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} and other orbitally degenerate V-based spinels, [@Khomskii05] as well as the Verwey transition in Fe$_{3}$O$_{4}$ [@Shapiro75]. The T$_{2}$ acoustic phonon anomalies we observe in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} are along the same direction as the strong magnetic exchange resulting from orbital ordering resulting from the Jahn Teller distortion, but we emphasise the softening and dampening is most prominent in the high temperature cubic phase. While the analogy with the Peierls transition in one dimensional systems might be compelling, to our knowledge there is no report of structural dimerisation in either ZnV$_{2}$O$_{4}$ or [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{}. We speculate that the acoustic phonon anomaly in [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} may indicate the close proximity to such a structural phase without the actual formation of long-range structural order. It may even be indicative of spatially localised structural distortions that would increase the acoustic phonon lifetime in a similar manner to that reported in disordered piezoelectrics. [@Stock12] Supporting this, we note that additional structures are known to compete with the low temperature tetragonal phase. [@Suchomel12] In conclusion, we identify a softening and damping of the TA2 phonon modes over a range of intermediate wavevectors. Neutron measurements of the acoustic phonons at low momentum transfers do not show any observable softening on the THz timescale in contrast to expectations of a uniform softening of the mode due to a change in the elastic constants. The combined linewidth broadening and also softening indicates a coupling between the acoustic TA2 phonon and another degree of freedom, analogous to electron-phonon coupling observed in metallic compounds with a structural instability. The results are suggestive of coupling to charge fluctuations predicted by theory. We thank R. Schwikowski and A. Mantwill for technical support. Financial support from the EPSRC is gratefully acknowledged. This work is based on experiments performed at the Swiss spallation neutron source SINQ, Paul Scherrer Institute (PSI), Villigen, Switzerland. The project has received funding from the European Union’s Seventh Framework Programme for research, technological development and demonstration under the NMI3-II Grant number 283883. The authors gratefully acknowledge the financial support provided by JCNS to perform the neutron scattering measurements at the Institute Laue-Langevin (ILL), Grenoble, France. This work is furthermore based upon experiments performed at the TRISP instrument operated by MPG and the MIRA instrument at the Heinz Maier-Leibnitz Zentrum (MLZ), Garching, Germany. Experiments at the ISIS Pulsed Neutron and Muon Source were supported by a beamtime allocation from the Science and Technology Facilities Council. This work was part of the Ph.D. thesis of T. Weber [@PhDWeber]. [^1]: Wheeler *et al.* [@Wheeler10] has suggested the space groups of [[$\textrm{Mg}\textrm{V}_2\textrm{O}_4$]{}]{} to be $\mathrm{F\bar{4}3m}$ (cubic) and $\mathrm{I\bar{4}m2}$ (tetragonal) based on the measurement of weak Bragg peaks which are absent in the space group $\mathrm{Fd\bar{3}m}$

--- author: - 'Chenggang Yan, Biao Gong, Yuxuan Wei, and Yue Gao, ' bibliography: - 'IEEEabrv.bib' - 'paper.bib' title: | Deep Multi-View Enhancement Hashing for\ Image Retrieval --- the explosive growth of image data, the efficient large-scale image retrieval algorithms such as approximate nearest neighbour (ANN) search [@arya1998optimal; @gong2011iterative; @zhang2014supervised] which balances the retrieval time-consuming and retrieval efficiency on the large-scale dataset attract increasing attentions. In the field of large-scale image retrieval, learning-to-hash [@7915742] is a kind of emerging and highly efficient ANN search approach which aims to automatically learn optimal hash functions and generate image hash codes. The nearest neighbor is obtained by calculating the Hamming distance of these hash codes. ![Illustration of the key idea of D-MVE-Hash. The core part of D-MVE-Hash called stability evaluation is used to obtain view-relation matrix in step1. Dotted arrows indicate the path of back propagation. The memory network is used to memorize the view-relation matrix which is calculated by the stability evaluation method during training. In hash codes generation step (e.g., step2), we directly use the memory network to produce the view-relation matrix. After feeding the generated multi-view and basic binary codes with the view-relation matrix to following fusion and concat layers, D-MVE-Hash outputs fusion binary representation of images for retrieval.[]{data-label="graph:mmrnet"}](graphmmrnet.pdf) In recent years, several convolution artificial neural network hash methods [@liu2012supervised; @zhen2016spectral; @weiss2009spectral; @cao2017hashnet; @cao2018deep] which show significant improvement in the image retrieval task have been proposed. CNNH [@xia2014supervised] is a typical representative of these methods. Parallel to that, some impressive researches, such as [@lai2015simultaneous] propoesd by Lai *et al.* which learns a well image representation tailored to hashing as well as a set of hash functions, highly improve the retrieval precision and let the efficient large-scale image retrieval enter the next stage. However, these methods only focus on learning optimal hash functions from single-view data (i.e., obtaining image features from a single convolution feature space). Recently, a number of multi-view hashing methods [@shen2015multi; @liu2015multiview; @zhang2017semi; @xie2017dynamic] which mainly depend on spectral, graph and deep learning techniques to achieve data structure preserving encoding have been proposed for efficient similarity search. In a general sense, the hash method with multiple global features is so-called multi-view hashing. Other than this, the multi-view is also well known as multiple angles cameras of 3D model [@feng2018gvcnn]. For heterogeneous multimedia features such as multi-feature hashing of video data with local and global visual features [@song2013effective; @8744407] and other orthogonal or associated features, researchers use multi/cross-modal hashing to solve the complexity of the fusion problem of multiple modalities. It is disparate from the multi-view hashing. In this paper, we define the multi-view as multi-angle representations in different visual feature spaces. The supplement of additional information is one of the most prominent contribution for these multi-view methods. Recently, an increasing number of researches explicitly or implicitly capture the relations among different views [@shen2016semi; @jia2019deep] and modalities [@hu2018collective; @xu2017learning] to enhance the multi/cross-view/modal approaches. Compared to these methods, the view-relation matrix we proposed can directly be employed in original feature view spaces. It avoids information missing during the process of subspace mapping resulting in a better capture of relations. Furthermore, our approach optimizes the feature extraction network by both objective function and view relation calculation process to obtain the best multi-view features which are used to calculate the matrix. Our motivation to design deep multi-view enhancement hashing (D-MVE-Hash) arises from two aspects: (1) By splicing and normalizing the fluctuations of different image features under multiple feature views, we obtain the volatility matrix which regulates the view relevance based on fluctuation strength to produce a quality view-relation matrix. The view-independent and robustness are the special properties of this view-relation matrix. We have conducted detailed ablation experiments in Sec. \[sec:exper3\] to support this conclusion. (2) Since typical multi-view hash image retrieval algorithm still needs to manually extract image features [@xie2017dynamic], which makes the multi-view information and the deep network less impactful in learning-to-hash and image retrieval. For this reason, we want to design an end-to-end architecture with data fusion methods which are carried out in the Hamming space. In this work, a supervised multi-view hashing method called D-MVE-Hash is proposed for accurate and efficient image retrieval based on multiple visual features, multi-view hash and deep learning. Inspiring by multi-modal and cross-modal hashing methods [@Jiang_2017_CVPR; @hu2018collective], we excavate multiple view associations and make this potential knowledge dominant to explore the deep integration of the multi-view information. To this end, we design a view-independent and robustness view-relation matrix which is calculated from the continuously optimized features of each view to better capture the relation among views. Fig. \[graph:mmrnet\] and Fig. \[graph:newframwork\] are the illustration of the key idea and model framework. To cope with learning-to-hash, we retain the relaxation and constraint methods similar to [@zhu2016deep; @cao2017hashnet; @chen2018deep]. D-MVE-Hash automatically learns the optimal hash functions through iterative training. After that, we can achieve high-speed image retrieval using the image hash codes which are generated by the trained D-MVE-Hash. In summary, the main contributions of our work are two-fold: - We propose a general flexible end-to-end multi-view feature enhancement framework for image retrieval. This framework can fuse arbitrary view data combination with a unified architecture by fusion methods. The effectiveness of the framework is mainly empowered by a view-independent and robustness view-relation matrix using the fluctuations of different image features under multiple feature views. - Without loss of generality, we comprehensively evaluate the proposed method on three different datasets and implement detailed ablation experiments for endorsing the properties of our D-MVE-Hash. Extensive experiments demonstrate the superiority of the proposed D-MVE-Hash, as compared to several state-of-the-art hash methods in image retrieval task. Related Work ============ Approximate Nearest Searching with Hashing ------------------------------------------ The image retrieval mentioned in this paper refers to content-based visual information retrieval [@banerjee2015using; @8662712; @8361043]. This process can be simply expressed as: feeding the unlabeled original images to a deep net architecture or other retrieval methods to get images which are similar or belong to the same category of the inputs. It is the typical similarity searching problem. The similarity searching of multimedia data such as collections of images (e.g, views) or 3D point clouds [@Qi_2017_CVPR; @biyao2020] usually requires the compression processing. Similarity searching (or proximity search) is achieved by means of nearest neighbor finding [@arya1998optimal; @zhu2019eff; @zhu2019PR]. Hashing is an efficient method for nearest neighbor search in large-scale data spaces by embedding high-dimensional feature descriptors into a similarity preserving Hamming space with a low dimension. In this paper, we do research on the supervised hash learning algorithm [@DBLP:conf/cvpr/ShenSLS15; @DBLP:journals/tip/ShenSSHTS15] which uses tagged datasets. Compared with unsupervised hashing algorithm [@DBLP:journals/pami/ShenXLYHS18] such as LSH [@gionis1999similarity], KLSH [@kulis2009kernelized] and ITQ [@gong2012iterative], supervised methods could obtain more compact binary representaion and generally achieve better retrieval performance [@cao2017hashnet; @cao2018deep]. LSH [@gionis1999similarity] implements a hash retrieval by generating some hash functions (which is called the family $\mathcal{H}$ of functions) with local sensetive properties and applying some random choice methods to transform the input image and construct a hash table. KLSH’s [@kulis2009kernelized] main technical contribution is to formulate the random projections necessary for LSH [@gionis1999similarity] in kernal space. These sensible hashing methods show that the local clustering distance of the constrained binary code plays an important role in the overall optimization. ITQ [@gong2012iterative] finds a rotation of zero-centered data so as to minimize the quantization error of mapping this data to the vertices of a zero-centered binary hypercube. KSH [@liu2012supervised] maps the data to compact binary codes whose Hamming distance are minimized on similar pairs and simultaneously maximized on dissimilar pairs. All these traditional methods reveal the value of hash learning in retrieval. However, since the deep neural network have shown strong usability in various fields, researchers began to consider introducing the advantages of neural network and convolution operation into hash learning and proposed deep hash learning. Hashing with CNN ---------------- Deep learning based image retrieval methods have shown superior performance compared with the methods using traditional handcraft descriptors. The convolutional neural network [@lecun2015deep] is one of the most famous and efficacious deep learning based image processing method for various tasks such as image retrieval and classification. Recently, introducing deep learning into hashing methods yields breakthrough results on image retrieval datasets by blending the power of deep learning [@lecun2015deep]. CNNH [@xia2014supervised] proposed by Xia *et al.* is the first end-to-end framework that used the pairwise similarity matrix and deep convolutional network to learn the hash functions. Xia *et al.* propose a deep convolutional network tailored to the learned hash codes in $H$ and optionally the discrete class labels of the images. The reconstuction error is minimized during training. In [@lai2015simultaneous], Lai *et al.* make the CNNH [@xia2014supervised] become an “one-stage” supervised hashing method with a Triplet Ranking Loss. Zhu *et al.* [@zhu2016deep] proposed the DHN which uses a pairwise cross-entropy loss $L$ for similarity-preserving learning and a pairwise quantization loss $Q$ for controlling hashing quality. HashNet [@cao2017hashnet] attacks the ill-posed gradient problem in optimizing deep networks with non-smooth binary activations by continuation method. DMDH [@chen2018deep] transforms the original binary optimization into differentiable optimization problem over hash functions through series expansion to deal with the objective discrepancy caused by relaxation. Multi-view Hashing {#sec:relamultivihash} ------------------ The poor interpretability of deep learning makes it difficult to further optimize the model in a target manner. We have noticed that some multi-view-based hashing methods [@chen2018collaborative; @xie2017dynamic; @zhang2011composite; @liu2015multiview; @shen2015multi; @zhang2017semi] have emerged in recent years. This type of method processes images by traditional means and extracts various image features for enhancing the deep learning hashing method. The dispersion of hash codes is one of the reasons why such operations are effective. Since the data form among features is very different, it can be seen as viewing images from different views. The observation of an object should be multi-angled and multi-faceted. Whereas a quantity of multi-view hashing method emphasizes the importance of the multi-view space, ignoring the role of convolution and the relationship among views. Using kernel functions and integrated nonlinear kernel feature is a common way to achieve that, and we can use a weighting vector to constrain different kernels. However, the weighting vector which can gather relationships between views is often pre-set or automatically optimized as a parameter, which is not reasonable enough. In [@zhang2011composite], Zhang *et al.* used graphs of multi-view features to learn the hash codes, and each view is assigned with a weight for combination. Kim *et al.* proposed Multi-view anchor graph hashing [@kim2013multi] which concentrates on a low-rank form of the averaged similarity matrix induced by multi-view anchor graph. In [@shen2015multi], Shen *et al.* set $ \mu_m $ measures the weight of the $ m $-th view in the learning process. Multiview Alignment Hashing (MAH) [@liu2015multiview] seeks a matrix factorization to effectively fuse the multiple information sources meanwhile discarding the feature redundancy. Xie *et al.* proposed Dynamic Multi-View Hashing (DMVH) [@xie2017dynamic] which augments hash codes according to dynamic changes of image, and each view is assigned with a weight. Based on the above, we conducted in-depth research. Deep Multi-View Enhancement hashing {#sec:dmvhhash} ===================================  In this section, the details of our proposed D-MVE-Hash are introduced. We first briefly introduce the definition of multi-view problems and give the detailed description of MV-Hash. Following, we discuss the enhancement process and propose three different data fusion methods. In the last part of the section, We introduce the joint learning and the overall architecture of D-MVE-Hash. Problem Definition and MV-Hash {#sec:dmvhhash1} ------------------------------ Suppose $\mathbf{O}=\{o_i\}^N_{i=0}$ is a set of objects, and the corresponding features are $\{\mathbf{X}^{(m)}=[x^{(m)}_1,\cdots,x^{(m)}_N] \in \mathbb{R}^{d_m \times N} \}^M_{m=1}$, where $d_m$ is the dimension of the m-th view, $M$ is the number of views, and $N$ is the number of objects. We also denote an intergrated binary code matrix $\mathbf{B}=\{b_i\}^N_{i=1} \in \{-1,1\}^{q \times N} $, where $b_i$ is the binary code associated with $o_i$, and $q$ is the code length. We formulate a mapping function $\mathcal{F}(\mathbf{O})=[F_1(\mathbf{X}^{(1)}),\cdots,F_M(\mathbf{X}^{(M)})]$, where the function $F_m$ can convert a bunch of similar objects into classification scores in different views. Then, we define the composition of the potential expectation hash function $\varphi :\mathbf{X} \rightarrow \mathbf{B}$ as follow: $$\begin{aligned} \varphi(\mathbf{X})=[ & \varphi_1(\mathcal{E}(\mathcal{F}(\mathbf{X}_1,\cdots,\mathbf{X}_N)),\mathbf{X}^{(1)}), \cdots ,\\ & \varphi_m(\mathcal{E}(\mathcal{F}(\mathbf{X}_1,\cdots,\mathbf{X}_N)),\mathbf{X}^{(m)})] \end{aligned},$$ Before starting stability evaluation, we pre-train each view network in the tagged dataset for classification task. Using the following loss function: $$\label{equ:clsloss} \mathcal{L}^p(x, i) = -\log\frac{exp(x[i])}{\sum_j exp(x[j])}.$$ The criterion of Eq \[equ:clsloss\] expects a class index in the range \[0, class-numbers\] as the target for each value of a 1D tensor of size mini-batch. The input $x$ contains raw, unnormalized scores for each class. For instance, $x[0]$ is outputted by the classifier to measure the probability that the input image belongs to the first class. $x[1]$ is the prediction score of the second class. $x[i]$ is the prediction score of the ground truth. Specific to image data, given $N$ images $\mathbf{I}=\{i_1,...,i_N\}$. Set $\mathbf{Q}=\mathcal{F}(\mathbf{I})$ in which $\mathcal{F}$ means the testing process. The dimension of $\mathbf{Q}$ is $M\times N\times C$, where $M$ is the number of views, $N$ is the number of images, $C$ is the number of classes. $Q_{mc}$, which is actually $Q_{mc\cdot}$, omits the third dimension represented by ‘$\cdot$’ and stands for a one-dimensional vector rather than a number. $\mathcal{E}(\mathcal{F})$ is as follow: $$\label{equ:core}\begin{aligned} \mathcal{E}_m(\mathbf{Q})= & +\sum_{m=1}^M \max_c \sqrt{\mathcal{V}(\mathbf{Q}_{mc})}\\ & -\frac{1}{N} \sum_{c=1}^C \sqrt{\mathcal{V}(\mathbf{Q}_{mc})} \end{aligned},$$ and the $\mathcal{V}(\mathbf{Q}) = \frac{1}{N} \sum_{n=1}^N(\mathbf{Q}-\mu)$, where the $\mu$ is arithmetic mean. $\mathcal{E}$ is expressed as $[\mathcal{E}_1,\cdots,\mathcal{E}_M]$. Then we do a simple normalization of $\mathcal{E}$: $$\log\{\mathcal{E}_m(\mathbf{Q})\}=\frac{\log (\mathcal{E}_m(\mathbf{Q})+\vert\min(\mathcal{E}(\mathbf{Q}))\vert +1)}{\log (\max (\vert\mathcal{E}(\mathbf{Q})\vert)+\vert\min(\mathcal{E}(\mathbf{Q}))\vert+1)}.$$ Then we consider training multi-view binary code generation network with view-relation information. At the beginning, we ponder the case of a pair of images $i_1,i_2$ and corresponding binary network outputs $b_1,b_2 \in \mathbf{B}$, which can relax from $ \{-1,+1\}^q $ to $[-1,+1]^q$ [@weiss2009spectral]. We define $y=1$ if they are similar, and $y=-1$ otherwise. The following formula is the loss function of the m-th view: $$\label{equ:7} \begin{aligned} \mathcal{L}_m(b_1, b_2,y)= - & (y-1)\ \max (a-\Vert b_1-b_2\Vert^2_2,0)\\+ & (y+1)\Vert b_1-b_2\Vert^2_2\\+ & \alpha (\Vert\vert b_1 \vert -1 \Vert_1 + \Vert\vert b_2 \vert -1 \Vert_1) \end{aligned},$$ where the $\Vert\cdot\Vert_1$ is the L1-norm, $\vert\cdot\vert$ is the absolute value operation, $\alpha>0$ is a margin threshold parameter, and the third term in Equation \[equ:7\] is a regularizer term which is used to avoid gradient vanishing. More generally, we have the image input $\mathbf{I}=\{i_1,...,i_N\}$ and output $\mathbf{B}^{(m)}=\{b_1^{(m)},...,b_N^{(m)}\}$ in the multi-view space. In order to get the equation representation in matrix form, we substitute $\mathbf{B}$ into $\mathcal{L}_m$ formula, then complement the regular term and similarity matrix to get following global objective function: $$\label{equ:9} \begin{array}{c}\vspace{0.1cm} \mathcal{L}(\mathbf{I},\mathbf{Y})=\\ \mathbf{Y}_{ij} \cdot \log\{\mathcal{E}_m(\mathcal{F}(\mathbf{I})\} \cdot (\rho(\mathbf{I})+\alpha\Vert\varphi(\mathbf{I})'-\mathbf{1}_{K_{N*q}}\Vert) \end{array}.$$ The matrix merged by $\varphi(\mathbf{I})$ is denoted as $\rho(\mathbf{I})$ refer to the form of the second term of multiplication in Equation \[equ:8\]. The $\delta(\cdot)$ in Equation \[equ:8\] is the $maximum$ operation. The $\oplus$ is inner product. $$\label{equ:8} \begin{aligned} \left[\begin{array}{c} y(b_i,b_j)=+1\\ y(b_i,b_j)=-1 \end{array}\right] \cdot \left[\begin{array}{c} \delta((\mathbf{B}\oplus\mathbf{B})\cdot(\mathbf{B}\oplus\mathbf{B})')\\ (\mathbf{B}\oplus\mathbf{B})\cdot(\mathbf{B}\oplus\mathbf{B})' \end{array}\right] \end{aligned}.$$ In order to show the effect of view stability evaluation in Equation \[equ:9\] intuitively, we rewrite the overall loss function as Equation \[equ:10\]. That is, we want to highlight the position of view-relation matrix $\mathbf{E}$ which is the output of $\mathcal{E}(\mathcal{F}(\mathbf{I}))$ in the overall loss function. $$\label{equ:10} \begin{aligned} \mathcal{L}(\mathbf{E},\mathbf{B})= \sum_{n=1}^N \sum_{m=1}^M \mathbf{E}_{nm} \sum_{i=1}^N \mathcal{L}_m(\mathbf{B}_n^{(m)},\mathbf{B}_i^{(m)},y) \end{aligned}.$$ With this objective function, the network is trained using back-propagation algorithm with mini-batch gradient descent method. Meanwhile, since the view-relation matrix $\mathbf{E}$ is directly multiplied by $\mathcal{L}_m$ in Equation \[equ:10\], the view relationship information can affect the direction of gradient descent optimization. Enhancement and Fusion ---------------------- The next stage is integrating multi-view and view-relation information into the traditional global feature in the Hamming space. Fig. \[graph:loss\] is the illustration of the data enhancement process which is actually divided into two parts. Firstly, we use some hash mapping constaint rules which are occur simultaneously in single-view and multi-view spaces to embed features into the identical Hamming space. Secondly, we propose replication fusion, view-code fusion and probability view pooling which are all carried out in the Hamming space to accomplish the multi-view data, view-relation matrix and traditional global feature integration. In Fig. \[graph:loss\], the solid and dotted line respectively indicate the strong and weak constraint. For example, as for the green points: 1) The distance constraint (i.e., the clustering) is stronger than $\pm 1$ constraint (i.e., the hash dispersion) when points are within the $\alpha$ circle. Therefore, we connect the green points by solid lines, and connect the green dots with $\pm 1$ circle by dotted lines; 2) The distance constraint is weaker than $\pm 1$ constraint when points are between the $\alpha$ circle and $\pm 1$ circle. Therefore, we connect the green points by dotted lines, and connect the green dots with $\pm 1$ circle by solid lines; 3) The distance constraint and $\pm 1$ constraint are both the strong constraints when points are outside the $\pm 1$ circle. Similar operations also occur on the red points. Compared to the green points, the clustering direction of the red points is opposite. \[sec:dmvhhash2\] ![Illustration of the data enhancement process.The left half of the figure shows the double-sample hash mapping constraint rules (e.g., Equation \[equ:7\]) which are used to embed features into the identical Hamming space.[]{data-label="graph:loss"}](graphloss.pdf) ### Replication Fusion {#sec:f1} Replication fusion is a relatively simple solution which relies on parameters. We sort $\mathbf{E}$ to find important views, and strengthen the importance of a view by repeating the binary code of the corresponding view in multi-view binary code. Specifically, the basic binary code is denoted as $\mathbf{B}$. The intermediate code (multi-view binary code) is denoted as $\mathcal{H}$. We set fusion vector $v$ to guide the repetition of multi-view binary code under various views. Equation \[equ:11\] represents the encoding process. $$\label{equ:11} \begin{aligned} \mathbf{H}= [\mathbf{B}\ ,\ \mathop{\phi}\limits_{j=1}^M(\mathop{\phi}\limits_{i=1}^M((\mathcal{H})_i,v_i\cdot \mathop{\mathcal{S}}\limits_{d=0}(\mathbf{E}))_j,1)] \end{aligned},$$ where $\mathbf{H}$ represents the input binary code of the fusion layer. $\phi(\cdot)$ from $1$ to $M$ is the self-join operation of the vector. The second parameter in $\phi(\cdot)$ represents the number of self-copying. $\mathcal{S}$ is a sort function in $d$ dimension. The advantage of this fusion method is that it can convert $\mathbf{E}$ into a discrete control vector, therefore $\mathbf{E}$ only determines the order between views. The strength of enhancement or weakening is manually controlled by fusion vector. ### View-code Fusion {#sec:f2} View-code fusion considers the most primitive view-relation matrix and the least artificial constraints. Specifically, we want to eliminate fusion vector which is used to ensure that the dimensions of input data are unified in Fusion-R because of the dynamic view-relation matrix. At first, the entire binary string $\mathbf{H}$ is encoded into head-code ($\mathbf{H}_h$), mid-code ($\mathbf{H}_m$), and end-code ($\mathbf{H}_e$). $\mathbf{H}_h$ is the same as Fusion-R. $\mathbf{H}_m$ directly uses the product of binary code length and the coefficient of corresponding view as the current code segment repetition time. This operation produces a series of vacant bytes (i.e., $\mathbf{H}_e$) which are not equal in length. Second, we assign a specific and different coden called view-code which is a random number belonging to $[-1, 1]$ in each view. Compared with $\mathbf{H}_m$, $\mathbf{H}_e$ uses view-code instead of multi-view binary code. So that it can be completely filled regardless of the dynamic view-relation matrix and code length. The advantage of view-code fusion is that it fully utilizes the information contained in view-relation matrix. We find that view-code fusion is limited by view stability evaluation in our experiments, which means it can exceed replication fusion when the number of views increases. ### Probability View Pooling {#sec:f3} We propose probability view pooling with view-relation matrix as a multi-view fusion method. Traditional pooling operation selects maximum or mean as the result of each pooling unit. The view pooling is a dimensionality reduction method which use element-wise maximum operation across the views to unify the data of multiple views into one view [@feng2018gvcnn]. Since pooling operation inevitably cause information loss, we need to expand the length of multi-view binary code to retain multi-view information as much as possible before probability view pooling. Then the view probability distribution is generated according to $\mathbf{E}$. In each pooling filter, a random number sampled from the view probability distribution activates the selected view. The code segment of this view is used for traditional pooling operation. It ensures that sub-binaries of high-priority views are more likely to appear during the fusion process. Joint Learning {#sec:optim} -------------- In this section, we introduce a multi-loss synergistic gradient descent optimization for the proposed model. D-MVE-Hash is optimized based on $\mathcal{L}^p$ and $\mathcal{L}^c$ at first. The former is apply for view stability evaluation, and the latter is used to extract the basic binary code. At this stage, our loss function is $\mathcal{L}^p + \mathcal{L}^c$. Then we use $\mathcal{L}$ to train the backbone of D-MVE-Hash (including fusion part) and use $\mathcal{L}^w$ which is explained in the last paragraph of this section to train the memory network. At this stage, our loss function is $\mathcal{L} + \mathcal{L}^w$. As a consequence, the formula for segment optimization is as follows: $$\begin{aligned} \min_\Theta\mathcal{L}=\left\{ \begin{array}{l}\vspace{0.1cm} \mathcal{L}_1=\mathcal{L}^p + \mathcal{L}^c\\ \mathcal{L}_2=\mathcal{L}\ \ +\mathcal{L}^w \end{array} \right. \end{aligned}.$$ With the purpose of avoid losing gradient, all segmentation maps are not allowed, therefore the output is controlled within $[-1,1]$ (by using the third item in original $\mathcal{L}$). Finally, the $sgn(\cdot)$ function is used to map the output to the Hamming space during testing, thus we can measure the Hamming distance of the binary code. We also introduce the memory network into the proposed D-MVE-Hash to avoid excessive computing resources of the view stability evaluation method. It is noted that the memory network is a simplification of the view stability evaluation method. The memory network only focuses on the original image input and the view-relation matrix. It can learn such transformation through multiple iterations during training. However, since we use pre-converted hash codes for retrieval, the complexity of the model structure and the time complexity of the retrieval are separate. This means D-MVE-Hash still maintains the inherent advantages of hashing which is a very important and definitely the fastest retrieval method since such Hamming ranking only requires O(1) time for each query. More specifically, the memory network learns the view-relation matrix $\mathbf{E}$ in setp1, and then in step2, we can get view-relation matrix $\mathbf{E}$ by this module without using stability evaluation method. The structure of memory network is a multi-layer convolutional neural network (e.g. VGG, ResNet, DenseNet, etc.), but its output layer is relative to view-relation matrix $\mathbf{E}$. And the loss function during training is $\mathcal{L}^w = \{l_1,\dots,l_N\}^\top$, $l_n = \left( I_n - \mathbf{E}_n \right)^2$. Fig. \[graph:mmrnet\] shows the different states and association of D-MVE-Hash between two steps. In general, we design such a structure mainly for engineering considerations rather than performance improvement. In our actual model training, whether or not the memory network is used does not have much influence on retrieval performance. Experiments {#sec:exper} =========== In this section, we provide experiments on several public datasets, and compare with the state-of-the-art hashing methods. Multi-view hashing methods is also within our scope of comparison. Following [@liu2015multiview], we obtain the 2D images multi-view information through RGB color space color histogram, HSV color space color histogram, texture [@ojala2002multiresolution], and hog [@dalal2005histograms]. The implementation of D-MVE-Hash is based on PyTorch 1.1.0 and scikit-image 0.13.1 framework. Each fully connection layer uses Dropout Batch Normalization [@srivastava2014dropout] to avoid overfitting. The activation function is ReLu and the size of hidden layer is $4096 \times 4096$. We use mini-batch stochastic gradient decent (SGD) with 0.9 momentum. The learning rate is 0.001. The parameter in Eqn.(\[equ:7\]): $a=2$ and $\alpha=0.01$, and parameter $v=(1,4,8,16)$. Following the standard retrieval evaluation method [@cao2018deep] in the Hamming space, which consists of two consecutive steps: (1) Pruning, to return data points within Hamming radius 2 for each query using hash table lookups; (2) Scanning, to re-rank the returned data points in ascending order if their distances to each query using continuous codes. Three image datasets are used to evaluate our approach: CIFAR-10 [@krizhevsky2009learning] ,NUS-WIDE [@nus-wide-civr09] and MS-COCO [@lin2014microsoft]. CIFAR-10 consists of 60,000 $32 \times 32$ color images in 10 classes, with 6,000 images per class. The dataset is split to training set of 50,000 images and testing set of 10,000 images. NUS-WIDE includes a set of images crawled from Flickr, together with their associated tags, as well as the ground-truth for 81 concepts for these images. We randomly sample 4,000 images as training set, 1,000 images as test query. MS-COCO is composed of 82,783 training images and 40,504 validation images, as well as 80 semantic concepts. $\mathbf{E}$ view1 view2 view3 view4 Time(h) mAP -------------- ------- ------- ------- ------- --------- -------- 0.954 0.7588 1.399 0.7513 1.413 0.7699 1.390 0.7551 1.374 0.7854 1.664 0.8275 1.659 0.8214 1.629 0.8244 1.639 0.8229 1.613 0.8268 1.659 0.8315 1.659 0.8258 1.657 0.8237 1.666 0.8319 1.672 0.8352 1.711 0.8401 : Ablation experiments (64 bits, +R, CIAFR-10). Time(h) is the time for training process (Batch-size 2). The GPU is NVIDIA GeForce GTX 1080Ti 1481-1582MHz 11GB (CUDA 9.0.176 cuDNN 7.1.2)[]{data-label="tab:ablation"} [c|cccc|cccc|cccc]{}\ & & &\ \ \ & 16 bits& 32 bits &48 bits & 64 bits & 16 bits& 32 bits &48 bits & 64 bits & 16 bits& 32 bits &48 bits & 64 bits\ \[-0.9ex\]\ KSH [@liu2012supervised] &0.4368 &0.4585 &0.4012 &0.3819 &0.5185 &0.5659 &0.4102 &0.0608 &0.5010 &0.5266 &0.5981 &0.5004\ MvDH [@shen2018multiview] &0.3138 &0.3341 &0.3689 &0.3755 &0.4594 &0.4619 &0.4861 &0.4893 &0.5638 &0.5703 &0.5912 &0.5952\ CMH [@chen2018collaborative] &0.4099 &0.4308 &0.4411 &0.4841 &0.5650 &0.5653 &0.5813 &0.5910 &0.5126 &0.5340 &0.5455 &0.6034\ CNNH [@xia2014supervised] &0.5512 &0.5468 &0.5454 &0.5364 &0.5843 &0.5989 &0.5734 &0.5729 &0.7001 &0.6649 &0.6719 &0.6834\ HashNet [@cao2017hashnet] &0.7476 &0.7776 &0.6399 &0.6259 &0.6944 &0.7147 &0.6736 &0.6190 &0.7310 &0.7769 &0.7896 &0.7942\ DCH [@cao2018deep] &0.7901 &0.7979 &0.8071 &0.7936 &0.7401 &0.7720 &0.7685 &0.7124 &0.8022 &0.8432 &0.8679 &0.8277\ AGAH [@gu2019adversary] &0.8095 &0.8134 &0.8195 &0.8127 &0.6718 &0.6830 &0.7010 &0.7096 &0.8320 &0.8352 &0.8456 &0.8467\ \[-0.7ex\]\ **MV-Hash** &0.5061 &0.5035 &0.5339 &0.5370 &0.6727 &0.6836 &0.7141 &0.7150 &0.5902 &0.5941 &0.6268 &0.6339\ **D-MVE-Hash (+P)** &0.7234 &0.7535 &0.7982 &0.7712 & &0.7271 &0.7281 &0.7317 &0.7358 &0.7782 &0.7924 &0.8062\ **D-MVE-Hash (+C)** &**0.8422** & & & &0.6906 & & & &**0.8552** &**0.9023** & &\ **D-MVE-Hash (+R)** & &**0.8336** &**0.8501** &**0.8401** &**0.7883** &**0.8002** &**0.8057** &**0.8073** & & &**0.8735** &**0.8892**\ \[tabl:cnnbasemapcomp\] Ablation Experiment for View-relation Matrix $\mathbf{E}$ {#sec:exper3} --------------------------------------------------------- In this section, we evaluate the D-MVE-Hash in the absence of views and view-relation matrix. As can be seen from the results, since we adopt dominant view relations rather than implication relations, preserving view relationships by generating partial views with the broken $\mathbf{E}$ is feasible. Our D-MVE-Hash does not degenerate into normal image hashing when multi-view information is partially removed. Another special property is robustness. For the purpose of that, we obtain the volatility matrix which regulates the view relevance based on fluctuation strength to produce a quality view-relation matrix. Aditionaly, we use the mathematics (Equation \[equ:core\]) to reduce the interference of irrelevant views on the matrix $\mathbf{E}$. In ablation experiments, we degenerate D-MVE-Hash into a non-multi-view method as the baseline for subsequent comparisons. Then we evaluate our model under partial views with the broken $\mathbf{E}$. At last, we use the intact model which achieves gains of 8.13% when the complete matrix $\mathbf{E}$ is used, compared with the baseline. For the time complexity, our model takes about 1.711(h) to train our D-MVE-Hash with the complete multi-view information and view-relation calculation, which is 0.757(h) slower than the baseline. Comparisons and Retrieval Performance {#sec:exper2} ------------------------------------- We compare the retrieval performance of D-MVE-Hash and MV-Hash with several classical single/multi-view hashing method: KSH [@liu2012supervised], MvDH [@shen2018multiview], CMH[@chen2018collaborative], and recent deep methods: CNNH [@xia2014supervised], HashNet [@cao2017hashnet], DCH [@cao2018deep] and AGAH [@gu2019adversary]. All methods in the Tab. \[tabl:cnnbasemapcomp\] are supervised. As shown in these results, we have following observations: - Compared with classical hashing methods, for example, MV-Hash obtains gains of **9.62**%, **7.27**%, **9.28**% and **5.29**% when 16 bits, 32 bits, 48 bits and 64 bits hash codes are used, compared with CMH. Similar results can be observed in other experiments. - Compared with deep and multi-view methods, for example, D-MVE-Hash achieves gains of **5.21**%, **4.6**%, **3.57**% and **4.30**% when 16 bits, 32 bits, 48 bits and 64 bits hash codes are used for retrieval, compared with DCH. As shown in Fig. \[graph:doubleradiu\], when code length increases, similar objects fall into larger hamming radius. On one hand, the hash codes without enhancement are unable to cover the neighborhood of long codes, which causes the worse performance. On the other hand, our method is able to utilize multi-view information, clustering similar objects to smaller hamming radius. As a result, the performance of our method does not worsen when the retrieval hamming radius remains unchanged. In addition to that, the hash codes without enhancement also perform better when the code is short. Take the 32-bit code as an example. As shown in Fig. \[graph:len\], we use fixed length (16 bits) code as basic code to represent the binary code generated from non-multi-view features and the remaining length (16 bits in this case) to represent the binary code generated from multi-view features. Therefore, when the total code length becomes shorter, the code for multi-view is shorter and the misclassification increases, which harms the performance of the part of fixed length basic code. As a result, equal length (e.g., 32 bits) basic code without enhancement could outperform code with multi-view features. To get a full view of the model’s performance, we plot the receiver operating characteristic curves for different bits in Fig. \[graph:roc\]. Moreover, Fig. \[graph:lossfusion\] shows the change of the training loss with the increase of iterations. It turns out that our model can converge to a stable optimum point after about 50,000 training iterations. Fig. \[graph:visual\] is the visualization which shows that our D-MVE-Hash produces more relevant results. ![The top 10 retrieved images before and after enhancement.[]{data-label="graph:visual"}](graphvisual.pdf) Conclusion ========== In this paper, we present an efficient approach D-MVE-Hash to exploit multi-view information and the relation of views for hash code generation. The sub-module MV-Hash is a multi-view hash network which calculates view-relation matrix according to the stability of objects in different views. In our framework, we use three enhancement methods to merge view-relation matrix and variety of binary codes learned from single/multi-view spaces into the backbone network. Control experiments indicate that fusion methods has significant contribution to the proposed framework. In addition, we design the memory network to avoid excessive computing resources on view stability evaluation during retrieval. Experiment results and visualization results have demonstrated the effectiveness of D-MVE-Hash and MV-Hash on the tasks of image retrieval.

--- abstract: 'We estimate, in a number field, the number of elements and the maximal number of linearly independent elements, with prescribed bounds on their valuations. As a by-product, we obtain new bounds for the successive minima of ideal lattices. Our arguments combine group theory, ramification theory, and the geometry of numbers.' address: - 'Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, Budapest H-1364, Hungary' - MTA Rényi Intézet Lendület Groups and Graphs Research Group - MTA Rényi Intézet Lendület Automorphic Research Group - 'Institute for Advanced Study, Princeton NJ, USA' - 'Central European University, Nador u. 9, Budapest H-1051, Hungary' author: - Mikołaj Fraczyk - Gergely Harcos - Péter Maga title: Counting bounded elements of a number field --- [^1] Introduction ============ It was a decisive moment in the history of mathematics when Minkowski [@M] realized that certain geometric ideas are very powerful in tackling difficult arithmetic problems. In particular, Minkowski [@M] proved that in a number field $k$ of degree $d>1$ and discriminant $\Delta$, every ideal class can be represented by an integral ideal of norm less than $|\Delta|^{1/2}$. His proof relied on two ideas. First, the natural embedding $k\hookrightarrow k\otimes_{\mathbb{Q}}{\mathbb{R}}$ allows one to regard the ring of integers ${\mathfrak{o}}$ as a lattice in ${\mathbb{R}}^d$ of covolume $|\Delta|^{1/2}$. Second, a lattice in ${\mathbb{R}}^d$ contains a nonzero lattice point in a convex body symmetric about the origin[^2], as long as the volume of the body exceeds $2^d$ times the covolume of the lattice. The second idea was extended by Blichfeldt [@B2] and van der Corput [@C] to exhibit more lattice points in larger convex bodies. It leads to the following estimate that we state partly for motivation, partly as a technical ingredient for our investigations. For a modern exposition of the quoted results, see [@GL Ch. 2, §5.1 & §7.2]. \[thm1\] Let ${\mathfrak{n}}\subset{\mathfrak{o}}$ be a nonzero ideal, and let ${\mathcal{B}}\subset k\otimes_{\mathbb{Q}}{\mathbb{R}}$ be a convex body symmetric about the origin. Then $$|{\mathfrak{n}}\cap{\mathcal{B}}|{\geqslant}\frac{\operatorname{vol}({\mathcal{B}})}{2^d|\Delta|^{1/2}[{\mathfrak{o}}:{\mathfrak{n}}]}.$$ Blichfeldt [@B2] also established an upper bound of similar quality in the case when ${\mathfrak{n}}\cap{\mathcal{B}}$ contains $d$ linearly independent vectors. \[thm1b\] Let ${\mathfrak{n}}\subset{\mathfrak{o}}$ be a nonzero ideal, and let ${\mathcal{B}}\subset k\otimes_{\mathbb{Q}}{\mathbb{R}}$ be a convex body symmetric about the origin. Assume that ${\mathfrak{n}}\cap{\mathcal{B}}$ contains $d$ linearly independent vectors. Then $$|{\mathfrak{n}}\cap{\mathcal{B}}|{\leqslant}\frac{(d+1)!\operatorname{vol}({\mathcal{B}})}{|\Delta|^{1/2}[{\mathfrak{o}}:{\mathfrak{n}}]}.$$ In fact Blichfeldt proved a more general result, namely Theorem \[thm5\] in Section \[sect3\]. The original source [@B2] is an account of an AMS Sectional Meeting held in 1920 (written by B. A. Bernstein), so it does not contain any proof. What is worse, we could only find sketches of the proof in the literature. Hence we include a detailed proof in Section \[sect3\], without claiming any originality. Our principal goal in this paper is to provide an upper bound for $|{\mathfrak{n}}\cap{\mathcal{B}}|$ in the complementary case when ${\mathfrak{n}}\cap{\mathcal{B}}$ does not contain $d$ linearly independent vectors. More precisely, with certain arithmetic applications in mind, we restrict ourselves to the special convex bodies considered by Minkowski [@M] in his seminal work. They are the archimedean analogues of ideal lattices, and they are defined as follows. As before, let $k$ be a number field of degree $d>1$. Let $\Sigma:=\operatorname{Hom}(k,{\overline{{\mathbb{Q}}}})$, and let $K$ be the compositum of the fields $\sigma(k)$ for $\sigma\in\Sigma$. Then $K/{\mathbb{Q}}$ is a finite Galois extension whose Galois group $G:=\operatorname{Gal}(K/{\mathbb{Q}})$ acts transitively and faithfully on $\Sigma$. In this way, $G$ is a transitive permutation group of degree $d$. Fixing an embedding ${\overline{{\mathbb{Q}}}}\hookrightarrow{\mathbb{C}}$, we can think of the elements of $\Sigma$ as the embeddings $\sigma:k\hookrightarrow{\mathbb{C}}$, and we can identify $k\otimes_{\mathbb{Q}}{\mathbb{R}}$ with the set of column vectors $(z_\sigma)\in{\mathbb{C}}^\Sigma$ satisfying $z_{{\overline{\sigma}}}={\overline{z_\sigma}}$ for all $\sigma\in\Sigma$. See [@N Ch. I, §5] for more details. Let $(B_\sigma)$ be a collection of positive numbers such that $B_{{\overline{\sigma}}}=B_\sigma$ for all $\sigma\in\Sigma$. We shall focus on convex bodies of the form $$\label{eq1} {\mathcal{B}}:=\left\{(z_\sigma)\in{\mathbb{C}}^\Sigma:\text{$z_{{\overline{\sigma}}}={\overline{z_\sigma}}$ and $|z_\sigma|{\leqslant}B_\sigma$ for all $\sigma\in\Sigma$}\right\},$$ and we note for later reference that $$\label{eq19} \operatorname{vol}({\mathcal{B}})\asymp_d\prod_{\sigma\in\Sigma}B_\sigma.$$ Here and later, the symbols $\ll_d$, $\gg_d$, $\asymp_d$ have their usual meaning in analytic number theory: $X\ll_d Y$ (resp. $Y\gg_d X$) means that $|X|{\leqslant}CY$ holds for an absolute constant $C>0$ depending only on $d$, while $X\asymp_d Y$ abbreviates $X\ll_d Y\ll_d X$. \[thm3b\] Let ${\mathfrak{n}}\subset{\mathfrak{o}}$ be a nonzero ideal, and let ${\mathcal{B}}\subset k\otimes_{\mathbb{Q}}{\mathbb{R}}$ be a convex body of the form . Let $m$ be the maximal number of linearly independent lattice vectors contained in ${\mathfrak{n}}\cap{\mathcal{B}}$. If $m<d$, then $$\label{eq2b}|{\mathfrak{n}}\cap{\mathcal{B}}|\ll_d|\Delta|^{\min\left(\frac{1}{2},\frac{m}{2d-2m}\right)}.$$ Further, if $m<d$ and $G$ is $2$-homogeneous[^3], then $$\label{eq3b}|{\mathfrak{n}}\cap{\mathcal{B}}|\ll_d|\Delta|^{\frac{m}{2d-2}}.$$ Theorems \[thm1b\] and \[thm3b\] yield a practical estimate for the number of elements of $k$ which are bounded in every archimedean and non-archimedean valuation of $k$. \[cor1b\] Let ${\mathfrak{n}}\subset{\mathfrak{o}}$ be a nonzero ideal, and let ${\mathcal{B}}\subset k\otimes_{\mathbb{Q}}{\mathbb{R}}$ be a convex body of the form . Then $$\label{eq4b} |{\mathfrak{n}}\cap{\mathcal{B}}|\ll_d|\Delta|^{1/2}+\frac{\operatorname{vol}({\mathcal{B}})}{|\Delta|^{1/2}[{\mathfrak{o}}:{\mathfrak{n}}]}.$$ By combining Theorems \[thm1\] and \[thm3b\], we see that if the volume of our convex body is sufficiently large compared to the covolume of our ideal lattice, then the intersection contains several linearly independent lattice vectors. \[cor1\] Let ${\mathfrak{n}}\subset{\mathfrak{o}}$ be a nonzero ideal, and let ${\mathcal{B}}\subset k\otimes_{\mathbb{Q}}{\mathbb{R}}$ be a convex body of the form . Let $m$ be the maximal number of linearly independent lattice vectors contained in ${\mathfrak{n}}\cap{\mathcal{B}}$. If $m<d$, then $$\label{eq2}\operatorname{vol}({\mathcal{B}})\ll_d|\Delta|^{\min\left(1,\frac{d}{2d-2m}\right)}[{\mathfrak{o}}:{\mathfrak{n}}].$$ Further, if $m<d$ and $G$ is $2$-homogeneous, then $$\label{eq3}\operatorname{vol}({\mathcal{B}})\ll_d|\Delta|^{\frac{d-1+m}{2d-2}}[{\mathfrak{o}}:{\mathfrak{n}}].$$ If $m=0$, then and are trivial, while and boil down to the Minkowski bound $\operatorname{vol}({\mathcal{B}})\ll_d|\Delta|^{1/2}[{\mathfrak{o}}:{\mathfrak{n}}]$. If $m=1$ or $m=d-1$, then and (resp. and ) are identical. For $2{\leqslant}m{\leqslant}d-2$, the bound is stronger than (resp. is stronger than ), but its scope is restricted by the assumption that $G$ is $2$-homogeneous. The list of finite $2$-homogeneous groups is known by the work of many people, in particular by the classification of finite simple groups. For further details and references, see [@K Prop. 3.1], [@C2 Th. 5.3], [@H p. 198]. \[cor2\] Let ${\mathfrak{n}}\subset{\mathfrak{o}}$ be a nonzero ideal, and let ${\mathcal{B}}\subset k\otimes_{\mathbb{Q}}{\mathbb{R}}$ be a convex body of the form . If ${\mathcal{B}}$ does not contain a lattice basis of ${\mathfrak{n}}$, then $\operatorname{vol}({\mathcal{B}})\ll_d|\Delta|[{\mathfrak{o}}:{\mathfrak{n}}]$. Interestingly, when $k$ is totally real, the conclusion of Corollary \[cor2\] also follows from a celebrated result of McMullen [@M3 Th. 4.1] proved by topological arguments. In another direction, when the radii $B_\sigma$ are equal, the conclusion of Corollary \[cor2\] says that the last successive minimum[^4] of ${\mathfrak{n}}$ is $\ll_d|\Delta|^{1/d}[{\mathfrak{o}}:{\mathfrak{n}}]^{1/d}$. For ${\mathfrak{n}}={\mathfrak{o}}$, this bound was deduced earlier by Bhargava et al. [@B Th. 1.6] with a more direct approach. We will return to these connections in Section \[sect5\]. In fact we can control, to some extent, all successive minima of ideal lattices. \[thm4b\] Let $\lambda_1{\leqslant}\dotsb{\leqslant}\lambda_d$ be the successive minima of a nonzero ideal ${\mathfrak{n}}\subset{\mathfrak{o}}$ embedded as a lattice in $k\otimes_{\mathbb{Q}}{\mathbb{R}}$. Then for all $m\in\{1,\dotsc,d-1\}$ we have $$\begin{aligned} \label{eq21} \lambda_1\cdots\lambda_m&\gg_d|\Delta|^{\max\left(0,\frac{m}{d}-\frac{1}{2}\right)}[{\mathfrak{o}}:{\mathfrak{n}}]^{\frac{m}{d}};\\ \label{eq20} \lambda_{m+1}\lambda_{m+2}\cdots\lambda_d&\ll_d|\Delta|^{\min\left(\frac{1}{2},1-\frac{m}{d}\right)}[{\mathfrak{o}}:{\mathfrak{n}}]^{1-\frac{m}{d}}.\end{aligned}$$ If $G$ is $2$-homogeneous, then the exponents of $|\Delta|$ in and can be improved to $\frac{m(m-1)}{2d(d-1)}$ and $\frac{(d-m)(d+m-1)}{2d(d-1)}$, respectively. The example $k={\mathbb{Q}}(p^{1/d})$ mentioned by Bhargava et al. below their [@B Th. 1.6] shows that the $2$-homogeneous case of Theorem \[thm4b\] cannot be improved in general. Indeed, if $p>d>1$ are prime numbers and ${\mathfrak{n}}={\mathfrak{o}}$, then $G\cong\mathrm{Aff}({\mathbb{F}}_d)\cong({\mathbb{Z}}/d{\mathbb{Z}})\rtimes({\mathbb{Z}}/d{\mathbb{Z}})^\times$ is sharply $2$-transitive, while $\lambda_m\asymp_d|\Delta|^\frac{m-1}{d(d-1)}$ holds for all $m\in\{1,\dotsc,d\}$. The last relation follows from the straightforward upper bound $\lambda_m\ll_d p^\frac{m-1}{d}$ combined with $|\Delta|\asymp_d p^{d-1}$ and Minkowski’s result quoted below. The same example also shows that Corollary \[cor2\] cannot be improved in general. In contrast, the sharpness of – and – is less clear to us. Theorem \[thm4b\] readily yields two-sided bounds for individual successive minima, extending the result of Bhargava et al. [@B Th. 1.6] mentioned in the previous paragraph. \[cor3\] Let $\lambda_1{\leqslant}\dotsb{\leqslant}\lambda_d$ be the successive minima of a nonzero ideal ${\mathfrak{n}}\subset{\mathfrak{o}}$ embedded as a lattice in $k\otimes_{\mathbb{Q}}{\mathbb{R}}$. Then for all $m\in\{1,\dotsc,d\}$ we have $$\begin{aligned} \label{eq26} \Delta^{\max\left(0,\frac{1}{d}-\frac{1}{2m}\right)}[{\mathfrak{o}}:{\mathfrak{n}}]^{\frac{1}{d}}&\ll_d\lambda_m \ll_d\Delta^{\min\left(\frac{1}{2d-2m+2},\frac{1}{d}\right)}[{\mathfrak{o}}:{\mathfrak{n}}]^{\frac{1}{d}}&&\text{in general};\\ \label{eq27} \Delta^{\frac{m-1}{2d(d-1)}}[{\mathfrak{o}}:{\mathfrak{n}}]^{\frac{1}{d}}&\ll_d\lambda_m \ll_d\Delta^{\frac{d+m-2}{2d(d-1)}}[{\mathfrak{o}}:{\mathfrak{n}}]^{\frac{1}{d}}&&\text{if $G$ is $2$-homogeneous}.\end{aligned}$$ To form an idea of the accuracy of , it is instructive to observe that the two sides differ by a factor of $\Delta^{\frac{1}{2d}}$. Moreover, the product of the left hand side over $m\in\{1,\dotsc,d\}$ equals $\Delta^{\frac{1}{4}}[{\mathfrak{o}}:{\mathfrak{n}}]$, while the same for the right hand side equals $\Delta^{\frac{3}{4}}[{\mathfrak{o}}:{\mathfrak{n}}]$. This should be compared with the product of the $\lambda_m$’s, which by Minkowski’s theorem [@GL p. 124, Th. 3] is $$\label{eq28}\lambda_1\cdots\lambda_d\asymp_d|\Delta|^\frac{1}{2}[{\mathfrak{o}}:{\mathfrak{n}}].$$ The proof of Theorem \[thm3b\] combines group theory, ramification theory, and the geometry of numbers. The main idea is to obtain an upper bound for $|{\mathfrak{n}}\cap{\mathcal{B}}|$ by projecting ${\mathfrak{n}}\cap{\mathcal{B}}$ onto well-chosen “coordinate subspaces” ${\mathbb{R}}^S$ of ${\mathbb{C}}^\Sigma$ for $S\subset\Sigma$, and then compare it with the lower bound of Theorem \[thm1\]. We make sure that the projections of ${\mathfrak{n}}\cap{\mathcal{B}}$ generate lattices in their ambient spaces ${\mathbb{R}}^S$, and then we succeed by bounding from below the product of covolumes of those lattices. The proof of Theorem \[thm4b\] is similar, but it focuses on successive minima in place of lattice point counts. In order to formulate the key arithmetic ingredient of both proofs, Theorem \[thm3\] below, we need to introduce further notation. For a nonzero prime ideal ${\mathfrak{p}}\subset{\mathfrak{o}}$ dividing a rational prime $p$, let $e_{\mathfrak{p}}$ (resp. $f_{\mathfrak{p}}$) denote the ramification index (resp. inertia degree) of the local field extension $k_{\mathfrak{p}}/{\mathbb{Q}}_p$. By [@N Ch. III, §2], the exponent of ${\mathfrak{p}}$ in the different ideal of ${\mathfrak{o}}$ equals $e_{\mathfrak{p}}-1$ when $p\nmid e_{\mathfrak{p}}$, and it lies between $e_{\mathfrak{p}}$ and $e_{\mathfrak{p}}-1+v_{\mathfrak{p}}(e_{\mathfrak{p}})$ when $p\mid e_{\mathfrak{p}}$ (which can only occur for $p{\leqslant}d$). Therefore, the *tame discriminant* ${\Delta_\mathrm{tame}}$, defined as $$\label{eq5} {\Delta_\mathrm{tame}}:=\prod_p p^{d-f_p}\quad\text{with}\quad f_p:=\sum_{{\mathfrak{p}}\mid p}f_{\mathfrak{p}},$$ divides the discriminant $\Delta$, and it satisfies $$\label{eq4} |\Delta|<2^{d^3}{\Delta_\mathrm{tame}}.$$ The last bound is rather crude, and it can be verified as follows. The ratio $\Delta/{\Delta_\mathrm{tame}}$ divides the norm of the ideal $\prod_{p{\leqslant}d}\prod_{{\mathfrak{p}}\mid p}{\mathfrak{p}}^{v_{\mathfrak{p}}(e_{\mathfrak{p}})}$, which is a divisor of the principal ideal $\prod_{p{\leqslant}d}\prod_{{\mathfrak{p}}\mid p}(e_{\mathfrak{p}})$. Therefore, $$\frac{|\Delta|}{{\Delta_\mathrm{tame}}}{\leqslant}\prod_{p{\leqslant}d}\prod_{{\mathfrak{p}}\mid p}e_{\mathfrak{p}}^d<\prod_{p{\leqslant}d}2^{d\sum_{{\mathfrak{p}}\mid p}e_{\mathfrak{p}}}<2^{d^3}.$$ \[thm3\] Let ${\mathfrak{n}}\subset{\mathfrak{o}}$ be a nonzero ideal, and let $m\in\{1,\dotsc,d\}$. For any $m$-subsets $X\subset{\mathfrak{n}}$ and $S\subset\Sigma$, $$\label{eq6} \prod_{g\in G}{\det}^2(\sigma(x))^{\sigma\in gS}_{x\in X}\quad\text{is divisible by}\quad{\Delta_\mathrm{tame}}^{|G|\max\left(0,\frac{2m}{d}-1\right)}[{\mathfrak{o}}:{\mathfrak{n}}]^{|G|\frac{2m}{d}}.$$ If $G$ is $2$-homogeneous, then the exponent of ${\Delta_\mathrm{tame}}$ can be improved to $|G|\frac{m(m-1)}{d(d-1)}$. Note that $d$ divides $|G|$, and also $\binom{d}{2}$ divides $G$ when $G$ is $2$-homogeneous, so the exponents of ${\Delta_\mathrm{tame}}$ and $[{\mathfrak{o}}:{\mathfrak{n}}]$ are nonnegative integers. The next theorem is very similar to the $2$-homogeneous case of Theorem \[thm3\]. We do not need it for the proof of Theorem \[thm3b\], but we present it for its intrinsic beauty and interest. \[thm4\] Let ${\mathfrak{n}}\subset{\mathfrak{o}}$ be a nonzero ideal, and let $m\in\{2,\dotsc,d\}$. For any $m$-subset $X\subset{\mathfrak{n}}$, $$\label{eq8} \prod_{\substack{S\subset\Sigma\\|S|=m}}{\det}^2(\sigma(x))^{\sigma\in S}_{x\in X}\quad\text{is divisible by}\quad{\Delta_\mathrm{tame}}^{\binom{d-2}{m-2}}[{\mathfrak{o}}:{\mathfrak{n}}]^{2\binom{d-1}{m-1}}.$$ The determinants in and are only defined up to a factor of $\pm 1$, because we have not specified any ordering on $X$ and $S$. However, their squares are well-defined. If $m=d$, then Theorems \[thm3\] and \[thm4\] follow from the fact that either $\det(\sigma(x))^{\sigma\in\Sigma}_{x\in X}$ is zero, or it equals the covolume of a full rank sublattice of ${\mathfrak{n}}$. Another relatively simple special case is when ${\mathfrak{n}}={\mathfrak{o}}$ and $X=\{1,x,\ldots,x^{m-1}\}$ for some $x\in{\mathfrak{o}}$. Then, Theorem \[thm4\] and the $2$-homogeneous case of Theorem \[thm3\] are consequences of the Vandermonde determinant formula and the definition of the (usual) discriminant $\Delta$ of $k$. Not surprisingly, we shall only use the divisibility conclusion when the participating determinants are nonzero. On the other hand, it seems to be an interesting and difficult problem to characterize the vanishing of these determinants. One result in this direction is Chebotarev’s theorem from 1926: if $p$ is a prime, $k$ is the $p$-th cyclotomic field, and the elements of $X$ are $p$-th roots of unity, then none of these determinants vanish (see [@T] for a proof and for useful references). Another result is the following simple observation: if $k$ contains a proper subfield $k'$ with $m=[k:k']$, and the $m$-subset $X\subset k$ is linearly dependent over $k'$, then there is an $m$-subset $S\subset\Sigma$ such that all embeddings $\sigma\in S$ coincide on $k'$, whence $\det(\sigma(x))^{\sigma\in S}_{x\in X}=0$. Motivated by this example, we ask the following question: Assume that $X\subset k$ and $S\subset\Sigma$ satisfy $|X|=|S|$ and $\det(\sigma(x))^{\sigma\in S}_{x\in X}=0$. Does there exist a subfield $k'$ of $k$ such that $X$ is linearly dependent over $k'$, and all embeddings $\sigma\in S$ coincide on $k'$? If $X$ is of size $m$ and $G$ is $m$-homogeneous (e.g. when $G=S_d$ or $G=A_d$), then the answer to this question is affirmative. Indeed, in this case, the vanishing of one $m\times m$ minor of $\det(\sigma(x))^{\sigma\in\Sigma}_{x\in X}$ implies the vanishing of all $m\times m$ minors, which can happen if and only if $X$ is linearly dependent over ${\mathbb{Q}}$. We are grateful to the referees for their careful reading and valuable comments. We also thank Péter Pál Pálfy and Gergely Zábrádi for helpful discussions. Non-archimedean investigations ============================== In this section, we prove Theorems \[thm3\] and \[thm4\]. The two sides of and are rational integers, hence it suffices to show, for every rational prime $p$, that the exponent of $p$ is at least as large on the left hand side as on the right hand side (with the convention that the $p$-exponent of zero is infinity). We fix $p$ and an embedding ${\overline{{\mathbb{Q}}}}\hookrightarrow{\overline{{\mathbb{Q}}_p}}$, then we can think of the elements of $\Sigma$ as the embeddings $\sigma:k\hookrightarrow{\overline{{\mathbb{Q}}_p}}$. For each $\sigma\in\Sigma$, there is a unique prime ideal ${\mathfrak{p}}\mid p$ and a unique ${\mathbb{Q}}_p$-linear extension $\tilde\sigma:k_{\mathfrak{p}}\hookrightarrow{\overline{{\mathbb{Q}}_p}}$ of $\sigma$. Denoting by $I_{\mathfrak{p}}$ the set of $\sigma$’s corresponding to a given ${\mathfrak{p}}$, the extension map $\sigma\mapsto\tilde\sigma$ is a bijection $I_{\mathfrak{p}}\overset{\sim}\to\operatorname{Hom}_{{\mathbb{Q}}_p}(k_{\mathfrak{p}},{\overline{{\mathbb{Q}}_p}})$ with inverse being the restriction map. In particular, $I_{\mathfrak{p}}$ is a $\operatorname{Gal}({\overline{{\mathbb{Q}}_p}}/{\mathbb{Q}}_p)$-orbit on $\Sigma$ of cardinality $[k_{\mathfrak{p}}:{\mathbb{Q}}_p]=e_{\mathfrak{p}}f_{\mathfrak{p}}$. Let $v_p$ be the unique additive valuation on ${\overline{{\mathbb{Q}}_p}}$ extending the normalized additive valuation on ${\mathbb{Q}}_p$, and let $v_{\mathfrak{p}}$ be the normalized additive valuation on $k_{\mathfrak{p}}$. By “normalized” we mean that $v_p({\mathbb{Q}}_p^\times)={\mathbb{Z}}$ and $v_{\mathfrak{p}}(k_{\mathfrak{p}}^\times)={\mathbb{Z}}$. Then we have the important identity $$\label{eq11} v_p(\tilde\sigma(x))=\frac{1}{e_{\mathfrak{p}}}v_{\mathfrak{p}}(x),\qquad\tilde\sigma\in\operatorname{Hom}_{{\mathbb{Q}}_p}(k_{\mathfrak{p}},{\overline{{\mathbb{Q}}_p}}),\qquad x\in k_{\mathfrak{p}}^\times.$$ See [@N Ch. II, §8] for more details. Let $l_{\mathfrak{p}}$ be the maximal unramified subextension of $k_{\mathfrak{p}}/{\mathbb{Q}}_p$, then $$[k_{\mathfrak{p}}:l_{\mathfrak{p}}]=e_{\mathfrak{p}}\qquad\text{and}\qquad [l_{\mathfrak{p}}:{\mathbb{Q}}_p]=f_{\mathfrak{p}}.$$ Identifying $I_{\mathfrak{p}}$ with $\operatorname{Hom}_{{\mathbb{Q}}_p}(k_{\mathfrak{p}},{\overline{{\mathbb{Q}}_p}})$ as above, we can break up $I_{\mathfrak{p}}$ into $f_{\mathfrak{p}}$ subsets $I_{{\mathfrak{p}},l}$ of equal size $e_{\mathfrak{p}}$ according to how $l_{\mathfrak{p}}$ gets embedded into ${\overline{{\mathbb{Q}}_p}}$. In the end, two elements of $\Sigma$ belong to the same subset $I_{{\mathfrak{p}},l}$ if and only if they induce the same non-archimedean valuation $|\cdot|_{\mathfrak{p}}$ on $k$ and their ${\mathbb{Q}}_p$-linear extensions agree on $l_{\mathfrak{p}}$; we shall call two such elements of $\Sigma$ *inertially equivalent*. The proofs of Theorems \[thm3\] and \[thm4\] rely on the key observation that the $p$-adic valuation of the participating determinants can be estimated in terms of the inertial equivalence classes $I_{{\mathfrak{p}},l}$. \[prop1\] Let ${\mathfrak{n}}\subset{\mathfrak{o}}$ be a nonzero ideal, and let $m\in\{1,\dotsc,d\}$. For any $m$-subsets $X\subset{\mathfrak{n}}$ and $S\subset\Sigma$, $$\label{eq9} v_p\Bigl({\det}^2(\sigma(x))^{\sigma\in S}_{x\in X}\Bigr){\geqslant}\sum_{{\mathfrak{p}}\mid p}\frac{1}{e_{\mathfrak{p}}}\sum_{l=1}^{f_{\mathfrak{p}}}s_{{\mathfrak{p}},l}\bigl(2v_{\mathfrak{p}}({\mathfrak{n}})+s_{{\mathfrak{p}},l}-1\bigr),$$ where $s_{{\mathfrak{p}},l}$ abbreviates $|S\cap I_{{\mathfrak{p}},l}|$, and $v_{\mathfrak{p}}({\mathfrak{n}})$ stands for the exponent of ${\mathfrak{p}}$ in ${\mathfrak{n}}$. We recall that $K$ is the compositum of the fields $\sigma(k)$ for $\sigma\in\Sigma$, and we write $\tilde K$ for the extension of ${\mathbb{Q}}_p$ generated by $K$. We denote by $\tilde d$ the degree $[\tilde K:{\mathbb{Q}}_p]$, and by $\tilde{\mathfrak{o}}$ the ring of integers of $\tilde K$. We shall think of $\tilde{\mathfrak{o}}^m$ as the set of column vectors of length $m$ with entries in $\tilde{\mathfrak{o}}$. The $m$-set $S\subset\Sigma$ is partitioned into the $s_{{\mathfrak{p}},l}$-sets $S_{{\mathfrak{p}},l}:=S\cap I_{{\mathfrak{p}},l}$. Accordingly, the $m\times m$ matrix $A:=(\sigma(x))^{\sigma\in S}_{x\in X}$ decomposes into the $s_{{\mathfrak{p}},l}\times m$ blocks $A_{{\mathfrak{p}},l}:=(\sigma(x))^{\sigma\in S_{{\mathfrak{p}},l}}_{x\in X}$. Strictly speaking, these matrices are only defined up to a permutation of the rows and the columns, but this ambiguity disappears once we choose an ordering of the rows and the columns. We shall assume that $\det A\neq 0$, for otherwise is trivial. The natural isomorphism from $\tilde{\mathfrak{o}}^m$ to $\prod_{\mathfrak{p}}\prod_l\tilde{\mathfrak{o}}^{s_{{\mathfrak{p}},l}}$ maps $A\tilde{\mathfrak{o}}^m$ into $\prod_{\mathfrak{p}}\prod_l A_{{\mathfrak{p}},l}\tilde{\mathfrak{o}}^m$, hence it induces a surjective homomorphism from $\tilde{\mathfrak{o}}^m/A\tilde{\mathfrak{o}}^m$ onto $\prod_{\mathfrak{p}}\prod_l(\tilde{\mathfrak{o}}^{s_{{\mathfrak{p}},l}}/A_{{\mathfrak{p}},l}\tilde{\mathfrak{o}}^m)$. In particular, $$v_p\bigl([\tilde{\mathfrak{o}}^m:A\tilde{\mathfrak{o}}^m]\bigr){\geqslant}\sum_{{\mathfrak{p}}\mid p}\sum_{l=1}^{f_{\mathfrak{p}}}v_p\bigl([\tilde{\mathfrak{o}}^{s_{{\mathfrak{p}},l}}:A_{{\mathfrak{p}},l}\tilde{\mathfrak{o}}^m]\bigr).$$ The left hand side equals $\tilde d\cdot v_p(\det A)$, hence will follow if we can show that $$\label{eq10} v_p\bigl([\tilde{\mathfrak{o}}^{s_{{\mathfrak{p}},l}}:A_{{\mathfrak{p}},l}\tilde{\mathfrak{o}}^m]\bigr){\geqslant}\frac{\tilde d}{e_{\mathfrak{p}}}s_{{\mathfrak{p}},l}\left(v_{\mathfrak{p}}({\mathfrak{n}})+\frac{s_{{\mathfrak{p}},l}-1}{2}\right).$$ Let us fix ${\mathfrak{p}}\mid p$ and $l\in\{1,\dotsc,f_{\mathfrak{p}}\}$. We shall assume that $S_{{\mathfrak{p}},l}$ is not empty, for otherwise is trivial. We write $$\label{eq14} t:=s_{{\mathfrak{p}},l}\qquad\text{and}\qquad B:=A_{{\mathfrak{p}},l}$$ to simplify notation, and we list the elements of $S_{{\mathfrak{p}},l}$ as $\{\sigma_1,\dotsc,\sigma_t\}$. By , we have $$\label{eq12} v_p(\tilde\sigma_i(x))=\frac{1}{e_{\mathfrak{p}}}v_{\mathfrak{p}}(x),\qquad i\in\{1,\dotsc,t\},\qquad x\in k_{\mathfrak{p}}^\times.$$ We also list the elements of $X$ as $\{x_1,\dotsc,x_m\}$ in such a way that $$v_{\mathfrak{p}}({\mathfrak{n}}){\leqslant}v_{\mathfrak{p}}(x_1){\leqslant}\dotsb{\leqslant}v_{\mathfrak{p}}(x_m).$$ In particular, $v_p$ is constant on each column of $$B=\begin{pmatrix} \sigma_1(x_1) & \cdots & \sigma_1(x_m) \\ \vdots & \ddots & \vdots \\ \sigma_t(x_1)& \cdots & \sigma_t(x_m) \end{pmatrix},$$ and it is non-decreasing from left to right. As the $\sigma_i$’s are inertially equivalent, their ${\mathbb{Q}}_p$-linear extensions $\tilde\sigma_i$ coincide on $l_{\mathfrak{p}}$, and we can identify $l_{\mathfrak{p}}$ with its image in $\tilde K$ via any of these embeddings. A nice feature resulting from this identification is that the $\tilde\sigma_i$’s are $l_{\mathfrak{p}}$-linear, not just ${\mathbb{Q}}_p$-linear. We are ready to prove . We shall use the fact that the left hand side of , which is $[\tilde{\mathfrak{o}}^t:B\tilde{\mathfrak{o}}^m]$ in our new notation , remains unchanged if we multiply $B$ by elements of ${\mathrm{GL}}_m(\tilde{\mathfrak{o}})$ on the right and by elements of ${\mathrm{GL}}_t(\tilde{\mathfrak{o}})$ on the left. Writing ${\mathfrak{o}}_{l_{\mathfrak{p}}}$ (resp. ${\mathfrak{o}}_{k_{\mathfrak{p}}}$) for the ring of integers of $l_{\mathfrak{p}}$ (resp. $k_{{\mathfrak{p}}}$), we shall also utilize the fact that the group of units ${\mathfrak{o}}_{l_{\mathfrak{p}}}^\times$ contains a full set of representatives for the nonzero residue classes modulo ${\mathfrak{p}}{\mathfrak{o}}_{k_{\mathfrak{p}}}$ in ${\mathfrak{o}}_{k_{\mathfrak{p}}}$. This is because the residue fields of $l_{{\mathfrak{p}}}$ and $k_{{\mathfrak{p}}}$ have equal cardinality $p^{f_{\mathfrak{p}}}$. First, we perform invertible elementary column operations over ${\mathfrak{o}}_{l_{\mathfrak{p}}}$ in order to increase the additive valuations of the columns of $B$. Specifically, we run the following algorithm: 1. Set $j=1$. 2. For each $j'\in\{j+1,\dotsc,m\}$, if $v_{\mathfrak{p}}(x_{j'})=v_{\mathfrak{p}}(x_j)$, then choose $w\in{\mathfrak{o}}_{l_{\mathfrak{p}}}^\times$ such that $v_{\mathfrak{p}}(x_{j'}-wx_j)>v_{\mathfrak{p}}(x_j)$ and replace $x_{j'}$ by $x_{j'}-wx_j$. 3. Reorder $(x_{j+1},\dotsc,x_m)$ in such a way that $v_{\mathfrak{p}}$ is non-decreasing on the new sequence. 4. Replace $j$ by $j+1$. 5. If $j<m$, then go to the second step; otherwise, finish. We end up with a matrix $$C=\begin{pmatrix} \tilde\sigma_1(y_1) & \cdots & \tilde\sigma_1(y_m) \\ \vdots & \ddots & \vdots \\ \tilde\sigma_t(y_1)& \cdots & \tilde\sigma_t(y_m) \end{pmatrix}$$ with $y_1,\dotsc,y_m\in{\mathfrak{o}}_{k_{\mathfrak{p}}}$ such that $$v_{\mathfrak{p}}({\mathfrak{n}}){\leqslant}v_{\mathfrak{p}}(y_1)<\dotsb<v_{\mathfrak{p}}(y_m).$$ In particular, $v_{\mathfrak{p}}(y_j){\geqslant}v_{\mathfrak{p}}({\mathfrak{n}})+j-1$ for all $j\in\{1,\dotsc,m\}$. Second, we perform invertible elementary row operations over $\tilde{\mathfrak{o}}$ to transform $C$ into $$D=\begin{pmatrix} z_{1,1} & z_{1,2} & \cdots &\ z_{1,t}&\cdots & z_{1,m}\\ 0 & z_{2,2} & \cdots & z_{2,t}& \cdots & z_{2,m}\\ \vdots & \ddots & \ddots & \vdots & \ddots & \vdots\\ 0 & \ldots & 0 & z_{t,t} &\ldots & z_{t,m}\\ \end{pmatrix}$$ with $z_{i,j}\in\tilde{\mathfrak{o}}$ such that (cf. ) $$v_p(z_{i,j}){\geqslant}\frac{1}{e_{\mathfrak{p}}}\bigl(v_{\mathfrak{p}}({\mathfrak{n}})+j-1\bigr),\qquad i{\leqslant}j.$$ In particular, $D\tilde{\mathfrak{o}}^m$ is a subgroup of $\tilde{\mathfrak{n}}_1\times\dotsb\times\tilde{\mathfrak{n}}_t$, where $$\tilde{\mathfrak{n}}_i:=\left\{z\in\tilde{\mathfrak{o}}:v_p(z){\geqslant}\frac{1}{e_{\mathfrak{p}}}\bigl(v_{\mathfrak{p}}({\mathfrak{n}})+i-1\bigr)\right\},\qquad i\in\{1,\dotsc,t\}.$$ This implies, using that $e_{\mathfrak{p}}$ divides the ramification degree of the local field extension $\tilde K/{\mathbb{Q}}_p$, $$\label{eq13} v_p\bigl([\tilde{\mathfrak{o}}^t:D\tilde{\mathfrak{o}}^m]\bigr){\geqslant}\sum_{i=1}^t v_p\bigl([\tilde{\mathfrak{o}}:\tilde{\mathfrak{n}}_i]\bigr) =\sum_{i=1}^t\frac{\tilde d}{e_{\mathfrak{p}}}\bigl(v_{\mathfrak{p}}({\mathfrak{n}})+i-1\bigr).$$ The inequalities and are equivalent, because their left hand sides are equal, and their right hand sides are also equal (cf. ). The proof of Proposition \[prop1\] is complete. For any $g\in G$, it follows from Proposition \[prop1\] that $$v_p\Bigl({\det}^2(\sigma(x))^{\sigma\in gS}_{x\in X}\Bigr){\geqslant}\sum_{{\mathfrak{p}}\mid p}\frac{1}{e_{\mathfrak{p}}}\sum_{l=1}^{f_{\mathfrak{p}}}\sum_{\sigma\in I_{{\mathfrak{p}},l}}1_{gS}(\sigma) \left(2v_{\mathfrak{p}}({\mathfrak{n}})+\sum_{\sigma'\in I_{{\mathfrak{p}},l}\setminus\{\sigma\}}1_{gS}(\sigma')\right).$$ We average both sides over $g\in G$, utilizing that $G$ acts transitively and faithfully on $\Sigma$. For any $\sigma\in\Sigma$, we obtain readily that $$\label{eq18} \frac{1}{|G|}\sum_{g\in G}1_{gS}(\sigma)=\frac{1}{|G|}\sum_{g\in G}1_{S}(g^{-1}\sigma)=\frac{|S|}{d}=\frac{m}{d}.$$ As a consequence, for any distinct $\sigma,\sigma'\in\Sigma$, we see that $$\label{eq7} \frac{1}{|G|}\sum_{g\in G}1_{gS}(\sigma)1_{gS}(\sigma'){\geqslant}\frac{1}{|G|}\sum_{g\in G}\bigl(1_{gS}(\sigma)+1_{gS}(\sigma')-1\bigr)=\frac{2m}{d}-1.$$ This bound is trivial when $m<d/2$, in which case we shall only use that the left hand side is nonnegative. Combining these inequalities and noting that $|I_{{\mathfrak{p}},l}|=e_{\mathfrak{p}}$, we infer that $$\frac{1}{|G|}\sum_{g\in G}v_p\Bigl({\det}^2(\sigma(x))^{\sigma\in gS}_{x\in X}\Bigr){\geqslant}\sum_{{\mathfrak{p}}\mid p}f_{\mathfrak{p}}\left(v_{\mathfrak{p}}({\mathfrak{n}})\frac{2m}{d}+(e_{\mathfrak{p}}-1)\max\left(0,\frac{2m}{d}-1\right)\right).$$ Now from $[{\mathfrak{o}}:{\mathfrak{p}}]=p^{f_{\mathfrak{p}}}$ it is clear that $$\sum_{{\mathfrak{p}}\mid p}f_{\mathfrak{p}}v_{\mathfrak{p}}({\mathfrak{n}})=v_p\bigl([{\mathfrak{o}}:{\mathfrak{n}}]\bigr),$$ while implies that $$\sum_{{\mathfrak{p}}\mid p}f_{\mathfrak{p}}(e_{\mathfrak{p}}-1)=d-f_p=v_p({\Delta_\mathrm{tame}}).$$ Therefore, the last inequality can be rewritten as $$\frac{1}{|G|}\sum_{g\in G}v_p\Bigl({\det}^2(\sigma(x))^{\sigma\in gS}_{x\in X}\Bigr){\geqslant}\frac{2m}{d}v_p\bigl([{\mathfrak{o}}:{\mathfrak{n}}]\bigr)+\max\left(0,\frac{2m}{d}-1\right)v_p({\Delta_\mathrm{tame}}).$$ The rational prime $p$ was arbitrary here, so we have proved . If $G$ is $2$-homogeneous, then we can improve to $$\frac{1}{|G|}\sum_{g\in G}1_{gS}(\sigma)1_{gS}(\sigma')=\frac{1}{|G|}\sum_{g\in G}1_{S}(g^{-1}\sigma)1_{S}(g^{-1}\sigma') =\frac{\binom{|S|}{2}}{\binom{d}{2}}=\frac{m(m-1)}{d(d-1)}.$$ As a result, we can replace $\max\left(0,\frac{2m}{d}-1\right)$ by $\frac{m(m-1)}{d(d-1)}$ in the subsequent argument, and hence also in . The proof of Theorem \[thm3\] is complete. For any $m$-subset $S\subset\Sigma$, it follows from Proposition \[prop1\] that $$v_p\Bigl({\det}^2(\sigma(x))^{\sigma\in S}_{x\in X}\Bigr){\geqslant}\sum_{{\mathfrak{p}}\mid p}\frac{1}{e_{\mathfrak{p}}}\sum_{l=1}^{f_{\mathfrak{p}}}\sum_{\sigma\in I_{{\mathfrak{p}},l}}1_{S}(\sigma) \left(2v_{\mathfrak{p}}({\mathfrak{n}})+\sum_{\sigma'\in I_{{\mathfrak{p}},l}\setminus\{\sigma\}}1_{S}(\sigma')\right).$$ We sum both sides over all $m$-subsets $S\subset\Sigma$, using that $$\begin{aligned} \sum_{\substack{S\subset\Sigma\\|S|=m}}1_S(\sigma)&=\binom{d-1}{m-1}\quad\text{for any $\sigma\in\Sigma$};\\ \sum_{\substack{S\subset\Sigma\\|S|=m}}1_S(\sigma)1_S(\sigma')&=\binom{d-2}{m-2}\quad\text{for any distinct $\sigma,\sigma'\in\Sigma$.}\end{aligned}$$ From here we proceed as in the proof of Theorem \[thm3\], and conclude $$\sum_{\substack{S\subset\Sigma\\|S|=m}}v_p\Bigl({\det}^2(\sigma(x))^{\sigma\in S}_{x\in X}\Bigr){\geqslant}2\binom{d-1}{m-1}v_p\bigl([{\mathfrak{o}}:{\mathfrak{n}}]\bigr)+\binom{d-2}{m-2}v_p({\Delta_\mathrm{tame}}).$$ The rational prime $p$ was arbitrary here, so the proof of Theorem \[thm4\] is complete. Archimedean investigations {#sect3} ========================== In this section, we prove Theorems \[thm3b\]–\[thm4b\] and Corollaries \[cor1b\]–\[cor3\]. We shall combine Theorems \[thm1\] and \[thm3\] with the following lesser known result of Blichfeldt [@B2], of which Theorem \[thm1b\] is a special case. \[thm5\] Let $\Lambda\subset{\mathbb{R}}^m$ be a lattice, and let ${\mathcal{C}}\subset{\mathbb{R}}^m$ be a convex body containing the origin. If $\Lambda\cap{\mathcal{C}}$ contains $m$ linearly independent lattice vectors, then $$\label{eq15} |\Lambda\cap{\mathcal{C}}|{\leqslant}m!\frac{\operatorname{vol}({\mathcal{C}})}{\det(\Lambda)}+m{\leqslant}(m+1)!\frac{\operatorname{vol}({\mathcal{C}})}{\det(\Lambda)}.$$ The second inequality is clear by $\operatorname{vol}({\mathcal{C}}){\geqslant}\det(\Lambda)/m!$, hence we focus on the first inequality. In this proof, a polytope (resp. simplex) will always mean a convex lattice polytope (resp. simplex) with vertices lying in $\Lambda$. For other terminology, we follow the book [@DRS]. Without loss of generality, ${\mathcal{C}}$ is bounded. Then, by the initial assumptions on ${\mathcal{C}}$, the convex hull of $\Lambda\cap{\mathcal{C}}$ is an $m$-dimensional polytope, which can be decomposed into $m$-simplices according to [@DRS Prop. 2.2.4]. The corresponding triangulation of $\Lambda\cap{\mathcal{C}}$ can be refined to a full triangulation by decomposing recursively the participating $m$-simplices into smaller $m$-simplices. Alternatively, one can obtain a full triangulation of $\Lambda\cap{\mathcal{C}}$ by ordering its elements in such a way that no point belongs to the convex hull of previous points, and then taking the placing/pushing triangulation for that ordering. We fix a full triangulation of $\Lambda\cap{\mathcal{C}}$, and we denote by ${\mathcal{T}}$ the set of $m$-simplices that participate in it. We define a graph on ${\mathcal{T}}$ by declaring that two elements of ${\mathcal{T}}$ are connected by an edge if and only if their intersection is an $(m-1)$-simplex. One can show that this graph is connected, which forces $$|{\mathcal{T}}|{\geqslant}|\Lambda\cap{\mathcal{C}}|-m.$$ For details, see [@DRS Th. 2.6.1], [@RS Th. 3.2], and their proofs. On the other hand, as ${\mathcal{C}}$ is convex and each element of ${\mathcal{T}}$ has volume at least $\det(\Lambda)/m!$, we also have $$\operatorname{vol}({\mathcal{C}}){\geqslant}\operatorname{vol}(\cup{\mathcal{T}}){\geqslant}\frac{\det(\Lambda)}{m!}|{\mathcal{T}}|.$$ Combining these two bounds, we get the first inequality of . As remarked earlier, the second inequality of is straightforward, so the proof of Theorem \[thm5\] is complete. If $m=0$, then and are trivial, so we shall assume that $0<m<d$. We write $V$ for the ${\mathbb{R}}$-span of ${\mathfrak{n}}\cap{\mathcal{B}}$, so that $V$ is an $m$-dimensional ${\mathbb{R}}$-subspace of $k\otimes_{\mathbb{Q}}{\mathbb{R}}$, and ${\mathfrak{n}}\cap V$ is an $m$-dimensional lattice in $V$. We fix a basis $X\subset{\mathfrak{n}}$ of ${\mathfrak{n}}\cap V$, and we think of its elements as the columns of the $d\times m$ complex matrix $M:=(\sigma(x))^{\sigma\in\Sigma}_{x\in X}$. Strictly speaking, $M$ is only defined up to a permutation of the rows and the columns, but this ambiguity disappears once we choose an ordering of $\Sigma$ and $X$. By construction, the columns of $M$ are linearly independent over ${\mathbb{R}}$, and we claim that they are also linearly independent over ${\mathbb{C}}$. Indeed, if $c:X\to{\mathbb{C}}$ satisfies $\sum_{x\in X}c(x)\sigma(x)=0$ for all $\sigma\in\Sigma$, then complex conjugating the equations and switching from $\sigma$ to ${\overline{\sigma}}$, we get that $\sum_{x\in X}{\overline{c(x)}}\sigma(x)=0$ for all $\sigma\in\Sigma$. As a result, the real and imaginary parts of $c(x)$ must vanish for all $x\in X$, which proves the claim. Hence $\operatorname{rank}(M)=m$, and there exists an $m$-subset $S\subset\Sigma$ such that $\det(\sigma(x))^{\sigma\in S}_{x\in X}\neq 0$. We fix $S\subset\Sigma$ along with $X\subset{\mathfrak{n}}$. For any Galois automorphism $g\in G$, the image of $\det(\sigma(x))^{\sigma\in S}_{x\in X}$ under $g$ equals $\det(\sigma(x))^{\sigma\in gS}_{x\in X}$. Therefore, these $m\times m$ minors of $M$ are nonzero, and by and Theorem \[thm3\] they satisfy $$\label{eq16} \prod_{g\in G}\left|\det(\sigma(x))^{\sigma\in gS}_{x\in X}\right|\gg_d |\Delta|^{|G|\max\left(0,\frac{m}{d}-\frac{1}{2}\right)}[{\mathfrak{o}}:{\mathfrak{n}}]^{|G|\frac{m}{d}}.$$ Moreover, the exponent of $|\Delta|$ can be improved to $|G|\frac{m(m-1)}{2d(d-1)}$ when $G$ is $2$-homogeneous. Fixing $g\in G$ for a moment, the multilinearity of the determinant shows that there is a choice of $\tilde\sigma\in\{\operatorname{Re}(\sigma),\operatorname{Im}(\sigma)\}$ for each $\sigma\in gS$ such that $$\label{eq17} \left|\det(\sigma(x))^{\sigma\in gS}_{x\in X}\right|{\leqslant}2^m\left|\det(\tilde\sigma(x))^{\sigma\in gS}_{x\in X}\right|.$$ The left hand side is positive, hence the right hand side is also positive. Let $f:{\mathbb{C}}^\Sigma\to{\mathbb{R}}^{gS}$ be the product of the ${\mathbb{R}}$-linear surjections $f_\sigma:{\mathbb{C}}\to{\mathbb{R}}$ given by $$f_\sigma(z):=\begin{cases} \operatorname{Re}(z),&\sigma\in gS\ \ \text{and}\ \ \tilde\sigma=\operatorname{Re}(\sigma);\\ \operatorname{Im}(z),&\sigma\in gS\ \ \text{and}\ \ \tilde\sigma=\operatorname{Im}(\sigma);\\ 0,&\sigma\not\in gS. \end{cases}$$ Tautologically, $\tilde\sigma=f_\sigma\circ\sigma$ holds for all $\sigma\in gS$, hence $f$ restricts to an ${\mathbb{R}}$-linear isomorphism $V\overset{\sim}\to{\mathbb{R}}^{gS}$, and $\Lambda:=f({\mathfrak{n}}\cap V)$ is a lattice in ${\mathbb{R}}^{gS}$ of covolume $\left|\det(\tilde\sigma(x))^{\sigma\in gS}_{x\in X}\right|$. In addition, ${\mathcal{C}}:=f({\mathcal{B}})$ is an $o$-symmetric convex body in ${\mathbb{R}}^{gS}$, which lies in the orthotope $\prod_{\sigma\in gS}[-B_\sigma,B_\sigma]$ by . Clearly, $\Lambda\cap{\mathcal{C}}$ contains $f({\mathfrak{n}}\cap{\mathcal{B}})$, which in turn contains $m$ linearly independent lattice vectors. Now we combine these observations with Theorem \[thm5\] and to infer that $$|{\mathfrak{n}}\cap{\mathcal{B}}|{\leqslant}|\Lambda\cap{\mathcal{C}}|{\leqslant}4^m(m+1)!\frac{\prod_{\sigma\in gS}B_\sigma}{\left|\det(\sigma(x))^{\sigma\in gS}_{x\in X}\right|}.$$ We keep the two sides of the last inequality, and take their geometric mean over $g\in G$. Using also , , , we obtain $$\label{eq23} |{\mathfrak{n}}\cap{\mathcal{B}}|\ll_d\frac{\operatorname{vol}({\mathcal{B}})^\frac{m}{d}}{|\Delta|^{\max\left(0,\frac{m}{d}-\frac{1}{2}\right)}[{\mathfrak{o}}:{\mathfrak{n}}]^{\frac{m}{d}}}.$$ Finally, we invoke Theorem \[thm1\] to estimate from above the right hand side in terms of the left hand side: $$\label{eq23b} |{\mathfrak{n}}\cap{\mathcal{B}}|\ll_d|{\mathfrak{n}}\cap{\mathcal{B}}|^\frac{m}{d}|\Delta|^{\min\left(\frac{m}{2d},\frac{1}{2}-\frac{m}{2d}\right)}.$$ This bound is equivalent to in the light of $0<m<d$. If $G$ is $2$-homogeneous, then the exponent of $|\Delta|$ can be improved to $\frac{m(m-1)}{2d(d-1)}$ in , and to $\frac{m(d-m)}{2d(d-1)}$ in , so that the resulting bound is equivalent to . The proof of Theorem \[thm3b\] is complete. If ${\mathfrak{n}}\cap{\mathcal{B}}$ contains $d$ linearly independent vectors, then follows from Theorem \[thm1b\]. If ${\mathfrak{n}}\cap{\mathcal{B}}$ does not contain $d$ linearly independent vectors, then follows from Theorem \[thm3b\]. The proof of Corollary \[cor1b\] is complete. In the light of Theorem \[thm1\], the bound follows from , while the bound follows from . The proof of Corollary \[cor1\] is complete. Assume that ${\mathcal{B}}$ does not contain a lattice basis of ${\mathfrak{n}}$. Then, by an observation of Mahler [@M2] (see also [@GL Ch. 2, §10.2]), the scaled body $\tfrac{1}{d}{\mathcal{B}}$ does not contain $d$ linearly independent lattice vectors from ${\mathfrak{n}}$. Hence, by Corollary \[cor1\], it follows that $$\operatorname{vol}({\mathcal{B}})\ll_d\operatorname{vol}(\tfrac{1}{d}{\mathcal{B}})\ll_d|\Delta|[{\mathfrak{o}}:{\mathfrak{n}}].$$ The proof of Corollary \[cor2\] is complete. We borrow several ideas from the proof of Theorem \[thm3b\] without further mention. Let $x_1,\dotsc,x_m\in{\mathfrak{n}}$ be linearly independent lattice vectors whose Euclidean norms in $k\otimes_{\mathbb{Q}}{\mathbb{R}}$ are the successive minima $\lambda_1,\dotsc,\lambda_m$, respectively. Let $X$ be the $m$-set $\{x_1,\dotsc,x_m\}\subset{\mathfrak{n}}$, and let $V$ be the ${\mathbb{R}}$-span of $X$. Then $V$ is an $m$-dimensional ${\mathbb{R}}$-subspace of $k\otimes_{\mathbb{Q}}{\mathbb{R}}$, and ${\mathfrak{n}}\cap V$ is an $m$-dimensional lattice in $V$ of successive minima $\lambda_1{\leqslant}\dotsb{\leqslant}\lambda_m$. In particular, the covolume of ${\mathfrak{n}}\cap V$ is $\asymp_d\lambda_1\cdots\lambda_m$. We fix an $m$-subset $S\subset\Sigma$ such that $\det(\sigma(x))^{\sigma\in S}_{x\in X}\neq 0$. For any $g\in G$, there exists an orthogonal projection $f$ of $k\otimes_{\mathbb{Q}}{\mathbb{R}}$ onto an $m$-subspace such that the covolume of $f({\mathfrak{n}}\cap V)$ is at least $2^{-m}\left|\det(\sigma(x))^{\sigma\in gS}_{x\in X}\right|$. Since the covolume of $f({\mathfrak{n}}\cap V)$ cannot exceed the covolume of ${\mathfrak{n}}\cap V$, we infer that $$\lambda_1\cdots\lambda_m\gg_d\left|\det(\sigma(x))^{\sigma\in gS}_{x\in X}\right|,\qquad g\in G.$$ Taking the geometric mean of both sides over $g\in G$, and using , we obtain . Taking the reciprocal of , and then multiplying both sides by , we arrive at . If $G$ is $2$-homogeneous, then the exponent of $|\Delta|$ in can be improved to $|G|\frac{m(m-1)}{2d(d-1)}$, and our argument yields the following variants of and : $$\begin{aligned} \label{eq21b} \lambda_1\cdots\lambda_m&\gg_d|\Delta|^\frac{m(m-1)}{2d(d-1)}[{\mathfrak{o}}:{\mathfrak{n}}]^{\frac{m}{d}};\\ \label{eq20b} \lambda_{m+1}\lambda_{m+2}\cdots\lambda_d&\ll_d|\Delta|^{\frac{(d-m)(d+m-1)}{2d(d-1)}}[{\mathfrak{o}}:{\mathfrak{n}}]^{1-\frac{m}{d}}.\end{aligned}$$ The proof of Theorem \[thm4b\] is complete. We observe that and are also valid for $m=d$, while and are also valid for $m=0$. Indeed, these special cases amount to . Now, taking the $m$-th root of and readily yields the lower bound of and . Similarly, taking the $(d-m)$-th root of and readily yields the upper bound of and with $m+1$ in place of $m$. The proof of Corollary \[cor3\] is complete. Connections to the work of McMullen [@M3] and Bhargava et al. [@B] {#sect5} ================================================================== If the number field $k$ is totally real, then we can identify the ${\mathbb{R}}$-algebra $k\otimes_{\mathbb{Q}}{\mathbb{R}}$ with the set of column vectors $(z_\sigma)\in{\mathbb{R}}^\Sigma$. The multiplicative group $({\mathbb{R}}^\Sigma)^\times$ acts on ${\mathbb{R}}^\Sigma$ by multiplication, hence so does its subgroup $$A:=\biggl\{(a_\sigma)\in(0,\infty)^\Sigma:\prod_{\sigma\in\Sigma}a_\sigma=1\biggr\}.$$ Let us consider the induced action of $A$ on the space of lattices of ${\mathbb{R}}^\Sigma$. Geometrically, the space of lattices can be described as ${\mathrm{GL}}({\mathbb{R}}^\Sigma)/{\mathrm{GL}}({\mathbb{Z}}^\Sigma)$, and the induced action of $A$ is given by left multiplication by positive diagonal matrices of determinant $1$. In particular, this action is continuous and preserves the covolume. The group of totally positive units ${\mathfrak{o}}^\times_+$ is cocompact in $A$ (cf. Dirichlet’s unit theorem) and stabilizes the lattice ${\mathfrak{o}}$, hence the orbit $A{\mathfrak{o}}$ is compact. By a striking result of McMullen [@M3 Th. 4.1], the compactness of $A{\mathfrak{o}}$ implies the existence of $a\in A$ such that the successive minima of the lattice $a{\mathfrak{o}}$ are equal: $\mu_1=\dots=\mu_d$. As we shall explain in the next paragraph, this fact gives rise to a short alternative proof of Corollary \[cor2\] (when $k$ is totally real). We note in passing that Levin, Shapira, Weiss [@LSW Th. 1.1] have extended McMullen’s theorem to closed orbits of lattices; these orbits arise from direct sums of totally real number fields and their full rank additive subgroups [@SW Prop. 5.7]. Let $\mu$ be the common value of $\mu_1=\dots=\mu_d$, and let ${\mathcal{D}}$ be the closed Euclidean unit ball in ${\mathbb{R}}^\Sigma$ centered at the origin. Then $a{\mathfrak{o}}\cap\mu{\mathcal{D}}$ contains $d$ linearly independent vectors. Let ${\mathfrak{n}}\subset{\mathfrak{o}}$ be a nonzero ideal, and let ${\mathcal{B}}\subset{\mathbb{R}}^\Sigma$ be an orthotope of the form $\prod_{\sigma\in\Sigma}[-B_\sigma,B_\sigma]$. We claim that if ${\mathcal{B}}$ does not contain a lattice basis of ${\mathfrak{n}}$, then $$\label{eq24} \operatorname{vol}({\mathcal{B}}){\leqslant}(2d\mu)^d|\Delta|^{1/2}[{\mathfrak{o}}:{\mathfrak{n}}].$$ This is sufficient for the conclusion of Corollary \[cor2\], since $\mu^d=\mu_1\cdots\mu_d\asymp_d|\Delta|^{1/2}$. Let us assume that is false. Then $\operatorname{vol}(a\mu^{-1}d^{-1}{\mathcal{B}})>2^d|\Delta|^{1/2}[{\mathfrak{o}}:{\mathfrak{n}}]$, hence Theorem \[thm1\] guarantees the existence of a nonzero lattice point $x\in{\mathfrak{n}}\cap a\mu^{-1}d^{-1}{\mathcal{B}}$. By our initial remarks, $x{\mathfrak{o}}\cap xa^{-1}\mu{\mathcal{D}}$ contains $d$ linearly independent vectors, so by ${\mathfrak{n}}{\mathfrak{o}}\subset{\mathfrak{n}}$ and ${\mathcal{B}}{\mathcal{D}}\subset{\mathcal{B}}$ it follows that ${\mathfrak{n}}\cap d^{-1}{\mathcal{B}}$ also contains $d$ linearly independent vectors. Finally, by the earlier quoted observation of Mahler [@M2] (see also [@GL Ch. 2, §10.2]), we conclude that ${\mathcal{B}}$ contains a lattice basis of ${\mathfrak{n}}$. Corollary \[cor2\] can also be connected to the work of Bhargava et al. [@B] in multiple ways. Let $k$ be an arbitrary number field, and let $\lambda_1{\leqslant}\dotsb{\leqslant}\lambda_d$ be the successive minima of ${\mathfrak{o}}$ embedded as a lattice in $k\otimes_{\mathbb{Q}}{\mathbb{R}}$. Then [@B Th. 1.6] states that $$\label{eq25} \lambda_d\ll_d|\Delta|^{1/d}.$$ We claim that follows from Corollary \[cor2\], while a weaker version of Corollary \[cor2\] follows from . To justify the first claim, we set $B_\sigma:=\frac{1}{d+1}\lambda_d$ for all $\sigma\in\Sigma$ in . Clearly, ${\mathcal{B}}$ contains no lattice basis of ${\mathfrak{o}}$, hence $\operatorname{vol}({\mathcal{B}})\ll_d|\Delta|$ by Corollary \[cor2\], which is equivalent to by . To justify the second claim, we start from . Let ${\mathfrak{n}}\subset{\mathfrak{o}}$ be a nonzero ideal, and let ${\mathcal{B}}\subset k\otimes_{\mathbb{Q}}{\mathbb{R}}$ be a convex body of the form not containing a lattice basis of ${\mathfrak{n}}$. As ${\mathfrak{o}}\cap\lambda_d{\mathcal{D}}$ contains $d$ linearly independent vectors, we can proceed as in the previous paragraph but with $a\in A$ (resp. $\mu$) replaced by $1\in k$ (resp. $\lambda_d$). We deduce the following variant of : $$\operatorname{vol}({\mathcal{B}}){\leqslant}(2d\lambda_d)^d|\Delta|^{1/2}[{\mathfrak{o}}:{\mathfrak{n}}]\ll_d|\Delta|^{3/2}[{\mathfrak{o}}:{\mathfrak{n}}].$$ That is, alone implies a version of Corollary \[cor2\] in which $|\Delta|$ is replaced by $|\Delta|^{3/2}$. [10]{} M. Bhargava, A. Shankar, T. Taniguchi, F. Thorne, J. Tsimerman, Y. Zhao, *Bounds on $2$-torsion in class groups of number fields and integral points on elliptic curves*, [arXiv:1701.02458]{}, J. Amer. Math. Soc., to appear H. F. Blichfeldt, *Notes on geometry of numbers*, In: *The October meeting of the San Francisco Section*, Bull. Amer. Math. Soc. **27** (1921), 149–153. P. J. Cameron, *Finite permutation groups and finite simple groups*, Bull. London Math. Soc. **13** (1981), 1–22. J. G. van der Corput, *Verallgemeinerung einer Mordellschen Beweismethode in der Geometrie der Zahlen*, Acta Arith. **1** (1935), 62–66.; *Zweite Mitteilung*, ibid. **2** (1936), 145–146. J. A. De Loera, J. Rambau, F. Santos, *Triangulations: Structures for algorithms and applications*, Algorithms and Computation in Mathematics, Vol. 25, Springer-Verlag, Berlin, 2010. P. M. Gruber, C. G. Lekkerkerker, *Geometry of numbers*, 2nd edition, North-Holland Mathematical Library, Vol. 37, North-Holland Publishing Co., Amsterdam, 1987. M. Huber, *The classification of flag-transitive Steiner 3-designs*, Adv. Geom. **5** (2005), 195–221. W. M. Kantor, *Automorphism groups of designs*, Math. Z. **109** (1969), 246–252. M. Levin, U. Shapira, B. Weiss, *Closed orbits for the diagonal group and well-rounded lattices*, Groups Geom. Dyn. **10** (2016), 1211–1225. K. Mahler, *A theorem on inhomogeneous diophantine inequalities*, Proc. Kon. Ned. Akad. Wet. **41** (1938), 634–637. C. T. McMullen, *Minkowski’s conjecture, well-rounded lattices and topological dimension*, J. Amer. Math. Soc. **18** (2005), 711–734. H. Minkowski, *Über die positiven quadratischen Formen und über kettenbruchähnliche Algorithmen*, J. Reine Angew. Math. **107** (1891), 278–297. J. Neukirch, *Algebraische Zahlentheorie*, Springer-Verlag, Berlin, 1992. B. L. Rothschild, E. G. Straus, *On triangulations of the convex hull of $n$ points*, Combinatorica **5** (1985), 167–179. U. Shapira, B. Weiss, *On the Mordell-Gruber spectrum*, Int. Math. Res. Not. IMRN **2015**, *no.* 14, 5518–5559. T. Tao, *An uncertainty principle for cyclic groups of prime order*, Math. Res. Lett. **12** (2005), 121–127. [^1]: First author supported by ERC grant CoG-648017 and the MTA Rényi Intézet Lendület Groups and Graphs Research Group. Second and third author supported by NKFIH (National Research, Development and Innovation Office) grant K 119528 and by the MTA Rényi Intézet Lendület Automorphic Research Group. Third author also supported by the Premium Postdoctoral Fellowship of the Hungarian Academy of Sciences. [^2]: that is, a convex subset of ${\mathbb{R}}^d$ invariant under multiplication by $-1$ [^3]: that is, $G$ acts transitively on the $2$-element subsets of $\Sigma$ [^4]: we understand successive minima with respect to the closed Euclidean unit ball centered at the origin

--- abstract: 'We study the binary cold fission of $^{252}$Cf in the frame of a cluster model where the fragments are born to their respective ground states and interact via a double-folded potential with deformation effects taken into account up to multipolarity $\lambda=4$. The preformation factors were neglected. In the case when the fragments are assumed to be spherical or with ground state quadrupole deformation, the $Q$-value principle dictates the occurence of a narrow region around the double magic $^{132}$Sn, like in the case of cluster radioactivity. When the hexadecupole deformation is turned on, an entire mass-region of cold fission in the range 138$\div$156 for the heavy fragment arise, in agreement with the experimental observations. This fact suggests that in the above mentioned mass-region, contrary to the usual cluster radioactivity where the daughter nucleus is always a neutron/proton (or both) closed shell or nearly closed shell spherical nucleus, the clusterization mechanism seems to be strongly influenced by the hexadecupole deformations rather than the $Q$-value.' address: - ' $^{1)}$ Institute of Atomic Physics, Bucharest, P.O.Box MG-6, Romania' - ' $^{2)}$ Physics Department, Vanderbilt University, Nashville, TN 37235, USA' - ' $^{3)}$ Institut für Theoretische Physik der J.W.Goethe Universität, D-60054, Frankfurt am Main, Germany' author: - 'A. Săndulescu$^{1,2,3)}$, Ş. Mişicu$^{1)}$, F. C\^ arstoiu$^{1)}$, A. Florescu$^{1,2}$, and W. Greiner$^{2,3)}$' title: ' ROLE OF FRAGMENT HIGHER STATIC DEFORMATIONS IN THE COLD BINARY FISSION OF $^{252}$Cf' --- 0.25truecm [ **1. Introduction** ]{} .35truecm In the binary nuclear fission of actinide nuclei the fragments are usually formed in highly-excited states which subsequently decay to their ground-states by emitting neutrons and gamma rays. However a small fraction of these fragmentations will attain a very high kinetic energy $TKE$ which is very close to the corresponding binary decay energy $Q$. Since in this case the fragments are formed with excitations energies close to their ground-states no neutrons are emitted. Milton and Fraser [@Mil61] were the first who noticed that some of the fission fragments are produced at such high kinetic energies that the emerging nuclei are formed nearly in their ground-state. Later on Signarbeux et al. [@GUE78] confirmed the previous interpretation by determining the mass distributions of the primary fragments for the highest values of the kinetic energy. They concluded that even before the scission takes place we deal with a superposition of two fragments in their ground state, from which the [*cold fragmentation*]{} term emerged. An interesting remark they made was that the odd-even fluctuations of $Q$ due to nucleon pairing were not present also in the $TKE_{max}$ values. In their view this smoothing of the odd-even effect was a consequence of a pair-broken from one of the fragments. The probability for neutronless fission is 0.0021$\pm$0.0008 for $^{252}$Cf. In the last years the cold (neutronless) fission of many actinide nuclei into fragments with masses from $\approx$70 to $\approx$ 160 was an intensivelly studied phenomenon [@GB91; @HKB93; @Ben93; @Sch94; @Ham94; @Ter94; @San96]. An important step in the understanding of the cold fission phenomenon was the observation that the final nuclei are generated in their ground states or some low excited states, which prompted some authors to relate these decays to the spontaneous emission of light nuclei (cluster radioactivity) such as alpha particles and heavier clusters ranging from $^{14}$C to $^{34}$Si [@SG92]. The fragments emitted in binary cold decays are produced with very low or even zero internal excitation energy and consequently with very high kinetic energy $TKE = Q - TXE$. In order to achieve such large $TKE$ values, both fragments should have very compact shapes at the scission point and deformations close to those of their ground states [@GB91; @SFG89]. The first direct observation of cold (neutronless) binary fragmentations in the spontaneous fission of $^{252}$Cf was made by using the multiple Ge-detector Compact Ball facility at Oak Ridge National Laboratory [@Ham94; @Ter94], and more recently with the Gammasphere consisting of 72 detectors [@San96]. Using the triple-gamma coincidence technique, the correlations between the two fragments were observed unambiguously. In these cold fragmentations, some indications of a third light fragment such as $\alpha$, $^{6}$He and $^{10}$Be clusters [@Ram96; @San97; @Ram97], were also reported. In a recent series of publications [@MCGP96; @Gon97] the group of Tübingen reported some interesting results on the spontaneous decay of $^{252}$Cf using a twin ionization chamber. Two distinct mass regions of cold fission were observed : the first extending from the mass split 96/156 up to 114/138 and the second one comprising only a narrow mass range around the mass split 120/132. In the present paper, based on a cluster model similar to the cluster model used for cluster radioactivity, we estimated the relative isotopic yields for the spontaneous cold binary fission of $^{252}$Cf. These isotopic yields are given by the ratio of the penetrability through the potential barrier between the two final fragments for a given mass and charge splitting, over the sum of penetrabilities for all possible fragmentations. The corresponding barriers were evaluated using the double folding potential with M3Y nucleon-nucleon effective interactions and realistic ground state deformations including the octupole and hexadecupole ones [@San97]. We were mainly concerned with the study of the influence of the fragment deformations on the yields and we concluded that the occurence of the two mass-regions of cold fission is determined essentially by the ground state hexadecupole deformations. 0.5truecm 0.25truecm [ **2. Deformation Dependent Cluster Model** ]{} .25truecm In the present paper we consider a deformation dependent cluster model, similar to the one-body model used for the description of cluster radioactivity [@SG92]. The initial nucleus is assumed to be already separated into two parts, a heavy one and a light cluster, and the preformation factors for the fragments are not taken into account. An advantage of this model is that the barrier between the two fragments can be calculated quite accurately due to the fact that the touching configurations are situated inside of the barriers. The $Q$ values and the deformation parameters contain all nuclear shell and pairing effects of the corresponding fragments. The barriers were calculated using the double folding model for heavy ion interaction $$V_{F} ({\bf R}) = \int d{\bf r}_{1} d{\bf r}_{2}~ \rho_{1} ({\bf r}_{1}) \rho_{2} ({\bf r}_{2}) v({\bf s}) \eqno(1)$$ where $\rho_{1(2)}({\bf r})$ are the ground state one-body densities of the fragments and $v$ is the $NN$ effective interaction. The separation distance between two interacting nucleons is denoted by ${\bf s}={\bf r}_{1}+{\bf R}-{\bf r}_{2}$, where $R$ is the distance between the c.m. of the two fragments. We have choosen the $G$-matrix M3Y effective interaction which is representative for the so called local and density independent effective interactions [@BS97]. This interaction is particularly simple to use in folding models since it is parametrized as a sum of 3 Yukawa functions in each spin-isospin $(S,T)$ channel. In the present study the spin and spin-isospin dependent components have been neglected since for a lot of fragments involved in the calculation the ground state spins are unknown. The spin-spin component of the heavy-ion potential can be neglected here since it is of the order ${1\over {A_{1}A_{2}}}$. Only the isoscalar and isovector components have been retained in the present study for the central heavy ion interaction. The M3Y interaction is dominated by the one-nucleon knock-on exchange term, which leads to a nonlocal kernel. In the present case the nonlocal potential is reduced to a zero range pseudopotential $\hat{J}_{00} \delta({\bf s})$, with a strength depending slightly on the energy. We have used the common prescription [@BS97] $\hat{J}_{00}$ = -276 MeV$\cdot$fm$^{3}$ neglecting completely the small energy dependence. For example, the odd-even staggering in the $Q$-value for a fragmentation channel, which is tipically of the order $\Delta Q$=2 MeV, leads to a variation with $\Delta \hat{J}_{00}$=-0.005${\Delta Q}\over\mu$ MeV$\cdot$fm$^{3}$ with $\mu\approx$100. The one-body densities in (1) are taken as Fermi distributions in the intrinsic frame $$\rho({\bf r}) =\frac{\rho_0}{1+e^{\frac{r-c}{a}}}\eqno(2)$$ with $c=c_{0}(1+\sum_{\lambda\geq 2}\beta_{\lambda}Y_{\lambda 0}(\Omega))$. Only static axial symmetric deformations are considered. The half radius $c_{0}$ and the diffusivity $a$ are taken from the liquid drop model [@Mol95]. The normalization constant $\rho_0$ is determined by requiring the particle number conservation $$\int r^{2}dr~d\Omega\rho(r,\Omega) = A\eqno(3)$$ and then the multipoles are computed numerically $$\rho_{\lambda}(r) = \int d\Omega\rho(r,\Omega)Y_{\lambda 0}(\Omega). \eqno(4)$$ Once the multipole expansion of the density is obtained, the integral in (1) becomes $$V_{F}({\bf R},\omega_{1},\omega_{2}) = \sum_{\lambda_{1}\mu_{1}\lambda_{2}\mu_{2}} D_{\mu_{1}0}^{\lambda_{1}}(\omega_{1}) D_{\mu_{2}0}^{\lambda_{2}}(\omega_{2}) I_{\lambda_{1}\mu_{1}\lambda_{2}\mu_{2}} \eqno(5)$$ where [@CL92] $$I_{\lambda_{1}\mu_{1}\lambda_{2}\mu_{2}} = \sum_{\lambda_{3}\mu_{3}}B_{\lambda_{1}\mu_{1}\lambda_{2}\mu_{2}}^ {\lambda_{3}\mu_{3}} \int r_{1}^{2}dr_{1} r_{2}^{2}dr_{2} \rho_{\lambda_{1}}(r_{1}) \rho_{\lambda_{2}}(r_{2}) F_{\lambda_{1}\lambda_{2}\lambda_{3}}^{v}(r_{1},r_{2},R) \eqno(6)$$ and $$F_{\lambda_{1}\lambda_{2}\lambda_{3}}^{v}(r_{1},r_{2},R) = \int q^{2}dq{\tilde v}(q) j_{\lambda_{1}}(qr_{1})j_{\lambda_{2}}(qr_{2})j_{\lambda_{3}}(qr_{3}). \eqno(7)$$ Above, $D_{\mu 0}^{\lambda}(\omega)$ stands for the Wigner rotation matrix describing the orientation $\omega$ of the intrinsic symmetry axis with respect to the fixed frame, ${\tilde v}(q)$ denotes the Fourier transform of the interaction and $j_{\lambda}$ are the spherical Bessel functions. The matrix $B$ in (6) is defined in [@CL92] and contains selection rules for coupling angular momenta. Only $\lambda_{1}+\lambda_{2}+\lambda_{3}= $even, are allowed. When $\beta_{\lambda}\neq 0$, $\lambda=2,3,4$ for both fragments, the sum in (5) involves 36 terms for a nose-to-nose configuration with $\lambda_{3}\le 6$. For most of the fragmentation channels studied here, large quadrupole, hexadecupole, and occasionally octupole deformations are involved. Therefore a Taylor expansion method for obtaining the density multipoles turns out to be unsuitable. On the other hand, a large quadrupole deformation induces according to (4) nonvanishing multipoles with $\lambda$=4 and 6 even if $\beta_4$=$\beta_6$=0. Therefore for a correct calculation of (4), a numerical method with a truncation error of order O$(h^7)$ is needed in order to ensure the orthogonality of spherical harmonics with $\lambda\le 6$. Performing the integrals (6) and (7) we have used a numerical method with a truncation error of the order O$(h^9)$. All short range wavelength ($q\le 10$ fm$^{-1}$) have been included and particular care has been taken to ensure the convergence of the integrals with respect to the integration step and the range of integration. At the scission configuration two coaxial deformed fragments in contact at their tips were assumed. For quadrupole deformations we choose two coaxial prolate spheroids due to the fact that the prolate shapes are favoured in fission. It is known that for each oblate minimum always corresponds another prolate minimum. For pear shapes, i.e. fragments with quadrupole and octupole deformations, we choose opposite signs for the octupole deformations, i.e. nose-to-nose configurations (see Fig.1). For hexadecupole deformations we choose only positive signature, because it leads to a lowering of the barriers in comparison with negative ones and consequently they are much more favoured in fission (see Fig.2). In order to ilustrate the influence of deformations on the barriers we displayed in Fig.3 the M3Y-folding multipoles for $^{106}$Mo and $^{146}$Ba with all deformations included. The octupole component is large in the interior but gives negligible contribution in the barrier region in contrast to the hexadecupole one. Next, in Fig.4 we are illustrating for the same partners the cumulative effect of high rank multipoles on the barrier. 0.5truecm 0.25truecm [ **3. Cold Fission Binary Isotopic Yields**]{} .25truecm We should like to stress again that in our simple cluster model the preformation factors for different channels are neglected, i.e. we use the same assault frequency factor $\nu$ for the collisions with the fission barrier for all fragmentations. It is generally known that the general trends in alpha decay of heavy nuclei are very well described by barrier penetrabilities, the preformation factors becoming increasingly important only in the vicinity of the double magic nucleus $^{208}$Pb. On the other hand the cold binary fragmentation of $^{252}$Cf was also reasonably well described using constant preformation factors [@San96; @FSCG93]. However in this case too, as we shall see later, around the double-magic nucleus $^{132}$Sn the preformation turn out to be of capital importance. Eventually, as the experimental data become more accurate we would be able to extract some fragment preformation factors and discuss the related nuclear structure effects. In the laboratory frame of reference the $z$-axis was taken as the initial fissioning axis of the two fragments, with the origin at their point of contact. The potential barriers $V_{F} - Q_{LH}$ between the two fragments are high but rather thin with a width of about 2 to 3 fm. As an illustration, we show in Fig.5 a typical barrier between $^{146}$Ba and $^{106}$Mo, as a function of the distance $R_{LH}$ between their center of mass. Here $Q_{LH}$ is the decay energy for the binary fragmentation of $^{252}$Cf. For the two fragments, the exit point from their potential barrier is at $R_{LH}$ typically between 16 and 17 fm (see Fig.5) which supports our cluster model. The penetrabilities through the double-folded potential barrier between the two fragments were calculated by using the WKB approximation $$P = \exp \left\lbrace -{2 \over \hbar} \int_{s_{i}}^{s_{o}} \sqrt{~ 2 \mu~ [~V_{F}(s)-Q_{LH}~]~}~~ds \right\rbrace \eqno(8)$$ where $s$ is the relative distance, $\mu$ is the reduced mass and $s_{i}$ and $s_{o}$ are the inner and outer turning points, defined by $V_{F}( s_{i} ) = V_{F}( s_{o} ) = Q_{LH} $. The barriers were computed with the LDM parameters $a_p=a_n$=0.5 fm, $r_{0p}=r_{0n}=(R-{1\over R})A^{-1/3}~$fm with $R=1.28A^{1/3}+0.8A^{-1/3}-0.76$. Accurate knowledge of $Q$ values is crucial for the calculation, since the WKB penetrabilities are very sensitive to them. We obtained the $Q$ values from experimental mass tables [@WAH88], and for only a few of the fragmentations the nuclear masses were taken from the extended tables of Möller et al. [@Mol95] computed using a macroscopic-microscopic model. Let us consider for the beginning only the relative isotopic yields corresponding to true cold (neutronless) binary fragmentations in which all final nuclei are left in their ground state. These relative isotopic yields are given by the expression ($A_{1}=A_{L},A_{2}=A_{H}$) $$Y ( A_{1}, Z_{1} ) = { P ( A_{1}, Z_{1} ) \over \sum_{A_{1} Z_{1}} P ( A_{1}, Z_{1} ) } ~~\cdot \eqno(9)$$ As we mentioned above the fragment deformations were choosed to be the ground state deformations of Möller et al.[@Mol95], computed in the frame of the macroscopic-microscopic model. In Fig.6 we represented separately these deformations for the light $A_{L}$ and heavy $A_{H}$ fragments for odd and even charge $Z$. We can see that the light fragments, have mainly quadrupole deformations in contrast to the heavy fragments, which have all types of deformations. The octupole deformations are non-zero in a small heavy fragment mass number region 141 $\leq A_{H}\leq 148$. The fragments with mass number $A_{L}\leq 92$ and $A_{H}\leq 138$ are practically spherical. The computed M3Y-fission barriers heights, for different assumptions: no deformations, including the quadrupole ones, including the quadrupole and octupole ones and for all deformations, together with the corresponding $Q$-values are represented in Fig.7 for odd $Z$ and even $Z$ separetely. We notice the large influence due to the quadrupole deformations but also the hexadecupole ones are lowering the barriers very much. The octupole deformations in the mass region $141\leq A_{H}\leq 148$ have a smaller effect as we expected. This is a illustration of the difference between cluster radioactivity, which is due only to the large $Q$-values and the cold fission which is due mainly to the lowering of the barriers due to the fragment deformations. Both processes are cold fragmentation phenomena. The computed yields in percents, for the splittings represented by their fragment deformation parameters in Fig.6 or by their barrier heights in Fig.7, are given in Fig.8 for spherical fragments ($\beta_{i}=0$), for quadrupole deformations ($\beta_{2}$) and for all deformations ($\beta_{2}+\beta_{3}+\beta_{4}$) at zero excitation energy. We can see that when the fragments are assumed to be spherical the splittings with the highest $Q$-values, which correspond to real spherical heavy fragments(see Fig.6), i.e. for charge combinations $Z_1/Z_2=$ 48/50, 47/51 and 46/52 are the predominant ones. As we mentioned before this situation is similar with the cluster radioactivity were the governing principle is the $Q$-value. Due to the staggering of $Q$-values (see Fig.7) the highest yields are for even-even splittings. By including the $\beta_2$ deformations few asymmetric splittings exists. For all deformations more asymmetric yields appear. Now the principal yields are for $Z_{1}/Z_{2}$= 38/60, 40/58, 41/57 and 42/56 along with 44/54, 46/52 and 47/51. This is due to the fact that the influence of the fragment deformations on the yields overcome the influence of $Q$-values in the more asymmetric region. This illustrate the fact that cold fission is a cold rearrangement process in which all deformations are playing the main role and not the $Q$-values. The staggering for odd $Z$ fragmentations like $Z_{1}/Z_{2}$ = 39/59, 41/57, 43/55, 45/53 and 47/51 or odd $N$ fragmentations is recognized at first glance. However, by the introduction of the density levels this staggering is reversed. The largest yields will be for odd $Z$ and/or $N$ fragmentations. In the next figure we represented the mass yields $Y_{A_{2}}=\sum_{Z_{2}}Y(A_{2},Z_{2})$ (Fig.9) for spherical fragments ($\beta_i$=0), for quadrupole deformations ($\beta_2\neq$0) and for all deformations ($\beta_i\neq$0). We can see in the spherical case that the main mass yields are centered around $A_{2}$=132. All these heavy fragments are spherical or nearly spherical (with a small prolate deformation) and have high-$Q$ values. Since other spherical fragments does not arise in the yields diagram it occurs that in the spherical case the $Q$-value is the dictating principle. When we turn on the quadrupole deformation a rearrangent in this spherical region takes place. The yield corresponding to $A_{2}$=132 is still important but the one for $A_{2}$=134 takes over although the maximum decay energy of the first mass split $Q_{max}$ is larger than that of the former. In this case the larger quadrupole deformation of the light partner decides the augmentation of the $A_{2}$=134 yield. When we include the higher multipole deformations, i.e. octupole and hexadecupole deformations the yields diagram will change drastically over the whole mass range. First of all, in the spherical region the mass-splitings yields $A_{2}$=132, 134 are lowered whereas their odd neighbours are augmented. Once again this is a consequence of the fact that the hexadecupole deformations of the odd light partners are slightly larger. But the most important change occurs in the mass region $A_{2}$=138$\div$156 where a whole bunch of splittings show up with yields greater than 0.01$\%$. This is, beyond any doubt, an effect due to the hexadecupole deformations. As can be infered from Fig.6 the above mentioned mass region is characterized by noticeable values of the hexadecupole deformation. Before adding the hexadecupole deformation this region was completely desertic whereas after the inclusion of $\beta_4$ the most pronounced peaks are $A_2$= 138, 140, 146, 150 and 154. It is the place to mention that the first mass region, in the cold fission of $^{252}$Cf reported in the paper of Gönnenwein et al.[@Gon97] coincides with the range obtained by us employing a deformation dependent cluster model. However in order to reproduce completely the experimental data we have to underline the elements that have to be supplied further in our model. First, in the spherical region, the experiment claim a mass region of cold fission centered around $A_2$=132, instead of $A_2$=134 as we obtained. However this misfit was to be expected since as we mentioned in the beginning of our paper we didn’t included the preformation factors. In the case of the doubly magic nucleus $^{132}$Sn this assumption proves to be unsatisfactory. As has been advocated by the Tübingen group [@Gon97] this is a possible manifestation of heavy-cluster decay. Therefore it is very likely that in this case the preformation factor, which multiplies the penetrability, is larger than for the neighbouring nuclei, which could then account for the discrepancies between our calculations and experimental data. However an encouraging experimental point which supports our calculations is the fact that the even masses 134 and 136 are accompanying the leading yield for 132. In fig.10 we compare the total yields for 132 (left side) and 134 (right side). We see that the $Z$-splitting corresponding to the spherical $^{134}$Te dominates in all the three cases, because, as we mentioned earlier its light partner has a sensitive quadrupole deformation and a non-vanishing hexadecupole one. Its $Z$ partner $^{134}$Sn has a smaller hexadecupole deformation. The same reasoning apply to $A_2$=136. Therefore it could be possible that in the case of these nuclei the deformation dictates the yield magnitude rather than the magic number in protons or neutrons. The experimental determination of the double fine structure in this region will, hopefully, clarify the situation. The [*hexadecupole deformed*]{} region, extending from 138 to 156, obtained in the frame of our cluster model, presents also some discrepancies compared to the experimental findings. The main problem that we faced here concerns the odd-even effect which seems to be very strong in this region according to the Tübingen group [@MCGP96]. The things can be understood as follows: In the vicinity of the ground state, the level densities of odd mass nuclei are much larger than for even nuclei and consequently it will be more probable to observe cold fission for odd-odd mass splits in comparison to even-even mass splits. Since in our present calculations the level density of fragments is not taken into account our results points to an enhancement of even-even mass splits with respect to the odd-odd mass splits. In a preceding paper [@San97] the effect of level density was incorporated in the calculation of yields by means of the Fermi Back-shifted Model valide also for small excitation energies. In order to get a rough idea of how the odd-even effect influence the yields, we simply shift the decay energy by the fictious ground-state position $\Delta$ taken from the global analysis of Dilg et al. [@DSVU73], $Q^{*}=Q-\Delta$. In fig.11 we represented the same thing like in Fig.10 but with the above mentioned shift in the $Q$-value. It is obvious from the inspection of this figure that except $A_{2}$=138 , the odd splittings take over, in agreement with the experimental data. It is worthwile to stress once again that in our view the mass region extending from 138 to 156 the hexadecupole deformation is the leading mechanism responsible for the cold fragmentation of $^{252}$Cf. The lowering of the barriers due to hexadecupole deformation increase dramatically the penetrabilities and eventually the yields. In figure 12 we represented the yields for the $Z$-splittings of $A_2$=143. Comparing the first two cases we see that the yields are almost unsensitive to quadrupole deformation. When the hexadecupole deformation is included the distribution changes, all the yields being shifted uniformly (in the log scale) towards magnitudes four times larger. It is worthwile to notice before ending this section that the octupole deformations are not inducing the tremendous changes that the hexadecupole does. .5truecm .75truecm [ **4. Discussions and Conclusions** ]{} .25truecm The deformation dependent cluster model which we used in this paper for calculating the isotopic yields associated to cold binary fission, predicts a large number of favored binary splittings in which one or both fragments are well deformed in their ground states. For cold binary fission the initial scission configurations are known : the fragment deformations should be essentially those of the ground state deformations. The main result obtained in our paper represents the theoretical confirmation of the existence of two distinct regions of $^{252}$Cf cold fission. The results indicate two different mechanisms. In the heavy mass region situated between 138 and 156, the hexadecupole deformation gives rise to a large number of splittings. Here the shell closure in neutrons or protons seems to not be involved. Although the shell effects should play an important role in the odd-even differences by enhancing the odd-odd mass splits with respect to the even-even one, our result emphasize that the fragments are emitted with the deformations corresponding to those of the ground state. In the spherical region our results give only a hint of the importance of the magic nucleus $^{132}$Sn which is susceptible to be produced in a heavy clusterization process, similar to that for light clusters [@SG77]. Here the decay mechanism should be similar to the light cluster radioactivity, the daughter nucleus $^{132}$Sn being traded for $^{208}$Pb and the heavy cluster $^{120}$Cd for $^{14}$C. The results reported in this paper are pointing to the importance of deformations included in the cold fission model since the $Q$-value seems to be no longer the absolute ruler of the process like in the case of cluster radioactivity. In the future the investigations should be extended in such a way to explain also the yields structure at finite excitation energy. 0.5truecm [99]{} 1.5truecm J.C.D.Milton and J.S.Fraser, Can.J.Phys. $\bf 40$, 1626 (1962). C.Guet, M.Ashgar, P.Perrin and C.Signarbieux, Nucl.Instr.and Meth. [**150**]{}, 189 (1978) F. Gönnenwein and B. Borsig, Nucl. Phys. $\bf A530$, 27 (1991). F.-J. Hambsch, H.-H. Knitter and C. Budtz-Jorgensen, Nucl. Phys.$\bf A554$, 209 (1993). A. Benoufella, G. Barreau, M. Asghar, P. Audouard, F. Brisard, T.P. Doan, M. Hussonnois, B. Leroux, J. Trochon and M.S. Moore, Nucl. Phys. $\bf A565$, 563 (1993). W. Schwab, H.-G. Clerc, M. Mutterer, J.P. Theobald and H. Faust, Nucl. Phys. $\bf A577$, 674 (1994). J.H. Hamilton et al., J. Phys. G: Nucl. Part. Phys. $\bf 20$, L85 (1994). G.M. Ter-Akopian et al., Phys. Rev. Lett. $\bf 73$, 1477 (1994). A. Săndulescu, A. Florescu, F. C\^ arstoiu, W. Greiner, J.H. Hamilton,A.V. Ramayya and B.R.S. Babu, Phys. Rev. $\bf C~~54$, 258 (1996). A. Săndulescu and W. Greiner, Rep. Progr. Phys. $\bf 55$, 1423 (1992). A. Săndulescu, A. Florescu and W. Greiner, J. Phys. G: Nucl. Part.Phys. $\bf 15$, 1815 (1989). A.V. Ramayya et al., Third Int. Conf. on Dynamical Aspects of Nuclear Fission (August 30-September 4) 1996, Casta-Papiernicka, Slovak Republic. A.Săndulescu, F.C\^ arstoiu, Ş.Mişicu, A. Florescu, A.V.Ramayya, J.H.Hamilton and W.Greiner, J. Phys. G : Nucl. Part. Phys. $\bf 24$, 181 (1997) A.V.Ramayya et al. (submitted to Phys.Rev.Lett.) A.M" oller, M.Cr" onni, F.G" onnenwein, G.Petrov, Int.Conference on Large Scale Collective Motion of Atomic Nuclei, Brolo, 1996. F.Gönnenwein, A.M" oller, M.Cr" onni, M.Hesse, M.Wöstheinrich, H.Faust, G.Fioni and S.Oberstedt, Il Nuovo Cimento (1997) (to be published). M.E. Brandan, G.R. Satchler , Phys. Rep. $\bf 285$, 143 (1997). P. Möller, J.R. Nix, W.D. Myers and W.J. Swiatecki, At.Data Nucl. Data Tables $\bf 59$, 185 (1995). F. C\^ arstoiu and R.J. Lombard, Ann. Phys. (N.Y.) $\bf 217$, 279 (1992). A. Florescu, A. Săndulescu, C. Cioacă and W. Greiner, J.Phys. G: Nucl. Part. Phys. $\bf 19$, 669 (1993). A.H. Wapstra, G. Audi and R. Hoeckstra, At. Data Nucl. Data Tables $\bf 39$, 281 (1988). W. Dilg, W. Schantl, H. Vonach and M. Uhl, Nucl.Phys. $\bf A217$, 269(1973). A. Săndulescu and W. Greiner, J. Phys. G : Nucl. Phys. $\bf 3$, L189 (1977). 7truecm **Figure Captions** 1truecm ${\bf Fig.~1.}$ Density plots of $^{106}$Mo and $^{146}$Ba fragments, placed at $R$=15 fm, considered with quadrupole and octupole deformations. In the upper part are represented the prolate-prolate, oblate-prolate positions and in the lower part two pear shapes nose to back and nose to nose. The positions are given by the deformation signs. 1.0truecm ${\bf Fig.~2.}$ Same as for Fig.1. The influence of different signs of hexadecupole deformations on $^{106}$Mo and $^{146}$Ba densities in the presence of large quadrupole and octupole deformations. The penetrability is maximized for $\beta_{4}>$0 configurations. 1.0truecm ${\bf Fig.~3.}$ The influence of the M3Y-folding multipoles on the barrier between $^{106}$Mo and $^{146}$Ba. Notice that the main effect is due to $\lambda_{3}=2$. The influence of $\lambda_{3}=3$ is large but less important in the barrier region compared with the induced deformations $\lambda_{3}=5$ and $\lambda_{3}=6$ 1.0truecm ${\bf Fig.~4.}$ The cumulative effect of high rank multipoles on the barrier between $^{106}$Mo and $^{146}$Ba. We considered the deformations $\beta_{3}$ and $\beta_{4}$ much larger than the real ones in order to illustrate the effect of deformations. 1.0truecm ${\bf Fig.~5.}$ The barrier between $^{146}$Ba and $^{106}$Mo as a function of the distance $R_{HL}$ between their centers of mass. By $Q_{LH}$ we denote the decay energy. 1.0truecm ${\bf Fig.~6.}$ The assumed $\beta_{2}$, $\beta_{3}$, $\beta_{4}$ ground state fragment deformations [@Mol95]. We can see that the light fragments $(Z_1,A_1)$ have mainly quadrupole deformations in contrast to the heavy fragments $(Z_2,A_2)$. The octupole deformations are existing in a small mass region 141$\leq A_2\leq$148 whereas the hexadecupole deformations are important in the region 138$\leq A_2\leq$158. The fragments with masses $A_1\leq$94 and $A_2\leq$138 are practically spherical. 1.0truecm ${\bf Fig.~7.}$ The barrier heights for all considered fragmentations channels represented for different charges $Z_1$ and mass numbers $A_1$ of the light fragment. 1.0truecm ${\bf Fig.~8.}$ The true cold fission yields in percents for all fragmentations channels computed with the LDM parameters, for spherical nuclei, with the inclusion of quadrupole deformations and with all deformations at zero excitation energy. 1.0truecm ${\bf Fig.~9.}$ The mass yields $Y_{A_2}=\sum_{Z_2}Y(A_2,Z_2)$ in percents, as a function of light fragment mass computed with LDM parameters. Calculations without deformations ($\beta_{2,3,4}$=0) enhance only the spherical region $ A_2 \leq$ 136; the inclusion of quadrupole deformations ($\beta_2\neq$0) enhances the yield with $A_2 =$134; for all deformations there are two main mass yields regions, i.e. 133$\leq A_2 \leq$136 and 138$\leq A_2 \leq$156. 1.0truecm ${\bf Fig.~10.}$ The yields for the $Z$-splittings of $A_2$=132, 134 in percents computed with LDM-parameters. 1.0truecm ${\bf Fig.~11.}$ The mass yields $Y_{A_2}=\sum_{Z_2}Y(A_2,Z_2)$ in percents, as a function of light fragment mass computed with LDM parameters with the decay energy modified $Q^{*}=Q-\Delta$. The odd-odd mass splitings are this time favoured. 1.0truecm ${\bf Fig.~12.}$ The yields for the $Z$-splittings of $A_2$=143 in percents computed with LDM-parameters. Calculations without deformations and with the inclusion of quadrupole deformation give nearly the same yields. The inclusion of hexadecupole deformation increase uniformly by 4 orders of magnitude the yields.

--- abstract: | We show a deterministic secure direct communication protocol using single qubit in mixed state. The security of this protocol is based on the security proof of BB84 protocol. And it can be realized with today’s technology. PACS numbers: 03.67.Dd, 03.67.Hk, 03.67.-a address: | Wuhan Institute of Physics and Mathematics, The Chinese Academy of\ Sciences, Wuhan, 430071, People’s Republic of China author: - 'CAI Qing-Yu ( )[^1] and LI Bai-Wen ( )' title: Deterministic secure communication without using entanglement --- Quantum key distribution (QKD) is a protocol which is $provably$ secure, by which private key bit can be created between two parties over a public channel. The key bits can then be used to implement a classical private key cryptosystem, to enable the parties to communicate securely. The basic idea behind QKD is that Eve cannot gain any information from the qubits transmitted from Alice to Bob without disturbing their states. First, the no-cloning theorem forbids Eve to perfectly clone Alice’s qubit. Secondly, in any attempt to distinguish between two non-orthogonal quantum states, information gain is only possible at the expense of introducing disturbance to the signal \[1\]. Based on the postulate of quantum measurement \[2\] and no-cloning theorem \[3\], different QKD protocols are presented \[4-7\]. However, these types of cryptographic schemes are usually nondeterministic. In Ref.\[8\], K. Boström and T. Felbinger presented a protocol, which allows for deterministic communication using entanglement. The basic idea of the ping-pong protocol is that one can encode the information locally on an EPR pair, but it has a nonlocal effect. In this paper, we show a secure communication protocol which is a deterministic secure direct communication protocol using single qubit in mixed state. This protocol is based on the property that non-orthogonal quantum states cannot be reliably distinguished \[2\]. One cannot simultaneously measure the polarization of a photon in the vertical-horizontal basis and simultaneously in the diagonal basis. It is well known one can prepare a photon in states $% \{|0>,|1>\}$ or $\{|\varphi _{0}>,|\varphi _{1}>\}$in its polarization degree of freedom, where $$\begin{aligned} |\varphi _{0} &>&=\frac{1}{\sqrt{2}}(|0>+|1>), \\ |\varphi _{1} &>&=\frac{1}{\sqrt{2}}(|0>-|1>).\end{aligned}$$ Denoting that $i\sigma _{y}=|0><1|-|1><0|$, it can be obtained: $$i\sigma _{y}|0>=-|1>,\text{ }i\sigma _{y}|1>=0,$$ and $$i\sigma _{y}|\varphi _{0}>=|\varphi _{1}>\text{, }i\sigma _{y}|\varphi _{1}>=-|\varphi _{0}>.$$ Suppose Alice want to obtain some information from Bob. First Alice selects state $|0>$ or $|\varphi _{0}>$ randomly with the probability $\frac{1}{2}$ every time. For an external person without Alice’s a prior knowledge, this qubit appears to be in a mixed state $\rho _{0}:$$$\rho _{0}=\frac{1}{2}|0><0|+\frac{1}{2}|\varphi _{0}><\varphi _{0}|.$$ Then Alice sends this qubit to Bob. Bob decides either to perform the operation $i\sigma _{y}$ on the travel qubit to encode the information ‘1’ or do nothing, i.e., to perform the operation $I=|0><0|+|1><1|$ to encode the information ‘0’. Then Bob sends the travel qubit back to Alice. Alice performs a measurement on this back qubit to gain the information Bob encoded. After Alice’s decoding measurement, she tells Bob she has received the back qubit through the public channel by one bit (This can be called as $% Alice^{\prime }s$ $receipt$. ). In this protocol, there are two communication modes, ‘$message$ $mode$’ and ‘$control$ $mode$’. By default, Bob and Alice are in message mode and the communication is described as above. With probability $c$, Bob switches the message mode to control mode. $% In$ $Control$ $Mode$. Instead of his encoding operation, Bob replaces the qubit he receives from Alice with a qubit that he completely randomly prepares in the state $|0>$ , $|1>$, $|\varphi _{0}>$ or $|\varphi _{1}>$ and sends this qubit to Alice. Alice performs her decoding measurement and in 50% of the case she uses the basis as Bob used. After Alice announces her receipt of the qubit, Bob announces that this has been a control run and he tells Alice which state he prepared. If Alice used the same basis as Bob and if she found a state different from Bob prepared then Eve is detected and the communication stops. This protocol can be described explicitly like this: > (1). Alice prepares one qubit in state $|0>$ or $|\varphi _{0}>$ randomly with record. > > (2). Alice sends this qubit to Bob. > > (3). Bob receives the travel qubit. He decides to the message mode (4m) or the control mode (4c) by chance. > > (4c). $Control$ $mode$. Bob replaces the qubit he receives from Alice with a qubit that he randomly prepares in the state $|0>$ , $|1>$, $|\varphi _{0}>$ or $|\varphi _{1}>$ and sends this qubit to Alice. Alice performs her decoding measurement. After Alice announces her receipt of the qubit, Bob announces that it is a control run this time and he tells Alice which state he prepared. If Alice used the same basis as Bob and if she found a state different from Bob prepared then Eve is detected and the communication stops. Else, Alice sends next qubit to Bob. > > (4m). $Message$ $mode$. Bob performs an operation on the travel qubit to encode information. He encodes the bit ‘0’ using by the operation $I$ and the bit ‘1’ by the operation $i\sigma _{y}$. Then Bob sends this travel qubit back to Alice. Alice measures the qubit to gain the message Bob encoded and sends her $receipt$ to Bob through public channel. > > (5). When all of Bob’s information is transmitted, this communication is successfully terminated. $Security$ $proof$. The basic idea behind QKD is the fundamental proposition: Eve can not gain any information from the qubits transmitted from Alice to Bob with out disturbing their state \[1\]. Consider that $% |\varphi >$ and $|\phi >$ are non-orthogonal quantum states Eve is trying to obtain information about. Without loss of generality that the process she uses to obtain information is to unitarily interact the state with an ancilla prepared in a standard state $|e_{0}>$. Assuming that this process does not disturb the states, then one obtains: $$\begin{aligned} U|\varphi &>&|e_{0}>=|\varphi >|e_{1}>, \\ U|\phi &>&|e_{0}>=|\phi >|e_{2}>.\end{aligned}$$ To acquire information about the different state, Eve would like $|e_{1}>$ and $|e_{2}>$ different. Since the inner products are preserved under unitary transformations, it must be that $$<\varphi |\phi ><e_{0}|e_{0}>=<\varphi |\phi ><e_{2}|e_{3}>.$$ Since $|\varphi >$ and $|\phi >$ are non-orthogonal, then it has $$<e_{2}|e_{3}>=<e_{0}|e_{0}>=1,$$ which implies that $|e_{2}>$ and $|e_{3}>$ must be identical. That distinguishing two non-orthogonal states would at least disturb one of them. To gain information, Eve has to know which operation Bob performed. First, she can attack the travel qubit in the line $A\rightarrow B$. And perform a measurement to acquire Bob’s information in line $B\rightarrow A$. Or she can take another strategy that she only performs a measurement after Bob’s operation to gain Bob’s information. No matter what strategy Eve uses, she has to attack the qubit in the line $B\rightarrow A$. We can see that our protocol in control mode is as the same as the BB84 protocol’s detection of Eve’s eavesdropping. Many works have been accomplished of the security proof of the BB84 protocol \[7, 9\]. Eve’s any attempt to eavesdrop the information will give a detection probability $d>0$. Taking into account the probability $c$ of a control run, the effective transmission rate is $r=1-c$. The probability of Eve’s eavesdropping one message transfer without being detected is \[8\] $$s(c,d)=\frac{1-c}{1-c(1-d)},$$ where $d(I_{0})$ is the detection probability in the control mode. After $n$ protocol run, the probability to successfully eavesdrop $I=nI_{0}(d)$, the probability to successfully eavesdrop becomes $$s(n,c,d)=(\frac{1-c}{1-c(1-d)})^{I/I_{0}}.$$ For $c>0$, $d>0$, this value decreases exponentially. In the limit $% n\rightarrow \infty $, we have $s\rightarrow 0$. So this protocol is asymptotically secure. In principle, the security can arbitrarily be improved by increasing the control parameter $c$ at cost of decreasing the transmission rate. In practice, we can use a small $c$, which will improves the efficiency of the communication. To realize a perfectly secure communication, we must abandon the direct transfer in favor of a key transfer \[8\]. Instead of transmitting the message directly to Alice, Bob will take a random sequence of $N$ bits from a secret random number generator. After a successful transmission, the random sequence is used as a shared secret key between Bob and Alice. Bob and Alice can choose classical privacy amplification protocols, which make it very hard to decode parts of the message with only some of the key bits given. So Eve has virtually no advantage in eavesdropping only a few bits. When Eve is detected, the transfer stops. Then Eve has nothing but a sequence of nonsense random bits \[13\]. In contrast to quantum key distribution protocol BB84 \[4\], our protocol provides a deterministic transmission of bits. It is possible to communicate the message directly from Bob to Alice. Essentially, this protocol is a special case of BB84 protocol. The essence of this protocol is that the communicators can freely select the message mode and the control mode. In BB84 protocol, when Alice and Bob want to transform $n$ bit message, it need about $4n$ qubit. On the other hand, comparing with the ‘ping-pong’ protocol, we use a single qubit to realize the deterministic secure direct communication instead of using entanglement. Also, there maybe a denial-of-service(DoS) attack in the line $A\rightarrow B$ \[14\]. But any method of message authentification can protect the protocol against man-in-the-middle attacks with a reliable public channel. In order to be practical and secure, a quantum key distribution scheme must be based on existing—or nearly existing—technology \[15\]. Experimental quantum key distribution was demonstrated for first time by Bennett, et al \[16\]. Since then, single photon source have been studied in recent years and a great variety of approaches has been proposed and implemented \[17-22\]. Today, several groups have shown that quantum key distribution is possible, even outside the laboratory. In principle, any two-level quantum system could be used to implement quantum cryptography (QC). In practice, all implementations have relied on photons. The reason is that their decoherence can be controlled and moderated. The technological challenges of the QC are the questions of how to produce single photons, how to transmit them, how to detect single photons, and how to exploit the intrinsic randomness of quantum processes to build random generators \[23\]. Considered the experimental feasibility, our protocol needs a single photon source and some linear optical elements and a single-photon detector. Recently, the full implementation of a quantum cryptography protocol using a stream of a single photon pulses generated by a stable and efficient source operating at room temperature was reported \[24\]. The single pulses are emitted on demand and the secure bit rate is 7700bits/s. And quantum logic operations using linear optical elements can be realized with today’s technology \[25\]. The implementation of the single-photon detection technology for quantum cryptography have been reported \[26\] and the values of $\sigma _{x},\sigma _{y},$ and $\sigma _{z}$ of a polarization qubit on a single photon can be ascertained \[27\].Considered the experimental feasibility, this protocol can be realized with today’s technology. It is explained that when this paper was completed, we see the protocol \[28\] presented by Deng $et$ $al$, which also is a secure direct communication, using Einstein-Podolsky-Rosen pair block. I thank Yuan-chuan Zou for useful discussion. This work is supported by the National Nature Science Foundation of China (Grant No. 10274094). References: =========== \[1\] Nielsen M. A. and Chuang I. L. 2000 Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, UK) \[2\] Peres A. 1998 Phys. Lett. A 128, 19; Duan L.-M. and Guo G.-C. 1998 Phys. Rev. Lett., 80, 4999-5002 \[3\] Dieks D. 1982 Phys. Lett. A, 92, 271-272; Wootters W. K. and Zurek W. H. 1982 Nature, 299, 802-803; Barnum H., Caves C. M. , Fuchs C. A., Jozsa R. and Schumacher B. 1996 Phys. Rev. Lett., 76, 2818-2821; Mor T. 1998 Phys. Rev. Lett., 80, 3137-3140 \[4\] Bennett C. H. and Brassard G. 1984, in $proceedings$ $of$ $the$ $IEEE$ $% International$ $Conference$ $on$ $Computers$, $Systems$ $and$ $% %TCIMACRO{\func{Si}} %BeginExpansion \mathop{\rm Si}% %EndExpansion gnal$ $\Pr oces\sin g$, Bangalor, India, (IEEE, New York), p175-179. \[5\] Ekert A. 1991 Phys. Rev. Lett. 67, 661 \[6\] Bruss D. 1998 Phys. Rev. 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B. 2002 Phys. Rev . Lett., 89, 137901 \[26\] Ribordy G., Gautier J. D., Zbinden H. and Gisin N. 1998 Appl. Opt. 37, 2272-2277 ; Bourennane M., Gibson F., Karlsson A., Hening A., Jonsson P., Tsegaye T., Ljunggren D. and Sundberg E. 1999 Opt. Express 4, 383; Bethune D. and Risk W. 2000 IEEE J, Quantum Electron. 36, 340 ; Hughes R., Morgan G. and Peterson C. 2000 J. Mod. Opt. 47, 533; Ribordy G., Gautier J.-D., Gisin N., Guinnard O. and Zbinden H. 2000 J. Mod. Opt. 47, 517 \[27\] Vaidman L., Aharonov Y. and Albert D. Z. 1987 Phys. Rev. Lett. 58, 1385 ; Schulz O., Steinh$\stackrel{..}{u}$bl R., Weber M., Englert B. G., Kurtsiefer C. and Weinfurter H. 2003 Phys. Rev. Lett. 90, 177901 \[28\] Deng F.-G., Long G. L., and Liu X.-S. 2003 Phys. Rev. A 68 042317 [^1]: Corresponding author. Tel: +862787199109(O); fax: +862787198200 E-mail: qycai@wipm.ac.cn

--- abstract: | A formalism for electronic-structure calculations is presented that is based on the functional renormalization group ([[FRG]{}]{}). The traditional [[FRG]{}]{} has been formulated for systems that exhibit a translational symmetry with an associated Fermi surface, which can provide the organization principle for the renormalization group (RG) procedure. We here advance an alternative formulation, where the RG-flow is organized in the energy-domain rather than in $k$-space. This has the advantage that it can also be applied to inhomogeneous matter lacking a band-structure, such as disordered metals or molecules. The energy-domain [[FRG]{}]{} ([[$\epsilon$FRG]{}]{}) presented here accounts for Fermi-liquid corrections to quasi-particle energies and particle-hole excitations. It goes beyond the state of the art [[$\text{GW}$]{}]{}-BSE, because in [[$\epsilon$FRG]{}]{} the Bethe-Salpeter equation (BSE)is solved in a self-consistent manner. An efficient implementation of the approach that has been tested against exact diagonalization calculations and calculations based on the density matrix renormalization group is presented. Similar to the conventional [[FRG]{}]{}, also the [[$\epsilon$FRG]{}]{} is able to signalize the vicinity of an instability of the Fermi-liquid fixed point via runaway flow of the corresponding interaction vertex. Embarking upon this fact, in an application of [[$\epsilon$FRG]{}]{} to the spinless disordered Hubbard model we calculate its phase-boundary in the plane spanned by the interaction and disorder strength. Finally, an extension of the approach to finite temperatures and spin $S{=}1/2$ is also given. author: - Christian - Ferdinand bibliography: - 'frg\_p1.bib' date: 'May 14, 2016' title: A functional renormalization group approach to electronic structure calculations for systems without translational symmetry --- Introduction ============ Correlation effects are the driving agent behind a great many of the phenomena that are comprising the contemporary physics of condensed matter systems. As long as interactions are not too strong, such correlation phenomena can be understood in terms of an effective single particle picture as it is provided, e.g., by the Fermi-liquid theory. In this weakly correlated limit, the density-functional theory (DFT) can yield useful, often quantitative results for the electronic structure of crystalline or molecular matter. Where DFT fails to be quantitative, post-DFT correction schemes have been introduced that can significantly improve the accuracy, in particular with respect to (charged) excitation energies. [@bookBechstedt15] As a particularly successful example, we mention the [[$\text{GW}$]{}]{}-approximation motivated by conventional diagrammatic perturbation theory. [@HedinPaper; @onida02; @vanSetten15] At low enough temperatures most real materials undergo a transition into a correlated low-temperature phase, such as a magnet or a superconductor. Such phenomena are usually at the verge of applicability of perturbative methods. Still, perturbation theory can be very useful, because it often signalizes the existence of such phase-transitions via divergent diagrams. In recent years a powerful method has been devised to deal with stronger correlations, the functional renormalization group ([[FRG]{}]{}), that has proven particularly successful in this respect. [@FrgReviewMetzner; @SalmhoferBook] It can be (roughly) thought of as a systematic extension of [[$\text{GW}$]{}]{}-theory and its Bethe-Salpeter-type generalizations. Because it monitors the RG-flow of a representative set of interaction vertices, [[FRG]{}]{} can predict in an unbiased way the leading Fermi liquid instabilities together with estimates for the corresponding phase boundaries. Beyond phase boundaries, the [[FRG]{}]{} is capable to predict a variety of other physical observables including Luttinger-liquid parameters,[@FRGLuttinger1] Fermi-liquid corrections,[@FRGFermiLiquidCorrection1; @FRGFermiLiquidCorrection2] and spin susceptibilities.[@FRGSpin; @FRGSpin2; @ClusterFRG] Correspondingly, the [[FRG]{}]{} has been applied to a variety of systems, e.g., the Hubbard model in various parameter regimes,[@FRGHubbard; @FRGHubbard2; @FRGHubbard3; @FRGHubbard4; @FRGHubbardKatanin; @FRGHubbard5; @FRGAttractiveHubbbardSF] single impurity models,[@FRGImpurity; @FRGImpurity2] spin–,[@FRGSpin; @FRGSpin2; @FRGSpin3; @iqbal16; @FRGSpin4; @ClusterFRG] and quantum critical systems,[@FRGAFCriticalPoint; @FRGFermiSurfaceReconstruction; @FRGDensityWaveMultiCritical] and superfluids.[@FRGSuperfluid; @FRGAttractiveHubbbardSF] For an overview we direct the reader to Refs. . Motivation underlying this work ------------------------------- Good progress has been made in electronic structure calculations for real materials as well as for model Hamiltonians. Still, we believe that there is room for improvement. With an eye on ab-initio calculations, we observe that it is still very challenging to accurately calculate, e.g., ionization energies and electron affinities of small molecules or atom clusters. Quantitative results from DFT can be obtained only via procedures, such as $\Delta$SCF, that rely on error cancellation. The [[$\text{G}_0\text{W}_0$]{}]{}-method in this respect seems more reliable; benchmarks for different implementations have recently become available. [@bruneval13; @koerbel14; @vanSetten15] The [[$\text{G}_0\text{W}_0$]{}]{}-approximation is not selfconsistent, however, and partly for this reason it comes in many flavors. The development and testing of self-consistent and computationally affordable [[$\text{GW}$]{}]{}-schemes is currently under way. [@rostgaard10; @koerbel14; @knight16; @vanSetten16] Even more challenging it is to calculate the dynamical response, e.g., the optical gap or the absorption spectrum. The traditional time-dependent DFT, such as TDLDA, tends to underestimate optical gaps in solids by $\sim eV$. Interestingly, it can quantitatively reproduce excitation gaps of small molecules when combined with long range functionals, especially if they are optimally tuned. [@kronik12; @refaely15] In combination with [[$\text{GW}$]{}]{}-theory one solves the Bethe-Salpeter equation to find the optical properties. Due to the computational complexity, one usually keeps only the simplest non-trivial vertex corrections ([[$\text{GW}{+}\text{BSE}$]{}]{}). The approach yields results often with a typical accuracy of a few hundred meV, see Ref. for a recent overview and Ref. for benchmarks. In some cases much larger deviations have been reported, however, calling for a further validation of [[$\text{GW}{+}\text{BSE}$]{}]{}.[@hirose15] State of the art [[$\text{GW}$]{}]{}-implementations can be found in many standard band structure codes, e.g., Refs. (i) In this situation it seems advisable to go a step forward and explore more complete approximation schemes that in principle could go significantly beyond the lowest order BSE-technology by incorporating, e.g., a self-consistent evaluation of screening in the presence of vertex corrections. The extended scheme would thus provide a laboratory for testing the current BSE-technology against a more accurate higher order method. Our work is underlying the idea that the [[FRG]{}]{} could be an interesting candidate for such a more advanced electronic structure theory. A certain limitation of the [[FRG]{}]{} in its current formulation is that it is applicable to homogeneous (clean) systems, only. It thus could form the basis for improved band structure calculations for crystalline matter, but it will be inapplicable to the more inhomogeneous systems that we are mostly interested in, here. Specifically, the program lined out before in (i) cannot be followed within the present framework of [[FRG]{}]{} for molecules or disordered metals. From a methodological point of view, we therefore consider it an interesting challenge modifying the traditional $k$-space [[FRG]{}]{} ([[$k$FRG]{}]{}) into a new tool – energy-domain [[FRG]{}]{} ([[$\epsilon$FRG]{}]{}) – that can also describe the phases and the corresponding transitions in weakly correlated, inhomogeneous matter. (ii) To elaborate on the perspective for the [[$\epsilon$FRG]{}]{}, we mention two research fields with prospective applications. (1) Quantum chemistry calculations could benefit from [[$\epsilon$FRG]{}]{} in a range of system sizes where high-precision calculations, e.g. the couple-cluster approach, are computationally not affordable any more. (2) The [[$\epsilon$FRG]{}]{} might prove a useful tool for investigating the effect that disorder has on those quantum phase transitions that have already been investigated in the clean limit. [@FrgReviewMetzner] Conversely, there is the intriguing prospect to study the effect that weak interactions have on disordered systems with wavefunctions that are localized due to quantum interference. [@evers2008] Motivated by (2), we here present an implementation of an [[$\epsilon$FRG]{}]{} that can operate on disordered model Hamiltonians. Our goal is to explore the potential of the approach as a higher-order method for studies of weakly correlated fermions in generic environments lacking translational symmetries. FRG for systems without translational symmetries – [[$\epsilon$FRG]{}]{} ------------------------------------------------------------------------ Consider a fermion system with a Hamiltonian that decomposes into a one-body and a two-body part, $$\hat H = \hat H_0 + \hat U.$$ The non-interacting part, $\hat H_0$, includes a static potential. It is considered generic in the sense that it does not exhibit translational symmetries; its single-particle eigenstates $\ket{\alpha}, \alpha=1,\ldots,N$ are far from plane waves. They can be thought of as wavefunctions of a strongly disordered metal or as molecular orbitals, e.g., of a generic organic molecule. We will leave the interacting part, $\hat U$, unspecified for the time being. ### Excursion: Hedin’s equations and [[FRG]{}]{} As was recognized by L. Hedin, in order to compute physical observables in the presence of two-body interactions, one can solve a set of self-consistent non-linear matrix equations for the exact (causal) Green’s function, the corresponding self-energies and vertex-functions. [@bechstedtBook15; @vignaleBook] Unfortunately, Hedin’s equations are impossible to solve exactly even with todays computational resourses for realistic system sizes. Difficulties arise because of (a) the complicated nature of the matrix-kernels and (b) the very large dimensions of the matrices involved, especially of the interaction vertex $\Gamma$. The ubiquitous approximation strategy therefore is truncating the matrix-equations so that the kernels simplify and reducing the effective matrix size by grading the many-particle Hilbert space. Eventually, also the [[FRG]{}]{} relies on such a truncation scheme. However, even the truncated set of equations is very difficult to solve. Partially, this is because the requirement of the solution being self-consistent. Here the idea of the renormalization group (RG) with the corresponding flow-equation comes in. Speaking in a lose manner, what corresponds to an iteration cycle in conventional solutions of self-consistency problems is in the framework of [[FRG]{}]{} replaced by a consecutive integration of a differential equation that establishes the RG-flow. The initializing guess of the iteration cycle corresponds to the initialization of the flow equation; the flow stops once the (self-consistent) fixed-point has been reached. Advantages of the RG-approach over self-consistency cycles are (a) that uncertainties related to the proper choice of the starting guess are removed and (b) there is a clear physical interpretation in terms of “runaway flow” even when the numerical integration breaks down, so the RG-flow cannot be followed all the way to the fixed-point. In contrast, the lack of convergence of a self-consistency cycle is much more difficult to interpret consistently. ### Mathematical challenges of [[FRG]{}]{} For the specific set of flow equations used in this work, we adopt the same truncation scheme for the RG-equations, Fig. \[fig:flow\_equations\], that also is underlying the traditional [[FRG]{}]{} for periodic systems ([[$k$FRG]{}]{}). At this stage the only difference is that with [[$\epsilon$FRG]{}]{} each line represents a (Matsubara) Green’s function deriving from a resolvent $G=({{\mathrm{i}}}\omega -H_0)^{-1}$ that is not diagonal in momentum ($k-$) space. Fig. \[fig:flow\_equations\] gives a graphical representation of a set of nonlinear (integro-)differential equations that represent a typical initial-value problem; the flowing energy-cutoff $\Lambda$ plays a role analogous to a time. Ideally, after integrating the equations from $\Lambda=\infty$ to $\Lambda=0$ an exact solution of the (truncated) vertex-equation has been found. ![Diagrammatic representation of the FRG flow equations for the self-energy $\Sigma^\Lambda$ and the vertex $\Gamma^\Lambda$. A vertical bar denotes the single-scale propagator $\mathcal{S}^\Lambda$, the other propagators are $\mathcal{G}^\Lambda$. As usual, external legs do not entail a propagator.[]{data-label="fig:flow_equations"}](figure1){width=".95\linewidth"} As we already mentioned, solving the truncated set of flow equations, Fig. \[fig:flow\_equations\], still poses a problem of formidable computational complexity. The difficulty arises from the fact that the vertex function, $\Gamma(\Omega)$, is represented as very large family of matrices with three continuous frequencies, $\Omega=(\omega_1, \omega_2,\omega_3)$, acting as family parameters. In addition, each matrix has four indices, every one of which explores, in principle, the basis set of the full single-particle Hilbert space. ### Established approximation strategies Two main simplification strategies can reduce the computational effort, making [[FRG]{}]{} feasible and competitive. We offer a short overview. #### Static (or adiabatic) approximation. The frequency-dependence of the vertex function is neglected, $\Gamma(\omega_1,\omega_2;\omega_1')\rightarrow\Gamma(0)$. This is analogous to the static screening approximation familiar from the traditional treatment of the BSE imposed on top of [[$\text{G}_0\text{W}_0$]{}]{}. [@hybertsen86; @strinati88; @rohlfing98; @rohlfing00] In [[FRG]{}]{} one also ignores the frequency dependency of the self-energy, $\Sigma$. As a consequence, $\Sigma_\text{{{FRG}}}$ turns into an energy-independent, hermitian correction to the reference Hamiltonian $\hat H_0$. The effective Hamiltonian matrix $H_\text{eff}=H_0+\Sigma_\text{{{FRG}}}$ defines the quasi-particle energies and wavefunctions. With respect to the static self-energy, the situation in [[FRG]{}]{} is completely analogous to the one in the quasi-particle self-consistent [[$\text{GW}$]{}]{}-theory ([[$\text{qpGW}$]{}]{}). [@vanSchilfgaarde06; @brunval06prb; @kotani07; @shishkin:235102; @vanSetten16] The advantage of [[FRG]{}]{} over this theory is, that vertex corrections are accounted for in [[FRG]{}]{} in a [*self-consistent*]{} manner. In the static approximation, the scaling of [[FRG]{}]{} with the dimension of the single-particle Hilbert space, $N$, is formally $N^6$ if one does not consider further symmetries such as translational invariance. It is thus roughly comparable to the scaling of high-precision methods in quantum chemistry, like the coupled cluster method (flavor CCSD).[@bartlett07] #### Clean systems: Fermi-surface projection for [[$k$FRG]{}]{}. In the clean case, $H_0$ exhibits a translational symmetry, so the number of independent matrix elements of $\Gamma(0)$ reduces significantly. Moreover, a Fermi-surface exists that helps to identify a hierarchical structure within the matrix elements of $\Gamma(0)$. In many cases, only matrix elements with wavevectors close to the Fermi surface dominate the physics of the system, so the vertex at momenta away from this surface may be replaced by the vertex with momenta projected onto it, drastically simplifying the calculation. ### “Active-space” approximation for [[$\epsilon$FRG]{}]{}. In the case of generic systems, there is no intrinsic symmetry guidance as towards how to simplify the matrix structure of $\Gamma(0)$. In particular, Fermi-surface projection is not feasible. The most important new conceptual step in [[$\epsilon$FRG]{}]{} as compared to [[$k$FRG]{}]{} will be to find an alternative to the common Fermi-surface projection. It should reduce the number of degrees of freedom that are kept explicit in the RG-calculation without invoking a momentum-space concept. In this work we propose and test an “active-space” approximation that can achieve this goal. #### How to choose the active space. Similar to the [[FRG]{}]{}-treatment of clean systems, we also work in the eigenstate basis $|\alpha\rangle$ of the non-interacting Hamiltonian $\hat H_0$. Then the vertex function takes a matrix representation $\Gamma_{\alpha_1,\alpha_2,\alpha_3,\alpha_4}(0)$. To simplify the flow equations, we will approximate this matrix by the bare interaction vertex, $U_{\alpha_1,\alpha_2,\alpha_3,\alpha_4}$, whenever one of the states $|\alpha_i\rangle, i=1,\ldots,4$ is outside a certain active space $\mathcal{H}_M$ of the full single-particle Hilbert space, $\mathcal{H}$. A natural choice of $\mathcal{H}_M$ corresponds to states with energy $\epsilon_\alpha$ in the vicinity of the chemical potential $\mu_\text{chem}$. The index $M$ indicates the size of the volume, which could be characterized by an energy scale or simply by the number of states that it contains. We will adopt the simplest choice associating $M$ with the number of states kept in $\mathcal{H}_M$. #### Computational scaling. The important computational aspect of the active-space concept is that it brings the nominal scaling of [[$\epsilon$FRG]{}]{} down to $M^4N^2 + M^2N^3$. The optimal choice of $M$ balances the computational effort against the required numerical accuracy of the calculational results. In our applications we found that typically $M=N/3$ is a reliable choice. It is implying a speedup of a factor $10^2$ for the applications that we have investigated. For the limit of large $N$, we argue that $M\sim N^{1/2}$ in two-dimensional systems, so that the net scaling of [[$\epsilon$FRG]{}]{} would be $N^4$. It thus formally scales comparable to current implementations of the [[$\text{GW}$]{}]{}-method. Application of [[$\epsilon$FRG]{}]{}: Disordered Hubbard model -------------------------------------------------------------- As a first application of the new formalism and in order to demonstrate what can be achieved with it, we have studied the 2D-spinless, repulsive Hubbard model with on-site disorder at half filling. At zero disorder, $W{=}0$, the model exhibits a charge-density wave, while at zero interaction, $U{=}0$, the ground state corresponds to an Anderson insulator. Our interest is in determining the phase boundary that separates the two phases in the situation where disorder and interaction compete. We have calculated it in the $U/W{-}$plane. Specifically, we can establish that at $W{>}0$ the Anderson-insulator survives as long as the interaction does not exceed a critical value, $U>U^{*}(W)>0$. Conclusion and Outlook ---------------------- As it is typical with higher-order methods, the computational bottleneck restricts the feasible system sizes. In our applications, we found it practical to work with a single particle Hilbert space containing $N{\sim}50{-}100$ states. Our preliminary tests indicate that substantially bigger system sizes of a few hundred states are realistically accessible, $N{\sim} 200{-}400$, after additional improvements in the code performance have been implemented. It is only the limit of very large values of $N$, though, where the superior scaling of [[$\epsilon$FRG]{}]{} will become effective, so that the method becomes favorable as compared to other well established techniques, such as CCSD or quantum-Monte-Carlo. Whether these system sizes actually can be reached, future research will tell. At present, [[$\epsilon$FRG]{}]{} is readily applicable to models of interacting fermions in low dimensions, which includes Hubbard models with spin and (attractive) interactions at different filling fractions, but also, e.g., small molecules. Organization of this paper -------------------------- The paper is organized in the following way. In section \[sec:methodology\] we give the main formalism including the formul[æ]{} needed to reconstruct physical observables, in particular densities and occupation numbers. Also the formul[æ]{} for the finite-temperature formalism are given there, so that also, e.g., the effect of heat could be studied. Section \[sec:impl\] provides the computational details of our specific implementation of the main formulæ. In the consecutive section \[sec:verification\] we test this implementation on 1D- and 2D-model systems of disordered fermions against numerically exact results from exact diagonalization and the density matrix renormalization group (DMRG) for small system sizes. To illustrate the potential of [[$\epsilon$FRG]{}]{}, we present in section \[sec:results\] an application to the disordered, spinless 2D-Hubbard model. We will calculate and discuss the phase boundary between the Anderson-insulator and the Mott-phase in the plane spanned by the disorder and interaction strength. General Methodology of [[$\epsilon$FRG]{}]{} {#sec:methodology} ============================================ In this section we will develop our [[$\epsilon$FRG]{}]{}-scheme. We will assume that it is practical to diagonalize the non-interacting Hamiltonian exactly, $$\hat H_0 \ket{\alpha} = \epsilon_{\alpha} \ket{\alpha},$$ yielding eigenstates $\{\ket{\alpha}\}$ with corresponding eigenenergies $\{\epsilon_\alpha\}$. This allows us to rewrite the full Hamiltonian in terms of the non-interacting eigenbasis, $$\hat H = \sum_{\alpha} \epsilon_{\alpha} \mathrm{\hat c}_\alpha^\dagger \mathrm{\hat c}_\alpha + \frac{1}{4} \sum_{\alpha\beta\gamma\delta} U_{\alpha\beta\gamma\delta} \mathrm{\hat c}_\alpha^\dagger \mathrm{\hat c}_\beta^\dagger \mathrm{\hat c}_\delta \mathrm{\hat c}_\gamma.$$ Here, $U_{\alpha\beta\gamma\delta}$ are the anti-symmetrized bare interaction matrix elements in the non-interacting eigenbasis. As discussed in Ref. , the FRG is a means to solve this interacting problem by introducing a cutoff into the bare propagator of the system. As the systems we want to study are inhomogeneous in nature, and hence the single-particle states are not easily classified systematically, we introduce a cutoff in frequency space (as opposed to momentum space), see Eq. (57) in Ref. , $$\mathcal{G}^{0,\Lambda}({\mathrm{i}}\omega) = \frac{\Theta^\Lambda(\omega)}{{\mathrm{i}}\omega - H_0 + {\mu_{\text{chem}}}}, \label{eq:definition:G0}$$ where $\Theta^\Lambda(\omega)$ vanishes at $\Lambda\to\infty$ and approaches $1$ at $\Lambda\to 0$; see below for a discussion of our choice for $\Theta^\Lambda(\omega)$. As a consequence of introducing the infrared cutoff, $\Lambda$, all other quantities of the system depend on $\Lambda$. If we take the limit of $\Lambda\to\infty$, it can be shown (see Eq (31) in Ref. ) that the self-energy vanishes and the effective interaction vertex $\Gamma$ is given by the matrix elements of the bare interaction, $U_{\cdot\cdot\cdot\cdot}$. On the other hand, taking the limit of $\Lambda\to 0$, we recover the original system without the introduced cutoff. There is now a continuous variable that connects the real system ($\Lambda\to 0$), where the physical quantities are not known *a priori*, with a trivial system ($\Lambda\to\infty$), where all quantities are known. Flow equations -------------- As is discussed in the literature[@FrgReviewMetzner; @SalmhoferBook], the derivatives of the vertex functions (self-energy, effective interaction, etc.) yield a set of flow equations; a full derivation of their most generic form may be found in Chapter. 4 of Ref. . Following Ref.  (Eq. (50)), we will adopt the generic formulation of the flow equations, $${\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\Sigma^\Lambda(x',x) = \sum_{y,y'} \mathcal{S}^\Lambda(y,y')\Gamma^\Lambda(x',y';x,y), \label{eq:fromref:flow:Sigma}$$ and for the vertex, Ref.  (Eq. (52)), $$\begin{aligned} & & {\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\Gamma^\Lambda(x_1',x_2';x_1,x_2) \nonumber \\ & & \hspace{2em} = \sum_{y_1,y_1'} \sum_{y_2,y_2'} \mathcal{G}^\Lambda(y_1,y_1') \mathcal{S}^\Lambda(y_2,y_2') \nonumber \\ & & \hspace{2em} \times\Big\{\Gamma^\Lambda(x_1',x_2';y_1,y_2) \Gamma^\Lambda(y_1',y_2';x_1,x_2) \nonumber \\ & & \hspace{2em} -\big[ \Gamma^\Lambda(x_1',y_2';x_1,y_1) \Gamma^\Lambda(y_1',x_2';y_2,x_2) \nonumber \\ & & \hspace{2em} \hphantom{\times}+(y_1 \leftrightarrow y_2, y_1' \leftrightarrow y_2')\big] \nonumber \\ & & \hspace{2em} +\big[ \Gamma^\Lambda(x_2',y_2';x_1,y_1) \Gamma^\Lambda(y_1',x_1';y_2,x_2) \nonumber \\ & & \hspace{2em} \hphantom{\times}+(y_1 \leftrightarrow y_2, y_1' \leftrightarrow y_2')\big]\Big\} \nonumber \\ & & \hspace{2em} -\sum_{y,y'} \mathcal{S}^\Lambda(y,y') \Gamma^{(6),\Lambda}(x_1',x_2',y';x_1,x_2,y). \label{eq:fromref:flow:Gamma}\end{aligned}$$ Here, $x$ and $y$ are combined indices for space and time coordinates. A diagrammatic representation of these equations is given in in Fig. \[fig:flow\_equations\]. Furthermore, we copy the definition of Ref.  (Eq. (47)) for the single-scale propagator, $$\mathcal{S}^\Lambda = - \mathcal{G}^\Lambda \left[ {\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\left( \mathcal{G}^{0,\Lambda} \right)^{-1} \right] \mathcal{G}^\Lambda. \label{eq:definition:SingleScalePropagator}$$ We next rewrite these quantities into our own nomenclature, where we work in Matsubara space. Furthermore, we separate the generic indices into Matsubara frequencies and Hilbert space indices, $x \rightarrow (\mu,\omega_n)$. We also drop the term with $\Gamma^{(6),\Lambda}$ in accordance with the standard truncation scheme for these equations,[@FrgReviewMetzner] where in the case of short-range interactions, power counting arguments establish the scheme’s validity. Since energy is conserved, the self-energy, the single-particle Green’s functions, the single-scale propagator and the vertex include the corresponding $\delta$-function, $$\begin{aligned} \Sigma^\Lambda_{\alpha\beta}(\omega_n;\omega_{n'}) & \to & T^{-1} \delta_{n,n'} \Sigma^\Lambda_{\alpha\beta}(\omega_n), \\ \mathcal{G}^{0,\Lambda}_{\alpha\beta}(\omega_n;\omega_{n'}) & \to & T^{-1} \delta_{n,n'} \mathcal{G}^{0,\Lambda}_{\alpha\beta}(\omega_n), \\ \mathcal{G}^\Lambda_{\alpha\beta}(\omega_n;\omega_{n'}) & \to & T^{-1} \delta_{n,n'} \mathcal{G}^\Lambda_{\alpha\beta}(\omega_n), \\ \mathcal{S}^\Lambda_{\alpha\beta}(\omega_n;\omega_{n'}) & \to & T^{-1} \delta_{n,n'} \mathcal{S}^\Lambda_{\alpha\beta}(\omega_n), \\ \Gamma^\Lambda_{\alpha\beta\gamma\delta}(\omega_n,\omega_{\tilde n};\omega_{n'},\omega_{\tilde n'}) & \to & T^{-1} \delta_{n+\tilde n,n'+\tilde n'}\times \nonumber \\ & & \Gamma^\Lambda_{\alpha\beta\gamma\delta}(\omega_n,\omega_{\tilde n};\omega_{n'}).\hspace{1em}\end{aligned}$$ Inserting this into Eq. (\[eq:fromref:flow:Sigma\]) yields $$\begin{aligned} \hspace{-1em} {\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}T^{-1} \delta_{n,n'} \Sigma^\Lambda_{\alpha\beta}(\omega_n) & = & T^2 \sum_{\omega_m\omega_{m'}} \sum_{\mu\nu} \mathcal{S}^\Lambda_{\mu\nu}(\omega_m) \times \nonumber\\ & & \Gamma^\Lambda_{\alpha\nu\beta\mu}(\omega_n,\omega_{m'};\omega_{n'}) \times \nonumber\\ & & T^{-1} \delta_{m,m'} T^{-1}\delta_{n+m',n'+m},\end{aligned}$$ and after evaluating the sum over the Matsubara frequency $\omega_{m'}$, one arrives at $$\begin{aligned} \hspace{-1em}{\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\Sigma^\Lambda_{\alpha\beta}(\omega_n) & = & T \sum_{\omega_m} \sum_{\mu\nu} \mathcal{S}^\Lambda_{\mu\nu}(\omega_m) \times \nonumber \\ & & \Gamma^\Lambda_{\alpha\nu\beta\mu}(\omega_n,\omega_m;\omega_n). \label{eq:flow:SigmaWithOmegaAtFiniteT}\end{aligned}$$ Here, we have used that a $\delta_{n,n'}$ appears on both sides and have multiplied the equation by $T$. Proceeding in a similar way for the equation of the flow of the vertex, Eq. (\[eq:fromref:flow:Gamma\]), we arrive at $$\begin{aligned} {\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\Gamma^\Lambda_{\alpha\beta\gamma\delta}(\omega_n,\omega_{\tilde n};\omega_{n'}) & = & T \sum_{\omega_m\omega_{\tilde m}} \sum_{\mu\nu\rho\sigma} \mathcal{G}^\Lambda_{\rho\mu}(\omega_m) \mathcal{S}^\Lambda_{\sigma\nu}(\omega_{\tilde m}) \times \Big\{ \nonumber \\ & & \hspace{-3em} \hphantom{+ \big[} \Gamma^\Lambda_{\alpha\beta\rho\sigma}(\omega_n,\omega_{\tilde n};\omega_{m}) \Gamma^\Lambda_{\mu\nu\gamma\delta}(\omega_m,\omega_{\tilde m};\omega_{n'}) \delta^{(\text{c})}_{\tilde m} \nonumber \\ & & \hspace{-3em} + \big[ \Gamma^\Lambda_{\beta\nu\gamma\rho}(\omega_{\tilde n},\omega_{\tilde m};\omega_{n'}) \Gamma^\Lambda_{\mu\alpha\sigma\delta}(\omega_{m},\omega_{n};\omega_{\tilde m}) \delta^{(\text{ph},1)}_{\tilde m} \nonumber \\ & & \hspace{-3em} \hphantom{+ \big[} + \Gamma^\Lambda_{\beta\mu\gamma\sigma}(\omega_{\tilde n},\omega_{m};\omega_{n'}) \Gamma^\Lambda_{\nu\alpha\rho\delta}(\omega_{\tilde m},\omega_{n};\omega_{m}) \delta^{(\text{ph},2)}_{\tilde m} \big] \nonumber \\ & & \hspace{-3em} - \big[ \Gamma^\Lambda_{\alpha\nu\gamma\rho}(\omega_{n},\omega_{\tilde m};\omega_{n'}) \Gamma^\Lambda_{\mu\beta\sigma\delta}(\omega_{m},\omega_{\tilde n};\omega_{\tilde m}) \delta^{(\text{ph},3)}_{\tilde m} \nonumber \\ & & \hspace{-3em} \hphantom{+ \big[} + \Gamma^\Lambda_{\alpha\mu\gamma\sigma}(\omega_{n},\omega_{m};\omega_{n'}) \Gamma^\Lambda_{\nu\beta\rho\delta}(\omega_{\tilde m},\omega_{\tilde n};\omega_{m}) \delta^{(\text{ph},4)}_{\tilde m} \big] \Big\}, \label{eq:flow:GammaWithOmegaAtFiniteT}\end{aligned}$$ where $\delta^{(\text{c})}_{\tilde m}$ and $\delta^{(\text{ph},\cdot)}_{\tilde m}$ reflect the energy conservation of the vertex, e.g. $\delta^{\text{c}}_{\tilde m} = \delta_{n+\tilde n,m+\tilde m}.$ Formalism at Zero Temperature ----------------------------- For the most part, we will discuss the Formalism at $T = 0$. In that case, sums over Matsubara frequencies are replaced by integrals, $$T\sum_{\omega_n} \to (2\pi)^{-1} \int{\mathrm{d}}\omega, \label{eq:Matsubara:T0-transition}$$ and the Kronecker symbols will be replaced by $\delta$-functions, $$T^{-1} \delta_{n,n'} \to 2\pi\delta(\omega - \omega'). \label{eq:Matsubara:T0-transition:Kronecker}$$ Eqs. (\[eq:flow:SigmaWithOmegaAtFiniteT\],\[eq:flow:GammaWithOmegaAtFiniteT\]) now read $$\begin{aligned} {\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\Sigma^\Lambda_{\alpha\beta}(\omega) & = & \frac{1}{2\pi} \int{\mathrm{d}}\bar\omega \sum_{\mu\nu} \mathcal{S}^\Lambda_{\mu\nu}(\bar\omega) \Gamma^\Lambda_{\alpha\nu\beta\mu}(\omega,\bar\omega;\omega), \label{eq:flow:SigmaWithOmegaAtT0} \\ {\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\Gamma^\Lambda_{\alpha\beta\gamma\delta}(\omega,\tilde\omega;\omega') & = & \frac{1}{2\pi} \int{\mathrm{d}}\bar\omega{\mathrm{d}}\bar\omega' \sum_{\mu\nu\rho\sigma} \mathcal{G}^\Lambda_{\rho\mu}(\bar\omega) \mathcal{S}^\Lambda_{\sigma\nu}(\bar\omega') \times \Big\{ \Gamma^\Lambda_{\alpha\beta\rho\sigma}(\omega,\tilde\omega;\bar\omega) \Gamma^\Lambda_{\mu\nu\gamma\delta}(\bar\omega,\bar\omega';\omega') \delta^{(\text{c})}(\bar\omega') \nonumber \\ & & \hspace{-9em} + \big[ \Gamma^\Lambda_{\beta\nu\gamma\rho}(\tilde\omega,\bar\omega';\omega') \Gamma^\Lambda_{\mu\alpha\sigma\delta}(\bar\omega,\omega;\bar\omega') \delta^{(\text{ph},1)}(\bar\omega') + \Gamma^\Lambda_{\beta\mu\gamma\sigma}(\tilde\omega,\bar\omega;\omega') \Gamma^\Lambda_{\nu\alpha\rho\delta}(\bar\omega',\omega;\bar\omega) \delta^{(\text{ph},2)}(\bar\omega') \big] \nonumber \\ & & \hspace{-9em} - \big[ \Gamma^\Lambda_{\alpha\nu\gamma\rho}(\omega,\bar\omega';\omega') \Gamma^\Lambda_{\mu\beta\sigma\delta}(\bar\omega,\tilde\omega;\bar\omega') \delta^{(\text{ph},3)}(\bar\omega') + \Gamma^\Lambda_{\alpha\mu\gamma\sigma}(\omega,\bar\omega;\omega') \Gamma^\Lambda_{\nu\beta\rho\delta}(\bar\omega',\tilde\omega;\bar\omega) \delta^{(\text{ph},4)}(\bar\omega') \big] \Big\}, \label{eq:flow:GammaWithOmegaAtT0}\end{aligned}$$ where again, $\delta^{(\text{c})}(\bar\omega')$ and $\delta^{(\text{ph},\cdot)}(\bar\omega')$ reflect the energy conservation of the vertex, e.g., $\delta^{(\text{ph},1)}(\bar\omega') = \delta(\tilde\omega+\bar\omega'-\omega'-\bar\omega)$. We now proceed to take the static limit, i.e. by replacing the frequency dependence of the vertex and the self-energy by their static limit, e.g., $\Gamma(\omega,\omega';\bar\omega)\to\Gamma(0)$. For short-range interactions, power counting of the flow equations demonstrates that the dominant contribution for small $\Lambda$ comes from zero frequencies and states close to the Fermi energy. This approximation has been discussed extensively in Ref. . We arrive at $$\begin{aligned} {\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\Sigma^\Lambda_{\alpha\beta} & = & \frac{1}{2\pi} \int{\mathrm{d}}\bar\omega \sum_{\mu\nu} \mathcal{S}^\Lambda_{\mu\nu}(\bar\omega) \Gamma^\Lambda_{\alpha\nu\beta\mu}, \label{eq:flow:SigmaWithOmegaStatic} \\ {\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\Gamma^\Lambda_{\alpha\beta} & = & \frac{1}{2\pi} \int{\mathrm{d}}\bar\omega \sum_{\mu\nu\rho\sigma} \Big\{ \mathcal{G}^\Lambda_{\rho\mu}(\bar\omega) \mathcal{S}^\Lambda_{\sigma\nu}(-\bar\omega) \Gamma^\Lambda_{\alpha\beta\rho\sigma} \Gamma^\Lambda_{\mu\nu\gamma\delta} \nonumber \\ & & \hspace{-3em} + \mathcal{G}^\Lambda_{\rho\mu}(\bar\omega) \mathcal{S}^\Lambda_{\sigma\nu}(\bar\omega) \big[ \Gamma^\Lambda_{\beta\nu\gamma\rho} \Gamma^\Lambda_{\mu\alpha\sigma\delta} + \Gamma^\Lambda_{\beta\mu\gamma\sigma} \Gamma^\Lambda_{\nu\alpha\rho\delta} \big] \nonumber \\ & & \hspace{-3em} - \mathcal{G}^\Lambda_{\rho\mu}(\bar\omega) \mathcal{S}^\Lambda_{\sigma\nu}(\bar\omega) \big[ \Gamma^\Lambda_{\alpha\nu\gamma\rho} \Gamma^\Lambda_{\mu\beta\sigma\delta} + \Gamma^\Lambda_{\alpha\mu\gamma\sigma} \Gamma^\Lambda_{\nu\beta\rho\delta} \big] \Big\}. \label{eq:flow:GammaWithOmegaStatic}\end{aligned}$$ Note that the vertex $\Gamma^\Lambda$ is antisymmetric under exchange of the first or the last pair of indices, $$\Gamma^{\Lambda}_{\alpha\beta\gamma\delta} = - \Gamma^{\Lambda}_{\beta\alpha\gamma\delta} = - \Gamma^{\Lambda}_{\alpha\beta\delta\gamma} = \Gamma^{\Lambda}_{\beta\alpha\delta\gamma}. \label{eq:symmetry:Gamma}$$ Furthermore, one can easily show that in the static limit for finite system sizes the self-energy $\Sigma$ is hermitian. To further simplify these equations, we choose our cutoff $\Theta^\Lambda(\omega)$ to be a simple step function, $$\Theta^\Lambda(\omega) = \Theta(|\omega| - \Lambda),$$ such that its derivative is $${\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\Theta^\Lambda(\omega) = - \delta(|\omega| - \Lambda).$$ Since by construction the self-energy is not frequency dependent, the frequency integrals may now be solved analytically. For Eq. (\[eq:flow:SigmaWithOmegaStatic\]), we have to integrate $$\int{\mathrm{d}}\bar\omega \mathcal{S}^\Lambda_{\mu\nu}(\bar\omega).$$ Inserting Dyson’s equation into Eq. (\[eq:definition:SingleScalePropagator\]), we have $$\begin{aligned} \mathcal{S} & = & - \mathcal{G}\left({\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\left[\mathcal{G}^{0}\right]^{-1}\right)\mathcal{G} = - \mathcal{G}\left({\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\left[\mathcal{G}^{-1} + \Sigma\right]\right)\mathcal{G} \nonumber \\ & = & \mathcal{\dot G} - \mathcal{G}\dot\Sigma\mathcal{G},\end{aligned}$$ in matrix notation. We note that $\mathcal{G} = (\mathcal{Q} - \Theta\Sigma)^{-1}\Theta$, where we use the shorthand $\Theta = \Theta(|\omega|-\Lambda)$ and $\mathcal{Q} = {\mathrm{i}}\omega - H_0 + {\mu_{\text{chem}}}$. Using ${\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}A^{-1}(\Lambda) = - A^{-1}(\Lambda) \dot A(\Lambda) A^{-1}(\Lambda)$, simple algebra yields $$\mathcal{S} = - \delta\left(\mathds{1} + \frac{\Theta}{\mathcal{Q} - \Theta\Sigma} \Sigma\right) \frac{1}{\mathcal{Q} - \Theta\Sigma}. \label{eq:derivation:SingleScalePropagator:partialform}$$ Since the $\delta$ and $\Theta$ functions are to be taken at the same argument, we employ Morris’s Lemma[^1] to resolve this, $$\mathcal{S} = - \delta \int_0^1 {\mathrm{d}}t \left(\mathds{1} + t \frac{1}{\mathcal{Q} - t\Sigma} \Sigma\right) \frac{1}{\mathcal{Q} - t\Sigma}. \label{eq:derivation:SingleScalePropagator:integral}$$ Using the fact that $$\frac{{\mathrm{d}}}{{\mathrm{d}}t} \frac{1}{\mathcal{Q} - t\Sigma} = \frac{1}{\mathcal{Q} - t\Sigma} \Sigma \frac{1}{\mathcal{Q} - t\Sigma}$$ and partial integration, the second summand of the integral yields $$- \left[ \frac{t}{\mathcal{Q} - t\Sigma} \right]_0^1 + \int_0^1 {\mathrm{d}}t \frac{1}{\mathcal{Q} - t\Sigma},$$ where it can be seen that the remaining integral cancels the first summand of the integral in Eq. (\[eq:derivation:SingleScalePropagator:integral\]), so we arrive at $$\mathcal{S}^\Lambda(\omega) = - \frac{\delta(|\omega| - \Lambda)}{{\mathrm{i}}\omega - H_0 + {\mu_{\text{chem}}}- \Sigma^\Lambda}.$$ The frequency integral is now trivial, yielding $$\int{\mathrm{d}}\bar\omega \mathcal{S}^\Lambda(\bar\omega) = - \sum_{\bar\omega=\pm\Lambda} \frac{1}{{\mathrm{i}}\bar\omega - H_0 + {\mu_{\text{chem}}}- \Sigma^\Lambda}. \label{eq:result:SingleScalePropagatorAtT0}$$ As the following quantity will appear also in the flow equation for the vertex, we will define $$P^\Lambda_{\mu\nu}(\bar\omega) := \left.\frac{1}{{\mathrm{i}}\bar\omega - H_0 + {\mu_{\text{chem}}}- \Sigma^\Lambda}\right|_{\mu\nu}. \label{eq:definition:GenericPropagatorAtT0}$$ Inserting Eqs. (\[eq:result:SingleScalePropagatorAtT0\],\[eq:definition:GenericPropagatorAtT0\]) into Eq. (\[eq:flow:SigmaWithOmegaStatic\]), the flow equation for the self-energy now reads $${\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\Sigma^\Lambda_{\alpha\beta} = - \frac{1}{2\pi} \sum_{\mu\nu} \underbrace{\left( P^{\Lambda}_{\mu\nu}(\Lambda) + P^{\Lambda}_{\mu\nu}(-\Lambda)\right)}_{=:\Pi^{\Sigma,\Lambda}_{\mu\nu}} \Gamma^\Lambda_{\alpha\nu\beta\mu}. \label{eq:flow:Sigma}$$ When evaluating the flow equation for the vertex, Eq. (\[eq:flow:GammaWithOmegaStatic\]), one must take care that the arguments for the $\delta$ and $\Theta$ functions coincide, so one may not simply take the result derived for the single-scale propagator in the self-energy flow and apply it, but one rather uses the same kind of treatment of the $\delta$ and $\Theta$ functions for the entire expression, on a term by term basis. In the end, the flow equation for the vertex in the static limit reads, $$\begin{aligned} {\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\Gamma^\Lambda_{\alpha\beta\gamma\delta} & = & - \frac{1}{2\pi} \sum_{\mu\nu\rho\sigma} \sum_{\bar\omega=\pm\Lambda} \Big\{ \frac{1}{2} P^{\Lambda}_{\rho\mu}(-\bar\omega) P^{\Lambda}_{\sigma\nu}(\bar\omega) \Gamma^\Lambda_{\alpha\beta\rho\sigma} \Gamma^\Lambda_{\mu\nu\gamma\delta} \nonumber \\ & & \hspace{-3em} + P^{\Lambda}_{\rho\mu}(\bar\omega) P^{\Lambda}_{\sigma\nu}(\bar\omega) \left[ \Gamma^\Lambda_{\beta\nu\gamma\rho} \Gamma^\Lambda_{\alpha\mu\delta\sigma} - \Gamma^\Lambda_{\alpha\nu\gamma\rho} \Gamma^\Lambda_{\beta\mu\delta\sigma} \right] \Big\} \nonumber \\ & = & - \frac{1}{2\pi} \sum_{\mu\nu\rho\sigma} \Big\{ \Pi^{\text{c},\Lambda}_{\mu\nu\sigma\rho} \Gamma^\Lambda_{\nu\rho\gamma\delta} \Gamma^\Lambda_{\alpha\beta\sigma\mu} \nonumber \\ & & + \Pi^{\text{ph},\Lambda}_{\mu\nu\rho\sigma} \left[ \Gamma^\Lambda_{\beta\nu\gamma\rho} \Gamma^\Lambda_{\alpha\sigma\delta\mu} - \Gamma^\Lambda_{\alpha\nu\gamma\rho} \Gamma^\Lambda_{\beta\sigma\delta\mu} \right], \label{eq:flow:Gamma}\end{aligned}$$ where we have used the symmetries of $\Gamma$ to simplify the equations and abberviated $$\begin{aligned} \Pi^{\text{c},\Lambda}_{\mu\nu\sigma\rho} & := & P^{\Lambda}_{\mu\nu}(\Lambda) P^{\Lambda}_{\sigma\rho}(-\Lambda) \\ \Pi^{\text{ph},\Lambda}_{\mu\nu\sigma\rho} & := & P^{\Lambda}_{\mu\nu}(\Lambda) P^{\Lambda}_{\rho\sigma}(\Lambda) + P^{\Lambda}_{\mu\nu}(-\Lambda) P^{\Lambda}_{\rho\sigma}(-\Lambda).\end{aligned}$$ The full derivation may be found in Appendix \[app:T0StaticGamma\]. ### Initial conditions The initial conditions at $\Lambda\to\infty$ are given by $$\Sigma^{\Lambda\to\infty}_{\alpha\beta} = 0 \hspace{1em}\text{and}\hspace{1em} \Gamma^{\Lambda\to\infty}_{\alpha\beta\gamma\delta} = U_{\alpha\beta\gamma\delta}. \label{eq:initcond:SigmaGamma:Infty}$$ In order to solve the equations numerically, we need to choose an initial value $\Lambda_0$ that is still finite but larger than all other energy scales in the system. For $\Lambda > \Lambda_0$ one may assume a form of $({\mathrm{i}}\omega)^{-1}\mathds{1}$ for the propagator, allowing us to analytically integrate the flow equations from $\infty$ to $\Lambda_0$. In case of the flow equation for the vertex, power counting in $U$ and $\Lambda_0$ immediately yields $$\Gamma^{\Lambda_0} - U \sim - \int_\infty^{\Lambda_0} U U \frac{1}{\Lambda^2} {\mathrm{d}}\Lambda = \frac{1}{\Lambda_0} U U,$$ and hence $$|\Gamma^{\Lambda_0}-U|/|U| \sim |U|/\Lambda_0.$$ We therefore may simply use that $\Gamma^{\Lambda_0}$ does not differ from $\Gamma^{\Lambda\to\infty}$ for large enough $\Lambda_0$ and arrive at $$\Gamma^{\Lambda}_{\alpha\beta\gamma\delta} = U_{\alpha\beta\gamma\delta}, \qquad \Lambda > \Lambda_0. \label{eq:initcond:Gamma}$$ The same does not hold true for the flow equation for the self-energy, where the analytical integral gives a non-negligible contribution, $$\begin{aligned} \Sigma^{\Lambda_0}_{\alpha\beta} & = & - \frac{1}{2\pi} \sum_{\mu} U_{\alpha\mu\beta\mu} \lim_{\eta\to 0^{+}} \int_{\infty}^{\Lambda_0} \left( \frac{{\mathrm{e}}^{{\mathrm{i}}\Lambda\eta}}{{\mathrm{i}}\Lambda} - \frac{{\mathrm{e}}^{-{\mathrm{i}}\Lambda\eta}}{{\mathrm{i}}\Lambda} \right) {\mathrm{d}}\Lambda \nonumber \\ & = & - \frac{1}{\pi} \sum_{\mu} U_{\alpha\mu\beta\mu} \lim_{\eta\to 0^{+}} \eta \int_{\infty}^{\Lambda_0} {\mathrm{sinc}}(\eta\Lambda){\mathrm{d}}\Lambda \nonumber \\ & = & \frac{1}{\pi} \sum_{\mu} U_{\alpha\mu\beta\mu} \lim_{\eta\to 0^{+}} \left[ \int_0^\infty {\mathrm{sinc}}(x) {\mathrm{d}}x - \mathcal{O}(\eta) \right] \nonumber \\ & = & \frac{1}{2} \sum_{\mu} U_{\alpha\mu\beta\mu}. \label{eq:initcond:Sigma}\end{aligned}$$ Here we have explicitly included the required convergence factor ${\mathrm{e}}^{{\mathrm{i}}\omega 0^{+}}$ that appears in the Green’s function in imaginary frequency space. Systems with Spin ----------------- In Eqs. (\[eq:flow:Sigma\],\[eq:flow:Gamma\]), the indices represent generic states in the Hilbert space. We will now discuss the case where the system is fully $\mathrm{SU}(2)$ symmetric. Here, it is convenient to separate the orbital degrees of freedom from the spin degrees of freedom, $\alpha\to(\alpha,\sigma_1)$. Our derivation will follow Ref. , but we will discuss the generic case without the additional particle-hole symmetry. Single-particle quantities (self-energy, propagators) do not depend on the spin degree of freedom, $$\begin{aligned} \Sigma^\Lambda_{(\alpha,\sigma_1)(\beta,\sigma_2)} & = & \Sigma^{\text{s},\Lambda}_{\alpha\beta} \delta_{\sigma_1\sigma_2}, \label{eq:definition:Sigma:with-spin} \\ \mathcal{G}^\Lambda_{(\alpha,\sigma_1)(\beta,\sigma_2)} & = & \mathcal{G}^{\text{s},\Lambda}_{\alpha\beta} \delta_{\sigma_1\sigma_2}, \label{eq:definition:G:with-spin} \\ \mathcal{S}^\Lambda_{(\alpha,\sigma_1)(\beta,\sigma_2)} & = & \mathcal{S}^{\text{s},\Lambda}_{\alpha\beta} \delta_{\sigma_1\sigma_2}, \label{eq:definition:S:with-spin} \\ P^\Lambda_{(\alpha,\sigma_1)(\beta,\sigma_2)} & = & P^{\text{s},\Lambda}_{\alpha\beta} \delta_{\sigma_1\sigma_2}. \label{eq:definition:P:with-spin}\end{aligned}$$ The spin structure of the vertex is determined by the fact that two particles may either keep their spin or exchange it, and may thus be decomposed into $$\begin{aligned} \Gamma^\Lambda_{(\alpha,\sigma_1),(\beta,\sigma_2),(\gamma,\sigma_3),(\delta,\sigma_4)} & = & \hphantom{+} c^{\text{I},\Lambda}_{\alpha\beta\gamma\delta} \delta_{\sigma_1\sigma_3} \delta_{\sigma_2\sigma_4} \nonumber\\ & & + c^{\text{II},\Lambda}_{\alpha\beta\gamma\delta} \delta_{\sigma_1\sigma_4} \delta_{\sigma_2\sigma_3},\end{aligned}$$ where $c^{\text{I}}$ and $c^{\text{II}}$ are the coefficients for each of these processes. Using the antisymmetry of $\Gamma$, Eq. (\[eq:symmetry:Gamma\]), we may exchange $(\gamma,\sigma_3)$ with $(\delta,\sigma_4)$, $$\begin{aligned} & & \Gamma^\Lambda_{(\alpha,\sigma_1),(\beta,\sigma_2),(\gamma,\sigma_3),(\delta,\sigma_4)} = - \Gamma^\Lambda_{(\alpha,\sigma_1),(\beta,\sigma_2),(\delta,\sigma_4),(\gamma,\sigma_3)} \nonumber \\ & & \hspace{4em} = - c^{\text{I},\Lambda}_{\alpha\beta\delta\gamma} \delta_{\sigma_1\sigma_4} \delta_{\sigma_2\sigma_3} - c^{\text{II},\Lambda}_{\alpha\beta\delta\gamma} \delta_{\sigma_1\sigma_3} \delta_{\sigma_2\sigma_4}.\end{aligned}$$ By comparing the coefficients of the Kronecker-$\delta$s, we may identify $$c^{\text{I},\Lambda}_{\alpha\beta\gamma\delta} = - c^{\text{II},\Lambda}_{\alpha\beta\delta\gamma} := - \Gamma^{\text{s},\Lambda}_{\alpha\beta\delta\gamma},$$ and hence write the vertex as $$\begin{aligned} \Gamma^\Lambda_{(\alpha,\sigma_1),(\beta,\sigma_2),(\gamma,\sigma_3),(\delta,\sigma_4)} & = & \hphantom{-} \Gamma^{\text{s},\Lambda}_{\alpha\beta\gamma\delta} \delta_{\sigma_1\sigma_4} \delta_{\sigma_2\sigma_3} \nonumber\\ & & - \Gamma^{\text{s},\Lambda}_{\alpha\beta\delta\gamma} \delta_{\sigma_1\sigma_3} \delta_{\sigma_2\sigma_4}. \label{eq:definition:Gamma:with-spin}\end{aligned}$$ Using the symmetry of $\Gamma^\Lambda$, one can see that $\Gamma^{\text{s},\Lambda}$ is still symmetric under exchange of both pairs of indices, $$\Gamma^{\text{s},\Lambda}_{\alpha\beta\gamma\delta} = \Gamma^{\text{s},\Lambda}_{\beta\alpha\delta\gamma},$$ but in general it is not antisymmetric with respect to the exchange of a single pair of indices. Instead, one may identify the part of $\Gamma^{\text{s},\Lambda}$ that is antisymmetric under exchange of $\alpha$ and $\beta$ with the triplet channel of the vertex, whereas the part that is symmetric under the exchange of $\alpha$ and $\beta$ represents the singlet channel. Inserting Eqs. (\[eq:definition:Sigma:with-spin\],\[eq:definition:P:with-spin\],\[eq:definition:Gamma:with-spin\]) into Eq. (\[eq:flow:Sigma\]), we have $$\begin{aligned} {\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\Sigma^{\text{s},\Lambda}_{\alpha\beta} \delta_{\sigma_1\sigma_2} & = & - \frac{1}{2\pi} \sum_{\mu\nu} \sum_{\sigma_3} \Pi^{\Sigma,\text{s},\Lambda}_{\mu\nu} \big( \Gamma^{\text{s},\Lambda}_{\alpha\nu\beta\mu} \delta_{\sigma_1\sigma_3} \delta_{\sigma_3\sigma_2} \nonumber \\ & & \hspace{6em} - \Gamma^{\text{s},\Lambda}_{\alpha\nu\mu\beta} \delta_{\sigma_1\sigma_2} \delta_{\sigma_3\sigma_3} \Big) \nonumber \\ & = & - \frac{1}{2\pi} \sum_{\mu\nu} \Pi^{\Sigma,\text{s},\Lambda}_{\mu\nu} \big( \Gamma^{\text{s},\Lambda}_{\alpha\nu\beta\mu} - 2 \Gamma^{\text{s},\Lambda}_{\alpha\nu\mu\beta} \big) \delta_{\sigma_1\sigma_2} \nonumber,\end{aligned}$$ and hence $${\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\Sigma^{\text{s},\Lambda}_{\alpha\beta} = - \frac{1}{2\pi} \sum_{\mu\nu} \Pi^{\Sigma,\text{s},\Lambda}_{\mu\nu} \big( \Gamma^{\text{s},\Lambda}_{\alpha\nu\beta\mu} - 2 \Gamma^{\text{s},\Lambda}_{\alpha\nu\mu\beta} \big). \label{eq:flow:Sigma:WithSpin}$$ Here, we have defined $$\Pi^{\Sigma,\text{s},\Lambda}_{\mu\nu} := P^{\text{s},\Lambda}_{\mu\nu}(\Lambda) + P^{\text{s},\Lambda}_{\mu\nu}(-\Lambda)$$ in analogy to the definition in Eq. \[eq:flow:Sigma\], as we will do with $\Pi^{\text{ph},\text{s},\Lambda}_{\mu\nu}$ and $\Pi^{\text{c},\text{s},\Lambda}_{\mu\nu}$ in the following. To obtain the flow equation for $\Gamma^{\Lambda,s}$, we must insert Eqs. (\[eq:definition:Sigma:with-spin\],\[eq:definition:P:with-spin\],\[eq:definition:Gamma:with-spin\]) into Eq. (\[eq:flow:Gamma\]). To simplify our notation, we will use $\delta^{34}_{12} = \delta_{\sigma_1\sigma_2}\delta_{\sigma_3\sigma_4}$. For the first term with $\Pi^{\text{c},\text{s},\Lambda}$, we have $$\begin{aligned} & & - \frac{1}{2\pi} \sum_{\mu\nu\rho\sigma} \sum_{\sigma_5\sigma_6} \Pi^{\text{c},\text{s},\Lambda}_{\mu\nu\sigma\rho} \times \nonumber \\ & & \hspace{1em} \Big( \Gamma^{\text{s},\Lambda}_{\nu\rho\gamma\delta} \delta^{63}_{54} - \Gamma^{\text{s},\Lambda}_{\nu\rho\delta\gamma} \delta^{64}_{53} \Big) \times \nonumber \\ & & \hspace{1em} \Big( \Gamma^{\text{s},\Lambda}_{\alpha\beta\sigma\mu} \delta^{26}_{15} - \Gamma^{\text{s},\Lambda}_{\alpha\beta\mu\sigma} \delta^{25}_{16} \Big).\end{aligned}$$ Multiplying out the main product, there are four terms of combinations of $\Gamma^{\text{s},\Lambda}$ that appear, $$\begin{aligned} \sum_{\sigma_5\sigma_6} \Gamma^{\text{s},\Lambda}_{\nu\rho\gamma\delta} \Gamma^{\text{s},\Lambda}_{\alpha\beta\sigma\mu} \delta^{63}_{54} \delta^{26}_{15} & = & \Gamma^{\text{s},\Lambda}_{\nu\rho\gamma\delta} \Gamma^{\text{s},\Lambda}_{\alpha\beta\sigma\mu} \delta^{23}_{14}. \\ \sum_{\sigma_5\sigma_6} - \Gamma^{\text{s},\Lambda}_{\nu\rho\delta\gamma} \Gamma^{\text{s},\Lambda}_{\alpha\beta\sigma\mu} \delta^{64}_{53} \delta^{26}_{15} & = & - \Gamma^{\text{s},\Lambda}_{\nu\rho\delta\gamma} \Gamma^{\text{s},\Lambda}_{\alpha\beta\sigma\mu} \delta^{24}_{13},\end{aligned}$$ $$\begin{aligned} \sum_{\sigma_5\sigma_6} - \Gamma^{\text{s},\Lambda}_{\nu\rho\gamma\delta} \Gamma^{\text{s},\Lambda}_{\alpha\beta\mu\sigma} \delta^{63}_{54} \delta^{25}_{16} & = & - \Gamma^{\text{s},\Lambda}_{\nu\rho\gamma\delta} \Gamma^{\text{s},\Lambda}_{\alpha\beta\mu\sigma} \delta^{24}_{13}, \\ \sum_{\sigma_5\sigma_6} \Gamma^{\text{s},\Lambda}_{\nu\rho\delta\gamma} \Gamma^{\text{s},\Lambda}_{\alpha\beta\mu\sigma} \delta^{64}_{53} \delta^{25}_{16} & = & \Gamma^{\text{s},\Lambda}_{\nu\rho\delta\gamma} \Gamma^{\text{s},\Lambda}_{\alpha\beta\mu\sigma} \delta^{23}_{14}.\end{aligned}$$ On the other hand, the left hand side of the flow equation reads $${\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\left( \Gamma^{\text{s},\Lambda}_{\alpha\beta\gamma\delta} \delta_{14}^{23} - \Gamma^{\text{s},\Lambda}_{\alpha\beta\delta\gamma} \delta_{13}^{24} \right).$$ We may thus look at the products that contain $\delta_{14}^{23}$ to obtain the first term of the flow equation for $\Gamma^{\text{s},\Lambda}$, $$\begin{aligned} & & - \frac{1}{2\pi} \sum_{\mu\nu\rho\sigma} \Pi^{\text{c},\Lambda}_{\mu\nu\sigma\rho} \Big( \Gamma^{\text{s},\Lambda}_{\nu\rho\gamma\delta} \Gamma^{\text{s},\Lambda}_{\alpha\beta\sigma\mu} + \Gamma^{\text{s},\Lambda}_{\nu\rho\delta\gamma} \Gamma^{\text{s},\Lambda}_{\alpha\beta\mu\sigma} \Big) \delta_{14}^{23}. \label{eq:spinflow:cooper:1423part}\end{aligned}$$ We may now proceed in doing the same for the particle-hole channel, $$\begin{aligned} & & - \frac{1}{2\pi} \sum_{\mu\nu\rho\sigma} \sum_{\sigma_5\sigma_6} \Pi^{\text{ph},\text{s},\Lambda}_{\mu\nu\rho\sigma} \times \nonumber \\ & & \Big\{ \big( \Gamma^{\text{s},\Lambda}_{\alpha\nu\gamma\rho} \delta^{16}_{53} - \Gamma^{\text{s},\Lambda}_{\alpha\nu\rho\gamma} \delta^{13}_{56} \big) \big( \Gamma^{\text{s},\Lambda}_{\beta\sigma\delta\mu} \delta^{25}_{64} - \Gamma^{\text{s},\Lambda}_{\beta\sigma\mu\delta} \delta^{24}_{65} \big) \nonumber \\ & & + \big( \Gamma^{\text{s},\Lambda}_{\beta\nu\gamma\rho} \delta^{26}_{53} - \Gamma^{\text{s},\Lambda}_{\beta\nu\rho\gamma} \delta^{23}_{56} \big) \big( \Gamma^{\text{s},\Lambda}_{\alpha\sigma\delta\mu} \delta^{15}_{64} - \Gamma^{\text{s},\Lambda}_{\alpha\sigma\mu\delta} \delta^{14}_{65} \big) \Big\}.~~\end{aligned}$$ Of the eight products that appear, we again pick out those that appear with a $\delta_{14}^{23}$, where we use that $$\begin{aligned} \sum_{\sigma_5\sigma_6} \delta^{16}_{53} \delta^{25}_{64} = \delta_{14}^{23}, & \hspace{2em} & \sum_{\sigma_5\sigma_6} \delta^{23}_{56} \delta^{14}_{65} = 2 \delta_{14}^{23}, \\ \sum_{\sigma_5\sigma_6} \delta^{26}_{53} \delta^{14}_{65} = \delta_{14}^{23}, & \hspace{2em} & \sum_{\sigma_5\sigma_6} \delta^{23}_{56} \delta^{15}_{64} = \delta_{14}^{23},\end{aligned}$$ so that we arrive at $$\begin{aligned} & & \hspace{-2em} - \frac{1}{2\pi} \sum_{\mu\nu\rho\sigma} \Pi^{\text{ph},\text{s},\Lambda}_{\mu\nu\rho\sigma} \times \nonumber \\ & & \hspace{1em} \Big( 2 \Gamma^{\text{s},\Lambda}_{\beta\nu\rho\gamma} \Gamma^{\text{s},\Lambda}_{\alpha\sigma\mu\delta} + \Gamma^{\text{s},\Lambda}_{\alpha\nu\gamma\rho} \Gamma^{\text{s},\Lambda}_{\beta\sigma\delta\mu} \nonumber \\ & & \hspace{1em} - \Gamma^{\text{s},\Lambda}_{\beta\nu\rho\gamma} \Gamma^{\text{s},\Lambda}_{\alpha\sigma\delta\mu} - \Gamma^{\text{s},\Lambda}_{\beta\nu\gamma\rho} \Gamma^{\text{s},\Lambda}_{\alpha\sigma\mu\delta} \Big) \delta_{14}^{23}.~~~~ \label{eq:spinflow:ph:1423part}\end{aligned}$$ Adding Eq.  and Eq. , the flow equation for $\Gamma^{\text{s},\Lambda}$ now reads $$\begin{aligned} & & {\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\Gamma^{\text{s},\Lambda}_{\alpha\beta\delta\gamma} = - \frac{1}{2\pi} \sum_{\mu\nu\rho\sigma} \Big\{ \nonumber \\ & & \hspace{2em} \hphantom{+} \Pi^{\text{c},\text{s},\Lambda}_{\mu\nu\sigma\rho} \big( \Gamma^{\text{s},\Lambda}_{\nu\rho\gamma\delta} \Gamma^{\text{s},\Lambda}_{\alpha\beta\sigma\mu} + \Gamma^{\text{s},\Lambda}_{\nu\rho\delta\gamma} \Gamma^{\text{s},\Lambda}_{\alpha\beta\mu\sigma} \big) \nonumber \\ & & \hspace{1em} + \Pi^{\text{ph},\text{s},\Lambda}_{\mu\nu\rho\sigma} \big( 2 \Gamma^{\text{s},\Lambda}_{\beta\nu\rho\gamma} \Gamma^{\text{s},\Lambda}_{\alpha\sigma\mu\delta} + \Gamma^{\text{s},\Lambda}_{\alpha\nu\gamma\rho} \Gamma^{\text{s},\Lambda}_{\beta\sigma\delta\mu} \nonumber \\ & & \hspace{5em} - \Gamma^{\text{s},\Lambda}_{\beta\nu\rho\gamma} \Gamma^{\text{s},\Lambda}_{\alpha\sigma\delta\mu} - \Gamma^{\text{s},\Lambda}_{\beta\nu\gamma\rho} \Gamma^{\text{s},\Lambda}_{\alpha\sigma\mu\delta} \big) \Big\}.\end{aligned}$$ Finite Temperature ------------------ For completeness, we also derive the form of the flow equations at finite temperature. In this case, using a sharp $\Theta$-function is ill-suited. Instead, we utilize the cutoff suggested in Ref. , hence we replace $\Theta(|\omega|-\Lambda)$ by $\chi^\Lambda(\omega_n)$, which is given by $$\chi^\Lambda(\omega_n) = \left\{ \begin{array}{ll} 0, & |\omega_n| \le \Lambda - \pi T, \\ \frac{1}{2} + \frac{|\omega_n| - \Lambda}{2\pi T} & \Lambda - \pi T \le |\omega_n| \le \Lambda + \pi T, \\ 1, & \Lambda + \pi T \le |\omega_n|, \end{array} \right.$$ and its derivative with respect to $\Lambda$ is then given by $$-(\partial_\Lambda \chi^\Lambda(\omega_n)) = \left\{ \begin{array}{ll} \frac{1}{2\pi T} & \Lambda - \pi T \le |\omega_n| \le \Lambda + \pi T, \\ 0 & \text{otherwise}. \end{array} \right.$$ We note that $\chi^\Lambda(\omega_n) \to \Theta(|\omega|-\Lambda)$ as $T\to 0$. The full Green’s function is now given by $$\mathcal{G}^\Lambda(\omega_n) = \frac{\chi^\Lambda(\omega_n)}{{\mathrm{i}}\omega_n - H_0 + {\mu_{\text{chem}}}- \chi^\Lambda(\omega_n) \Sigma^\Lambda(\omega_n)},$$ whereas the single-scale propagator, Eq. \[eq:definition:SingleScalePropagator\], reads $$\begin{aligned} \mathcal{S}^\Lambda(\omega_n) & = & \frac{\partial_\Lambda \chi^\Lambda(\omega_n)}{{\mathrm{i}}\omega_n - H_0 + {\mu_{\text{chem}}}- \chi^\Lambda(\omega_n) \Sigma^\Lambda(\omega_n)} \times \nonumber \\ & & \hspace{1.5em} \big( {\mathrm{i}}\omega_n - H_0 + {\mu_{\text{chem}}}\big) \times \nonumber \\ & & \hspace{1.5em} \frac{1}{{\mathrm{i}}\omega_n - H_0 + {\mu_{\text{chem}}}- \chi^\Lambda(\omega_n) \Sigma^\Lambda(\omega_n)}.\end{aligned}$$ With this form of a cutoff function, the Matsubara sums may be evaluated analytically. Since Matsubara frequencies have a distance of $2\pi T$ from each other, the derivative of the cutoff is only nonzero for a two Matsubara frequencies, whose magnitude are that closest to the parameter $\Lambda$. Any sum with a single derivative of $\chi^\Lambda$ may hence be evaluated as $$T \sum_{n} -(\partial_\Lambda \chi^\Lambda(\omega_n)) f(\omega_n) = \frac{1}{2\pi} \sum_{|\omega_n| \approx \Lambda} f(\omega_n).$$ This structure is very similar to the situation at $T = 0$, where we have $$\frac{1}{2\pi} \int{\mathrm{d}}\omega \delta(|\omega|-\Lambda) f(\omega) = \frac{1}{2\pi} \sum_{|\omega| = \Lambda} f(\omega).$$ Again we adopt the static limit and define $P^{T,\Lambda}(\omega_n)$ as $$P^{T,\Lambda}(\omega_n) := \frac{1}{{\mathrm{i}}\omega_n - H_0 + {\mu_{\text{chem}}}- \chi^\Lambda(\omega_n) \Sigma^\Lambda},$$ and $P^{'T,\Lambda}, P^{''T,\Lambda}$ as $$\begin{aligned} P^{'T,\Lambda}(\omega_n) & := & P^{T,\Lambda}(\omega_n) ({\mathrm{i}}\omega_n - H_0 + {\mu_{\text{chem}}}) P^{T,\Lambda}(\omega_n), \nonumber \\ & & \\ P^{''T,\Lambda}(\omega_n) & := & P^{T,\Lambda}(\omega_n) \chi^\Lambda(\omega_n),\end{aligned}$$ the flow equation for the self-energy now reads $$\begin{aligned} {\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\Sigma^\Lambda_{\alpha\beta} & = & - \frac{1}{2\pi} \sum_{|\omega_n| \approx \Lambda} P^{'T,\Lambda}_{\mu\nu}(\omega_n) \Gamma^\Lambda_{\alpha\nu\beta\mu}. \label{eq:flow:Sigma:finiteT}\end{aligned}$$ Setting all external frequencies to zero and dropping the frequency dependence of the vertex, its flow equation is now given by $$\begin{aligned} {\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\Gamma^\Lambda_{\alpha\beta\gamma\delta} & = & - \frac{1}{2\pi} \sum_{|\omega_n| \approx \Lambda} \sum_{\mu\nu\rho\sigma} \Big\{ P^{''T,\Lambda}_{\mu\nu}(\omega_n) P^{'T,\Lambda}_{\rho\sigma}(-\omega_n) \times \nonumber \\ & & \hspace{2em} \Gamma^\Lambda_{\alpha\beta\sigma\mu} \Gamma^\Lambda_{\nu\rho\gamma\delta} + \nonumber \\ & & \hspace{4em} P^{''T,\Lambda}_{\mu\nu}(\omega_n) P^{'T,\Lambda}_{\rho\sigma}(\omega_n) \times \nonumber \\ & & \hspace{1em} \big[ \Gamma^\Lambda_{\beta\nu\gamma\rho} \Gamma^\Lambda_{\alpha\mu\delta\sigma} - \Gamma^\Lambda_{\alpha\mu\gamma\sigma} \Gamma^\Lambda_{\beta\nu\delta\rho} \nonumber \\ & & \hspace{2em} + \Gamma^\Lambda_{\beta\mu\gamma\sigma} \Gamma^\Lambda_{\alpha\nu\delta\rho} - \Gamma^\Lambda_{\alpha\nu\gamma\rho} \Gamma^\Lambda_{\beta\mu\delta\sigma} \big] \Big\}. \label{eq:flow:Gamma:finiteT}\end{aligned}$$ We note that $\chi^\Lambda(\omega_n) \to \frac{1}{2}$ for $\omega_n \to \Lambda$, so if taking the limit $T \to 0$ (and applying the symmetries of the vertex) one recovers Eq. (\[eq:flow:Gamma\]). Observables and Correlators --------------------------- ### Single-particle observables Single-particle observables may be expressed by the Green’s function, which is given by $$\mathcal{G}({\mathrm{i}}\omega) = \frac{1}{{\mathrm{i}}\omega - H_0 + \mu - \Sigma}{\mathrm{e}}^{{\mathrm{i}}\omega 0^+}. \label{eq:obs:G}$$ The convergence factor ${\mathrm{e}}^{{\mathrm{i}}\omega 0^+}$ is explicitly required here. In the following we will summarize (trivial) statements that follow from employing the static limit. For example, the density matrix for the occupancy of single-particle states, $\rho_{ij}$, is given by $$\rho_{ij} = \sum_{\alpha\beta} V^{\mathrm{rn}}_{i\alpha} \left[ \frac{1}{2\pi} \int_{-\infty}^{\infty} {\mathrm{d}}\omega\, \mathcal{G}_{\alpha\beta}({\mathrm{i}}\omega) {\mathrm{e}}^{{\mathrm{i}}\omega 0^+} \right] V^{\mathrm{rn},-1}_{\beta j},$$ where $$V^{\mathrm{rn}}_{i\alpha} = \braket{\alpha|i}$$ and $\ket{i}$ is one out of $N$ basis-vectors spanning the single-particle Hilbert space $\mathcal{H}$. The frequency integral may be calculated analytically by going into the basis where $\mathcal{G}$ is diagonal, i.e. the eigenbasis of $H_0 + \Sigma$. We will denote indices in that basis by a tilde, e.g. $\tilde\mu$ and the eigenvalues of $H_0 + \Sigma$ with $\tilde\epsilon_{\tilde\mu}$. (As $\Sigma$ is hermitian in the static limit, $\tilde\epsilon_{\tilde\mu}$ are real.) The basis transform from that basis into the basis chosen for observables will be denoted by $V^{\rm{ri}}_{i\tilde\mu}$. The integral may now be performed analytically, closing the integration loop around the left half-plane, $$\begin{aligned} \rho_{ij} & = & \sum_{\tilde\mu} V^{\mathrm{ri}}_{i\tilde\mu} \left[ \frac{1}{2\pi} \int_{-\infty}^{\infty} {\mathrm{d}}\omega\, \frac{{\mathrm{e}}^{{\mathrm{i}}\omega 0^+}}{{\mathrm{i}}\omega - \tilde\epsilon_{\tilde\mu} + {\mu_{\text{chem}}}} \right] V^{\mathrm{ri},-1}_{\tilde\mu j} \nonumber \\ & = & \sum_{\tilde\mu}^{\text{occ.}} V^{\mathrm{ri}}_{i\tilde\mu} V^{\mathrm{ri},-1}_{\tilde\mu j}, \label{eq:obs:density-matrix}\end{aligned}$$ where the summation is now only performed over states below the chemical potential. (Occupied states.) In order to obtain the result at finite temperature, $T > 0$, we must replace the integral by a Matsubara sum, performing the inverse of Eq. (\[eq:Matsubara:T0-transition\]). The sum may be performed analytically, using the well-known relation $$T \sum_{\omega_n} \frac{1}{{\mathrm{i}}\omega_n - \xi} = n_{\mathrm{F}}(\xi),$$ where $n_{\mathrm{F}}$ is the Fermi function. We now obtain $$\begin{aligned} \rho_{ij} & = & \sum_{\tilde\mu} V^{\mathrm{ri}}_{i\tilde\mu} \left[ T \sum_{\omega_n} \frac{1}{{\mathrm{i}}\omega_n - \tilde\epsilon_{\tilde\mu} + {\mu_{\text{chem}}}} \right] V^{\mathrm{ri},-1}_{\tilde\mu j} \nonumber \\ & = & \sum_{\tilde\mu} V^{\mathrm{ri}}_{i\tilde\mu} n_{\mathrm{F}}(\tilde\epsilon_{\tilde\mu} - {\mu_{\text{chem}}}) V^{\mathrm{ri},-1}_{\tilde\mu j}, \label{eq:obs:density-matrix:finiteT}\end{aligned}$$ which reproduces Eq. (\[eq:obs:density-matrix\]) for $T \to 0$. Another single-particle quantity of interest is the (normalized) density of states (DOS), which may be calculated from the imaginary part of the retarded Green’s function after Wick rotation. As we work in the static limit for the self-eenergy, the Wick rotation is trivial and yields the following expression for the density of states at $T = 0$, $$\begin{aligned} \rho(\epsilon) = - \frac{1}{2\pi N} \Im \sum_{\tilde\mu} \frac{1}{\epsilon - \tilde\epsilon_{\tilde\mu} + {\mu_{\text{chem}}}+ {\mathrm{i}}0}.\end{aligned}$$ Finally, in systems with spin rotational invariance the single-particle Green’s function is diagonal in spin space and the previously discussed quantities simply acquire a factor of 2. ### Correlator of Occupancy Numbers ($T = 0$) Two-particle observables may be rewritten in terms of single- and two-particle Green’s functions. In the case of spinless Fermions the correlator of occupancy numbers, $\mathcal{C}^{\mathrm{dd}}_{ij}$, may be rewritten as $$\begin{aligned} \hspace{-1em} \mathcal{C}^{\mathrm{dd}}_{ij} & = & \big<\mathrm{\hat n}_i^{\vphantom{\dagger}} \mathrm{\hat n}_j^{\vphantom{\dagger}}\big> = \big<\mathrm{\hat c}_i^\dagger \mathrm{\hat c}_i^{\vphantom{\dagger}} \mathrm{\hat c}_j^\dagger \mathrm{\hat c}_j^{\vphantom{\dagger}}\big> = \big<\mathrm{\hat c}_j^\dagger \mathrm{\hat c}_i^\dagger \mathrm{\hat c}_i^{\vphantom{\dagger}} \mathrm{\hat c}_j^{\vphantom{\dagger}}\big> + \big<\mathrm{\hat c}_i^\dagger \mathrm{\hat c}_i^{\vphantom{\dagger}}\big> \delta_{ij} \nonumber \\ & = & \mathcal{C}^{\mathrm{dd},(2)}_{ij} + \big<\mathrm{\hat n}_i^{\vphantom{\dagger}}\big> \big<\mathrm{\hat n}_j^{\vphantom{\dagger}}\big> - \big<\mathrm{\hat c}_i^\dagger \mathrm{\hat c}_j^{\vphantom{\dagger}}\big> \big<\mathrm{\hat c}_j^\dagger \mathrm{\hat c}_i^{\vphantom{\dagger}}\big> + \big<\mathrm{\hat n}_i^{\vphantom{\dagger}}\big> \delta_{ij},\end{aligned}$$ where $\mathcal{C}^{\mathrm{dd},(2)}_{ij}$ is the part of the correlation function arising from the connected two-particle Green’s function and thus the vertex. In the case of spinful Fermions, the correlator includes a sum over the spin degrees of freedom, $$\mathcal{C}^{\mathrm{dd}}_{ij} = \sum_{\sigma\sigma'} \big<\mathrm{\hat n}_{i\sigma} \mathrm{\hat n}_{j\sigma'}\big>.$$ For systems that obey the full $\mathrm{SU}(2)$ symmetry, it reads $$\begin{aligned} \mathcal{C}^{\mathrm{dd}}_{ij} & = & \mathcal{C}^{\mathrm{dd},(2)}_{ij} + 4 \big<\mathrm{\hat n}_{i\sigma}^{\vphantom{\dagger}}\big> \big<\mathrm{\hat n}_{j\sigma}^{\vphantom{\dagger}}\big> - 2 \big<\mathrm{\hat c}_{i\sigma}^\dagger \mathrm{\hat c}_{j\sigma}^{\vphantom{\dagger}}\big> \big<\mathrm{\hat c}_{j\sigma}^\dagger \mathrm{\hat c}_{i\sigma}^{\vphantom{\dagger}}\big> \nonumber \\ & & + 2 \big<\mathrm{\hat n}_{i\sigma}^{\vphantom{\dagger}}\big> \delta_{ij}, \label{eq:obs:ddcorr:spin}\end{aligned}$$ where $\sigma$ is an arbitrary spin index that is not summed over, as the single-particle quantities are proportional to $\delta_{\sigma\sigma'}$. We will first derive the expression for $\mathcal{C}^{\mathrm{dd},(2)}_{ij}$ for the spinless case at $T = 0$. Since we are looking at static quantities, but our formalism is derived in Matsubara frequency space, we must perform a Fourier transform, $$\begin{aligned} \mathcal{C}^{\mathrm{dd},(2)}_{ij} & = & \int\frac{{\mathrm{d}}\omega_1}{2\pi} \int\frac{{\mathrm{d}}\omega_2}{2\pi} \int\frac{{\mathrm{d}}\omega_3}{2\pi} \int\frac{{\mathrm{d}}\omega_4}{2\pi} \times \nonumber \\ & & \mathcal{G}^{(2,c)}_{ijij}({\mathrm{i}}\omega_1, {\mathrm{i}}\omega_2, {\mathrm{i}}\omega_3, {\mathrm{i}}\omega_4), \label{eq:TwoParticleGF:connected:fourier}\end{aligned}$$ where $\mathcal{G}^{(2,c)}$ is the two-particle connected Green’s function. Using the well-known relation between the two-particle connected Green’s function and the vertex, $$\raisebox{-2.3em}{\includegraphics[height=5em]{inline_g2}}$$ we arrive at $$\begin{aligned} & & \mathcal{G}^{(2,c)}_{ijij}({\mathrm{i}}\omega_1, {\mathrm{i}}\omega_2, {\mathrm{i}}\omega_3, {\mathrm{i}}\omega_4) = - 2\pi \sum_{\alpha\beta\gamma\delta} \sum_{\alpha'\beta'\gamma'\delta'} V^{\mathrm{rn}}_{i\alpha'} V^{\mathrm{rn}}_{j\beta'} \times \nonumber \\ & & \hspace{1em} \mathcal{G}_{\alpha'\alpha}({\mathrm{i}}\omega_1) \mathcal{G}_{\beta'\beta}({\mathrm{i}}\omega_2) \Gamma_{\alpha\beta\gamma\delta} \delta({\mathrm{i}}\omega_1 + {\mathrm{i}}\omega_2 - {\mathrm{i}}\omega_3 - {\mathrm{i}}\omega_4) \times \nonumber \\ & & \hspace{1em} \mathcal{G}_{\gamma\gamma'}({\mathrm{i}}\omega_3) \mathcal{G}_{\delta\delta'}({\mathrm{i}}\omega_4) V^{\mathrm{rn},-1}_{\gamma' i} V^{\mathrm{rn},-1}_{\delta' j}. \label{eq:TwoParticleGF:connected:frequency:initial}\end{aligned}$$ In order to solve the frequency integral analytically, we again transform into the eigenbasis of $H_0 + \Sigma$. Eq. (\[eq:TwoParticleGF:connected:frequency:initial\]) now reads $$\begin{aligned} & & \mathcal{G}^{(2,c)}_{ijij}({\mathrm{i}}\omega_1, {\mathrm{i}}\omega_2, {\mathrm{i}}\omega_3, {\mathrm{i}}\omega_4) = - 2\pi \sum_{\alpha\beta\gamma\delta} \sum_{\tilde\mu\tilde\nu\tilde\rho\tilde\sigma} V^{\mathrm{ri}}_{i\tilde\mu} V^{\mathrm{ri}}_{j\tilde\nu} \times \nonumber \\ & & \hspace{1em} \mathcal{G}_{\tilde\mu\tilde\mu}({\mathrm{i}}\omega_1) \mathcal{G}_{\tilde\nu\tilde\nu}({\mathrm{i}}\omega_2) V^{\mathrm{in}}_{\tilde\mu\alpha} V^{\mathrm{in}}_{\tilde\nu\beta} \times \nonumber \\ & & \hspace{1em} \Gamma_{\alpha\beta\gamma\delta} \delta({\mathrm{i}}\omega_1 + {\mathrm{i}}\omega_2 - {\mathrm{i}}\omega_3 - {\mathrm{i}}\omega_4) \times \nonumber \\ & & \hspace{1em} V^{\mathrm{in},-1}_{\gamma\tilde\rho} V^{\mathrm{in},-1}_{\delta\tilde\sigma} \mathcal{G}_{\tilde\rho\tilde\rho}({\mathrm{i}}\omega_3) \mathcal{G}_{\tilde\sigma\tilde\sigma}({\mathrm{i}}\omega_4) V^{\mathrm{ri},-1}_{\tilde\rho i} V^{\mathrm{ri},-1}_{\tilde\sigma j}. \label{eq:TwoParticleGF:connected:frequency:ibasis}\end{aligned}$$ For any given $\tilde\mu,\tilde\nu,\tilde\rho,\tilde\sigma$, we have for the frequency-dependent part $$\begin{aligned} & & 2\pi \int\frac{{\mathrm{d}}\omega_1}{2\pi} \int\frac{{\mathrm{d}}\omega_2}{2\pi} \int\frac{{\mathrm{d}}\omega_3}{2\pi} \int\frac{{\mathrm{d}}\omega_4}{2\pi} \mathcal{G}_{\tilde\mu\tilde\mu}({\mathrm{i}}\omega_1) \mathcal{G}_{\tilde\nu\tilde\nu}({\mathrm{i}}\omega_2) \times \nonumber \\ & & \hspace{1em} \mathcal{G}_{\tilde\rho\tilde\rho}({\mathrm{i}}\omega_3) \mathcal{G}_{\tilde\sigma\tilde\sigma}({\mathrm{i}}\omega_4) \delta({\mathrm{i}}\omega_1 + {\mathrm{i}}\omega_2 - {\mathrm{i}}\omega_3 - {\mathrm{i}}\omega_4) \nonumber \\ & = & \int\frac{{\mathrm{d}}\omega_1}{2\pi} \int\frac{{\mathrm{d}}\omega_2}{2\pi} \int\frac{{\mathrm{d}}\omega_3}{2\pi} \mathcal{G}_{\tilde\mu\tilde\mu}({\mathrm{i}}\omega_1) \mathcal{G}_{\tilde\nu\tilde\nu}({\mathrm{i}}\omega_2) \times \nonumber \\ & & \hspace{1em} \mathcal{G}_{\tilde\rho\tilde\rho}({\mathrm{i}}\omega_3) \mathcal{G}_{\tilde\sigma\tilde\sigma}({\mathrm{i}}(\omega_1 + \omega_2 - \omega_3)). \label{eq:TwoParticleGF:connected:frequency:nodelta}\end{aligned}$$ Using the convention that $\tilde\epsilon_{\tilde\mu}$ is the $\tilde\mu$-th eigenvalue of $H_0 + \Sigma$, we may now write $$\mathcal{G}_{\tilde\mu\tilde\mu}({\mathrm{i}}\omega_1) = \frac{1}{{\mathrm{i}}\omega_1 - \tilde\epsilon_{\tilde\mu} + {\mu_{\text{chem}}}} =: \frac{1}{{\mathrm{i}}\omega_1 - \tilde\xi_{\tilde\mu}}. \label{eq:TwoParticleGF:Gdef}$$ All occurring integrals are of similar form and may be solved by simply closing the integration loop around the left complex half-plane, $$\int \frac{{\mathrm{d}}\omega}{2\pi} \frac{1}{{\mathrm{i}}\omega - z} \frac{1}{{\mathrm{i}}\omega - \xi} = \frac{g(z,\xi)}{z-\xi}.$$ The exact result of the integral will depend on the position of each of the poles {$z$, $\xi$} relative to the integration loop. If they are either both inside or both outside, the integral gives zero (either the residues cancel or there are no poles inside the loop), there is only a contribution if there is just a single pole inside the loop. The residue is always $\pm(z-\xi)^{-1}$. Therefore, we define $g(z,\xi)$ to keep track of the correct sign. It may be represented as $$\begin{aligned} g(z,\xi) & = & - g(\xi,z) \nonumber \\ & = & \Theta_{\Re}(-z)\Theta_{\Re}(\xi) - \Theta_{\Re}(z)\Theta_{\Re}(-\xi),~~\end{aligned}$$ where $\Theta_{\Re}(z)$ is the Heaviside step function of the real part of $z$. Performing the first integral over $\omega_1$, we have $$\begin{aligned} & & \int\frac{{\mathrm{d}}\omega_1}{2\pi} \frac{1}{{\mathrm{i}}\omega_1 - \tilde\xi_{\tilde\mu}} \frac{1}{{\mathrm{i}}\omega_1 - (\tilde\xi_{\tilde\sigma} - {\mathrm{i}}\omega_2 + {\mathrm{i}}\omega_3)} \nonumber \\ & = & \frac{g(\tilde\xi_{\tilde\mu}, \tilde\xi_{\tilde\sigma} + {\mathrm{i}}(\omega_3 - \omega_2))}{\tilde\xi_{\tilde\mu} - \tilde\xi_{\tilde\sigma} + {\mathrm{i}}\omega_2 - {\mathrm{i}}\omega_3}.\end{aligned}$$ The expression $g(\tilde\xi_{\tilde\mu}, \tilde\xi_{\tilde\sigma} + {\mathrm{i}}(\omega_3 - \omega_2))$ may be simplified further, since for real $\omega_{2,3}$, it is equal to $g(\tilde\xi_{\tilde\mu}, \tilde\xi_{\tilde\sigma})$.[^2] Applying this result sequentially, the integral in Eq. (\[eq:TwoParticleGF:connected:frequency:nodelta\]) has the result $$\begin{aligned} & & \frac{g(\tilde\xi_{\tilde\mu}, \tilde\xi_{\tilde\sigma}) g(\tilde\xi_{\tilde\sigma} - \tilde\xi_{\tilde\mu}, \tilde\xi_{\tilde\nu}) g(\tilde\xi_{\tilde\mu} + \tilde\xi_{\tilde\nu} - \tilde\xi_{\tilde\sigma}, \tilde\xi_{\tilde\rho})}{\tilde\xi_{\tilde\mu} + \tilde\xi_{\tilde\nu} - \tilde\xi_{\tilde\rho} - \tilde\xi_{\tilde\sigma}}.\end{aligned}$$ Further simplification is possible: if $\Re \tilde\xi_{\tilde\mu} > 0$, then $\Re \tilde\xi_{\tilde\sigma}$ must be less than zero, or the contribution vanishes. In that case, it follows that $\Re(\tilde\xi_{\tilde\sigma} - \tilde\xi_{\tilde\mu}) < 0$, and we may deduce in the same way that $\Re \tilde\xi_{\tilde\nu}$ should be greater than zero. Finally, $\Re(\tilde\xi_{\tilde\mu} + \tilde\xi_{\tilde\nu} - \tilde\xi_{\tilde\sigma}) > 0$ leads to the conclusion that $\Re \tilde\xi_{\tilde\rho} < 0$. On the other hand, if $\Re \tilde\xi_{\tilde\mu} < 0$, the analogous argument can be made with flipped inequalities. The only non-zero contributions arise from combinations where the real parts of $\tilde\xi_{\tilde\mu}$ and $\tilde\xi_{\tilde\nu}$ have the same sign, but have the opposite sign to both $\tilde\xi_{\tilde\rho}$ and $\tilde\xi_{\tilde\sigma}$. Using this result, Eq. (\[eq:TwoParticleGF:connected:fourier\]) now reads $$\begin{aligned} \mathcal{C}^{\mathrm{dd},(2)}_{ij} & = & \sum_{\alpha\beta\gamma\delta} \left[ \sum_{\substack{\tilde\mu,\tilde\nu\in\mathcal{H}_e\\\tilde\rho,\tilde\sigma\in\mathcal{H}_h}} - \sum_{\substack{\tilde\mu,\tilde\nu\in\mathcal{H}_h\\\tilde\rho,\tilde\sigma\in\mathcal{H}_e}} \right] \times \nonumber \\ & & \hspace{1em} V^{\mathrm{ri}}_{i\tilde\mu} V^{\mathrm{ri}}_{j\tilde\nu} \frac{1}{\tilde\epsilon_{\tilde\mu} + \tilde\epsilon_{\tilde\nu} - \tilde\epsilon_{\tilde\rho} - \tilde\epsilon_{\tilde\sigma}} V^{\mathrm{ri},-1}_{\tilde\rho i} V^{\mathrm{ri},-1}_{\tilde\sigma j} \times \nonumber \\ & & \hspace{1em} V^{\mathrm{in}}_{\tilde\mu\alpha} V^{\mathrm{in}}_{\tilde\nu\beta} \Gamma_{\alpha\beta\gamma\delta} V^{\mathrm{in},-1}_{\gamma\tilde\rho} V^{\mathrm{in},-1}_{\delta\tilde\sigma}, \label{eq:TwoParticleGF:connected:T0:result}\end{aligned}$$ where $\mathcal{H}_e$ is the subspace where $\tilde\epsilon - {\mu_{\text{chem}}}< 0$ (“electrons”) and $\mathcal{H}_h$ the subspace where $\tilde\epsilon + {\mu_{\text{chem}}}> 0$ (“holes”). ### Correlator of Occupancy Numbers ($T > 0$) At finite temperatures $T > 0$, the result is very similar. To derive it, we need to replace the integrals in Eq. (\[eq:TwoParticleGF:connected:frequency:nodelta\]) by Matsubara sums according to the inverse of Eqs. (\[eq:Matsubara:T0-transition\],\[eq:Matsubara:T0-transition:Kronecker\]), $$\begin{aligned} & & T\sum_{\omega_n} T\sum_{\omega_m} T\sum_{\omega_{n'}} \mathcal{G}_{\tilde\mu\tilde\mu}({\mathrm{i}}\omega_n) \mathcal{G}_{\tilde\nu\tilde\nu}({\mathrm{i}}\omega_m) \times \nonumber \\ & & \hspace{1em} \mathcal{G}_{\tilde\rho\tilde\rho}({\mathrm{i}}\omega_{n'}) \mathcal{G}_{\tilde\sigma\tilde\sigma}({\mathrm{i}}(\omega_n + \omega_m - \omega_{n'})). \label{eq:TwoParticleGF:connected:finiteT:frequency:nodelta}\end{aligned}$$ Inserting Eq. (\[eq:TwoParticleGF:Gdef\]) into this expression, we may now perform the Matsubara sums analytically, which are of the form $$T\sum_{\omega_n} \frac{1}{{\mathrm{i}}\omega_n - z} \frac{1}{{\mathrm{i}}\omega_n - \xi} = \frac{n_{\mathrm{F}}(z) - n_{\mathrm{F}}(\xi)}{z - \xi},$$ where $n_{\mathrm{F}}$ is the Fermi function. We note that due to its periodicity we have $n_{\mathrm{F}}(\tilde\xi \pm {\mathrm{i}}\omega_{n'}) = n_{\mathrm{F}}(\tilde\xi)$ if $\omega_{n'}$ is a Matsubara frequency, so we may simplify the numerator again. Eq. (\[eq:TwoParticleGF:connected:finiteT:frequency:nodelta\]) is thus equal to $$\begin{aligned} & & [n_{\mathrm{F}}(\tilde\xi_{\tilde\mu}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\sigma})] [n_{\mathrm{F}}(\tilde\xi_{\tilde\sigma} - \tilde\xi_{\tilde\mu}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\nu})] \nonumber \times \\ & & \qquad \frac{ [n_{\mathrm{F}}(\tilde\xi_{\tilde\mu} + \tilde\xi_{\tilde\nu} - \tilde\xi_{\tilde\sigma}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\rho})] }{\tilde\xi_{\tilde\mu} + \tilde\xi_{\tilde\nu} - \tilde\xi_{\tilde\rho} - \tilde\xi_{\tilde\sigma}}.\end{aligned}$$ Therefore, we have $$\begin{aligned} \mathcal{C}^{\mathrm{dd},(2)}_{ij} & = & \sum_{\alpha\beta\gamma\delta}\sum_{\tilde\mu\tilde\nu\tilde\rho\tilde\sigma} V^{\mathrm{ri}}_{i\tilde\mu} V^{\mathrm{ri}}_{j\tilde\nu} \frac{1}{\tilde\epsilon_{\tilde\mu} + \tilde\epsilon_{\tilde\nu} - \tilde\epsilon_{\tilde\rho} - \tilde\epsilon_{\tilde\sigma}} \times \nonumber \\ & & \hspace{1em} [n_{\mathrm{F}}(\tilde\xi_{\tilde\sigma}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\mu})] [n_{\mathrm{F}}(\tilde\xi_{\tilde\sigma} - \tilde\xi_{\tilde\mu}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\nu})] \times \nonumber \\ & & \hspace{1em} [n_{\mathrm{F}}(\tilde\xi_{\tilde\mu} + \tilde\xi_{\tilde\nu} - \tilde\xi_{\tilde\sigma}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\rho})] V^{\mathrm{ri},-1}_{\tilde\rho i} V^{\mathrm{ri},-1}_{\tilde\sigma j} \times \nonumber \\ & & \hspace{1em} V^{\mathrm{in}}_{\tilde\mu\alpha} V^{\mathrm{in}}_{\tilde\nu\beta} \Gamma_{\alpha\beta\gamma\delta} V^{\mathrm{in},-1}_{\gamma\tilde\rho} V^{\mathrm{in},-1}_{\delta\tilde\sigma}. \label{eq:TwoParticleGF:connected:finiteT:result} \end{aligned}$$ For orbitals far away from the Fermi energy, $|\tilde\xi| \gg T$, this expression goes over into the expression for $T = 0$ and we arrive at Eq. (\[eq:TwoParticleGF:connected:T0:result\]) again. ### Correlator of Occupancy Numbers (Systems with spin) In systems with spin we must also sum over two spin indices when calculating $\mathcal{C}^{\mathrm{dd},(2)}_{ij}$. We replace all orbital indices in Eq. (\[eq:TwoParticleGF:connected:finiteT:result\]) by pairs of orbital and spin indices, $\alpha\to(\alpha,\sigma)$. For systems with $\mathrm{SU}(2)$ symmetry all single-particle quantities are diagonal in spin space, so after performing sums over all the relevant Kronecker-$\delta$s, we have $$\begin{aligned} \mathcal{C}^{\mathrm{dd},(2)}_{ij} & = & \sum_{\sigma\sigma'} \sum_{\alpha\beta\gamma\delta}\sum_{\tilde\mu\tilde\nu\tilde\rho\tilde\sigma} V^{\mathrm{ri}}_{i\tilde\mu} V^{\mathrm{ri}}_{j\tilde\nu} \frac{1}{\tilde\epsilon_{\tilde\mu} + \tilde\epsilon_{\tilde\nu} - \tilde\epsilon_{\tilde\rho} - \tilde\epsilon_{\tilde\sigma}} \times \nonumber \\ & & \hspace{.5em} [n_{\mathrm{F}}(\tilde\xi_{\tilde\sigma}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\mu})] [n_{\mathrm{F}}(\tilde\xi_{\tilde\sigma} - \tilde\xi_{\tilde\mu}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\nu})] \times \nonumber \\ & & \hspace{.5em} [n_{\mathrm{F}}(\tilde\xi_{\tilde\mu} + \tilde\xi_{\tilde\nu} - \tilde\xi_{\tilde\sigma}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\rho})] V^{\mathrm{ri},-1}_{\tilde\rho i} V^{\mathrm{ri},-1}_{\tilde\sigma j} \times \nonumber \\ & & \hspace{.5em} V^{\mathrm{in}}_{\tilde\mu\alpha} V^{\mathrm{in}}_{\tilde\nu\beta} \Gamma_{(\alpha,\sigma)(\beta,\sigma')(\gamma,\sigma)(\delta,\sigma')} V^{\mathrm{in},-1}_{\gamma\tilde\rho} V^{\mathrm{in},-1}_{\delta\tilde\sigma}. \label{eq:TwoParticleGF:connected:finiteT:spin:inserted}\end{aligned}$$ Inserting Eq. (\[eq:definition:Gamma:with-spin\]), we may perform the summation over the remaining spin indices and arrive at $$\begin{aligned} \mathcal{C}^{\mathrm{dd},(2)}_{ij} & = & \sum_{\alpha\beta\gamma\delta}\sum_{\tilde\mu\tilde\nu\tilde\rho\tilde\sigma} V^{\mathrm{ri}}_{i\tilde\mu} V^{\mathrm{ri}}_{j\tilde\nu} \frac{1}{\tilde\epsilon_{\tilde\mu} + \tilde\epsilon_{\tilde\nu} - \tilde\epsilon_{\tilde\rho} - \tilde\epsilon_{\tilde\sigma}} \times \nonumber \\ & & \hspace{.5em} [n_{\mathrm{F}}(\tilde\xi_{\tilde\sigma}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\mu})] [n_{\mathrm{F}}(\tilde\xi_{\tilde\sigma} - \tilde\xi_{\tilde\mu}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\nu})] \times \nonumber \\ & & \hspace{.5em} [n_{\mathrm{F}}(\tilde\xi_{\tilde\mu} + \tilde\xi_{\tilde\nu} - \tilde\xi_{\tilde\sigma}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\rho})] V^{\mathrm{ri},-1}_{\tilde\rho i} V^{\mathrm{ri},-1}_{\tilde\sigma j} \times \nonumber \\ & & \hspace{.5em} V^{\mathrm{in}}_{\tilde\mu\alpha} V^{\mathrm{in}}_{\tilde\nu\beta} \big[ 2\Gamma^{\text{s}}_{\alpha\beta\gamma\delta} - 4\Gamma^{\text{s}}_{\alpha\beta\delta\gamma} \big] V^{\mathrm{in},-1}_{\gamma\tilde\rho} V^{\mathrm{in},-1}_{\delta\tilde\sigma}. \label{eq:TwoParticleGF:connected:finiteT:spin:result}\end{aligned}$$ At $T = 0$, the result is analogously given by $$\begin{aligned} \mathcal{C}^{\mathrm{dd},(2)}_{ij} & = & \sum_{\alpha\beta\gamma\delta} \left[ \sum_{\substack{\tilde\mu,\tilde\nu\in\mathcal{H}_e\\\tilde\rho,\tilde\sigma\in\mathcal{H}_h}} - \sum_{\substack{\tilde\mu,\tilde\nu\in\mathcal{H}_h\\\tilde\rho,\tilde\sigma\in\mathcal{H}_e}} \right] \times \nonumber \\ & & V^{\mathrm{ri}}_{i\tilde\mu} V^{\mathrm{ri}}_{j\tilde\nu} \frac{1}{\tilde\epsilon_{\tilde\mu} + \tilde\epsilon_{\tilde\nu} - \tilde\epsilon_{\tilde\rho} - \tilde\epsilon_{\tilde\sigma}} V^{\mathrm{ri},-1}_{\tilde\rho i} V^{\mathrm{ri},-1}_{\tilde\sigma j} \times \nonumber \\ & & V^{\mathrm{in}}_{\tilde\mu\alpha} V^{\mathrm{in}}_{\tilde\nu\beta} \big[ 2\Gamma^{\text{s}}_{\alpha\beta\gamma\delta} - 4\Gamma^{\text{s}}_{\alpha\beta\delta\gamma} \big] V^{\mathrm{in},-1}_{\gamma\tilde\rho} V^{\mathrm{in},-1}_{\delta\tilde\sigma}. \label{eq:TwoParticleGF:connected:T0:spin:result}\end{aligned}$$ ### Spin-Spin Correlator In contrast to the expectation value of ${\mathbf{S}}_i$, the expectation value of ${\mathbf{S}}_i \cdot {\mathbf{S}}_j$ does not automatically vanish in systems with $\mathrm{SU}(2)$ symmetry. Using $${\mathbf{\hat S}}_i = \sum_{\sigma\sigma'} \mathrm{\hat c}^{\dagger}_{i\sigma} \vec\tau_{\sigma\sigma'} \mathrm{\hat c}^{\vphantom{\dagger}}_{i\sigma'},$$ where $\vec\tau$ are the Pauli matrices and the identity $$\sum_{k=0}^{3} \tau^k_{\sigma\sigma'} \tau^k_{\bar\sigma\bar\sigma'} = 2 \delta_{\sigma\bar\sigma'} \delta_{\sigma'\bar\sigma},$$ we may write $$\begin{aligned} \mathcal{C}^{\mathrm{ss}}_{ij} & := & \big< {\mathbf{S}}_i \cdot {\mathbf{S}}_j \big> \nonumber \\ & = & \sum_k \sum_{\sigma\sigma'} \sum_{\bar\sigma\bar\sigma'} \tau^k_{\sigma\sigma'} \tau^k_{\bar\sigma\bar\sigma'} \big< \mathrm{\hat c}^{\dagger}_{i\sigma} \mathrm{\hat c}^{\vphantom{\dagger}}_{i\sigma'} \mathrm{\hat c}^{\dagger}_{j\bar\sigma} \mathrm{\hat c}^{\vphantom{\dagger}}_{j\bar\sigma'}\big> \nonumber \\ & = & 2 \sum_{\sigma\sigma'} \big< \mathrm{\hat c}^{\dagger}_{i\sigma} \mathrm{\hat c}^{\vphantom{\dagger}}_{i\sigma'} \mathrm{\hat c}^{\dagger}_{j\sigma'} \mathrm{\hat c}^{\vphantom{\dagger}}_{j\sigma} \big> - \big<\mathrm{\hat n}_i^{\vphantom{\dagger}} \mathrm{\hat n}_j^{\vphantom{\dagger}}\big> \nonumber \\ & = & 2 \sum_{\sigma\sigma'} \big< \mathrm{\hat c}^{\dagger}_{i\sigma} \mathrm{\hat c}^{\dagger}_{j\sigma'} \mathrm{\hat c}^{\vphantom{\dagger}}_{j\sigma} \mathrm{\hat c}^{\vphantom{\dagger}}_{i\sigma'} \big> - \big<\mathrm{\hat n}_i^{\vphantom{\dagger}} \mathrm{\hat n}_j^{\vphantom{\dagger}}\big> - 4 \delta_{ij} \big< \mathrm{\hat n}_{i}^{\vphantom{\dagger}} \big> \nonumber \\ & = & \mathcal{C}^{\mathrm{ss},(2)}_{ij} - \big<\mathrm{\hat n}_i^{\vphantom{\dagger}} \mathrm{\hat n}_j^{\vphantom{\dagger}}\big> - 4 \delta_{ij} \big< \mathrm{\hat n}_{i}^{\vphantom{\dagger}} \big> \nonumber \\ & & + 2 \sum_{\sigma\sigma'} \Big[ \big< \mathrm{\hat c}^{\dagger}_{j\sigma'} \mathrm{\hat c}^{\vphantom{\dagger}}_{j\sigma} \big> \big< \mathrm{\hat c}^{\dagger}_{i\sigma} \mathrm{\hat c}^{\vphantom{\dagger}}_{i\sigma'} \big> - \big< \mathrm{\hat c}^{\dagger}_{i\sigma} \mathrm{\hat c}^{\vphantom{\dagger}}_{j\sigma} \big> \big< \mathrm{\hat c}^{\dagger}_{j\sigma'} \mathrm{\hat c}^{\vphantom{\dagger}}_{i\sigma'} \big> \Big] \nonumber \\ & = & \mathcal{C}^{\mathrm{ss},(2)}_{ij} + 4 \big<\mathrm{\hat n}_{i\sigma}^{\vphantom{\dagger}}\big> \big<\mathrm{\hat n}_{j\sigma}^{\vphantom{\dagger}}\big> - 8 \big<\mathrm{\hat c}_{i\sigma}^\dagger \mathrm{\hat c}_{j\sigma}^{\vphantom{\dagger}}\big> \big<\mathrm{\hat c}_{j\sigma}^\dagger \mathrm{\hat c}_{i\sigma}^{\vphantom{\dagger}}\big> \nonumber \\ & & \hphantom{\mathcal{C}^{\mathrm{ss},(2)}_{ij}} - \big<\mathrm{\hat n}_i^{\vphantom{\dagger}} \mathrm{\hat n}_j^{\vphantom{\dagger}}\big> - 4 \delta_{ij} \big< \mathrm{\hat n}_{i}^{\vphantom{\dagger}} \big>.\end{aligned}$$ Inserting Eq. (\[eq:obs:ddcorr:spin\]), several terms cancel and we arrive at $$\mathcal{C}^{\mathrm{ss}}_{ij} = \mathcal{C}^{\mathrm{ss},(2)}_{ij} - \mathcal{C}^{\mathrm{dd},(2)}_{ij} - 6 \Big( \big<\mathrm{\hat c}_{i\sigma}^\dagger \mathrm{\hat c}_{j\sigma}^{\vphantom{\dagger}}\big> \big<\mathrm{\hat c}_{j\sigma}^\dagger \mathrm{\hat c}_{i\sigma}^{\vphantom{\dagger}}\big> + \delta_{ij} \big< \mathrm{\hat n}_{i}^{\vphantom{\dagger}} \big> \Big).$$ The expression for $\mathcal{C}^{\mathrm{ss},(2)}_{ij}$ may be derived in the same manner as the expression for $\mathcal{C}^{\mathrm{dd},(2)}_{ij}$. At finite temperatures, it reads $$\begin{aligned} \mathcal{C}^{\mathrm{ss},(2)}_{ij} & = & 2 \sum_{\alpha\beta\gamma\delta}\sum_{\tilde\mu\tilde\nu\tilde\rho\tilde\sigma} V^{\mathrm{ri}}_{i\tilde\mu} V^{\mathrm{ri}}_{j\tilde\nu} \frac{1}{\tilde\epsilon_{\tilde\mu} + \tilde\epsilon_{\tilde\nu} - \tilde\epsilon_{\tilde\rho} - \tilde\epsilon_{\tilde\sigma}} \times \nonumber \\ & & \hspace{.5em} [n_{\mathrm{F}}(\tilde\xi_{\tilde\sigma}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\mu})] [n_{\mathrm{F}}(\tilde\xi_{\tilde\sigma} - \tilde\xi_{\tilde\mu}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\nu})] \times \nonumber \\ & & \hspace{.5em} [n_{\mathrm{F}}(\tilde\xi_{\tilde\mu} + \tilde\xi_{\tilde\nu} - \tilde\xi_{\tilde\sigma}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\rho})] V^{\mathrm{ri},-1}_{\tilde\rho i} V^{\mathrm{ri},-1}_{\tilde\sigma j} \times \nonumber \\ & & \hspace{.5em} V^{\mathrm{in}}_{\tilde\mu\alpha} V^{\mathrm{in}}_{\tilde\nu\beta} \big[ 4\Gamma^{\text{s}}_{\alpha\beta\gamma\delta} - 2\Gamma^{\text{s}}_{\alpha\beta\delta\gamma} \big] V^{\mathrm{in},-1}_{\gamma\tilde\rho} V^{\mathrm{in},-1}_{\delta\tilde\sigma}. \label{eq:sscorr:finiteT:spin:result}\end{aligned}$$ As one is often interested in both the occupation number and spin correlators, we note that the expression for the difference between $\mathcal{C}^{\mathrm{ss},(2)}_{ij}$ and $\mathcal{C}^{\mathrm{dd},(2)}_{ij}$ simplifies slightly, $$\begin{aligned} \mathcal{C}^{'\mathrm{ss},(2)}_{ij} & = & \mathcal{C}^{\mathrm{ss},(2)}_{ij} - \mathcal{C}^{\mathrm{dd},(2)}_{ij} \nonumber \\ & = & 6 \sum_{\alpha\beta\gamma\delta}\sum_{\tilde\mu\tilde\nu\tilde\rho\tilde\sigma} V^{\mathrm{ri}}_{i\tilde\mu} V^{\mathrm{ri}}_{j\tilde\nu} \frac{1}{\tilde\epsilon_{\tilde\mu} + \tilde\epsilon_{\tilde\nu} - \tilde\epsilon_{\tilde\rho} - \tilde\epsilon_{\tilde\sigma}} \times \nonumber \\ & & \hspace{.5em} [n_{\mathrm{F}}(\tilde\xi_{\tilde\sigma}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\mu})] [n_{\mathrm{F}}(\tilde\xi_{\tilde\sigma} - \tilde\xi_{\tilde\mu}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\nu})] \times \nonumber \\ & & \hspace{.5em} [n_{\mathrm{F}}(\tilde\xi_{\tilde\mu} + \tilde\xi_{\tilde\nu} - \tilde\xi_{\tilde\sigma}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\rho})] V^{\mathrm{ri},-1}_{\tilde\rho i} V^{\mathrm{ri},-1}_{\tilde\sigma j} \times \nonumber \\ & & \hspace{.5em} V^{\mathrm{in}}_{\tilde\mu\alpha} V^{\mathrm{in}}_{\tilde\nu\beta} \Gamma^{\text{s}}_{\alpha\beta\gamma\delta} V^{\mathrm{in},-1}_{\gamma\tilde\rho} V^{\mathrm{in},-1}_{\delta\tilde\sigma}. \label{eq:sscorr:finiteT:spin:result:v2}\end{aligned}$$ At $T = 0$, the expression reads $$\begin{aligned} \mathcal{C}^{'\mathrm{ss},(2)}_{ij} & = & 6 \sum_{\alpha\beta\gamma\delta} \left[ \sum_{\substack{\tilde\mu,\tilde\nu\in\mathcal{H}_e\\\tilde\rho,\tilde\sigma\in\mathcal{H}_h}} - \sum_{\substack{\tilde\mu,\tilde\nu\in\mathcal{H}_h\\\tilde\rho,\tilde\sigma\in\mathcal{H}_e}} \right] \times \nonumber \\ & & \hspace{1em} V^{\mathrm{ri}}_{i\tilde\mu} V^{\mathrm{ri}}_{j\tilde\nu} \frac{1}{\tilde\epsilon_{\tilde\mu} + \tilde\epsilon_{\tilde\nu} - \tilde\epsilon_{\tilde\rho} - \tilde\epsilon_{\tilde\sigma}} V^{\mathrm{ri},-1}_{\tilde\rho i} V^{\mathrm{ri},-1}_{\tilde\sigma j} \times \nonumber \\ & & \hspace{.5em} V^{\mathrm{in}}_{\tilde\mu\alpha} V^{\mathrm{in}}_{\tilde\nu\beta} \Gamma^{\text{s}}_{\alpha\beta\gamma\delta} V^{\mathrm{in},-1}_{\gamma\tilde\rho} V^{\mathrm{in},-1}_{\delta\tilde\sigma}. \label{eq:sscorr:T0:spin:result}\end{aligned}$$ Reducting the Hilbert space size: Active-space approximation ([[ASA]{}]{}) {#sec:orbital-reduction} -------------------------------------------------------------------------- The flow equations for the self-energy and the vertex, even in their simplest form Eqs. (\[eq:flow:Sigma\], \[eq:flow:Gamma\]), are still computationally challenging. In translationally invariant systems simplifications arise, because the vertex only depends on three momenta, the fourth given by momentum conservation. Moreover, one only tracks momenta near the Fermi surface: The Brillouin zone is divided into patches each containing a single tracked momentum and the interaction vertex $\Gamma$ is only calculated at these momenta. Whenever it needs to be evaluated for other momenta, the other momentum is replaced by the tracked one located within the same patch (coarse graining).[@FRGHubbard] In the absence of periodicity, this kind of patching is not possible, since there is no well-defined concept of a Fermi surface. ![Active space: Selection of $M$ orbitals (green) around the chemical potential, ${\mu_{\text{chem}}}$, for which the vertex will be renormalized in the active space approximation ([[ASA]{}]{}). The self-energy will still be renormalized for all $N$ orbitals, including the remaining (purple) ones.[]{data-label="fig:state_selection"}](figure2){width=".95\linewidth"} For systems without translational symmetries, we here propose an approach alternative to Fermi-surface patching for reducing the number of explicit degrees of freedom. Similar to the patching scheme, we define an “active space” $\mathcal{H}_M$ of (effective) single-particle states near the chemical potential that are kept. In our case $\mathcal{H}_M$ simply contains the $M$ orbitals closest to the chemical potential, ${\mu_{\text{chem}}}$, (e.g. half above and half below); see Fig. \[fig:state\_selection\] for details. We will refer to this approach in a loose manner of speaking as “active-space approximation” ([[ASA]{}]{}). Within [[ASA]{}]{} external indices of the flow equation for the vertex, Eq. (\[eq:flow:Gamma\]), only refer to a reduced number of states. In summations over the full single-particle Hilbert space, we adopt the approximation scheme $$\Gamma^\Lambda_{\alpha\beta\gamma\delta} \rightarrow \left\{ \begin{array}{ll} \Gamma^\Lambda_{\alpha\beta\gamma\delta} & \{\alpha,\beta,\gamma,\delta\} \subseteq \mathcal{H}_M \\ U_{\alpha\beta\gamma\delta} & \text{otherwise} \end{array} \right., \label{eq:definition:replacement-gamma}.$$ To simplify the notation, in the following we label states from the active space $\mathcal{H}_M$ with barred indices, e.g. $\bar\alpha$, whereas states from the full set of orbitals are denoted without bars, e.g. $\alpha$. We comment on the choice for $M$ at a given system size. As long as mostly the states close to the Fermi energy are important for screening (as is also assumed in applications of the FRG for translationally invariant systems), we can argue that the number of states necessarily kept in $\mathcal{H}_M$, $M$, should grow sub-linearly with the total number of orbitals, $N$. We remind ourselves that in a translationally invariant system, the Fermi surface has dimensionality $(d-1)$ within the $d$-dimensional Brillouin zone. Since the number of states in the Brillouin zone grows as $L^d$, but the number of states on a surface within that space grows as $L^{d-1}$, we suggest the number of states required should be proportional to $L^{d-1}$, which can be rewritten as $L^{d-1} = (L^d)^{(d-1)/d} = N^{1-1/d}$. To the extent that $M$ scales the same also for generic systems, we have $M\sim N^{1-1/d}$, implying $M\sim N^{1/2}$ in 2D. In Sec. \[sec:verification\] we will establish the efficacy of the [[ASA]{}]{} and also revisit the system size scaling. Runaway Flow {#sec:frg:runaway-flow} ------------ At present, one of the main applications of [[$k$FRG]{}]{} is the study of phase diagrams, because an unbiased view of competing instabilities of the system is provided. In parameter regimes where the system shows a phase transition, the instabilities pertaining to the new phase lead to “runaway flow”: at a critical scale, $\Lambda_{\text{c}}$, the integration of the RG-equations exhibits matrix elements of the interaction vertex that diverge. The physical nature of the instability reveals itself in what matrix element actually shows the strongest divergence. This property of the FRG has been used very successfully to study the phase diagram of a multitude of systems, for an overview see Ref. . With [[$\epsilon$FRG]{}]{} one needs to keep in mind that the eigenstate representation is not based on plane waves. Therefore, the physics of individual vertex-elements may not be as transparent as it is in the clean case. Hence, it can be helpful to calculate two-particle correlators at $\Lambda_{\text{c}}$ to support interpretations of the precise nature of the instability. We mention that cases exist in which competing order parameters influence each other (such as antiferromagnetism and $d$-wave superconductivity). Strategies how to deal with this situation have been developed within [[$k$FRG]{}]{}. Ideally, one should continue the flow to $\Lambda \to 0$ to obtain information about the true phase diagram of the system. This may be done in principle, e.g., by introducing an infinitesimal symmetry-breaking term that grows under the RG-flow, as has been done for superconductivity[@SalmhoferBrokenSymmetry]. Alternatively, one may calculate the flow for the combined Bose-Fermi system, where fermions were decoupled via a Hubbard-Stratonovich transformation.[@WetterichHubbard] Implementation {#sec:impl} ============== We implement the FRG procedure in C++, using the Eigen linear algebra library[@EigenLibrary] for matrix products and the HDF5 file format [@Hdf5Library] for storage. We employ the OpenMP 3.1 standard [@OpenMP3.1] for parallelization. Computational Details {#sec:impl:efficient-traces} --------------------- The computational complexity of the self-energy flow, Eqs. (\[eq:flow:Sigma\],\[eq:flow:Sigma:WithSpin\],\[eq:flow:Sigma:finiteT\]), is given by ${\ensuremath{\mathcal{O}\big(N^4\big)}}$ – two loops for each of the outer indices, two loops for the contraction with the non-diagonal single-scale propagator. At first glance the flow of the vertex, e.g., Eq. (\[eq:flow:Gamma\]) appears to have a complexity of ${\ensuremath{\mathcal{O}\big(N^8\big)}}$. However, one may define intermediate products, $I^{\text{c},\pm}, I^{\text{ph},\pm}$, $$\begin{aligned} I^{\text{c},+}_{\mu\rho\bar\gamma\bar\delta} & = & \sum_{\nu} P^{\Lambda,s}_{\mu\nu}(\Lambda) \Gamma^\Lambda_{\nu\rho\bar\gamma\bar\delta} \label{eq:definition:intermediate:cooper:plus} \\ I^{\text{c},-}_{\bar\alpha\bar\beta\rho\mu} & = & \sum_{\sigma} P^{\Lambda,s}_{\rho\sigma}(-\Lambda) \Gamma^\Lambda_{\bar\alpha\bar\beta\sigma\mu} \label{eq:definition:intermediate:cooper:minus} \\ I^{\text{ph},\pm}_{\bar\alpha\nu\bar\gamma\sigma} & = & \sum_{\rho} \Gamma^\Lambda_{\bar\alpha\nu\bar\gamma\rho} P^{\Lambda,s}_{\rho\sigma}(\pm\Lambda), \label{eq:definition:intermediate:particle-hole}\end{aligned}$$ where each of these partial diagrams has a complexity of ${\ensuremath{\mathcal{O}\big(N^5\big)}}$. The flow equation for the vertex now reads $$\begin{aligned} {\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}\Gamma^\Lambda_{\bar\alpha\bar\beta\bar\gamma\bar\delta} & = & - \frac{1}{2\pi} \sum_{\mu\rho} \left\{ I^{\text{c},+}_{\mu\rho\bar\gamma\bar\delta} I^{\text{c},-}_{\bar\alpha\bar\beta\rho\mu} \right.\nonumber\\ & & \left. \hspace{2em} + I^{\text{ph},+}_{\bar\alpha\mu\bar\gamma\rho} I^{\text{ph},+}_{\bar\beta\rho\bar\delta\mu} + I^{\text{ph},-}_{\bar\alpha\mu\bar\gamma\rho} I^{\text{ph},-}_{\bar\beta\rho\bar\delta\mu} \right.\nonumber\\ & & \left. \hspace{2em} - I^{\text{ph},+}_{\bar\beta\mu\bar\gamma\rho} I^{\text{ph},+}_{\bar\alpha\rho\bar\delta\mu} - I^{\text{ph},-}_{\bar\beta\mu\bar\gamma\rho} I^{\text{ph},-}_{\bar\alpha\rho\bar\delta\mu} \right\}, \label{eq:flow:Gamma:with-intermediates}\end{aligned}$$ with a computational complexity of ${\ensuremath{\mathcal{O}\big(N^6\big)}}$. In the case of $M < N$, using the replacement in Eq. (\[eq:definition:replacement-gamma\]), this reduces to ${\ensuremath{\mathcal{O}\big(N^3 N^3\big)}}$ for the calculation of the intermediates and to ${\ensuremath{\mathcal{O}\big(N^2 M^4\big)}}$ for the trace. Repeating our argument from Sec. \[sec:orbital-reduction\] that $M \propto \sqrt{N}$, we expect a scaling of ${\ensuremath{\mathcal{O}\big(N^4\big)}}$ for two-dimensional systems. ### Efficient Trace Evaluation In order to evaluate the temporary products for the flow of the vertex, Eqs. (\[eq:definition:intermediate:cooper:plus\],\[eq:definition:intermediate:cooper:minus\],\[eq:definition:intermediate:particle-hole\]), it is advantageous to rewrite the expression in terms of a matrix product, e.g. $$I^{\text{c},+}_{\mu,(\rho\bar\gamma\bar\delta)} = \sum_{\nu} P^{\Lambda,s}_{\mu\nu}(\Lambda) \Gamma^\Lambda_{\nu,(\rho\bar\gamma\bar\delta)},$$ where we interpret $(\rho\bar\gamma\bar\delta)$ as a single index, because modern generic matrix-matrix multiplication (GEMM) kernels are highly optimized and perform far better than a simple sum. For the cases where we calculate the renormalization of the vertex for all states, this is trivial. Note that for some equations one needs to retain a copy of the vertex with transposed indices to be able to do this. Since our implementation is typically not constrained by the available memory but rather the available processing power, this tradeoff is advantageous. ![\[fig:impl:gemm-statesel\] The subdivision of the GEMM kernel for the intermediate product $I^{\text{c},+}$ in the $\nu$ and $\rho$ indices. The regions one to five in the diagram correspond to the terms of Eqs. (\[eq:impl:gemmsplit:tV\],\[eq:impl:gemmsplit:tU0s\],\[eq:impl:gemmsplit:tU0l\],\[eq:impl:gemmsplit:tUs\],\[eq:impl:gemmsplit:tUl\]), respectively.](figure3){height="6cm"} It is trickier to approximate the vertex according to Eq. . Instead of rewriting the entire expression in terms of a GEMM kernel, we need to perform the loop on the external indices explicitly. We may then split the resulting matrix product into five parts. Taking for example Eq.  and using that $\mathcal{H}_M$ is the subset of states for which the vertex is renormalized, we have $$\begin{aligned} I^{\text{c},+}_{\mu\rho\bar\gamma\bar\delta} & = & \hphantom{+} \hspace{-1em}\sum_{\bar\nu\in \mathcal{H}_M} P^{\Lambda,s}_{\mu\bar\nu}(\Lambda) \Gamma^\Lambda_{\bar\nu\rho\bar\gamma\bar\delta} \hspace{1em}[\rho\in \mathcal{H}_M] \label{eq:impl:gemmsplit:tV} \\ & & \hspace{-1em} + \sum_{\bar\nu\in \mathcal{H}_M} P^{\Lambda,s}_{\mu\bar\nu}(\Lambda) U_{\bar\nu\rho\bar\gamma\bar\delta} \hspace{1em}[\rho < \min(\mathcal{H}_M)] \label{eq:impl:gemmsplit:tU0s} \\ & & \hspace{-1em} + \sum_{\bar\nu\in \mathcal{H}_M} P^{\Lambda,s}_{\mu\bar\nu}(\Lambda) U_{\bar\nu\rho\bar\gamma\bar\delta} \hspace{1em}[\rho > \max(\mathcal{H}_M)] \label{eq:impl:gemmsplit:tU0l} \\ & & \hspace{-1em} + \sum_{\nu < \min(\mathcal{H}_M)} P^{\Lambda,s}_{\mu\nu}(\Lambda) U_{\nu\rho\bar\gamma\bar\delta} \label{eq:impl:gemmsplit:tUs} \\ & & \hspace{-1em} + \sum_{\nu > \max(\mathcal{H}_M)} P^{\Lambda,s}_{\mu\nu}(\Lambda) U_{\nu\rho\bar\gamma\bar\delta}. \label{eq:impl:gemmsplit:tUl}\end{aligned}$$ We assume here that the non-interacting states are ordered in energy. The five subexpressions may then be written in terms of GEMM kernels with rectangular blocks of the matrices $P^{\Lambda,s}$ and $U_{\cdot\cdot\bar\gamma\bar\delta}$. Figure \[fig:impl:gemm-statesel\] shows the division into these terms in the plane of $\nu$ and $\rho$ indices. There are no standard kernels for trace evaluation, e.g. Eq. , hence we implement that directly in terms of a loop. ### Parametrization of the Flow Equations We use an exponential parametrization for the flow equations, Eqs. (\[eq:flow:Sigma\],\[eq:flow:Gamma:with-intermediates\]), $$\Lambda = \Lambda_0 \mathrm{e}^{-l \Delta s}, \hspace{2em} l \in \mathds{N},$$ where $\Lambda_0$ is the initial $\Lambda$ at which the flow starts and $l$ is our discretizing iteration number. This parametrization has the advantage that it captures the physics close to the Fermi energy well, as the integration mesh gets denser, while still being relatively fast in reaching that point. Both flow equations are of the form $${\frac{{\mathrm{d}}}{{\mathrm{d}}\Lambda}}A(\Lambda) = -\frac{1}{2\pi} B(\Lambda).$$ allowing for a trivial discretization, $$A(\Lambda(l+1)) = A(\Lambda(l)) + \frac{\Lambda(l)\Delta s}{2\pi} B(\Lambda(l)),$$ assuming that $\Delta s$ is sufficiently small. In the following calculations we have chosen the parameters $\Delta s = 0.02$ and $\Lambda_0 = 40$. Unless we encounter a divergence in the flow, we stop as soon as $\Lambda < 10^{-4}$ (giving a total of $l_{\text{max}} = 645$ iterations). Chemical Potential ------------------ We would like to keep the number of particles fixed to study the system at a given filling fraction. Since our flow modifies the real part of the self-energy, we need to constantly adjust the chemical potential during the renormalization procedure. At $T = 0$ we diagonalize the matrix $H_0 + \Sigma^\Lambda$ to obtain the updated quasi-particle energies for a given $\Lambda$ (including the initial $\Lambda_0$, since $\Sigma^{\Lambda_0} \neq 0$). We choose our chemical potential to be $${\mu_{\text{chem}}}^\Lambda = \frac{1}{2}(\tilde\epsilon^\Lambda_{N_{\text{e}}+1} + \tilde\epsilon^\Lambda_{N_{\text{e}}}), \label{eq:impl:muchem:T0}$$ where $\tilde\epsilon^\Lambda_{N_{\text{e}}}$ is the energy of the highest occupied quasi-particle state and $\tilde\epsilon^\Lambda_{N_{\text{e}}}$ the energy of the lowest unoccupied quasi-particle state. At $T > 0$ the value of ${\mu_{\text{chem}}}^\Lambda$ follows as usual from the solution to the equation $$N_{\text{e}} = \sum_{\tilde\epsilon^\Lambda_{\tilde\alpha} < {\mu_{\text{chem}}}^\Lambda} n_{\mathrm{F}}(\tilde\epsilon_{\tilde\alpha} - {\mu_{\text{chem}}}^\Lambda), \label{eq:impl:muchem:finiteT:tosolve}$$ where $N_{\text{e}}$ is the number of electrons and $\tilde\epsilon^\Lambda_{\tilde\alpha}$ are the quasi-particle energies for a given $\Lambda$, i.e. the eigenvalues of $H_0 + \Sigma^\Lambda$. Correlators {#sec:impl:correlators} ----------- Starting from Eq. , we first transform the vertex into the $\Lambda$-dependent quasi-particle basis, $$\tilde\Gamma^\Lambda_{\tilde\mu\tilde\nu\tilde\rho\tilde\sigma} = \sum_{\alpha\beta\gamma\delta} V^{\mathrm{in}}_{\tilde\mu\alpha} V^{\mathrm{in}}_{\tilde\nu\beta} \Gamma^{\Lambda}_{\alpha\beta\gamma\delta} V^{\mathrm{in},-1}_{\gamma\tilde\rho} V^{\mathrm{in},-1}_{\delta\tilde\sigma}.$$ We exploit fast matrix multiplication routines to perform these basis transforms. As these routines require us to group either the three left- or rightmost indices together, we first transform the vertex in $\alpha$ and $\delta$, then transpose it to have $\beta$ as the first index and $\gamma$ as the last index, and apply the final pair of transformations, yielding the following sequence of steps: $$\begin{aligned} \Gamma^{\Lambda,(1)}_{\tilde\mu\beta\gamma\delta} & = & \sum_{\alpha} V^{\mathrm{in}}_{\tilde\mu\alpha} \Gamma^{\Lambda}_{\alpha\beta\gamma\delta} \\ \Gamma^{\Lambda,(2)}_{\tilde\mu\beta\gamma\tilde\sigma} & = & \sum_{\delta} \Gamma^{\Lambda,(1)}_{\alpha\beta\gamma\delta} V^{\mathrm{in},-1}_{\delta\tilde\sigma} \\ \Gamma^{\Lambda,(3)}_{\beta\tilde\mu\tilde\sigma\gamma} & = & \Gamma^{\Lambda,(2)}_{\tilde\mu\beta\gamma\tilde\sigma} \\ \Gamma^{\Lambda,(4)}_{\tilde\nu\tilde\mu\tilde\sigma\gamma} & = & \sum_{\tilde\nu} V^{\mathrm{in}}_{\tilde\nu\beta} \Gamma^{\Lambda,(3)}_{\beta\tilde\mu\tilde\sigma\gamma} \\ \tilde\Gamma^\Lambda_{\tilde\mu\tilde\nu\tilde\rho\tilde\sigma} & = & \sum_{\tilde\rho} \Gamma^{\Lambda,(4)}_{\tilde\nu\tilde\mu\tilde\sigma\gamma} V^{\mathrm{in},-1}_{\gamma\tilde\rho}\end{aligned}$$ We do not need to transpose the final result because of the symmetry of $\Gamma$. If our “active space” approximation ([[ASA]{}]{}) is used, Eq. , we employ rectangular submatrices of the $V^{\mathrm{in}}$, since $\Gamma$ is only of size $\mathds{C}^{M^4}$ but $\tilde\Gamma$ needs to be of size $\mathds{C}^{N^4}$. Within [[ASA]{}]{} a decomposition similar to the one used in the flow equations, Eqs. (\[eq:impl:gemmsplit:tV\]-\[eq:impl:gemmsplit:tUl\]), is not useful here, as a single matrix multiplication already decomposes into 5 products. Instead, we transform the entire bare interaction, $U$, in the full Hilbert space, and additionally transform $\Gamma - U$ in the activate space and add the results together in the end. We then proceed to multiply the transformed vertex by the energy denominator of Eq. , $$\tilde\Gamma^{\Lambda,\text{div}}_{\tilde\mu\tilde\nu\tilde\rho\tilde\sigma} = \tilde\Gamma^\Lambda_{\tilde\mu\tilde\nu\tilde\rho\tilde\sigma} \frac{1}{\tilde\epsilon_{\tilde\mu} + \tilde\epsilon_{\tilde\nu} - \tilde\epsilon_{\tilde\rho} - \tilde\epsilon_{\tilde\sigma}}. \label{eq:impl:ddcorr:div}$$ Finally, we need to transform to the target basis and select the proper orbitals. At $T = 0$, we have $$\begin{aligned} \mathcal{C}^{\mathrm{dd-pre},(2)}_{i\tilde\nu\tilde\sigma} & = & \left[ \sum_{\substack{\tilde\mu\in\mathcal{H}_e\\\tilde\rho\in\mathcal{H}_h}} - \sum_{\substack{\tilde\mu\in\mathcal{H}_h\\\tilde\rho\in\mathcal{H}_e}} \right] V^{\mathrm{ri}}_{i\tilde\mu} \tilde\Gamma^{\Lambda,\text{div}}_{\tilde\mu\tilde\nu\tilde\rho\tilde\sigma} V^{\mathrm{ri},-1}_{\tilde\rho i}, \label{eq:impl:ddcorr:pre-final} \\ \mathcal{C}^{\mathrm{dd},(2)}_{ij} & = & \left[ \sum_{\substack{\tilde\nu\in\mathcal{H}_e\\\tilde\sigma\in\mathcal{H}_h}} - \sum_{\substack{\tilde\nu\in\mathcal{H}_h\\\tilde\sigma\in\mathcal{H}_e}} \right] V^{\mathrm{ri}}_{j\tilde\nu} \mathcal{C}^{\mathrm{dd-pre},(2)}_{i\tilde\nu\tilde\sigma} V^{\mathrm{ri},-1}_{\tilde\sigma j}. \nonumber \\ & & \label{eq:impl:ddcorr:final}\end{aligned}$$ Because we transform into the basis of the quasi-particles for a given $\Lambda$, the transformation matrices $V^{\mathrm{in}}$ are $\Lambda$-dependent and the contribution from the bare interaction, $U$, cannot be calculated just once initially. This means that for each $\Lambda$ the density-density correlator incurs a cost of ${\ensuremath{\mathcal{O}\big(N^5\big)}}$. Eq.  has a complexity of ${\ensuremath{\mathcal{O}\big(N^4\big)}}$ and Eq.  a complexity of ${\ensuremath{\mathcal{O}\big(N^5\big)}}$. This cannot be simplified further without additional approximations, making it the most expensive object to calculate. Fortunately, the density-density-correlator is not actually required for the flow of the vertex or the self-energy. Therefore, unless we see a divergence in our flow in $\Lambda$, we calculate it only once at the very end of the flow. In case a divergence is seen, we perform a backtracking procedure: while we don’t store the vertex for all iteration steps, we do keep it for the last $n_{\text{bt}}$ iterations. Once we detect a divergence, we reset the system to the current iteration minus $n_{\text{bt}}$ steps (typically 10) and calculate the density-density correlator at that iteration step and proceed to the next iteration again. This is performed for a total of $n_{\text{dv}} \le n_{\text{bt}}$ iterations (typically 1 or 2), where we don’t need to recalculate the flow but can just use the known self-energy and the vertex. Verification - Tests on the spinless Hubbard model {#sec:verification} ================================================== ![\[fig:verification:w0.1-u0.01:density:cmp\] Comparison of the particle density, $n({\mathbf{r}})$, calculated from ED ($n_\text{ED}$) and [[$\epsilon$FRG]{}]{} ($n_\text{{{$\epsilon$FRG}}}$) for a single disorder realization at $U = 0.01$ and $W = 0.1$. Left: Normalized relative deviation $(n_\text{ED}-n_0)/n_0U$, where $n_0$ denotes the density for the same disorder realization at $U{=}0$. Right: $(n_\text{ED}-n_\text{{{$\epsilon$FRG}}})/n_0U$. ](figure4){width=".95\linewidth"} In this section we test our implementation applying it to disordered spinless Hubbard model. We compare results from [[$\epsilon$FRG]{}]{} for the quasiparticle energies and the particle density to the exact diagonalization (ED) in 2D and to the density matrix renormalization group (DMRG) in 1D. The corresponding Hamiltonian reads $$\begin{aligned} \hat H & = & -t \sum_{<ij>} \mathrm{\hat c}_i^\dagger \mathrm{\hat c}_j + \sum_{i} \delta\epsilon_i \mathrm{\hat n}_i + U \sum_{<ij>} \mathrm{\hat n}_i \mathrm{\hat n}_j, \label{eq:verification:hamiltonian}\label{e124}\end{aligned}$$ where $t$ is the hopping parameter, $U$ the interaction strength and the $\delta\epsilon_i$ the on-site energies, which are chosen at random from a box distribution with width $W$ centered around $\epsilon = 0$. [^3] In all calculations we will be working at half-filling. All energies will be measured in units of $t$. [[$\epsilon$FRG]{}]{} vs. ED for square lattices {#sec:disorder:2d} ------------------------------------------------ In this section we test our implementation of the [[$\epsilon$FRG]{}]{} equations. To this end, we work with small systems, so ED is feasible and there is no need to apply the [[ASA]{}]{}. Specifically, we consider the model Hamiltonian of Eq.  on a $4{\times}4$ square lattice with $N{=}16$ sites and periodic boundary conditions at half filling, $\nu=1/2$. The details of our ED-implementation are given in App. \[app:ED\]. #### Density ![\[fig:verification:w5-u0.1:density:cmp\]\[f5\] Plot similar to Fig. \[fig:verification:w0.1-u0.01:density:cmp\] with $U = 0.1$ and $W = 5$.](figure5){width=".95\linewidth"} Fig. \[fig:verification:w0.1-u0.01:density:cmp\] (left) displays the interaction induced shift of the particle density as it is obtained for a typical disorder realization at very weak interactions and disorder $U{=}0.01, W{=}0.1$. To highlight the density response, we have divided the relative displacement by $U$. We obtain a checkerboard pattern that we interpret as a precursor to the system ordering in a charge-density wave (CDW). In the absence of disorder there is a two-fold degeneracy associated with the placement of the wave. The pattern is visible in our calculation due to the disorder which breaks this symmetry. As seen from Fig. \[fig:verification:w0.1-u0.01:density:cmp\] (right) the density response to very small values of $U$ is reproduced by the [[$\epsilon$FRG]{}]{} reasonably well with a typical error of about 30%. A comparison at stronger interaction and disorder is given in Fig. \[f5\] where $U{=}0.1$ and where the disorder potential of the previous realization has been recycled, but multiplied with a factor of fifty corresponding to $W=5.0$. #### Quasiparticle energies We also compare the spectral properties, i.e. the quasiparticle energies, for both systems, see Fig. \[fig:verification:qpe\]. [^4] The ordinate shows the energies of the corresponding non-interacting system, i.e. of $\hat H_0$. At low disorder, $W = 0.1$, the degeneracies of the clean system are only slightly lifted, hence the crosses in Fig. \[fig:verification:qpe\] appear in groups. The vertical spreading of these groups is seen to be larger than for the case with stronger disorder, $W{=}5$. We attribute the larger error for the near-degenerate situation to the fact that our formulation of the [[$\epsilon$FRG]{}]{} assumes that $\hat H_0$ is non-degenerate and becomes singular, otherwise. We observe that the normalized deviations between [[$\epsilon$FRG]{}]{} and ED are approximately independent of the interaction strength $U$. For the occupied states below the chemical potential, ${\mu_{\text{chem}}}\approx 0$, the error depends very weakly on energy with a typical error smaller than 5%. In contrast, the deviations keep growing for the unoccupied levels reaching values of 20% near the band edge. ![\[fig:verification:qpe\] Comparison of the quasiparticle energies obtained with [[$\epsilon$FRG]{}]{} and ED, normalized by the interaction-induced shift, $(\epsilon_{\text{ED}} - \epsilon_{\text{{{$\epsilon$FRG}}}})/(\epsilon_{\text{ED}} - \epsilon_0)$, for the same systems as in Fig. \[fig:verification:w0.1-u0.01:density:cmp\] (crosses) and Fig. \[f5\] (circles), respectively.](figure6){width=".8\linewidth"} [Active Space Approximation]{}([[ASA]{}]{}) ------------------------------------------- As has been discussed in Sec. \[sec:orbital-reduction\], we will consider the renormalized vertex within an active space of $M < N$ states. In this section we test the sensitivity of $n({\mathbf{r}})$ and the spectral function to variation of $M$. To this end we will use a $6{\times}6$ square lattice with periodic boundary conditions, so $N{=}36$. Each system is calculated twice, with the full $M{=}36$, and with $M{=}16$. ![\[fig:verification:36:density\]\[f6\] Testing the [[ASA]{}]{} via density calculations at $N{=}36$ with $U{=}0.1$ for a given disorder realization at $W=1.0$. Left: $(n_\text{{{$\epsilon$FRG}}}{-}n_0)/U n_0$. Right: $(n_\text{{{$\epsilon$FRG}}}{-}n_\text{{{ASA}}})/U n_0$ where $M=16$ has been used in the [[ASA]{}]{}-calculation. ](figure7){width=".95\linewidth"} The real space density at $U = 0.01$ for a specific disorder realization at $W=1$ is shown in Fig. \[fig:verification:36:density\]. We see that there is a very good agreement between the density profiles of both methods, validating our approach at least for small system sizes and moderate interaction strengths. We also compare the quasi-particle energies as obtained from [[$\epsilon$FRG]{}]{} for both choices of $M{=}36, 16$. Fig. \[fig:verification:36:w1-u0.01:dos\] shows the normalized difference of both spectra. As can be seen, the overall performance of [[ASA]{}]{} is acceptable with a relative error of about 0.5% for quasi-particle energies close to the Fermi level. Remarkably, the error does not exceed 1% even for states outside of the active space. [[$\epsilon$FRG]{}]{} vs. DMRG for chains ----------------------------------------- As a second, independent line of testing we also compare the results from [[$\epsilon$FRG]{}]{} with DMRG calculations. To this end we consider the same Hamiltonian as before, but now the geometry represents a short chain of $L = 16$ sites. In the [[$\epsilon$FRG]{}]{} we keep $N{=}16{=}M$. At a given, fixed disorder configuration with $W = 0.2$ we compare the particle density for two different interaction strengths, $U = 0.2$ and $U = 1.5$. ![\[fig:verification:36:w1-u0.01:dos\]\[f7\] Difference of the quasi-particle energies obtained with [[ASA]{}]{} ($\epsilon_\text{{{ASA}}}$, $M{=}16$) and without ($\epsilon_\text{{{$\epsilon$FRG}}}$, $N{=}M{=}36$) normalized by the interaction induced shift: $(\epsilon_\text{{{$\epsilon$FRG}}}-\epsilon_\text{{{ASA}}})/(\epsilon_\text{{{$\epsilon$FRG}}}{-}\epsilon_0)$. The same sample was used as in the previous Fig. \[fig:verification:36:density\]. ](figure8){width="0.85\linewidth"} ![\[fig:verification:1d:lowU\]\[f8\] Normalized interaction induced density response, $(n_\text{X}-n_0)/U$, of a $16$ site chain obtained with DMRG (empty squares an circles) and FRG (crosses) at $U = 0.2$ and $U = 1.5$; $n_0$ denotes the non-interacting density. ](figure9){width=".85\linewidth"} Fig. \[fig:verification:1d:lowU\] displays the response of the density when switching on $U$ as obtained with both methods. At smaller interaction values, $U = 0.2$, the [[$\epsilon$FRG]{}]{} reproduces the DMRG results quantitatively with errors in the percent-regime. When the interaction reaches values of the order of the band-width, $2t$, larger deviations occur reaching values of up to 50%. The systematic overshooting that is observed in the data, we tentatively attribute to a lack of screening related to the static approximation. Detecting the CDW state with [[$\epsilon$FRG]{}]{} and [[ASA]{}]{} {#sec:results:cdw-frg} ------------------------------------------------------------------ ![\[fig:verification:w0.1-u0.1:flow-frg\]\[f9\] RG flow of the norm of $\Gamma^\Lambda$ for $U{=}W{=}0.1$ on a lattice $4\times 4$ (left) and $U{=}0.01, W{=}0.001, N{=}36$ on a lattice $6{\times}6$ (right). The right panel shows in addition to full [[$\epsilon$FRG]{}]{} ($M=36$) also [[ASA]{}]{}-data with $M{=}16$ demonstrating that the critical value $\Lambda_c$ is a very robust indicator of runaway flow.](figure10){width=".9\linewidth"} As we pointed out in section \[sec:frg:runaway-flow\], the [[FRG]{}]{} formalism signalizes the presence of an instability of the Fermi-liquid via runaway flow of certain elements of the interaction vertex $\Gamma$. Therefore, a matrix-norm, e.g., $$|\Gamma^\Lambda| = M^{-4} \sqrt{\sum_{\bar\alpha\bar\beta\bar\gamma\bar\delta} (\Gamma^\Lambda_{\bar\alpha\bar\beta\bar\gamma\bar\delta})^2},$$ is a reliable indicator of a nearby instability. Fig. \[fig:verification:w0.1-u0.1:flow-frg\] shows how this norm flows under the action of the RG. It is seen to diverge, e.g., at $\Lambda_{\text{c}}\approx 0.04$ for $U{=}0.1$. Ideally, to pinpoint the nature of the instability, one would investigate which one of the matrix elements of $\Gamma$ diverges so as to predict the nature of the instability. Since we here expect a CDW, we omit this step and just check that this interpretation is indeed consistent with the [[$\epsilon$FRG]{}]{} results. At first sight one might suspect that it would be sufficient to this end calculating the particle density $n({\mathbf{r}})$ and ensuring that it indeed exhibits the checkerboard pattern. However, this perspective is slightly misleading. In the presence of runaway flow, we cannot evaluate the density at $\Lambda=0$, but only at $\Lambda\gtrsim \Lambda_c$ where the ground-state does not yet fully exhibit the broken symmetry. Therefore, instead of calculating $n({\bf r})$ one rather evaluates the density-correlator at $\Lambda = \Lambda_{\text{c}}$, $$\mathcal{D}({{\mathbf{k}}}) = N^{-1} \sum_{{\mathbf{x}} {\mathbf{x}}'} \mathrm{e}^{{\mathrm{i}}{\mathbf{k}} ({\mathbf{x}} - {\mathbf{x}}')} \mathcal{C}^{\mathrm{dd},(2)}_{i=(x,y),j=(x',y')}, \label{eq:verification:ddcorr:ft}$$ where $\mathcal{C}^{\mathrm{dd},(2)}$ may be calculated according to Eq. (\[eq:TwoParticleGF:connected:T0:result\]). The result for $\mathcal{D}$ is displayed in Fig. \[f10\] (left column) for two different values of interactions and disorder. The peak at the correct ordering wavenumber $({\mathbf{Q}}{=}\pi,\pi)$ of the density response is already clearly visibly foreshadowing the upcoming ordered phase. ![ \[fig:verification:w0.01-u0.01:ddcorr-frg\]\[f10\] Left column: The density-density correlator as defined in Eq.  calculated at $\Lambda_{\text{c}}$ at $U{=}W{=}0.01$ (upper row, system in Fig. \[f9\]) and at $U{=}W{=}5$ (lower row). The peak indicates the CDW instability with wave-vector $({\mathbf{Q}}{=}\pi,\pi)$. Right column: Respective densities $n({\mathbf{r}})$ from exact diagonalization (ED) exhibiting the correspondig pinned CDW.](figure11a "fig:"){width=".45\linewidth"} ![ \[fig:verification:w0.01-u0.01:ddcorr-frg\]\[f10\] Left column: The density-density correlator as defined in Eq.  calculated at $\Lambda_{\text{c}}$ at $U{=}W{=}0.01$ (upper row, system in Fig. \[f9\]) and at $U{=}W{=}5$ (lower row). The peak indicates the CDW instability with wave-vector $({\mathbf{Q}}{=}\pi,\pi)$. Right column: Respective densities $n({\mathbf{r}})$ from exact diagonalization (ED) exhibiting the correspondig pinned CDW.](figure11b "fig:"){width=".45\linewidth"} ![ \[fig:verification:w0.01-u0.01:ddcorr-frg\]\[f10\] Left column: The density-density correlator as defined in Eq.  calculated at $\Lambda_{\text{c}}$ at $U{=}W{=}0.01$ (upper row, system in Fig. \[f9\]) and at $U{=}W{=}5$ (lower row). The peak indicates the CDW instability with wave-vector $({\mathbf{Q}}{=}\pi,\pi)$. Right column: Respective densities $n({\mathbf{r}})$ from exact diagonalization (ED) exhibiting the correspondig pinned CDW.](figure11c "fig:"){width=".45\linewidth"} ![ \[fig:verification:w0.01-u0.01:ddcorr-frg\]\[f10\] Left column: The density-density correlator as defined in Eq.  calculated at $\Lambda_{\text{c}}$ at $U{=}W{=}0.01$ (upper row, system in Fig. \[f9\]) and at $U{=}W{=}5$ (lower row). The peak indicates the CDW instability with wave-vector $({\mathbf{Q}}{=}\pi,\pi)$. Right column: Respective densities $n({\mathbf{r}})$ from exact diagonalization (ED) exhibiting the correspondig pinned CDW.](figure11d "fig:"){width=".45\linewidth"} To give further evidence of the correct prediction of charge ordering, we also calculate the real space density. Since due to runaway flow this cannot be done with [[$\epsilon$FRG]{}]{}, we again employ the ED. As expected, the resulting densities – shown in Fig. \[f10\] (right column) – exhibit the checkerboard pattern. ![\[fig:verification:36:ddcorr:36\]\[f11\] [\[fig:verification:36:ddcorr\] The density-density correlator as obtained from [[$\epsilon$FRG]{}]{} at $\Lambda_{\text{c}}$ with (left, $M{=}16$) and without (right, $N{=}M{=}36$) [[ASA]{}]{} for $W = 0.001$, $U = 0.01$ on a $6{\times}6$-lattice. The peak is well exposed in both plots, so the CDW-nature of the ordering phase is reliably reproduced by [[ASA]{}]{}. ]{}](figure12){width=".9\linewidth"} We have already demonstrated that $\Lambda_\text{c}$ is properly reproduced within [[ASA]{}]{}. As a final step in this section we show that this is also the case for the density response $\mathcal{D}$. In Fig. \[f11\] we compare two calculations with full [[$\epsilon$FRG]{}]{}, $N{=}M{=}36$ and with [[ASA]{}]{} ($M{=}16$) for a system with very weak disorder and interaction. As is seen there, the ordering peak is quantitatively reproduced by the active-space approximation to the [[$\epsilon$FRG]{}]{}. Application – Phase-diagram of spinless disordered Hubbard model {#sec:results} ================================================================ As a relevant application of our method, we determine the phase diagram of the spinless Hubbard model on a square lattice with periodic boundary conditions. For two limiting cases the phases of the model are well known. In the absence of disorder, $W{=}0$, the ground state exhibits the charge-density wave (CDW) at any finite value of $U{>}0$;[@RGShankar] it already made its appearance in the previous section. On the other hand, in the absence of interaction, $U{=}0$, the system becomes an Anderson insulator (AI) for any finite disorder $W > 0$.[@AndersonScaling] The purpose of this investigation is to determine the phase-boundary in the general case, $U,W > 0$, as is indicated in the [[$\epsilon$FRG]{}]{} by runaway flow. Our tests on small systems so far have indicated, that with disorder, $W>0$, a minimum value of the interaction, $U^*(W)$, is required for the system to form a CDW ground state. This is in contrast to the clean case where for any $U>0$ a charge density order is established, at least for large enough systems. We evaluate $U^{*}(W)$ with the [[$\epsilon$FRG]{}]{}. ![\[fig:ucrit-plot\]\[f12\] Critical interaction $U^{*}$ beyond which [[$\epsilon$FRG]{}]{} predicts CDW-ordering plotted over the inverse system size $1/L^2$ for multiple different values of the disorder strength $W$. The results have been averaged over $5$ disorder configurations and $1\sigma$-error bars are given. ](figure13){width=".9\linewidth"} Note, that $U^{*}$ will somewhat vary between different disorder realization and may, in addition, exhibit a dependency on the system size $L$. To deal with this, we apply the following strategy: for a fixed system size and disorder realization, we scan over $U$ and thus obtain $U^{*}$ for this specific sample. We repeat the run for more samples with different disorder realizations keeping the same disorder strength $W$ thus finding the average $U^{*}(W,L)$. Finally, to account for finite size effects we analyze the behavior of $U^{*}(L,W)$ for varying system sizes. Results – Phase diagram ----------------------- Fig. \[fig:ucrit-plot\] displays $U^{*}(W,L)$ after averaging over five disorder configurations for $L{\times} L$-lattices with $L = 4,6,8$. For $L=6,8$ we have used [[ASA]{}]{} with $M = 16$ states in both cases. Our data indicates that except at very large disorder values, $W{=}3$, $U^{*}$ appears to remain largely insensitive to variations of the (lateral) system size by a factor of two. We take this as an indication that $U^{*}$ will indeed remain finite even at large system sizes. Thus encouraged we take the data at $L{=}8$ as an estimate for the phase boundary $U^{*}(W)$ at $L\to\infty$. Fig. \[fig:pdres\] shows the resulting phase diagram. #### Computational details. {#computational-details. .unnumbered} We found it practical to work with a single particle Hilbert space containing $N{\sim}50{-}100$ states. For example, with a single-particle Hilbert space consisting of $N = 64$ states and the active space consisting of $M{=}32$ states, a single calculation on 8 CPU cores takes less than 24 hours. Discussion ---------- ![\[fig:pdres\]\[f13\] The phase-diagram for the spinless disordered Hubbard model in 2D as calculated with [[$\epsilon$FRG]{}]{}. ](figure14){width=".85\linewidth"} ### Stability arguments and quasistatic approximation Due to the quasi-static approximation the [[$\epsilon$FRG]{}]{}-self-energy is hermitian and energy-independent. On this level of approximation, the interaction is dealt with by replacing the non-interacting Hamiltonian $H_0$ with an effective quasi-particle (qp) Hamiltonian $H_\text{qp}$. The latter deviates from $H_0$ by a renormalized kinetic energy term, a renormalized effective potential that can, in general, carry off-diagonal entries. After these preliminaries, one expects that the Anderson-localized phase (at $U{=}0$) is seen to be stable within quasi-static [[$\epsilon$FRG]{}]{} against introducing a small repulsive interaction. [^5] After all, the renormalized Hamiltonian $\hat H_\text{qp}$ is still a generic representative of the orthogonal symmetry class and hence should exhibit conventional behavior. A similar stability argument also applies to the ordered phase: the leading effect of weak disorder is pinning of the charge-density wave (CDW). The wave is destroyed only when strong fluctuations of the local potential allow for lattice defects, where two neighboring lattice sites are occupied. For box-distributed on-site potentials, isolated defects can occur only when $W\sim U$. As a consequence, one expects $U^{*}(W)\sim W$ at weak disorder $W$, which is consistent with the phase-boundary seen in Fig. \[f13\]. Remarkably, at interaction strengths comparable to the band-width, $U\gtrsim 1$, the disorder strength necessary to destroy the CDW appears to be considerably smaller than $U$. We hypothesize that we here witness the onset of a collective effect in which several particles can optimize their energy with respect to the disorder potential at the expense of very few particles that built up a defect line thus producing a phase-separation. #### Physics beyond the quasi-static approximation. {#physics-beyond-the-quasi-static-approximation. .unnumbered} The quasi-static approximation ignores the energy exchange between the quasi-particles that of course also is included in the model Hamiltonian, Eq. . Effects of dephasing and many-body localization[@nandkishore15] are beyond its scope. Therefore, we consider it likely that the phase seen as (conventional) Anderson-insulator by the (quasi-static) [[$\epsilon$FRG]{}]{} is missing aspects of dynamical physics that dominate essential properties of the phase at non-vanishing temperature. What implications this may have on the (zero-temperature) phase-boundary between the CDW and the Anderson insulator remains to be seen. ### Relation to earlier work The spin$-1/2$ Hubbard model enjoyed considerable attention in recent years, because physical realizations can be found not only within condensed matter systems but also in cold atomic gases, see Refs.  for very recent results. In principle, also the spinless model, Eq. , that we deal with in this work could find a cold-atom realization which, however, would require the application of a strong homogeneous in-plane magnetic field. This could be one reason, why the spinless model, Eq. , has received considerably less attention over the years. Numerical investigations of the spinless model have been concentrating on its quantum glass variant that deviates from Eq.  replacing the short-range interaction by a long-range Coulomb interaction. [@vojta98; @benenti99; @berkovits01] An analytical treatment of the model, Eq. , has been given by Vlaming et al., Refs. . The authors employed the Bethe lattice where an exact solution can be given in the limit of infinite branching number. The physical picture developed there for the zero temperature limit is in qualitative agreement with our own findings. More recently, Foster and Ludwig studied the model, Eq. , with (complex) off-diagonal disorder focussing on the effect of interactions on the Gade-fixpoint.[@foster08] In that case the non-interacting reference state is not an insulator but a (critical) metal that – according to perturbative RG – is unstable against weak repulsive interactions. Conclusion and Outlook {#sec:outlook} ====================== The main purpose of this work was a methodological one: to develop, implement and test a variant of the traditional functional renormalization group (FRG) method that is applicable to generic systems, such as molecules or disordered metal grains, which are lacking translational invariance. Within the new approach ([[$\epsilon$FRG]{}]{}), the renormalization of the interaction vertex occurs only for matrix elements with single-particle states that are situated in an energy shell around the Fermi-energy ([*active space*]{}). The method is computationally efficient provided this shell can be taken smaller than the (non-interacting) bandwidth. We argue that the scaling with the size of the single-particle Hilbert space $N$ should be $N^4$ for 2D-lattice systems which compares favorably well with the typical $N^6$ scaling of competing methods, such as CCSD(T). Specifically, calculations with $N{=}64$ and an active space of size $M{=}32$ require less than 24h on 8 CPU-cores. An explicit implementation of [[$\epsilon$FRG]{}]{} has been coded for the spinless Hubbard model in 1D and 2D in the presence of on-site disorder. A comparison to (numerically exact) calculations employing the diagonalization of small systems suggests that the accuracy of [[$\epsilon$FRG]{}]{} concerning quasiparticle energies typically is below $20\%$ in relative error to the interaction-induced shift, as compared to the non-interacting system. Similarly, the interaction induced shift in the ground-state density is recovered quantitatively at small interaction strength $U$ with an error that increases to $\sim$50% if $U$ reaches the band-width. At its current development stage, the [[$\epsilon$FRG]{}]{} is readily applicable to models of interacting fermions in low dimensions, which includes Hubbard clusters with spin and (attractive) interactions at different filling fractions, but also, e.g., small molecules. Our preliminary tests suggests that with the current formalism system sizes of, e.g., $N{=}256$ are already within reach. Significantly bigger system sizes might be attainable, after additional improvements in the code performance have been implemented. As an example we mention the numerical integration of the flow-equations that at present is done in the simplest possible discretization scheme. Also, the flow equations are well-suited for parallelization on distributed memory systems, allowing for a significant increase in the number of CPU cores used in a single calculation. To give a perspective, we mention that the molecules in the GW100 test set have been described with a QZVP-basis set requiring ca. 800 basis function for the biggest species, the amino-acids Guanin and Adenin.[@vanSetten15] We hope that this work helps paving the way for electronic-structure calculations beyond the present paradigm of GW-BSE. Admittedly before the envisioned applications to real systems, an efficient [[$\epsilon$FRG]{}]{}-implementation should be installed that is also prepared for dealing with long-range interactions. Here, we see at present the biggest bottleneck to be overcome in future research. Perhaps additional motivation to overcome this obstacle could come from the fact that we have also given formul[æ]{} for the finite-temperature formalism in this work, so that the effect of heat could be included. Acknowledgements {#acknowledgements .unnumbered} ================ We thank S. Bera, A. D. Mirlin, J. Reuther, J. Schmalian, M. van Setten and P. W[ö]{}lfle for inspiring discussions. We are indebted to A. D. Mirlin for supporting our project in an early stage. Support has also been received from the DFG under grants EV30/7-1, EV30/11-1 and EV30/12-1. and from the Landesgraduiertenförderung of the state of Baden-Württemberg. The DMRG results shown here have been provided by F. Weiner using the Schmitteckert-code. We acknowledge the support provided by computational resources of the Institute of Nanotechnology (INT) and the Steinbuch Centre for Computing (SCC), both at the Karlsruhe Institute of Technology (KIT). Flow equations for $\Gamma$ in the static limit {#app:T0StaticGamma} =============================================== Here we will derive the flow equation for $\Gamma$ in the static limit, Eq. (\[eq:flow:Gamma\]), analogous to the derivation for the self-energy. Starting at Eq. (\[eq:flow:GammaWithOmegaStatic\]), looking at the first term, $$\int{\mathrm{d}}\bar\omega \sum_{\mu\nu\rho\sigma} \mathcal{G}^\Lambda_{\rho\mu}(\bar\omega) \mathcal{S}^\Lambda_{\sigma\nu}(-\bar\omega) \Gamma^\Lambda_{\alpha\beta\rho\sigma} \Gamma^\Lambda_{\mu\nu\gamma\delta},$$ it can be seen that by exchanging all traced indices in both vertices that appear, and then renaming the summation indices, the formula may be rewritten as $$\int{\mathrm{d}}\bar\omega \sum_{\mu\nu\rho\sigma} \mathcal{S}^\Lambda_{\rho\mu}(-\bar\omega) \mathcal{G}^\Lambda_{\sigma\nu}(\bar\omega) \Gamma^\Lambda_{\alpha\beta\rho\sigma} \Gamma^\Lambda_{\mu\nu\gamma\delta},$$ which is just an exchange of both propagators. Utilizing this, we may write it formulated in terms of matrix products, $$\frac{1}{2} \mathrm{tr}\, \int{\mathrm{d}}\bar\omega \big[ \mathcal{S'} \Gamma^{\mathrm{T}} \mathcal{G}^{\mathrm{T}} \Gamma + \mathcal{G} \Gamma^{\mathrm{T}} \mathcal{S'}^{\mathrm{T}} \Gamma \big].$$ We note that the frequency of the single-scale propagator is negative here, which we denote with prime for $\mathcal{Q}$ and $\Sigma$; the $\Theta$ and $\delta$-functions only depend on the modulus. Inserting Eq. (\[eq:derivation:SingleScalePropagator:partialform\]) and using the same representation for $\mathcal{G}$, we can separate four terms, $$\begin{aligned} \hspace{-1em} & & -\frac{\delta}{2} \frac{1}{\mathcal{Q'} - \Theta\Sigma'} \Gamma^{\mathrm{T}} \left( \frac{\Theta}{\mathcal{Q} - \Theta\Sigma } \right)^{\mathrm{T}} \Gamma, \label{eq:derivation:flow:Gamma:cooper:term1} \\ \hspace{-1em} & & -\frac{\delta}{2} \frac{\Theta}{\mathcal{Q} - \Theta\Sigma } \Gamma^{\mathrm{T}} \left( \frac{1}{\mathcal{Q'} - \Theta\Sigma'} \right)^{\mathrm{T}} \Gamma, \label{eq:derivation:flow:Gamma:cooper:term2} \\ \hspace{-1em} & & -\frac{\delta}{2} \frac{1}{\mathcal{Q'} - \Theta\Sigma'} \Sigma' \frac{\Theta}{\mathcal{Q'} - \Theta\Sigma'} \Gamma^{\mathrm{T}} \left( \frac{\Theta}{\mathcal{Q} - \Theta\Sigma } \right)^{\mathrm{T}} \Gamma, \label{eq:derivation:flow:Gamma:cooper:term3} \\ \hspace{-1em} & & -\frac{\delta}{2} \frac{\Theta}{\mathcal{Q} - \Theta\Sigma} \Gamma^{\mathrm{T}} \left( \frac{\Theta}{\mathcal{Q'} - \Theta\Sigma' } \right)^{\mathrm{T}} \hspace{-.5em} \Sigma'^{\mathrm{T}} \hspace{-.3em} \left( \frac{1}{\mathcal{Q'} - \Theta\Sigma' } \right)^{\mathrm{T}} \Gamma. \label{eq:derivation:flow:Gamma:cooper:term4}\end{aligned}$$ Since all of these terms occur underneath an integral over $\int{\mathrm{d}}\bar\omega \delta(|\omega|-\Lambda)$, we may switch primes within each term, and we note for future use that the terms of Eqs. (\[eq:derivation:flow:Gamma:cooper:term1\],\[eq:derivation:flow:Gamma:cooper:term2\]) are equal to each other. We now apply Morris’s Lemma again. In both other terms, Eqs. (\[eq:derivation:flow:Gamma:cooper:term3\],\[eq:derivation:flow:Gamma:cooper:term4\]), we can rewrite them in terms of derivatives w.r.t. the integration variable $t$, $$\begin{aligned} & & -\frac{\delta}{2} \int_0^1 t^2 \left( \frac{{\mathrm{d}}}{{\mathrm{d}}t} \frac{1}{\mathcal{Q'} - t \Sigma'} \right) \Gamma^{\mathrm{T}} \left( \frac{1}{\mathcal{Q} - t \Sigma} \right)^{\mathrm{T}} \Gamma {\mathrm{d}}t, \label{eq:derivation:flow:Gamma:cooper:term3:morris} \\ & & -\frac{\delta}{2} \int_0^1 t^2 \frac{1}{\mathcal{Q} - t \Sigma} \Gamma^{\mathrm{T}} \left( \frac{{\mathrm{d}}}{{\mathrm{d}}t} \frac{1}{\mathcal{Q'} - t \Sigma'} \right)^{\mathrm{T}} \Gamma {\mathrm{d}}t. \label{eq:derivation:flow:Gamma:cooper:term4:morris}\end{aligned}$$ Partial integration of Eq. \[eq:derivation:flow:Gamma:cooper:term3:morris\] yields $$\begin{aligned} & & - \frac{\delta}{2} \left[ t^2 \frac{1}{\mathcal{Q'} - t \Sigma'} \Gamma^{\mathrm{T}} \left( \frac{1}{\mathcal{Q} - t \Sigma} \right)^{\mathrm{T}} \Gamma \right]_0^1 \nonumber \\ & & + \frac{\delta}{2} \int_0^1 2t \frac{1}{\mathcal{Q'} - t \Sigma'} \Gamma^{\mathrm{T}} \left( \frac{1}{\mathcal{Q} - t \Sigma} \right)^{\mathrm{T}} \Gamma {\mathrm{d}}t \nonumber \\ & & + \frac{\delta}{2} \int_0^1 t^2 \frac{1}{\mathcal{Q} - t \Sigma} \Gamma^{\mathrm{T}} \left( \frac{{\mathrm{d}}}{{\mathrm{d}}t} \frac{1}{\mathcal{Q'} - t \Sigma'} \right)^{\mathrm{T}} \Gamma {\mathrm{d}}t.\end{aligned}$$ One sees that the second term cancels Eqs. (\[eq:derivation:flow:Gamma:cooper:term1\],\[eq:derivation:flow:Gamma:cooper:term2\]) and the third term cancels Eq. (\[eq:derivation:flow:Gamma:cooper:term4:morris\]), leaving the result $$- \frac{\delta}{2} \frac{1}{\mathcal{Q'} - \Sigma'} \Gamma^{\mathrm{T}} \left( \frac{1}{\mathcal{Q} - \Sigma} \right)^{\mathrm{T}} \Gamma,$$ which can be rewritten in terms of the index notation as $$- \frac{1}{2} \sum_{\bar\omega=\pm\Lambda} \sum_{\mu\nu\rho\sigma} P^{\Lambda}_{\rho\mu}(-\bar\omega) P^{\Lambda}_{\sigma\nu}(\bar\omega) \Gamma^\Lambda_{\alpha\beta\rho\sigma} \Gamma^\Lambda_{\mu\nu\gamma\delta}.$$ We note that if one were to keep the frequency dependence of the vertex and the self-energy, two cases need to be distinguished: for the case where all external frequencies are zero, the same derivation applies, so our result holds there. For the case where at least some external frequencies are non-zero, the arguments for the $\delta$ and $\Theta$ functions differ, so one may directly insert Eq. \[eq:result:SingleScalePropagatorAtT0\] into the flow equations for the vertex. An analogous treatment is possible for the other four terms in Eq. (\[eq:flow:GammaWithOmegaStatic\]). The other terms may be written as $$\mathrm{tr}\, \int{\mathrm{d}}\bar\omega \big[ \mathcal{S} \Gamma_{\alpha\cdot\delta\cdot} \mathcal{G} \Gamma_{\beta\cdot\gamma\cdot} + \mathcal{G} \Gamma_{\alpha\cdot\delta\cdot} \mathcal{S} \Gamma_{\beta\cdot\gamma\cdot} - [\alpha\leftrightarrow\beta] \big].$$ Looking at the first two terms, they may be divided in the same mannger as in Eqs. (\[eq:derivation:flow:Gamma:cooper:term1\],\[eq:derivation:flow:Gamma:cooper:term2\],\[eq:derivation:flow:Gamma:cooper:term3\],\[eq:derivation:flow:Gamma:cooper:term4\]), without the factor $1/2$, and with the same frequency for the single-scale and the regular propagator. This yields the result $$- \sum_{\bar\omega=\pm\Lambda} \sum_{\mu\nu\rho\sigma} P^{\Lambda}_{\rho\mu}(\bar\omega) P^{\Lambda}_{\sigma\nu}(\bar\omega) \Gamma^\Lambda_{\beta\nu\gamma\rho} \Gamma^\Lambda_{\alpha\mu\delta\sigma} + [\alpha\leftrightarrow\beta].$$ Putting this all together, one arrives at Eq. (\[eq:flow:Gamma\]). Implementation Details {#app:implementation} ====================== Chemical Potential for $T > 0$ ------------------------------ Our algorithm to solve this equation for ${\mu_{\text{chem}}}$ works in three stages: obtain an initial guess for ${\mu_{\text{chem}}}$, ${\mu_{\text{chem}}}^{(0)}$, (trivially) obtain a second guess, ${\mu_{\text{chem}}}^{(1)}$, with ${\mathrm{sgn}}(N_{\text{e}}({\mu_{\text{chem}}}^{\Lambda,(1)}) - N_{\text{e}}) = - {\mathrm{sgn}}(N_{\text{e}}({\mu_{\text{chem}}}^{\Lambda,(0)}) - N_{\text{e}})$ and then use the secant algorithm [@Numerical77] to iteratively find the final ${\mu_{\text{chem}}}$. The initial guess is taken to be the same as for $T = 0$, Eq. , since at low temperatures the value is a very good approximation. We then calculate $${\mu_{\text{chem}}}^{\Lambda,(0)} + {\mathrm{sgn}}(N_{\text{e}}({\mu_{\text{chem}}}^{(0)}) - N_{\text{e}}) \frac{\Delta}{4} i,$$ where $\Delta$ is the mean level spacing of the system and $i$ is an integer that starts at $1$ and is incremented until the condition ${\mathrm{sgn}}(N_{\text{e}}({\mu_{\text{chem}}}^{(1)}) - N_{\text{e}}) = - {\mathrm{sgn}}(N_{\text{e}}({\mu_{\text{chem}}}^{(0)}) - N_{\text{e}})$ is satisfied. In practice $i = 1$ or $i = 2$ will already be sufficient, which is why $\Delta/4$ is a good empirical choice here.[^6] Both initial guesses are then used as input for the secant algorithm. Since $N_{\text{e}}(\epsilon)$ is monotonous and the value searched for is encompassed with both guesses, convergence will be quite fast ($10$ to $20$ iterations in practice). We consider the chemical potential to be converged if the relative error of the number of electrons, $$\left|\frac{N_{\text{e}}({\mu_{\text{chem}}}^{(i)}) - N_{\text{e}}}{N_{\text{e}}({\mu_{\text{chem}}}^{(i)}) - N_{\text{e}}}\right|,$$ is larger than the square root of the machine precision. While the smallest possible error here would be of the order of $\hat\epsilon N$, with $\hat\epsilon$ being the machine precision and $N$ the number of orbitals in the system, the energies $\tilde\epsilon_{\tilde\alpha}$ only have a precision of $\sqrt{\hat\epsilon}$ due to the diagonalization procedure. Parallelization --------------- We will now discuss how we exploit parallelization in our implementation. We use a scheme based on a shared memory architecture, OpenMP [@OpenMP3.1]. It is in principle possible to utilize distributed memory methods, such as MPI (Message Passing Interface, [@MPI]), which allow the usage of far more processor cores for the same calculation. The intermediate products offer a trivial way to parallelize: it is possible to use a parallel version of the GEMM kernel to calculate the matrix products. In the case we track the renormalization of the entire vertex, this would likely be the most efficient avenue. In our case, however, the effective matrix size that is fed into the GEMM kernel is relatively small (we want to calculate the vertex for as few states as possible), so it is unlikely that using a parallel matrix product kernel will scale well even for a low amount of processors. Instead, we parallelize the loops over the two outer indices in the intermediate products and perform serialized matrix products on each processor. This is trivially possible, since the calculations are independent of each other for any given pair of external indices. Similarly, for the evaluation of the trace, we parallelize the loops over all four external indices and have each processor evaluate the trace for a given set of external indices serially. Restarting ---------- Calculations for larger systems may take a relatively long time. In case of technical difficulties, we implement a restarting procedure that allows us to continue a calculation at the point where it last stopped. We save the initial $\Lambda$, the step size, the number of selected states $M$, the chosen target $\Lambda$. Furthermore, we keep the last self-energy and vertex as well as the number of the last iteration to complete. These quantities suffice to reproduce the calculation at a later point in time. ED Implementation {#app:ED} ================= In Sec. \[sec:disorder:2d\] we compare the FRG to exact diagonalization. In the following we provide edtails on how we implemented ED as a reference method. In our implementation, we construct the full $N_{\text{e}}$-particle Hilbert space. Its dimension is $\left( \begin{array}{c} N \\ N_{\text{e}} \end{array} \right)$ and grows exponentially with the number of orbitals $N$. We systematically construct the basis states of that space and implement the action of the full many-body Hamiltonian on that basis (we do *not* explicitly construct the matrix elements of the Hamiltonian itself). An iterative eigensolver for sparse problems is employed to calculate the full many-body ground state for a given system. We utilize the standard ARPACK package [@Arpack] in direct mode.[^7] For simple observables, such as the density, we may then simply calculate expectation values with respect to the many-body ground state, $$\left< \mathrm{\hat n}_i \right> = \bra{0} \mathrm{\hat c}_i^\dagger \mathrm{\hat c}_i \ket{0}.$$ We also want to calculate the single-particle density of states, $\rho(\epsilon)$. This is given by the expectation value $$\begin{aligned} \rho(\epsilon) & = & -\frac{1}{\pi} \Im\, {\mathrm{tr}}_{ij} \left< \mathrm{\hat c}_i \frac{1}{\epsilon - \hat H + E_0 + i\eta} \mathrm{\hat c}_j^\dagger \right> \nonumber \\ & & \hphantom{\frac{1}{\pi} \Im\, {\mathrm{tr}}_{ij}} + \left< \mathrm{\hat c}_j^\dagger \frac{1}{\epsilon + \hat H - E_0 + i\eta} \mathrm{\hat c}_i \right>,\end{aligned}$$ which we arrive at by Fourier transforming the definition of the retarded Green’s function. This expressions contains the inverse of a very large matrix, which needs to be done for every single energy at which the density of states is to be evaluated at. Furthermore, directly inverting such a large matrix is only possible using iterative algorithms, which would again have to be applied for every single energy. We therefore follow an alternative approach as outlined in the PhD thesis of Alexander Braun [@AlexBraunPhD]. One may expand the denominator in terms of Chebyshev polynomials $T_n(x)$, such that we get $$\begin{aligned} c^{(+)}_{ij,n} & = & \bra{0} \mathrm{\hat c}_i T_n\big(a(\hat H - E_0 - b)\big) \mathrm{\hat c}_j^\dagger \ket{0}, \\ c^{(-)}_{ij,n} & = & \bra{0} \mathrm{\hat c}_i^\dagger T_n\big(a(\hat H - E_0 - b)\big) \mathrm{\hat c}_j \ket{0},\end{aligned}$$ where $E_0$ is the ground state energy. The variables $a$ and $b$ are scaling factors that arise due to the fact that the Chebyshev polynomials are only well-defined in the interval $[-1, 1]$, so the Hamiltonian needs to be scaled to fit into that range. We note that since we are calculating expectations in the Hilbert spaces for $N_{\text{e}}+1$ and $N_{\text{e}}-1$ particles, we need to take into account the extremal eigenvalues of the Hamiltonian in those spaces. To make sure we don’t suffer from numerical artifacts, we scale the argument of the Chebyshev polynomials into the interval $[-0.9, 0.9]$.[^8] This gives us $$\begin{aligned} \delta & = & 0.1, \hspace{2em}\text{(distance to interval boundaries)} \nonumber \\ a & = & \frac{2 (1 - \delta)}{(\epsilon_{\text{max}} - E_0) - (\epsilon_{\text{min}} - E_0)}, \\ b & = & \frac{(\epsilon_{\text{max}} - E_0) + (\epsilon_{\text{min}} - E_0)}{2} - \delta,\end{aligned}$$ where $\epsilon_{\text{min,max}}$ are the extremal many-body eigenvalues of the system with $N_{\text{e}}+1$ ($N_{\text{e}}-1$) particles and $E_0$ is the ground state energy for $N_{\text{e}}$ particles. We may then rewrite the single-particle retarded Green’s function in terms of these coefficients, $$\begin{aligned} \mathcal{G}_{ij}(\omega) & = & a \sum_{n=0}^{\infty} \Big( \alpha_n^{+}\big( a(\omega + i\eta \mp b) \big) c^{(+)}_{ij,n} \nonumber \\ & & \hspace{3em} - \alpha_n^{-}\big( a(\omega + i\eta \mp b) \big) c^{(-)}_{ji,n} \Big).\end{aligned}$$ The density of states is then given by the imaginary part of this expression traced over the real space indices, which is why we only need to calculate the diagonal part of this expression. If we terminate the expansion at a finite $n$, the formula remains only valid for finite $\eta$, with $$\eta \gtrsim \frac{1}{a n_{\text{max}}}. \label{eq:verification:ed-eta}$$ For further discussion on this topic we would like to defer to Alexander Braun’s thesis. [@AlexBraunPhD] [^1]: $\delta(x)f(\Theta(x)) \to \delta(x)\int_0^1 f(t) {\mathrm{d}}t$, see Ref. . [^2]: Note that while closing the integrals over $\omega_{2,3}$, those frequencies may obtain an imaginary part, but since semi-circle contour parts have a vanishing contribution to the integral itself, this may be ignored. [^3]: In 2D this model could be realized in terms of a strongly screened two-dimensional electron gas with a strong in-plane magnetic field. This would polarize all of the spins due to the Zeemann effect, but have no orbital contribution. [^4]: To obtain the quasiparticle energies in the ED case, we calculate the spectral function utilizing the truncated Chebyshev expansion discussed in App. \[app:ED\], where we have kept $10^5$ Chebyshev moments. With an artifical broadening ($2\cdot 10^{-3}t$) to ensure the validity of the truncation of the expansion, the resulting density of states has been fitted against Lorentzians (with a maximum relative error of the position always below $10^{-6}t$ for each peak). [^5]: We tacitly assume here that the short-range Hubbard term does not introduce long-range correlations in the matrix elements of $H_\text{qp}$. [^6]: We cut this scheme off at $i = 10$, since it is only used to accelerate the convergence of the secant algorithm, which is likely to also work if the second value does not satisfy the condition, albeit more slowly. [^7]: The shift-inverse mode is not required, since the eigenvalues we are interested in are taken from the spectrum edges, not the center. [^8]: Using exactly $[-1,1]$ does not work, since the polynomials are fixed at the boundaries of the interval. One needs to distance oneself at least by relative error in the eigenvalues from the boundary.

--- abstract: 'The problem of finding a triangulation of a convex three-dimensional polytope with few tetrahedra is $NP$-hard. We discuss other related complexity results.' author: - 'Alexander Below[^1]' - 'Jesús A. De Loera[^2]' - 'Jürgen Richter-Gebert[^3]' title: 'The Complexity of Finding Small Triangulations of Convex $3$-Polytopes.' --- [^1]: Institut für Theoretische Informatik, ETH-Zürich ([below@inf.ethz.ch]{}). [^2]: Dept. of Mathematics, Univ. of California-Davis ([deloera@math.ucdavis.edu]{}). [^3]: Institut für Theoretische Informatik, ETH-Zürich ([richter@inf.ethz.ch]{}).

--- author: - | $^a$, E. Ros$^{b,a}$, M. Kadler$^{c,d}$, M. Böck$^c$, J. Wilms$^c$, M. F. Aller$^e$, H. D. Aller$^e$, L. Fuhrmann$^a$, E. Angelakis$^a$, and I. Nestoras$^a$\ Max-Planck-Institut für Radioastronomie, Bonn, Germany\ Universitat de València, Spain\ Dr. Remeis-Sterwarte & ECAP, Germany\ CRESST/NASA GSFC & USRA, USA\ University of Michigan, USA\ E-mail: title: The broadband emission properties of AGN jets --- Introduction ============ Active Galactic Nuclei (AGN) are among the most energetic objects in the Universe. AGN dominate the extragalactic high-energy sky, and are very active at all wavelengths from radio to $\gamma$-rays. According to the unified model of AGN, it is believed that a super massive black hole is located in the center of host galaxy, and it fuels the whole system with matter accreted around the central engine. In the radio-loud scheme, an energetic jet is launched in the vicinity of the central engine following the warped magnetic field from the pole direction of the accretion disk. Blazars[^1] are AGN whose jets are pointing toward us, and dominate the radio and the high-energy sky. After the beginning of operations of the Large Area Telescope (LAT) on-board the *Fermi* $\gamma$-ray Space telescope in mid 2008, *Fermi*/LAT detected 709 AGN in the first 11 months, and 85% of them being blazars [@abdo10a]. The high-energy emission location in blazar systems is not yet well-understood. By using the very long baseline interferometry (VLBI) technique, we are able to resolve jet structure and trace component ejection at milli-arcsecond scales [@lister09]. Combining the VLBI with the high-energy observatories (e.g., *Fermi*/LAT), we might be able to probe the location of the high-energy emission. The **M**onitoring **O**f **J**ets in **A**ctive Galactic Nuclei with **V**LBA **E**xperiments (MOJAVE) program has been monitoring a radio-selected sample since the mid 1990s. The sample contains mostly blazars due to the selection criteria[^2] used [@lister09a]. In the *Fermi* one-year AGN catalog [@abdo10b], 63% of the MOJAVE sources were detected. It was found that the LAT-detected MOJAVE sources have higher brightness temperature and higher Doppler-boosting factors than the non-detected ones [@kovalev09], and the authors suggested that the parsec-scale radio core is likely to be the location of the radio and the $\gamma$-ray flares. It was also reported that the $\gamma$-ray bright quasars have faster jets [@lister09b]. In order to investigate the relation between the parsec-scale jets and high-energy emission, we are studying the broadband spectral energy distribution (SED) from the radio to the $\gamma$-ray of the MOJAVE sample. By comparing the SED properties of the statistical-complete MOJAVE sources with the VLBI parameters, we want to further understand the physical mechanisms ongoing in blazar jets. The Project =========== We constructed the broadband SED catalog of the complete sample of 135 MOJAVE sources using simultaneous observations from the radio to the $\gamma$-ray band [@chang10a; @chang10b]. In the radio band, we use: i) data with the 26-meter radio telescope at the University of Michigan Radio Astronomy Observatory (UMRAO) [@aller85; @aller03]; ii) data obtained with the ***F**ermi*-**G**ST **A**GN **M**ulti-frequency **M**onitoring **A**lliance (FGAMMA) program with the Effelsberg 100-m telescope, the IRAM/Pico Veleta 30-m telescope, and the APEX 12-m telescope. In the optical band, we use the *Swift* UV-Optical Telescope (UVOT) observations. In the X-ray band, we use the *Swift* X-ray Telescope (XRT) and the Burst Alert Telescope (BAT) results. In the $\gamma$-ray band, we use the *Fermi*/LAT one-year catalog results for the 85 sources which were detected in the catalog [@abdo10b]. For the remaining 50 sources, we used the upper-limits of *Fermi*/LAT to be published in Böck et al. (in prep.). We also included the historical data from the NASA/IPAC Extragalactic Database[^3], which, among others, include data with the EGRET[^4] onboard *CGRO*[^5]. For details of data acquisition, see Chang et al. [@chang10a; @chang10b]. In order to model the SEDs, we applied 2–4 degree polynomial fits to the two humps of the SED for each MOJAVE source as a first approach, and derived the peak values in the frequency and in the energy domain. Results ======= Figure \[fig:1SED\] shows the broadband SED of the MOJAVE source 1546$+$027. The frequency coverage of the SED is reasonably good, see also Chang (2010) for details [@chang10a; @chang10b]. As shown in Figure \[fig:1SED\], two second-order polynomial fits describe the double-humped SED reasonably good. From the fits, we estimated the peak values of the lower-energy and the higher-energy hump for most of the sources. $\nu_{\mathrm{peak}}^{\mathrm{low}}$, $\nu^{\mathrm{high}}_{\mathrm{peak}}$, $\nu$F$\nu^{\mathrm{low}}_{\mathrm{peak}}$, $\nu$F$\nu^{\mathrm{high}}_{\mathrm{peak}}$, except for few sources with poorer frequency coverage at the higher energy range. We study the distributions and correlations of the four SED parameters for the different classes of AGN in our sample: quasars, BL Lac Objects, and radio galaxies. We found that the distribution of $\nu_{\mathrm{peak}}^{\mathrm{low}}$ locates at a similar range for three classes of AGN. However, the ranges of the distribution of $\nu^{\mathrm{high}}_{\mathrm{peak}}$ differ: (1) the quasars occupy a much wider range compared to the other two categories, as well as compared to the distribution of $\nu_{\mathrm{peak}}^{\mathrm{low}}$ for quasars; (2) the BL Lac objects occupy higher range of $\nu_{\mathrm{peak}}^{\mathrm{low}}$ than the radio galaxies. While investigating the relations between apparent speed $\beta_{\mathrm{app}}$ and $\nu^{\mathrm{high}}_{\mathrm{peak}}$, we found that there is no source with high apparent speed having low values of $\nu^{\mathrm{high}}_{\mathrm{peak}}$, which might suggest a relationship between the two parameters. {width="80.00000%"} Future Work =========== The broadband SED catalog of the MOJAVE sample has been constructed [@chang10b]. Currently, we are studying the distributions of the SED parameters, as well as the correlations with respect to the radio (VLBI), optical, X-ray, and $\gamma$-ray properties. Present physical SED models can describe the higher energy range well, however, the fitting results are not satisfactory in the radio band. The aims of our correlation studies are to understand the nature of blazar emissions, and also to find out possible linked parameters, which will help to improve blazar SED models. The physical models are essential to understand the local conditions of blazar jets, such as magnetic field intensity, jet composition, bulk flow, etc. The results of the correlation study will be presented in a forthcoming publication (Chang et al. in prep.). [99]{} A.A. Abdo, M. Ackermann, M. Ajello et al., *Fermi Large Area Telescope First Source Catalog*, *ApJS* **188** (2010) 405 M.L. Lister, M.H. Cohen, D.C. Homan et al., *MOJAVE: Monitoring of Jets in Active Galactic Nuclei with VLBA Experiments. VI. Kinematics Analysis of a Complete Sample of Blazar Jets*, *AJ* **138** (2009) 1874–1892 M.L. Lister, H.D. Aller, M.F. Aller et al., *MOJAVE: Monitoring of Jets in Active Galactic Nuclei with VLBA Experiments. V. Multi-Epoch VLBA Images*, *AJ* **137** (2009) 3718–3729 A.A. Abdo, M. Ackermann, M. Ajello et al., *The First Catalog of Active Galactic Nuclei Detected by the Fermi Large Area Telescope*, *ApJ* **715** (2010) 429–457 Y.Y. Kovalev, H.D. Aller, M.F. Aller et al., *The Relation Between AGN Gamma-Ray Emission and Parsec-Scale Radio Jets*, *ApJ* **696** (2009) L17–L21 M.L. Lister, D.C. Homan, M. Kadler et al., *A Connection Between Apparent VLBA Jet Speeds and Initial Active Galactic Nucleus Detections Made by the Fermi Gamma-Ray Observatory*, *ApJ* **696** (2009) L22–L26 C.S. Chang, E. Ros, M. Kadler et al, *The Broadband Spectral Energy Distribution of the MOJAVE Sample*, *Proceedings of the Workshop “Fermi meets Jansky - AGN in Radio and Gamma-Rays”, Savolainen, T., Ros, E., Porcas, R.W. & Zensus, J.A. (eds.), MPIfR, Bonn, June 21-23 2010* H.D. Aller, M.F. Aller, G.E. Latimer et al., *Spectra and linear polarizations of extragalactic variable sources at centimeter wavelengths*, *ApJS* **59** (1985) 513–768 M.F. Aller, H.D. Aller, P.A. Hughes et al., *Pearson-Readhead Survey Sources. II. The Long-Term Centimeter-Band Total Flux and Linear Polarization Properties of a Complete Radio Sample*, *ApJ* **586** (2003) 33–51 C.S. Chang, *Active Galactic Nuclei throughout the Spectrum: M87, PKS2052$-$47, and the MOJAVE Sample*, *PhD thesis*, University of Cologne (2010), Chapter 4 [^1]: Usually, we use the term blazar to refer to BL Lac objects and flat spectrum radio quasars (FSRQ). [^2]: \(1) J2000.0 declination$\leq-$20$^{\circ}$; (2) galactic latitude |b|$\leq$2.5$^{\circ}$; (3) 15GHz VLBI flux density$\geq$1.5Jy. [^3]: `http://nedwww.ipac.caltech.edu/` [^4]: **E**nergetic **G**amma **R**ay **E**xperiment **T**elescope [^5]: **C**ompton **G**amma **R**ay **O**bservatory

--- abstract: 'Data collection under local differential privacy (LDP) has been mostly studied for homogeneous data. Real-world applications often involve a mixture of different data types such as key-value pairs, where the frequency of keys and mean of values under each key must be estimated simultaneously. For key-value data collection with LDP, it is challenging to achieve a good utility-privacy tradeoff since the data contains two dimensions and a user may possess multiple key-value pairs. There is also an inherent correlation between key and values which if not harnessed, will lead to poor utility. In this paper, we propose a locally differentially private key-value data collection framework that utilizes correlated perturbations to enhance utility. We instantiate our framework by two protocols PCKV-UE (based on Unary Encoding) and PCKV-GRR (based on Generalized Randomized Response), where we design an advanced Padding-and-Sampling mechanism and an improved mean estimator which is non-interactive. Due to our correlated key and value perturbation mechanisms, the composed privacy budget is shown to be less than that of independent perturbation of key and value, which enables us to further optimize the perturbation parameters via budget allocation. Experimental results on both synthetic and real-world datasets show that our proposed protocols achieve better utility for both frequency and mean estimations under the same LDP guarantees than state-of-the-art mechanisms.' author: - | [Xiaolan Gu]{}\ University of Arizona\ [xiaolang@email.arizona.edu]{} - | [Ming Li]{}\ University of Arizona\ [lim@email.arizona.edu]{} - | [Yueqiang Cheng[^^]{}]{}\ Baidu X-Lab\ [chengyueqiang@baidu.com]{} - | [Li Xiong]{}\ Emory University\ [lxiong@emory.edu]{} - | [Yang Cao]{}\ Kyoto University\ [yang@i.kyoto-u.ac.jp]{} bibliography: - 'mybibfile.bib' title: | **PCKV: Locally Differentially Private Correlated Key-Value\ Data Collection with Optimized Utility** --- Introduction ============ Differential Privacy (DP) [@dwork2006differential; @dwork2006calibrating] has become the *de facto* standard for private data release. It provides provable privacy protection, regardless of the adversary’s background knowledge and computational power [@chen2016private]. In recent years, Local Differential Privacy (LDP) has been proposed to protect privacy at the data collection stage, in contrast to DP in the centralized setting which protects data after it is collected and stored by a server. In the local setting, the server is assumed to be untrusted, and each user independently perturbs her raw data using a privacy-preserving mechanism that satisfies LDP. Then, the server collects the perturbed data from all users to perform data analytics or answer queries from users or third parties. The local setting has been widely adopted in practice. For example, Google’s RAPPOR [@erlingsson2014rappor] has been employed in Chrome to collect web browsing behavior with LDP guarantees; Apple is also using LDP-based mechanisms to identify popular emojis, popular health data types, and media playback preference in Safari [@apple2017learning]. ![A motivating example (movie rating system).[]{data-label="fig:model"}](model.pdf){width="3.4in"} Early works under LDP mainly focused on simple statistical queries such as frequency/histogram estimation on categorical data [@wang2017locally] and mean estimation of numerical data [@duchi2018minimax; @nguyen2016collecting; @ding2017collecting]. Later works studied more complex queries or structured data, such as frequent item/itemset mining of itemset data [@qin2016heavy; @wang2018locally], computing mean value over a single numeric attribute of multidimensional data [@wang2019collecting; @zhang2018calm; @ren2018textsf], and generating synthetic social graphs from graph data [@qin2017generating]. However, few of them studied the hybrid/heterogeneous data types or queries (e.g., both categorical and numerical data). Key-value data is one such example, which is widely encountered in practice. As a motivating example, consider a movie rating system (shown in Figure \[fig:model\]), each user possesses multiple records of movies (the keys) and their corresponding ratings (the values), that is, a set of key-value pairs. The data collector (the server) can aggregate the rating records from all users and analyze the statistical property of a certain movie, such as the ratio of people who watched this movie (frequency) and the average rating (value mean). Then, the server (or a third party) can provide recommendations by choosing movies with both high frequencies and large value means. The main challenges to achieve high utility for key-value data collection under LDP are two-fold: multiple key-value pairs possessed by each user and the inherent correlation between the key and value. For the former, if all the key-value pairs of a user are reported to the server, each pair will split the limited privacy budget $\epsilon$ (the larger $\epsilon$ is, the more leakage is allowed), which requires more noise/perturbation for each pair. For the latter, correlation means reporting the value of a key also discloses information about the presence of that key. If the key and value are independently perturbed each under $\epsilon$-LDP, overall it satisfies $2\epsilon$-LDP according to sequential composition, which means more perturbation is needed for both key and value to satisfy $\epsilon$-LDP overall. Intuitively, jointly perturbing key and value by exploiting such correlation may lead to less overall leakage; however, it is non-trivial to design such a mechanism that substantially improves the budget composition. Recently, Ye et al. [@ye2019privkv] are the first to propose PrivKVM to estimate the frequency and mean of key-value data. Because of key-value correlation, they adopt an interactive protocol with multiple rounds used to iteratively improve the estimation of a key’s mean value. The mean estimation in PrivKVM is shown to be unbiased when the number of iterations is large enough. However, it has three major limitations. First, multiple rounds will enlarge the variance of mean estimation (as the privacy budget is split in each iteration) and reduce the practicality (since users need to be online). Second, they use a sampling protocol that samples an index from the domain of all keys to address the first challenge, which does not work well for a large key domain (explained in Sec. \[sec:PrivKVM\]). Third, although their mechanism considers the correlation between key and value, it does not lead to an improved budget composition for LDP (discussed in Sec. \[sec:perturbation\]). In this paper, we propose a novel framework for Locally Differentially Private Correlated Key-Value (PCKV) data collection with a better utility-privacy tradeoff. It enhances PrivKVM in four aspects, where the first three address the limitations of PrivKVM, and the last one further improves the utility based on optimized budget allocation. First, we propose an improved mean estimator which only needs a single-round. We divide the calibrated sum of values of a certain key by the calibrated frequency of that key (whose expectation is the true frequency of keys), unlike PrivKVM which uses uncalibrated versions of both (value sum and frequency) that is skewed by inputs from the fake keys and their values. To fill the values of fake keys, we only need to randomly generate values with zero mean (which do not change the expectation of estimated value sum), eliminating the need to iteratively estimate the mean for fake value generation. Although the division of two unbiased estimators is not unbiased in general, we show that it is a consistent estimator (i.e., the bias converges to 0 when the number of users increases). We also propose an improved estimator to correct the outliers when estimation error is large under a small $\epsilon$. Second, we adapt an advanced sampling protocol called Padding-and-Sampling [@wang2018locally] (originally used in itemset data) to sample one key-value pair from the local pairs that are possessed by the user to make sure most of sampled data are useful. Such an advanced sampling protocol can enhance utility, especially for a large domain size. Third, as a byproduct of uniformly random fake value generation (when a non-possessed key is reported as possessed), we show that the proposed correlated perturbation strategy consumes less privacy budget overall than the budget summation of key and value perturbations, by deriving a tighter bound of the composed privacy budget (Theorem \[thm:LDP\_UE\] and Theorem \[thm:LDP\_GRR\]). It can provide a better utility-privacy tradeoff than using the basic sequential composition of LDP which assumes independent mechanisms. Note that PrivKVM directly uses sequential composition for privacy analysis. Fourth, since the Mean Square Error (MSE) of frequency and mean estimations in our scheme can be theoretically analyzed (in Theorem \[thm:estimation\]) with respect to the two privacy budgets of key and value perturbations, it is possible to find the optimized budget allocation with minimum MSE under a given privacy constraint (budget). However, the MSEs depend on the true frequency and value mean that are unknown in practice. Thus, we derive near-optimal privacy budget allocation and perturbation parameters in closed-form (Lemma \[lem:budget\_UE\] and Lemma \[lem:budget\_GRR\]) by minimizing an approximate upper bound of the MSE. Our near-optimal allocation is shown (in both theoretical and empirical) to outperform the naive budget allocation with an equal split. Main contributions are summarized as follows: \(1) We propose the PCKV framework with two mechanisms PCKV-UE and PCKV-GRR under two baseline perturbation protocols: Unary Encoding (UE) and Generalized Randomized Response (GRR). Our scheme is non-interactive (compared with PrivKVM) as the mean of values is estimated in one round. We theoretically analyze the expectation and MSE and show its asymptotic unbiasedness. \(2) We adapt the Padding-and-Sampling protocol [@wang2018locally] for key-value data, which handles large domain better than the sampling protocol used in PrivKVM. \(3) We show the budget composition of our correlated perturbation mechanism, which has a tighter bound than using the sequential composition of LDP. \(4) We propose a near-optimal budget allocation approach with closed-form solutions for PCKV-UE and PCKV-GRR under the tight budget composition. The utility-privacy tradeoff of our scheme is improved by both the tight budget composition and the optimized budget allocation. \(5) We evaluate our scheme using both synthetic and real-world datasets, which is shown to have higher utility (i.e., less MSE) than existing schemes. Results also validate the correctness of our theoretical analysis and the improvements of the tight budget composition and optimized budget allocation. Related Work ============ The main task of local differential privacy techniques is to analyze some statistic information from the data that has been perturbed by users. Erlingsson et al. [@erlingsson2014rappor] developed RAPPOR satisfying LDP for Chrome to collect URL click counts. It is based on the ideas of Randomized Response [@warner1965randomized], which is a technique for collecting statistics on sensitive queries when a respondent wants to retain confidentiality. In the basic RAPPOR, they adopt unary encoding to obtain better performance of frequency estimation. Wang et al. [@wang2017locally] optimized the parameters of basic RAPPOR by minimizing the variance of frequency estimation. There are a lot of works that focus on complex data types and complex analysis tasks under LDP. Bassily and Smith [@bassily2015local] proposed an asymptotically optimal solution for building succinct histograms over a large categorical domain under LDP. Qin et al. [@qin2016heavy] proposed a two-phase work named LDPMiner to achieve the heavy hitter estimation (items that are frequently possessed by users) over the set-valued data with LDP, where each user can have any subset of an item domain with different length. Based on the work of LDPMiner, Wang et al. [@wang2018locally] studied the same problem and proposed a more efficient framework to estimate not only the frequent items but also the frequent itemsets. To the best of our knowledge, there are only two works on key-value data collection under LDP. Ye et al. [@ye2019privkv] are the first to propose PrivKV, PrivKVM, and PrivKVM$^{+}$, where PrivKVM iteratively estimates the mean to guarantee the unbiasedness. PrivKV can be regarded as PrivKVM with only one iteration. The advanced version PrivKVM$^{+}$ selects a proper number of iterations to balance the unbiasedness and communication cost. Sun et al. [@sun2019conditional] proposed another estimator for frequency and mean under the framework of PrivKV and several mechanisms to accomplish the same task. They also introduced conditional analysis (or the marginal statistics) of key-value data for other complex analysis tasks in machine learning. However, both of them use the naive sampling protocol and neither of them analyzes the tighter budget composition caused by the correlation between perturbations nor considers the optimized budget allocation. Preliminaries ============= Local Differential Privacy -------------------------- In the centralized setting of differential privacy, the data aggregator (server) is assumed to be trusted who possesses all users’ data and perturbs the query answers. However, this assumption does not always hold in practice and may not be convincing enough to the users. In the local setting, each user perturbs her input $x$ using a mechanism $\mathcal{M}$ and uploads $y=\mathcal{M}(x)$ to the server for data analysis, where the server can be untrusted because only the user possesses the raw data of herself; thus the server has no direct access to the raw data. For a given $\epsilon\in\mathbb{R}^{+}$, a randomized mechanism $\mathcal{M}$ satisfies $\epsilon$-LDP if and only if for any pair of inputs $x,x^{\prime}$, and any output $y$, the probability ratio of outputting the same $y$ should be bounded $$\begin{aligned} \label{equ:def_LDP} \frac{\Pr(\mathcal{M}(x)=y)}{\Pr(\mathcal{M}(x^{\prime})=y)} \leqslant e^{\epsilon} \end{aligned}$$ Intuitively, given an output $y$ of a mechanism, an adversary cannot infer with high confidence (controlled by $\epsilon$) whether the input is $x$ or $x^{\prime}$, which provides plausible deniability for individuals involved in the sensitive data. Here, $\epsilon$ is a parameter called *privacy budget* that controls the strength of privacy protection. A smaller $\epsilon$ indicates stronger privacy protection because the adversary has lower confidence when trying to distinguish any pair of inputs $x,x^{\prime}$. A very good property of LDP is sequential composition, which guarantees the overall privacy for a sequence of mechanisms that satisfy LDP. \[thm:seq\_compo\] If a randomized mechanism $\mathcal{M}_i: \mathcal{D}\rightarrow\mathcal{R}_i$ satisfies $\epsilon_i$-LDP for $i=1,2,\cdots,k$, then their sequential composition $\mathcal{M}: \mathcal{D}\rightarrow\mathcal{R}_1\times\mathcal{R}_2\times\cdots\times\mathcal{R}_k$ defined by $\mathcal{M}=(\mathcal{M}_1,\mathcal{M}_2,\cdots,\mathcal{M}_k)$ satisfies $(\sum_{i=1}^k\epsilon_i)$-LDP. According to sequential composition, a given privacy budget for a computation task can be split into multiple portions, where each portion corresponds to the budget for a sub-task. Mechanisms under LDP {#sec:LDP mechanism} -------------------- **Randomized Response.** Randomized Response (RR) [@warner1965randomized] is a technique developed for the interviewees in a survey to return a randomized answer to a sensitive question so that the interviewees can enjoy plausible deniability. Specifically, each interviewee gives a genuine answer with probability $p$ or gives the opposite answer with probability $q=1-p$. In order to satisfy $\epsilon$-LDP, the probability is selected as $p=\frac{e^\epsilon}{e^\epsilon+1}$. RR only works for binary data, but it can be extended to apply for the general category set $\{1,2,\cdots,d\}$ by Generalized Randomized Response (GRR) or Unary Encoding (UE). **Generalized Randomized Response.** The perturbation function in Generalized Randomized Response (GRR) [@wang2017locally] is $$\begin{aligned} \Pr(\mathcal{M}(x)=y)= \begin{cases} p=\frac{e^\epsilon}{e^\epsilon+d-1}, & \text{if } y=x\\ q=\frac{1-p}{d-1}, & \text{if } y\neq x \end{cases}\end{aligned}$$ where $x,y\in\{1,2,\cdots,d\}$ and the values of $p$ and $q$ guarantee $\epsilon$-LDP of the perturbation (because $\frac{p}{q}=e^{\epsilon}$). **Unary Encoding.** The Unary Encoding (UE) [@wang2017locally] converts an input $x=i$ into a bit vector $\mathbf{x}=[0,\cdots,0,1,0,\cdots,0]$ with length $d$, where only the $i$-th position is 1 and other positions are 0s. Then each user perturbs each bit of $\mathbf{x}$ independently with the following probabilities ($q\leqslant0.5\leqslant p$) $$\begin{aligned} \Pr(\mathbf{y}[k]=1)= \begin{cases} p, & \text{if } \mathbf{x}[k]=1\\ q, & \text{if } \mathbf{x}[k]=0 \end{cases}\quad (\forall k=1,2,\cdots,d)\end{aligned}$$ where $\mathbf{y}$ is the output vector with the same size as vector $\mathbf{x}$. It was shown in [@wang2017locally] that this mechanism satisfies LDP with $\epsilon=\ln\frac{p(1-q)}{(1-p)q}$. The selection of $p$ and $q$ under a given privacy budget $\epsilon$ varies for different mechanisms. For example, the basic RAPPOR [@erlingsson2014rappor] assigns $p=\frac{e^{\epsilon/2}}{e^{\epsilon/2}+1}$ and $q=1-p$, while the Optimized Unary Encoding (OUE) [@wang2017locally] assigns $p=\frac{1}{2}$ and $q=\frac{1}{e^{\epsilon}+1}$, which is obtained by minimizing the approximate variance of frequency estimation. **Frequency Estimation for GRR, RAPPOR and OUE.** After receiving the perturbed data from all users (with size $n$), the server can compute the observed proportion of users who possess the $i$-th item (or $i$-th bit), denoted by $f_i$. Since the perturbation is biased for different items (or bit-0 and bit-1), the server needs to estimate the observed frequency by an unbiased estimator $\hat{f}_i=\frac{f_i-q}{p-q}$, whose Mean Square Error (MSE) equals to its variance [@wang2017locally] $$\begin{aligned} \text{MSE}_{\hat{f}_i}=\text{Var}[\hat{f}_i]=\frac{q(1-q)}{n(p-q)^2}+\frac{f_i^{*}(1-p-q)}{n(p-q)}\end{aligned}$$ where $f_i^{*}$ is the ground truth of the frequency for item $i$. Key-Value Data Collection under LDP =================================== Problem Statement ----------------- **System Model.** Our system model (shown in Figure \[fig:model\]) involves one data server and a set of users $\mathcal{U}$ with size $|\mathcal{U}|=n$. Each user possesses one or multiple key-value pairs $\langle k,v\rangle$, where $k\in\mathcal{K}$ (the domain of key) and $v\in\mathcal{V}$ (the domain of value). We assume the domain size of key is $d$, i.e., $\mathcal{K}=\{1,2,\cdots,d\}$, and domain of value is $\mathcal{V}=[-1,1]$ (any bounded value space can be linearly transformed into this domain). The set of key-value pairs possessed by a user is denoted as $\mathcal{S}$ (or $\mathcal{S}_u$ for a specific user $u\in\mathcal{U}$). After collecting the perturbed data from all users, the server needs to estimate the frequency (the proportion of users who possess a certain key) and the value mean (the averaged value of a certain key from the users who possess such key), i.e., $$\begin{aligned} f_k^{*}=\frac{\sum_{u\in\mathcal{U}}\mathds{1}_{\mathcal{S}_u}(\langle k,\cdot\rangle)}{n},\quad m_k^{*} = \frac{\sum_{u\in\mathcal{U},\langle k,v\rangle\in\mathcal{S}_u}v}{n\cdot f_k^{*}}\end{aligned}$$ where $\mathds{1}_{\mathcal{S}_u}(\langle k,\cdot\rangle)$ is 1 when $\langle k,\cdot\rangle\in\mathcal{S}_u$ and is 0 otherwise. **Threat Model.** We assume the server is untrusted and each user only trusts herself because the privacy leakage can be caused by either unauthorized data sharing or breach due to hacking activities. Therefore, the adversary is assumed to have access to the output data of all users and know the perturbation mechanism adopted by the users. Note that we assume all users are honest in following the perturbation mechanism, thus we do not consider the case that some users maliciously upload bad data to fool the server. **Objectives and Challenges.** Our goal is to estimate frequency and mean with high accuracy (i.e., small Mean Square Error) under the required privacy constraint (i.e., satisfying $\epsilon$-LDP). However, the task is not trivial for key-value data due to the following challenges: **(1)** Considering each user can possess multiple key-value pairs (the number of pairs can be different for users), if each user uploads multiple pairs, then each pair needs to consume budget, leading to a smaller budget and larger noise in each pair. On the other hand, if simply sampling an index $j$ from the domain and uploading the key-value pair regarding the $j$-th key (which is used in PrivKVM [@ye2019privkv]), we cannot make full use of the original pairs. Therefore, an elaborately designed sampling protocol is necessary in order to estimate the frequency and mean with high accuracy. **(2)** Due to the correlation between key and value in a key-value pair, the perturbation of key and value should be correlated. If a user reports a key that does not exist in her local data, she has to generate a fake value to guarantee the indistinguishability; however, how to generate the fake value without any prior knowledge and how to eliminate the influence of fake values on the mean estimation are challenging tasks. **(3)** Considering the key and value are perturbed in a correlated manner, the overall perturbation mechanism may not leak as much information as two independent perturbations do (by sequential composition). Therefore, precisely quantifying the actually consumed privacy budget can improve the privacy-utility tradeoff of the overall key-value perturbation. PrivKVM {#sec:PrivKVM} ------- To the best of our knowledge, PrivKVM [@ye2019privkv] is the only published work on key-value data collection in the LDP setting (note that another existing work [@sun2019conditional] is a preprint). It utilizes one iteration for frequency estimation and multiple iterations to approximately approach the unbiased mean estimation. We briefly describe it as follows. Assume the total privacy budget is $\epsilon$, and the number of iterations is $c$. In the *first iteration*, each user randomly samples an index $j$ from the key domain $\mathcal{K}$ with uniform distribution (note that $j$ does not contain any private information). If the user processes key $k=j$ with value $v$, then she perturbs the key-value pair $\langle 1,v\rangle$; if not, the user perturbs the key-value pair $\langle 0,\tilde{v}\rangle$, where $\tilde{v}$ is initialized as 0 in the first iteration. In both cases, the input is perturbed with key-budget $\frac{\epsilon}{2}$ and value-budget $\frac{\epsilon}{2c}$. Then, each user uploads the index $j$ and one perturbed key-value pair $\langle 0,\cdot\rangle$ or $\langle 1,\cdot\rangle$ to the server and the server can compute the estimated frequency $f_k$ and mean $m_k~(k\in\mathcal{K})$ after collecting the perturbed data from all users, where the counts of output values will be corrected before estimation when outliers occur. In the *remaining iterations*, each user perturbs her data with a similar way but $\tilde{v}=m_k$ (the estimated mean of the previous round) and the budget for key perturbation is 0. Then, the server updates the mean $m_k$ in the current iteration. By multiple rounds of interaction between users and the server, the mean estimation is approximately unbiased, and the sequential composition guarantees LDP with privacy budget $\frac{\epsilon}{2}+\frac{\epsilon}{2c}\cdot c=\epsilon$. There are three limitations of PrivKVM. \(1) To achieve approximate unbiasedness, PrivKVM needs to run multiple rounds. This requires all users online during all rounds, which is impractical in many application scenarios. Also, the multiple iterations only guarantee the convergence of expectation of mean estimation (i.e., the bias theoretically approaches zero when $c\rightarrow\infty$), but the variance of mean estimation will be very large for a large $c$ because the budget $\frac{\epsilon}{2c}$ (for value perturbation in each round) is very small. Note that the estimation error depends on both bias and variance. \(2) The sampling protocol in PrivKVM may not work well for a large domain. When the domain size $d=|\mathcal{K}|$ is very large (such as millions) and each user only has a relatively small number of key-value pairs (such as less than 10), uniformly sampling an index from the large key domain $\mathcal{K}$ makes users rarely upload the information of the keys that they possess, resulting in a large variance of frequency and mean estimations. Also, when the number of users $n$ is not very large compared with domain size (such as $n<2d$), some keys may not be sampled, then the mean estimation does not work for such keys because of no samples. \(3) Although PrivKVM considers the correlation between key and value, it does not lead to an improved budget composition for LDP, which will be discussed in Sec. \[sec:perturbation\]. Proposed Framework and Mechanisms ================================= {width="3.4in"} \[fig:Diagram\] The overview of our PCKV framework is shown in Figure \[fig:Diagram\], where two specific mechanisms are included. The first one is PCKV-UE, which outputs a bit vector, and the second one is PCKV-GRR, which outputs a key-value pair. Note that the two mechanisms have similar ideas but steps are slightly different. In step , the system sets up some environment parameters (such as the total budget $\epsilon$ and domain size $d$), which can be used to allocate the privacy budget for key and value perturbations and compute the perturbation probabilities in mechanisms, where the optimized privacy budget allocation is discussed in Sec. \[sec:allocation\]. In step and step , each user samples one key-value pair from her local data and privately perturbs it, where the sampling protocol is discussed in Sec. \[sec:sampling\] and the perturbation mechanisms (PCKV-UE and PCKV-GRR) are proposed in Sec. \[sec:perturbation\]. The perturbation of value depends on the perturbation of key, which is utilized to improve the privacy budget composition. In step and step , the server aggregates the perturbed data from all users and estimates the frequency and mean, shown in Sec. \[sec:calibratoin\]. Sampling Protocol {#sec:sampling} ----------------- This subsection corresponds to step in Figure \[fig:Diagram\]. Considering each user may possess multiple key-value pairs, if the user perturbs and uploads all pairs, then each pair would consume the budget and the noise added in each pair becomes too large. Therefore, a promising solution is to upload the perturbed data of one pair (by sampling) to the server, which can avoid budget splitting. As analyzed in Sec. \[sec:PrivKVM\], the sampling protocol used in PrivKVM does not work well for a large domain. In this paper, we use an advanced protocol called Padding-and-Sampling [@wang2018locally] to improve the performance. The Padding-and-Sampling protocol [@wang2018locally] is originally used for itemset data, where each user samples one item from possessed items rather than sampling an index from the domain of all items. To make the sampling rate the same for all users, each user first pads her items into a uniform length $\ell$ by some dummy items from a domain of size $\ell$. Although there may still exist unsampled items, this case occurs only for infrequent items, thus the useful information of frequent items still can be reported with high probability. The set of key-value pairs $\mathcal{S}$, padding length $\ell$ Sampled key-value pair $\langle k,v\rangle$, where $k\in\mathcal{K}^{\prime}$ and $v\in\{1,-1\}$. Randomly draw $B\sim$ Bernoulli$(\eta)$, where $\eta=\frac{|\mathcal{S}|}{\max\{|\mathcal{S}|,\ell\}}$. Randomly sample one key-value pair $\langle k,v^{*}\rangle$ from $\mathcal{S}$ with discrete uniform distribution. `//sample a non-dummy key-value pair` Set $v^{*}=0$ and randomly draw $k$ from $\{d+1,\cdots,d^{\prime}\}$ with discrete uniform distribution. `//sample a dummy key-value pair` Discretize the value: $v\leftarrow 1$ w.p. $\frac{1+v^{*}}{2}$ or $v\leftarrow -1$ w.p. $\frac{1-v^{*}}{2}$ Return $x=\langle k,v\rangle$. \[alg:PS\] **Our Sampling Protocol.** The original Padding-and-Sampling protocol is designed for itemset data and does not work for key-value data. Thus, we modify it to handle the key-value data, shown in Algorithm \[alg:PS\], where $d^{\prime}=d+\ell$, $\mathcal{K}^{\prime}=\{1,2,\cdots,d^{\prime}\}$, and parameter $\eta=\frac{|\mathcal{S}|}{\max\{|\mathcal{S}|,\ell\}}$ represents the probability of sampling the non-dummy key-value pairs. The main differences are two-fold. First, we sample one key-value pair instead of one item, and if a dummy key is sampled, we assign a fake value $v^{*}=0$. Second, after sampling, the value is discretized into $1$ or $-1$ for the value perturbation to implement randomized response based mechanism, where the discretization in Line-7 guarantees the unbiasedness because $\mathbb{E}[v]=\frac{1+v^{*}}{2}-\frac{1-v^{*}}{2}=v^{*}$. By using Algorithm \[alg:PS\], the large domain size does not affect the probability of sampling a possessed key because it samples from key-value pairs possessed by users. Also, even when the user size is less than the domain size, the frequent keys still have larger probabilities to be sampled by users while only the infrequent keys may not be sampled. Therefore, the two problems of naive sampling protocol in PrivKVM (discussed in Sec \[sec:PrivKVM\]) can be solved by our advanced one. For the selection of $\ell$, a smaller $\ell$ will underestimate the frequency thus lead to a large bias, while a larger one will enlarge the variance [@wang2018locally]. Thus, it should balance the tradeoff between bias and variance. A baseline strategy of selecting a good $\ell$ was proposed in [@wang2018locally] for itemset data. They set $\ell$ as the 90th percentile of the length of inputs, where the length distribution is privately estimated from a subset of users. Note that the users are partitioned into multiple groups, where each group participates in only one task (the pre-task to estimate length distribution or the main task to estimate frequency); thus $\epsilon$-LDP in each group guarantees $\epsilon$-LDP for the whole group of users. However, how to select an optimal partition ratio for length distribution estimation (more users in this task can provide more accurate length estimation but leads to fewer users for the main task which impacts frequency and mean estimation) and how to select an optimal percentile (a larger percentile leads to less bias but more variance) are non-trivial tasks. Therefore, in this paper, we select some reasonable $\ell$ for different datasets in experiments for comparing with PrivKVM (which uses naive sampling protocol) and leave the strategies of finding the optimized partition ratio and percentile for estimating $\ell$ as future work. Perturbation Mechanisms {#sec:perturbation} ----------------------- ![Perturbation of $k$-th element ($\forall k\in\mathcal{K}^{\prime}$) in PCKV-UE.[]{data-label="fig:perturbation"}](PCKV.pdf){width="3.1in"} This subsection corresponds to step in Figure \[fig:Diagram\]. By Algorithm \[alg:PS\], each user samples one key-value pair $x=\langle k,v\rangle$ as the input of perturbation, where the domain is $k\in\mathcal{K}^{\prime},v\in\{1,-1\}$. If a non-possessed key is reported as possessed (in PCKV-UE), we need to generate fake value. If the original key is perturbed into another one (in PCKV-GRR), the original value is useless for the mean estimation since the original key is not reported to the server. In both cases, we can generate value with discrete uniform distribution to avoid influence of values of different keys. We will show that such a strategy can provide a tighter composition (in Theorem \[thm:LDP\_UE\] and Theorem \[thm:LDP\_GRR\]), which is reflected as a smaller total budget of the composed perturbation than sequential composition. By combining the above idea with sampling protocol for key-value data (Algorithm \[alg:PS\]) and two basic LDP mechanisms (UE and GRR) in Sec. \[sec:LDP mechanism\], we obtain two mechanisms under the PCKV framework: PCKV-UE and PCKV-GRR. **PCKV-UE Mechanism.** In Unary Encoding (UE), the original input is encoded as a bit vector, where only the input-corresponding bit is 1 and other bits are 0s, then each bit flips with specified probabilities to generate the output vector. For key-value data, denote the element in $k$-th position (regarding the key $k$) as $\langle i,v\rangle$ with domain $\{\langle1,1\rangle,\langle1,-1\rangle,\langle0,0\rangle\}$, i.e., the sampled pair $x=\langle k,v \rangle$ is encoded as a vector $\mathbf{x}$, where only the $k$-th element is $\langle 1,\pm 1\rangle$ and others are $\langle 0,0 \rangle$. Then, the perturbation of key $i\rightarrow i^{\prime}$ in each element can be implemented by $1\rightarrow 1$ w.p. $a$ or $0\rightarrow 1$ w.p. $b$ (where $b\leqslant0.5\leqslant a$). For value perturbation $v\rightarrow v^{\prime}$, we discuss three cases: Case 1. If $1\rightarrow 1$, then the value is maintained ($v^{\prime}=v$) with probability $p$ or flipped ($v^{\prime}=-v$) with probability $1-p$. Case 2. If $1\rightarrow 0$ or $0\rightarrow 0$, then the output value can be set to $v^{\prime}=0$ because the key $k$ is reported as not possessed. Case 3. If $0\rightarrow 1$, then the fake value $v^{\prime}=1$ or $v^{\prime}=-1$ are assigned with probability 0.5 respectively. The discretization and perturbation of PCKV-UE are shown in Figure \[fig:perturbation\]. For brevity, we use three states $\{1,-1,0\}$ to represent the key-value pairs $\{\langle1,1\rangle,\langle1,-1\rangle,\langle0,0\rangle\}$ in each position of output vector $\mathbf{y}$. If the sampled pair is $x=\langle k,1\rangle$, then only the $k$-th element of the encoded vector $\mathbf{x}$ is 1 (other elements are 0s), and the probability of $\mathbf{y}[k]=1$ is $$\begin{aligned} &\Pr(\mathbf{y}[k]=1|x=\langle k,1\rangle)=\Pr(\mathbf{y}[k]=1|\mathbf{x}[k]=1)=ap\end{aligned}$$ Similarly, we can compute the perturbation probabilities of other elements in all possible cases, shown in Algorithm \[alg:PCKV-UE\]. The set of key-value pairs $\mathcal{S}$, perturbation probabilities $a,b$ and $p$, where $a,p\in[\frac{1}{2},1)$ and $b\in(0,\frac{1}{2}]$. Vector $\mathbf{y}\in\{1,-1,0\}^{d^{\prime}}$, where $d^{\prime}=d+\ell$. Sample one key-value pair $x=\langle k,v\rangle$ from $\mathcal{S}$ by Algorithm \[alg:PS\]. Independently perturb the $k$-th element and other elements ($\forall i\in\mathcal{K}^{\prime}\backslash k$) $$\begin{aligned} \mathbf{y}[k]= \begin{cases} v, & \text{w.p.}\quad a\cdot p \\ -v, & \text{w.p.}\quad a\cdot(1-p) \\ 0, & \text{w.p.} \quad 1-a \end{cases},\qquad \mathbf{y}[i]= \begin{cases} 1, & \text{w.p.}\quad b/2 \\ -1, & \text{w.p.}\quad b/2 \\ 0, & \text{w.p.}\quad 1-b \end{cases} \end{aligned}$$ Return vector $\mathbf{y}$. \[alg:PCKV-UE\] **Privacy Analysis of PCKV-UE.** In PCKV-UE, the key is perturbed by Unary Encoding (UE) with budget $\epsilon_1=\ln\frac{a(1-b)}{b(1-a)}$ (refer to Sec. \[sec:LDP mechanism\]), and the value is perturbed by Randomized Response (because the discretized value is $1$ or $-1$) with budget $\epsilon_2=\ln\frac{p}{1-p}$ (then $p=\frac{e^{\epsilon_2}}{e^{\epsilon_2}+1}$). Also, the key and value are perturbed in a correlated manner. That is, the value perturbation mechanism depends on both the input key and perturbed key of a user. Intuitively, *correlated perturbation may leak less information than independent perturbation*, i.e., the total privacy budget $\epsilon$ can be less than the summation $\epsilon_1+\epsilon_2$. The following theorem shows the tight budget composition of our PCKV-UE mechanism. \[thm:LDP\_UE\] Assume the privacy budgets for key and value perturbations in PCKV-UE (Algorithm \[alg:PCKV-UE\]) are $\epsilon_1$ and $\epsilon_2$ respectively, i.e., the perturbation probabilities $a,b,p$ satisfies $$\begin{aligned} \frac{a(1-b)}{b(1-a)}=e^{\epsilon_1},\quad p=\frac{e^{\epsilon_2}}{e^{\epsilon_2}+1}\end{aligned}$$ then PCKV-UE satisfies LDP with privacy budget $$\begin{aligned} \label{equ:epsilon_UE} \epsilon=\max\left\{\epsilon_2,~\epsilon_1+\ln[2/(1+e^{-\epsilon_2})]\right\} \end{aligned}$$ where $\epsilon\leqslant (\epsilon_1+\epsilon_2)$ because of $\epsilon_1\geqslant0$ and $\frac{2}{1+e^{-\epsilon_2}}\leqslant e^{\epsilon_2}$. See Appendix \[apx:thm:LDP\_UE\]. **Interpretation of Theorem \[thm:LDP\_UE\].** For two different key-value pairs $\langle k_1,v_1\rangle$ and $\langle k_2,v_2\rangle$, where $v_1,v_2\in\{1,-1\}$, the probability ratio of reporting the same output vector $\mathbf{y}$ should be bounded to guarantee LDP. If $k_1=k_2=k$, then the probability ratio only depends on the perturbation of the $k$-th elements $v_1$ and $v_2$ (because other elements are the same, then the corresponding probabilities are canceled out in the ratio), thus the upper bound of the probability ratio is $\frac{ap}{a(1-p)}=e^{\epsilon_2}$. If $k_1\neq k_2$, the ratio depends on both $k_1$-th and $k_2$-th elements, thus the upper bound is $\frac{ap}{b/2}\cdot\frac{1-b}{1-a}=\frac{2e^{\epsilon_1+\epsilon_2}}{e^{\epsilon_2}+1}=e^{\epsilon_1+\ln[2/(1+e^{-\epsilon_2})]}$ (in the case of $\mathbf{y}[k_1]=v_1,\mathbf{y}[k_2]=0$ or $\mathbf{y}[k_1]=0,\mathbf{y}[k_2]=v_2$). Finally, the total privacy budget is the log of the maximum value of the upper bounds in the two cases. **Using Theorem \[thm:LDP\_UE\] to Allocate Budget.** Due to the non-linear relationship among $\epsilon$, $\epsilon_1$, and $\epsilon_2$ in Theorem \[thm:LDP\_UE\], the budget allocation of PCKV-UE is not direct as $\epsilon_2=\epsilon-\epsilon_1$. We discuss the budget allocation in PCKV-UE as follows. Assume $\epsilon>0$ is a given total privacy budget for composed key-value perturbation. According to (\[equ:epsilon\_UE\]), both $\epsilon_1$ and $\epsilon_2$ are less or equal to $\epsilon$. If $\epsilon_1=\epsilon$, we have $\epsilon_2=0$. If $\epsilon_2=\epsilon$, we have $\epsilon_1\leqslant\epsilon-\ln[2/(1+e^{-\epsilon})]=\ln[(e^{\epsilon}+1)/2]$. Therefore, $\epsilon_1$ and $\epsilon_2$ can be allocated by (with respect to a variable $\theta$) $$\begin{aligned} \label{equ:epsilon_1&2} \epsilon_1=\ln \theta,\quad \epsilon_2=\ln\frac{1}{2\theta e^{-\epsilon}-1},\quad \text{for }\frac{e^{\epsilon}+1}{2}\leqslant \theta <e^{\epsilon}\end{aligned}$$ where $\epsilon_1$ reaches its maximum value when given $\epsilon$ and $\epsilon_2$. The optimized budget allocation, i.e., finding the optimal $\theta$ in , will be discussed in Sec. \[sec:allocation\]. **No Tight Budget Composition for PrivKVM.** One may ask if PrivKVM can also be tightly composed like PCKV-UE. Indeed, when the key is perturbed from $0\rightarrow0$ or $1\rightarrow0$ (corresponding to our Case 2) the reported value must be 0. However, for the case of $0\rightarrow1$ (corresponding to our Case 3), the value is perturbed from the estimated mean (discretized as $1$ or $-1$) of the previous iteration with budget $\frac{\epsilon}{2c}$. Therefore, when the output is $\langle1,\cdot\rangle$ (for all rounds), the consumed budget of composed perturbation is $\frac{\epsilon}{2}+c\cdot\frac{\epsilon}{2c}=\epsilon$, which means no tighter composition for PrivKVM. The set of key-value pairs $\mathcal{S}$, perturbation probabilities $a,p\in[\frac{1}{2},1)$. one key-value pair $y^{\prime}=\langle k^{\prime},v^{\prime}\rangle$, where $k^{\prime}\in\mathcal{K}^{\prime}$ and $v^{\prime}\in\{1,-1\}$. Sample one key-value pair $\langle k,v\rangle$ from $\mathcal{S}$ by Algorithm \[alg:PS\]. Perturb $\langle k,v\rangle$ into $\langle k^{\prime},v^{\prime}\rangle$ (probability $b=\frac{1-a}{d^{\prime}-1}$) $$\begin{aligned} \langle k^{\prime},v^{\prime}\rangle= \begin{cases} \langle k,v \rangle,& \text{w.p.}\quad a\cdot p\\ \langle k,-v \rangle,& \text{w.p.}\quad a\cdot (1-p)\\ \langle i, 1 \rangle \quad (i\in\mathcal{K}^{\prime}\backslash k), & \text{w.p.}\quad b\cdot 0.5 \\ \langle i, -1 \rangle \quad (i\in\mathcal{K}^{\prime}\backslash k), & \text{w.p.}\quad b\cdot 0.5 \end{cases} \end{aligned}$$ Return $y^{\prime}=\langle k^{\prime},v^{\prime}\rangle$. \[alg:PCKV-GRR\] **PCKV-GRR Mechanism.** In GRR, the input is perturbed into another item with specified probabilities, where the input and output have the same domain. In PCKV-GRR, the key is perturbed by GRR with privacy budget $\epsilon_1$, i.e., $k\rightarrow k$ with probability $a=\frac{e^{\epsilon_1}}{e^{\epsilon_1}+d^{\prime}-1}$ and $k\rightarrow i~(i\neq k)$ with probability $b=\frac{1-a}{d^{\prime}-1}$. The value is perturbed by two cases: if $k\rightarrow i~(i\neq k)$, it is perturbed with privacy budget $\epsilon_2$; if $k\neq k^{\prime}$, it is randomly picked from $\{1,-1\}$ with probability 0.5 respectively (similar ideas as in PCKV-UE). The implementation of PCKV-GRR is shown in Algorithm \[alg:PCKV-GRR\]. **Privacy Analysis of PCKV-GRR.** Similar to PCKV-UE, the mechanism PCKV-GRR also consumes less privacy budget than $\epsilon_1+\epsilon_2$. Besides the tight composition obtained from the correlated perturbation, *PCKV-GRR would get additional privacy amplification benefit from Padding-and-Sampling*, though our sampling protocol is originally used to avoid privacy budget splitting (refer to Sec. \[sec:sampling\]). \[thm:LDP\_GRR\] Assume the privacy budgets for key and value perturbation of PCKV-GRR (Algorithm \[alg:PCKV-GRR\]) are $\epsilon_1$ and $\epsilon_2$ respectively, i.e., the perturbation probabilities $a,b$ and $p$ are $$\begin{aligned} \label{equ:abp_GRR} a = \frac{e^{\epsilon_1}}{e^{\epsilon_1}+d^{\prime}-1},\quad b = \frac{1}{e^{\epsilon_1}+d^{\prime}-1},\quad p = \frac{e^{\epsilon_2}}{e^{\epsilon_2}+1}\end{aligned}$$ then PCKV-GRR satisfies LDP with privacy budget $$\begin{aligned} \label{equ:epsilon_GRR} \epsilon=\ln\left(\frac{e^{\epsilon_1+\epsilon_2}+\lambda}{\min\{e^{\epsilon_1},(e^{\epsilon_2}+1)/2\}+\lambda}\right)\end{aligned}$$ where $\lambda=(\ell-1)(e^{\epsilon_2}+1)/2$. See Appendix \[apx:thm:LDP\_GRR\]. **Interpretation of Theorem \[thm:LDP\_GRR\].** According to (\[equ:epsilon\_GRR\]), the total budget $\epsilon$ is a decreasing function of $\lambda$, where $\lambda$ is an increasing function of $\ell$, indicating that a larger $\ell$ provides stronger privacy (smaller $\epsilon$) of PCKV-GRR under the given $\epsilon_1$ and $\epsilon_2$. Also, the above budget composition has two extreme cases. First, if $\ell=1$, then $\lambda=0$ and (\[equ:epsilon\_GRR\]) reduces to the budget composition of PCKV-UE in (\[equ:epsilon\_UE\]), which indicates that the two mechanisms obtain the same benefit (tight budget composition) by adopting the correlated perturbations. Second, if $\epsilon_2=0$, then $\lambda=\ell-1$ and (\[equ:epsilon\_GRR\]) reduces to $\epsilon=\ln\frac{e^{\epsilon_1}+\ell-1}{\ell}$, which is the corresponding result in [@wang2018locally] for itemset data. The intuitive reason for such consistency is that the key perturbation will consume all budget when $\epsilon_2=0$; thus, this special case of key-value perturbation can be regarded as item perturbation. **No Privacy Benefits from Padding-and-Sampling for PCKV-UE.** Since Theorem \[thm:LDP\_UE\] is independent of $\ell$ while Theorem \[thm:LDP\_GRR\] is dependent on $\ell$, PCKV-UE does not have the same privacy amplification benefits from Padding-and-Sampling as PCKV-GRR (both of which have been observed in [@wang2018locally] for itemset data collection). The main reason is that PCKV-UE outputs a vector that can contain multiple keys (i.e., multiple positions have 1). Take a toy example that only considers the perturbation of key (i.e., $\epsilon_2=0$) with domain $\mathcal{K}=\{1,2,3,4\}$ (then $d=|\mathcal{K}|=4$) and $\ell=2$, where the output domain is $\mathcal{Y}=\{0,1\}^{d+\ell}$. In the worst case that determines the upper bound of the probability ratio, we select two neighboring inputs $\mathcal{S}_1=\{1,2\}$ and $\mathcal{S}_2=\{3,4\}$ (note that LDP considers any set of keys as neighboring for one user) and output vector $\mathbf{y}=[110000]$. No matter which key is sampled from $\mathcal{S}_1$, the probability of reporting $\mathbf{y}$ is the same: $p^{*}=ab(1-b)^4$ (because $\mathbf{x}=[100000]$ or $[010000]$). Considering all sampling cases under sampling rate $\frac{1}{\max\{\ell,|\mathcal{S}_1|\}}$, we have $\Pr(\mathbf{y}|\mathcal{S}_1)=\frac{1}{\ell}p^{*}\cdot \ell=p^{*}$, which is independent of $\ell$. Similarly, $\Pr(\mathbf{y}|\mathcal{S}_2)=b^2(1-a)(1-b)^3$. Thus, the probability ratio is $\frac{\Pr(\mathbf{y}|\mathcal{S}_1)}{\Pr(\mathbf{y}|\mathcal{S}_2)}=\frac{a(1-b)}{b(1-a)}=e^{\epsilon_1}$, i.e., no privacy benefits from $\ell$. Note that for other $\mathcal{S}_1,\mathcal{S}_2$, and $\mathbf{y}$, the probability ratio might depend on $\ell$, but they are not the worst case that determines the upper bound. For PCKV-GRR, however, the output $y$ can be only one key. In the worst case, we select the above $\mathcal{S}_1$ and $\mathcal{S}_2$ but $y=\{1\}$. Then, $\Pr(y|\mathcal{S}_1)=\frac{1}{\ell}a +(1-\frac{1}{\ell})b$ because if $x=\{1\}$ (resp. $x=\{2\}$) is sampled from $\mathcal{S}_1$, the probability of reporting $y$ is $a$ (resp. $b$), where $a>b$. Also, $\Pr(y|\mathcal{S}_2)=\frac{1}{\ell}b\cdot \ell=b$ (no matter $x=\{3\}$ or $x=\{4\}$ is sampled, the probability of reporting $y$ is $b$). Thus, $\frac{\Pr(y|\mathcal{S}_1)}{\Pr(y|\mathcal{S}_2)}=1+\frac{a/b-1}{\ell}\leqslant\frac{a}{b}=e^{\epsilon_1}$, where a larger $\ell$ will reduce this ratio (i.e., privacy amplification). Theorem \[thm:LDP\_UE\] and Theorem \[thm:LDP\_GRR\] provide a tighter bound on the total privacy guarantee than the sequential composition ($\epsilon=\epsilon_1+\epsilon_2$). However, in practice, the budgets are determined in a reverse way: given $\epsilon$ (a constant), we need to allocate the corresponding $\epsilon_1$ and $\epsilon_2$ before any perturbation. In Sec. \[sec:allocation\], we will discuss the optimized privacy budget allocation (i.e., how to determine $\epsilon_1$ and $\epsilon_2$ when $\epsilon$ is given) by minimizing the estimation error that is analyzed in Sec. \[sec:calibratoin\]. In summary, both the tight budget composition and optimized budget allocation in our scheme will improve the privacy-utility tradeoff. Note that PrivKVM [@ye2019privkv] simply allocates the privacy budget with $\epsilon_1=\epsilon_2=\epsilon/2$ by sequential composition (Theorem \[thm:seq\_compo\]). Aggregation and Estimation {#sec:calibratoin} -------------------------- This subsection corresponds to step in Figure \[fig:Diagram\]. Intuitively, the value mean of a certain key can be estimated by the ratio between the summation of all true values and the count of values regarding this key; however, the fake values affect both the summation and the count. In PrivKVM [@ye2019privkv], since the count of values includes the fake ones, the mean of fake values should be close to the true mean to guarantee the unbiasedness of estimation. Therefore, a large number of iterations are needed to make the fake values approach the true mean. In our scheme, however, the fake values have expected zero summation because they are assigned as $-1$ or $1$ with probability 0.5 respectively. Therefore, we can use the estimated frequency to approach the count of truly existing values, thus only one round is needed. **Aggregation.** After all users upload their outputs to the server, the server will count the number of $1$’s and $-1$’s that supports $k\in\mathcal{K}$ in output, denoted as $n_1$ and $n_2$ respectively (the subscript $k$ is omitted for brevity). Since the outputs of the proposed two mechanisms have different formats, the server computes $n_1 = Count(\mathbf{y}[k]=1)$ and $n_2 = Count(\mathbf{y}[k]=-1)$ in PCKV-UE, or computes $n_1 = Count(y^{\prime}=\langle k,1\rangle)$ and $n_2 = Count(y^{\prime}=\langle k,-1\rangle)$ in PCKV-GRR. Then, $n_1$ and $n_2$ will be calibrated to estimate the frequency and mean of key $k\in\mathcal{K}$. **Baseline Estimation Method.** For frequency estimation, we use the estimator in [@wang2018locally] for itemset data, which is shown to be unbiased when each user’s itemset size is no more than $\ell$. Since $n_1+n_2$ is the observed count of users that possess the key, we have the following equivalent *frequency estimator* $$\begin{aligned} \label{equ:hat_f} \hat{f}_k=\frac{(n_1+n_2)/n-b}{a-b}\cdot\ell\end{aligned}$$ For mean estimation, since our mechanisms generate the fake values as $-1$ or $1$ with probability 0.5 respectively (i.e., the expectation is zero), they have no contribution to the value summation statistically. Therefore, we can estimate the value mean by dividing the summation with the count of real keys. According to Randomized Response (RR) in Sec. \[sec:LDP mechanism\], the calibrated summation is $\frac{n_1-n(1-p)}{2p-1}-\frac{n_2-n(1-p)}{2p-1}=\frac{n_1-n_2}{2p-1}$. The count of real keys which are still reported as possessed can be approximated by $n\hat{f}_ka/\ell$ because the sampling rate is $1/\ell$ and real keys are reported as possessed with probability $a$. Therefore, the corresponding *mean estimator* is $$\begin{aligned} \label{equ:hat_m} \hat{m}_k=\frac{(n_1-n_2)/(2p-1)}{n\hat{f}_ka/\ell}=\frac{(n_1-n_2)(a-b)}{ a(2p-1)(n_1+n_2-nb)}\end{aligned}$$ The following theorem analyzes the expectation and variance of our estimators in (\[equ:hat\_f\]) and (\[equ:hat\_m\]) when each user has no more than $\ell$ key-value pairs (the same condition as in [@wang2018locally]). \[thm:estimation\] If the padding length $\ell\geqslant |\mathcal{S}_u|$ for all user $u\in\mathcal{U}$; then, for frequency and mean estimators in (\[equ:hat\_f\]) and (\[equ:hat\_m\]) of $k\in\mathcal{K}$, $\hat{f}_k$ is unbiased, i.e., $\mathbb{E}[\hat{f}_k]=f_k^{*}$, and their expectation and variance are $$\begin{aligned} \label{equ:Var[f_k]} &\text{Var}[\hat{f}_k]= \frac{\ell^2 b(1-b)}{n(a-b)^2} + \frac{\ell\cdot f_k^{*}(1-a-b)}{n(a-b)}\\ \label{equ:E[m_k]} &\mathbb{E}[\hat{m}_k]\approx m_k^{*}\left[1+\frac{ (1-b-\delta)b }{n\delta^2}\right]\\ \label{equ:Var[m_k]} &\text{Var}[\hat{m}_k]\lesssim\frac{b+\delta}{n\gamma^2} + \frac{b(1-b)-\delta}{n\delta^2} \cdot{m_k^{*}}^2\end{aligned}$$ where parameters $\delta$ and $\gamma$ are defined by $$\begin{aligned} \label{equ:delta} \delta=(a-b)f_k^{*}/\ell,\quad \gamma=a(2p-1)f_k^{*}/\ell\end{aligned}$$ The variance in (\[equ:Var\[m\_k\]\]) is an approximate upper bound and the approximation in (\[equ:E\[m\_k\]\]) and (\[equ:Var\[m\_k\]\]) is from Taylor expansions. See Appendix \[apx:thm:estimation\_UE\]. Note that Theorem \[thm:estimation\] works for both PCKV-UE and PCKV-GRR. **Pros and Cons of the Baseline Estimator.** The baseline estimation method estimates frequency and mean by (\[equ:hat\_f\]) and (\[equ:hat\_m\]) respectively. According to (\[equ:E\[m\_k\]\]) and (\[equ:Var\[m\_k\]\]), for non-zero constants $\delta$ and $\gamma$, when the user size $n\rightarrow+\infty$, we have $\mathbb{E}[\hat{m}_k]-m_k^{*}=\frac{(1-b-\delta)bm_k^{*}}{n\delta^2}\rightarrow 0$ (i.e., the bias of $\hat{m}_k$ is progressively approaching 0) and $\text{Var}[\hat{m}_k]\rightarrow 0$, which means $\hat{m}_k$ converges in probability to the true mean $m_k^{*}$. However, when $\frac{1}{n(f^{*}_k/\ell)^2}$ is not small, the large bias and large variance would make the estimated mean $\hat{m}_k$ far away from the true mean, even out of the bound $[-1,1]$. Similarly, if $\text{Var}[\hat{f}_k]$ in (\[equ:Var\[f\_k\]\]) is not very small, then for $f_k^{*}\rightarrow 0$ or $f_k^{*}\rightarrow 1$, the estimated frequency $\hat{f}_k$ may also be outside the bound $[0,1]$. Hence, these outliers need further correction to reduce the estimation error. Outputs of all users, domain of keys $\mathcal{K}$, perturbation probabilities $a,b,p$ and padding length $\ell$. Frequency and mean estimation $\hat{f}_k$ and $\hat{m}_k$ for all $k\in\mathcal{K}$. Count the number of supporting $1$’s and $-1$’s for key $k$ in outputs from all users, denoted as $n_1$ and $n_2$. Compute $\hat{f}_k$ by (\[equ:hat\_f\]) and correct it into $[1/n,1]$. Compute $\hat{n}_1$ and $\hat{n}_2$ by (\[equ:hat\_n\]), and correct them into $[0,n\hat{f}_k/\ell]$. Compute $\hat{m}_k$ by (\[equ:hat\_m\_new\]). Return $\hat{f}_k$ and $\hat{m}_k$, where $k\in\mathcal{K}$. \[alg:Estimation\] **Improved Estimation with Correction.** Since the value perturbation depends on the output of key perturbation, we first correct the result of frequency estimation. Considering the corrected frequency cannot be 0 (otherwise the mean estimation will be infinity), we clip the frequency values using the range $[1/n,1]$, i.e., set the outliers less than $1/n$ to $1/n$ and outliers larger than $1$ to $1$. For the mean estimation, denote the true counts of sampled key-value pair $x=\langle k,1\rangle$ and $x=\langle k,-1\rangle$ (the output of Algorithm \[alg:PS\]) of all users as $n_1^{*}$ and $n_2^{*}$ respectively (the subscript $k$ is omitted for brevity). Then we have the following lemma for the estimation of $n_1^{*}$ and $n_2^{*}$. \[lem:hat\_n\] The unbiased estimators of $n_1^{*}$ and $n_2^{*}$ are $$\begin{aligned} \label{equ:hat_n} \begin{bmatrix} \hat{n}_1\\ \hat{n}_2 \end{bmatrix}=A^{-1} \begin{bmatrix} n_1-nb/2\\ n_2-nb/2 \end{bmatrix}, \text{ where } A=\left[\begin{smallmatrix} ap-\frac{b}{2} & a(1-p)-\frac{b}{2}\\ a(1-p)-\frac{b}{2} & ap-\frac{b}{2} \end{smallmatrix}\right]\end{aligned}$$ See Appendix \[apx:lem:hat\_n\]. Note that Lemma \[lem:hat\_n\] works for both PCKV-UE and PCKV-GRR. According to (\[equ:hat\_n\]), we have $$\begin{aligned} \hat{n}_1-\hat{n}_2= \begin{bmatrix} 1 & -1 \end{bmatrix} A^{-1} \begin{bmatrix} n_1-nb/2\\ n_2-nb/2 \end{bmatrix} =\frac{n_1-n_2}{a(2p-1)}\end{aligned}$$ then $\hat{m}_k$ in (\[equ:hat\_m\]) can be represented by $\hat{n}_1-\hat{n}_2$ and $\hat{f}_k$ in (\[equ:hat\_f\]) $$\begin{aligned} \label{equ:hat_m_new} \hat{m}_k =\ell(\hat{n}_1-\hat{n}_2)/(n\hat{f}_k) \end{aligned}$$ which means $n_1^{*}+n_2^{*}$ (the supporting number of $1$ and $-1$ for key $k\in\mathcal{K}$) is estimated by $n\hat{f}_k/\ell$. Therefore, $\hat{n}_1$ and $\hat{n}_2$ should be bounded by $[0,n\hat{f}_k/\ell]$. The aggregation and estimation mechanism (with correction) is shown in Algorithm \[alg:Estimation\], where the difference between PCKV-UE and PCKV-GRR is only on the aggregation step, which is caused by the different types of output (one is a vector, another is a key-value pair). Optimized Privacy Budget Allocation {#sec:allocation} ----------------------------------- In this section, we discuss how to optimally allocate budgets $\epsilon_1$ and $\epsilon_2$ given the total privacy budget $\epsilon$, which corresponds to step in Figure \[fig:Diagram\]. The budget composition (Theorem \[thm:LDP\_UE\] and Theorem \[thm:LDP\_GRR\]) provides the relationship among $\epsilon$, $\epsilon_1$, and $\epsilon_2$. Intuitively, when the total privacy budget $\epsilon$ is given, we can find the optimal $\epsilon_1$ and $\epsilon_2$ that satisfy the budget composition by solving an optimization problem of minimizing the combined Mean Square Error (MSE) of frequency and mean estimations, i.e., $\alpha\cdot \text{MSE}_{\hat{f}_k}+\beta\cdot \text{MSE}_{\hat{m}_k}$. However, from Theorem \[thm:estimation\], $\text{Var}[\hat{f}_k]$ and $\text{Var}[\hat{m}_k]$ depend on $f_k^{*}$ and $m_k^{*}$, whose true values or even the approximate values are unknown in the budget allocation stage (before any perturbation). Therefore, in the following, we simplify this optimization problem to obtain a practical budget allocation solution with closed-form. Note that a larger $\epsilon_1$ can benefit both frequency and mean estimations, but it restricts $\epsilon_2$ (which affects mean estimation) due to limited $\epsilon$. **Problem Simplification of Budget Allocation.** In this paper, we use Mean Square Error (MSE) to evaluate utility mechanisms, i.e., the less MSE the better utility. Note that the MSE of an estimator $\hat{\theta}$ can be calculated by the summation of variance and the square of its bias $$\begin{aligned} \label{equ:MSE} \text{MSE}_{\hat{\theta}}=\text{Var}[\hat{\theta}]+\text{Bias}^2=\text{Var}[\hat{\theta}]+(\mathbb{E}[\hat{\theta}]-\theta)^2\end{aligned}$$ When MSE is relatively large, the estimators will be corrected by the improved estimation in Algorithm \[alg:Estimation\]. Therefore, we mainly consider minimizing MSE when it is relatively small, i.e., $(2p-1)$ and $(a-b)$ are not very small, and $n$ (the number of users) is very large. Since $f_k^{*}\ll 1$ for most cases in real-world data, we have $\delta=(a-b)f_k^{*}/\ell\ll 1$. Denote $$\begin{aligned} \label{equ:gh} \mu=\frac{\ell^2}{n{f_k^{*}}^2},\quad g=\frac{b}{a^2(2p-1)^2},\quad h = \frac{(1-b)b}{(a-b)^2}\end{aligned}$$ The MSEs in Theorem \[thm:estimation\] can be approximated by $$\begin{aligned} \text{MSE}_{\hat{f}_k}&=\text{Var}[\hat{f}_k]\approx\ell^2\cdot h/n\\ \label{equ:MSE_m} \text{MSE}_{\hat{m}_k}&\approx\mu[g+(\mu h+1)h{m_k^{*}}^2] \approx\mu(g+h\cdot{m_k^{*}}^2)\end{aligned}$$ where $\mu\ll 1$ with a large $n$. Note that $\text{MSE}_{\hat{m}_k}$ dominates $\text{MSE}_{\hat{f}_k}$ because $\frac{\ell^2}{n}/\mu={f_k^{*}}^2\ll 1$. It is caused by the distinct sample size of the two estimations, i.e., frequency is estimated from all users (with user size $n$), while the value mean is estimated from the users who possess a certain key (with user size $nf_k^{*}$). Therefore, our objective function $\alpha\cdot \text{MSE}_{\hat{f}_k}+\beta\cdot \text{MSE}_{\hat{m}_k}$ mainly depends on $\text{MSE}_{\hat{m}_k}$ when $\alpha$ and $\beta$ are in the same magnitude. Motivated by this observation, we focus on minimizing $\text{MSE}_{\hat{m}_k}$ to obtain the optimized budget allocation. Note that $\text{MSE}_{\hat{f}_k}$ only depends on $\epsilon_1$ (the more $\epsilon_1$ the less $\text{MSE}_{\hat{f}_k}$), while $\text{MSE}_{\hat{m}_k}$ depends on both $\epsilon_1$ and $\epsilon_2$. However, if $\epsilon_1$ approaches to the maximum, which corresponds to the minimum $\text{MSE}_{\hat{f}_k}$, then $\epsilon_2=0$ and $\text{MSE}_{\hat{m}_k}\rightarrow \infty$. In the following, we discuss the optimized privacy budget allocation with minimum $\text{MSE}_{\hat{m}_k}$ in PCKV-UE and PCKV-GRR. **Budget Allocation of PCKV-UE.** In UE-based mechanisms, the Optimized Unary Encoding (OUE) [@wang2017locally] was shown to have the minimum MSE of frequency estimation under the same privacy budget. Accordingly, the OUE-based perturbation probabilities for key-value perturbation are $$\begin{aligned} \label{equ:abp_UE} a=1/2,\quad b=1/(e^{\epsilon_1}+1),\quad p=e^{\epsilon_2}/(e^{\epsilon_2}+1)\end{aligned}$$ where the values of $a$ and $b$ correspond to the minimum $\text{MSE}_{\hat{f}_k}$ under a given $\epsilon_1$ (budget for key perturbation). Furthermore, by minimizing $\text{MSE}_{\hat{m}_k}$, we have the following optimized budget allocation of PCKV-UE. \[lem:budget\_UE\] For a total privacy budget $\epsilon$, the optimized budget allocation for key and value perturbations can be approximated by $$\begin{aligned} \label{equ:budget_UE} \epsilon_1=\ln[(e^{\epsilon}+1)/2],\quad \epsilon_2=\epsilon\end{aligned}$$ See Appendix \[apx:lem:budget\_UE\]. **Interpretation of Lemma \[lem:budget\_UE\].** According to the budget allocation of PCKV-UE in , $\epsilon_1$ is an increasing function of $\theta$, while $\epsilon_2$ and the summation $\epsilon_1+\epsilon_2=\ln\frac{\theta}{2\theta e^{-\epsilon}-1}$ are decreasing functions of $\theta$. From , $\epsilon_1$ and $\epsilon_2$ are optimally allocated at $\theta=\frac{e^{\epsilon}+1}{2}$ (the minimum value), which corresponds to the maximum summation $\epsilon_1+\epsilon_2$. Moreover, under the optimized budget allocation, the two values in the max operation in equal to each other, i.e., $\epsilon_2=\epsilon_1+\ln[2/(1+e^{-\epsilon_2})]=\epsilon$, which indicates that the budgets are fully allocated. ![Comparison of $g$ and $h$ under three budget allocation methods for PCKV-UE, where $\text{MSE}_{\hat{m}_k}\approx\mu(g+h\cdot{m_k^{*}}^2)$.[]{data-label="fig:allocation"}](fig_allocation.pdf){width="3.3in"} **Comparison with Other Allocation Methods.** In order to show the advantage of our *optimized allocation* in (\[equ:budget\_UE\]), we compare it with two alternative methods. The first one is *naive allocation* with $\epsilon_1=\epsilon_2=\epsilon/2$ by sequential composition (which is used in PrivKVM). The second one is *non-optimized allocation* with $$\begin{aligned} \label{equ:non-optimized} \epsilon_1=\ln[(e^\epsilon+e^{\epsilon/2})/2],\quad \epsilon_2=\epsilon/2\end{aligned}$$ which sets $\epsilon_2$ as $\epsilon/2$ and computes $\epsilon_1$ by our tight budget composition (Theorem \[thm:LDP\_UE\]). Considering $\text{MSE}_{\hat{m}_k}\approx\mu(g+h\cdot{m_k^{*}}^2)$ in , we compare parameters $g$ and $h$ (with respect to $\epsilon$) under above three budget allocation methods, shown in Figure \[fig:allocation\]. We can observe that the optimized allocation has a much smaller $g$ than the other two, though a little bit larger $h$ than the non-optimized one, which is caused by the property that $h$ is a monotonically decreasing function of $\epsilon_1$, while $\epsilon_1$ and $\epsilon_2$ restrict each other. Note that in our optimized allocation, the decrement of $g$ dominates the increment of $h$. Thus, $\text{MSE}_{\hat{m}_k}$ in (\[equ:MSE\_m\]) will be greatly reduced since ${m_k^{*}}^2\leqslant 1$. **Budget Allocation of PCKV-GRR.** According to the budget composition (Theorem \[thm:LDP\_GRR\]) of PCKV-GRR, a larger padding length $\ell$ will further improve the privacy-utility tradeoff of key-value perturbation. Thus, given fixed total budget, the allocated budget for key (or value) perturbation can be larger (i.e., less noise will be added) under a larger $\ell$. The following lemma shows the optimized budget allocation (related to $\ell$) of PCKV-GRR with minimum $\text{MSE}_{\hat{m}_k}$. \[lem:budget\_GRR\] For a total privacy budget $\epsilon$, the optimized budget allocation for key and value perturbation can be approximated by $$\begin{aligned} \label{equ:budget_GRR} \epsilon_1=\ln\left[\ell\cdot(e^\epsilon-1)/2+1\right],\quad \epsilon_2=\ln\left[\ell\cdot(e^\epsilon-1)+1\right]\end{aligned}$$ See Appendix \[apx:lem:budget\_GRR\]. According to (\[equ:abp\_GRR\]) and (\[equ:budget\_GRR\]), with a given total budget $\epsilon$, the perturbation probabilities in PCKV-GRR are $$\begin{aligned} \label{equ:abp_GRR_opt} a=\frac{\ell(e^{\epsilon}-1)+2}{\ell(e^{\epsilon}-1)+2d^{\prime}},~ b=\frac{1-a}{d^{\prime}-1},~ p=\frac{\ell(e^{\epsilon}-1)+1}{\ell(e^{\epsilon}-1)+2}\end{aligned}$$ where $d^{\prime}=d+\ell$. Note that when $\ell=1$, the optimized budget allocation in reduces to the case of PCKV-UE in . ![Diagram of our optimized protocols (different types of arrows represent perturbations with different probabilities).[]{data-label="fig:optimized"}](optimized.pdf){width="3.4in"} **Interpretation of the Optimized Protocols.** Under the optimized budget allocation (Lemma \[lem:budget\_UE\] and Lemma \[lem:budget\_GRR\]), the perturbation probabilities of proposed protocols are shown in Figure \[fig:optimized\]. In optimized PCKV-UE, for two different input vectors $\mathbf{x}_1$ and $\mathbf{x}_2$ (encoded from the sampled key-value pairs), no matter they differ in one element (i.e., the sampled ones have the same key but different values) or differ in two elements (i.e., the sampled ones have different keys), the upper bound of the probability ratio of outputting the same vector $\mathbf{y}$ is the same, i.e., [$\frac{\textcircled{\scriptsize1}}{\textcircled{\scriptsize 2}}=\frac{\textcircled{\scriptsize 1}}{\textcircled{\scriptsize 5}}\cdot\frac{\textcircled{\scriptsize 4}}{\textcircled{\scriptsize 3}}=$]{} $e^{\epsilon}$ in Figure \[fig:optimized\]. In optimized PCKV-GRR, two of three different perturbation probabilities in Algorithm \[alg:PCKV-GRR\] equal with each other, i.e., $a(1-p)=b\cdot 0.5$ in the optimized solution. Also, the optimized PCKV-GRR can be regarded as the equivalent version of general GRR with doubled domain size (each key can have two different values), which can provide good utility on estimating the counts of $\langle k,1\rangle$ and $\langle k,-1\rangle$, say $n_{k1}$ and $n_{k2}$, where the mean of key $k$ can be estimated by $\frac{n_{k1} - n_{k2}}{n_{k1} + n_{k2}}$. From the previous analysis, PCKV-GRR can get additional benefit from sampling, thus it will outperform PCKV-UE for a large $\ell$. On the other hand, the performance of PCKV-UE is independent of the domain size $d$, thus it will have less MSE than PCKV-GRR when $d$ is very large. Therefore, the two mechanisms are suitable for different cases. By comparing parameters $g$ and $h$ in of PCKV-UE and PCKV-GRR respectively, for a smaller $\text{MSE}_{\hat{f}_k}$ (i.e., a smaller $h$), if $2(d-1)>\ell(4\ell-1)(e^\epsilon+1)$, then PCKV-UE is better; otherwise, PCKV-GRR is better. For a smaller $\text{MSE}_{\hat{m}_k}$ (i.e., a smaller $g$ approximately), if $2d>\ell\left(\frac{4\ell(e^\epsilon+1)}{e^\epsilon+3}-1\right)(e^\epsilon+1)$, then PCKV-UE is better; otherwise, PCKV-GRR is better. These can be observed in simulation results (Sec. \[sec:evaluation\]). Evaluation {#sec:evaluation} ========== In this section, we evaluate the performance of our proposed mechanisms (PCKV-UE and PCKV-GRR) and compare them with the existing mechanisms (PrivKVM [@ye2019privkv] and KVUE [@sun2019conditional]). We note that although KVUE [@sun2019conditional] is not formally published, we still implemented it with our best effort and included it for comparison purposes. **Mechanisms for Comparison.** In PrivKVM [@ye2019privkv], the number of iterations is set as $c=1$ because we observe that PrivKVM with a large number of iterations $c$ will have bad utility, which is caused by the small budget $\frac{\epsilon}{2c}$ and thus large variance of value perturbation in the last iteration (even though the result is theoretically unbiased). However, implementing PrivKVM with virtual iterations to predict the mean estimation of remaining iterations can avoid budget split [@ye2019privkv]. Thus, we also evaluate PrivKVM with one *real* iteration and five *virtual* iterations (`1r5v`). In [@sun2019conditional], multiple mechanisms are proposed to improve the performance of PrivKVM, where the most promising one is KVUE (which uses the same sampling protocol as in PrivKVM). Note that the original KVUE does not have corrections for mean estimation. For a fair comparison with PrivKVM, PCKV-UE, and PCKV-GRR (outliers are corrected in these mechanisms), we use the similar correction strategy used in PrivKVM for KVUE. **Datasets.** In this paper, we evaluate two existing mechanisms (PrivKVM [@ye2019privkv] and KVUE [@sun2019conditional]) and our mechanisms (PCKV-UE and PCKV-GRR) by synthetic datasets and real-world datasets. In synthetic datasets, the number of users is $n=10^6$, and the domain size is $d=100$, where each user only has one key-value pair (i.e., $\ell=1$), and both the possessed key of each user and the value mean of keys satisfy Uniform (or Gaussian) distribution. The Gaussian distribution is generated with $\mu=0,\sigma_\text{key}=50,\sigma_\text{mean}=1$, where samples outside the domain ($\mathcal{K}$ or $\mathcal{V}=[-1,1]$) are discarded. In real-world datasets, each user may have multiple key-value pairs, i.e., $\ell>1$ (how the selection of $\ell$ affects the estimation accuracy has been discussed in Sec. \[sec:sampling\]). Table \[tab:data\] summarizes the parameters of four real-world rating datasets (obtained from public data sources) with different domain sizes and data distributions. The item-rating corresponds to key-value, and all ratings are linearly normalized into $[-1,1]$. **Datasets** **\# Ratings** **\# Users** **\# Keys** **Selected $\ell$** ------------------------- ---------------- -------------- ------------- --------------------- E-commerce [@ecommerce] 23,486 23,486 1,206 1 Clothing [@clothing] 192,544 105,508 5,850 2 Amazon [@amazon] 2,023,070 1,210,271 249,274 2 Movie [@movie] 20,000,263 138,493 26,744 100 : Real-World Datasets \[tab:data\] **Evaluation Metric.** We evaluate both the frequency and mean estimation by the *averaged* Mean Square Error (MSE) among all keys or a portion of keys $$\begin{aligned} \text{MSE}_\text{freq} = \frac{1}{|\mathcal{X}|}\sum_{i\in \mathcal{X}}(\hat{f}_i-f_i^{*})^2,\quad \text{MSE}_\text{mean} = \frac{1}{|\mathcal{X}|}\sum_{i\in \mathcal{X}}(\hat{m}_i-m_i^{*})^2\end{aligned}$$ where $f_i^{*}$ and $m_i^{*}$ (resp. $\hat{f}_i$ and $\hat{m}_i$) are the true (resp. estimated) frequency and mean, and $\mathcal{X}$ is a subset of the domain $\mathcal{K}$ (the default $\mathcal{X}$ is $\mathcal{K}$). We also consider $\mathcal{X}$ as the set of top $N$ frequent keys (such as top 20 or top 50) because we usually only care about the estimation results of frequent keys. Also, infrequent keys do not have enough samples to obtain the accurate estimation of value mean. All MSE results are averaged with five repeats. Synthetic Data -------------- ![MSEs of synthetic data under two distributions, where the left is MSE of frequency estimation and the right is MSE of mean estimation. The theoretical MSEs (dashed lines) of PCKV-UE and PCKV-GRR are calculated by Theorem \[thm:estimation\]. When $\epsilon$ is small, the gap between empirical and theoretical results is caused by the correction in the improved estimation (Algorithm \[alg:Estimation\]), while our theoretical MSE is analyzed for the baseline estimation without correction.[]{data-label="fig:synthetic"}](fig1_UU.pdf "fig:"){width="3.2in"} (a) Uniform distribution (MSE is averaged of all keys) ![MSEs of synthetic data under two distributions, where the left is MSE of frequency estimation and the right is MSE of mean estimation. The theoretical MSEs (dashed lines) of PCKV-UE and PCKV-GRR are calculated by Theorem \[thm:estimation\]. When $\epsilon$ is small, the gap between empirical and theoretical results is caused by the correction in the improved estimation (Algorithm \[alg:Estimation\]), while our theoretical MSE is analyzed for the baseline estimation without correction.[]{data-label="fig:synthetic"}](fig1_GG.pdf "fig:"){width="3.2in"} (b) Gaussian distribution (MSE is averaged of all keys) ![MSEs of synthetic data under two distributions, where the left is MSE of frequency estimation and the right is MSE of mean estimation. The theoretical MSEs (dashed lines) of PCKV-UE and PCKV-GRR are calculated by Theorem \[thm:estimation\]. When $\epsilon$ is small, the gap between empirical and theoretical results is caused by the correction in the improved estimation (Algorithm \[alg:Estimation\]), while our theoretical MSE is analyzed for the baseline estimation without correction.[]{data-label="fig:synthetic"}](fig1_GG_top20.pdf "fig:"){width="3.2in"} (c) Gaussian distribution (MSE is averaged of top 20 frequent keys)\ **Overall Results.** The averaged MSEs of frequency and mean estimations are shown in Figure \[fig:synthetic\] (with domain size 100), where the MSE is averaged by all keys (Figure \[fig:synthetic\]a and \[fig:synthetic\]b) or the top 20 frequent keys (Figure \[fig:synthetic\]c). For frequency estimation, PrivKVM ($c=1$) and PrivKVM (`1r5v`) have the same MSE since the frequency is estimated by the first iteration. The proposed mechanisms (PCKV-UE and PCKV-GRR) have much less $\text{MSE}_{\hat{f}_k}$. For mean estimation, PrivKVM (`1r5v`) predicts the mean estimation of remaining iterations without splitting the budget, which improves the accuracy of PrivKVM ($c=1$) under larger $\epsilon$. The $\text{MSE}_{\hat{m}_k}$ of PrivKVM ($c=1$) does not decrease any more after $\epsilon=0.5$ since PrivKVM ($c=1$) always generates fake values as $v=0$. The PrivKVM (`1r5v`) with virtual iterations improves PrivKVM ($c=1$), but the estimation error is larger than other mechanisms. The $\text{MSE}_{\hat{m}_k}$ in PCKV-UE and PCKV-GRR is much smaller than other ones when $\epsilon$ is relatively large (e.g., $\epsilon>2$), thanks to the high accuracy of frequency estimation in this case. Also, the small gap between the theoretical and empirical results validate the correctness of our theoretical error analysis in Theorem \[thm:estimation\]. **Influence of Data Distribution.** By comparing the results of PCKV-UE and PCKV-GRR under different distributions in Figure \[fig:synthetic\], $\text{MSE}_{\hat{m}_k}$ of all keys in Gaussian distribution is larger than in Uniform distribution because the frequency of some keys is very small in Gaussian distribution. However, $\text{MSE}_{\hat{m}_k}$ of the top 20 frequent keys is much smaller because the frequent keys have higher frequencies. Note that the distribution has little influence on $\text{MSE}_{\hat{f}_k}$ in these mechanisms because the user size used in frequency estimation is always $n$, while the user size used in value mean estimation of $k\in\mathcal{K}$ is $nf_k^{*}$. {width="3.2in"} (a) Gaussian distribution (with $\epsilon=1$) {width="3.2in"} (b) Gaussian distribution (with $\epsilon=5$) \[fig:domain\_size\] ![Precision of top frequent keys estimation.[]{data-label="fig:success_top"}](fig4.pdf){width="3.3in"} **Influence of Domain Size.** The MSEs of frequency and mean estimation with respect to different domain size $d$ (where $\epsilon=1$ or $5$) are shown in Figure \[fig:domain\_size\]. We can observe that $\text{MSE}_{\hat{f}_k}$ is proportional to the domain size $d$ in PrivKVM, KVUE, and PCKV-GRR. Note that the reasons for the same observation are different. For PrivKVM and KVUE, the perturbation probabilities are independent of domain size, but the large domain size would make sampling protocol (randomly pick one index from the domain of keys) less possible to obtain the useful information. For PCKV-GRR, the large domain size does not influence the Padding-and-Sampling protocol, but it will decrease the perturbation probabilities $a$ and $b$ in and enlarge the estimation error. However, the large domain size does not affect the frequency estimation of PCKV-UE. For the result of mean estimation, we have similar observations. Note that $\text{MSE}_{\hat{m}_k}$ is not proportional to the domain size because the correction of mean estimation can alleviate the error. For PCKV-UE, the increasing $\text{MSE}_{\hat{m}_k}$ when $d<100$ is caused by the decreased true frequency when $d$ is increasing (note that $\sigma_{\text{key}}=50$ and samples outside the domain are discarded when generating the data). The prediction of PrivKVM (`1r5v`) with virtual iterations does not work well for a large domain size under small $\epsilon$. **Accuracy of Top Frequent Keys Selection.** To evaluate the success of the top frequent keys selection, we calculate the precision (i.e., the proportion of correct selections over all predicted top frequent keys) for different mechanisms, shown in Figure \[fig:success\_top\] (precision in this case is the same as recall). For the top 10 frequent keys under $\epsilon=3$, the precision of PCKV-UE is over $60\%$ even for a large $d$ (i.e., misestimation is at most 4 over the top 10 frequent keys). However, PrivKVM and KVUE incorrectly select almost all top 10 frequent keys when $d=2000$. For the top 20 frequent keys under $\epsilon=5$, PCKV-UE and PCKV-GRR can correctly estimate $95\%$ and $85\%$ respectively even for $d=2000$. **Comparison of Allocation Methods.** In our PCKV framework, the privacy-utility tradeoff is improved by both the tighter bound in budget composition (Theorem \[thm:LDP\_UE\] and Theorem \[thm:LDP\_GRR\]) and the optimized budget allocation (Lemma \[lem:budget\_UE\] and Lemma \[lem:budget\_GRR\]). In order to show the benefit of our optimized allocation, we compare the results of optimized method with two alternative allocation ones in Figure \[fig:PCKV\_allocation\], where the corresponding theoretical comparison has been discussed in Sec. \[sec:allocation\]. The naive allocation is $\epsilon_1=\epsilon_2=\epsilon/2$, and the non-optimized allocation with tighter bound is represented in (\[equ:non-optimized\]), which also works for PCKV-GRR when $\ell=1$. We can observe that for both PCKV-UE and PCKV-GRR, the allocation methods with tighter bound (non-optimized and optimized) outperform the naive one in the estimation accuracy of mean and frequency. Even though $\text{MSE}_{\hat{f}_k}$ in optimized allocation is slightly greater than the non-optimized one, it has much less $\text{MSE}_{\hat{m}_k}$. Note that the magnitude of $\text{MSE}_{\hat{f}_k}$ and $\text{MSE}_{\hat{m}_k}$ are different. For example, when $\epsilon=1$, the gap of $\text{MSE}_{\hat{f}_k}$ between non-optimized and optimized allocation in PCKV-UE is $4\times 10^{-6}$, but the gap of $\text{MSE}_{\hat{m}_k}$ between them is 0.08. These observations validate our theoretical analyses and discussions in Sec. \[sec:allocation\]. ![Comparison of three allocation methods in PCKV.[]{data-label="fig:PCKV_allocation"}](fig3_UU.pdf){width="3.3in"} \ \ \ Real-World Data --------------- The results of four types of real-world rating datasets are shown in Figure \[fig:real-world\], where the MSEs are averaged over the top 50 frequent keys. The parameters (number of ratings, users, and keys) are listed in Table \[tab:data\], where we select reasonable $\ell$ for evaluation to compare with existing mechanisms with naive sampling protocol (the advanced strategy of selecting an optimized $\ell$ is discussed in Sec. \[sec:sampling\]). Under the large domain size in real-world datasets, PrivKVM (`1r5v`) with virtual iterations does not work well, thus we only show the results of PrivKVM ($c=1$). Compared with the results of E-commerce dataset, the MSEs of Clothing dataset do not change very much because all algorithms can get benefits from the large $n$, which compensates the impacts from the larger $d$ or the larger $\ell$. Compared with the results of PCKV-UE in Clothing dataset, $\text{MSE}_{\hat{f}_k}$ in Amazon dataset is smaller (due to the large $n$) but $\text{MSE}_{\hat{m}_k}$ is larger (due to the small true frequencies). In the first three datasets, PCKV-UE has the best performance because $\ell$ is small and the large domain size does not impact its performance directly. In the Movie dataset, since PCKV-GRR can benefit more from a large $\ell$, it outperforms PCKV-UE in both frequency and mean estimation. Note that both PCKV-UE and PCKV-GRR have less MSEs compared with other mechanisms in Movie dataset. Since PCKV-UE and PCKV-GRR are suitable for different cases, in practice we can select PCKV-UE or PCKV-GRR by comparing the theoretical estimation error under specified parameters (i.e., $\epsilon,d$ and $\ell$) as discussed in Sec. \[sec:allocation\]. Conclusion ========== In this paper, a new framework called PCKV (with two mechanisms PCKV-UE and PCKV-GRR) is proposed to privately collect key-value data under LDP with higher accuracy of frequency and value mean estimation. We design a correlated key and value perturbation mechanism that leads to a tighter budget composition than sequential composition of LDP. We further improve the privacy-utility tradeoff via a near-optimal budget allocation method. Besides the tight budget composition and optimized budget allocation, the proposed sampling protocol and mean estimators in our framework also improve the accuracy of estimation than existing protocols. Finally, we demonstrate the advantage of the proposed scheme on both synthetic and real-world datasets. For future work, we will study how to choose an optimized $\ell$ in the Padding-and-Sampling protocol and extend the correlated perturbation and tight composition analysis to consider more general forms of correlation and other hybrid data types. Acknowledgments {#acknowledgments .unnumbered} =============== Yueqiang Cheng is the corresponding author (main work was done when the first author was a summer intern at Baidu X-Lab). The authors would like to thank the anonymous reviewers and the shepherd Mathias L[é]{}cuyer for their valuable comments and suggestions. This research was partially sponsored by NSF grants CNS-1731164 and CNS-1618932, JSPS grant KAKENHI-19K20269, AFOSR grant FA9550-12-1-0240, and NIH grant R01GM118609. Proof of Theorem \[thm:LDP\_UE\] {#apx:thm:LDP_UE} ================================ For a key-value set $\mathcal{S}$, denote the key-value pairs (raw data) are $\langle i,v_i^{*}\rangle$ for all $i\in\mathcal{S}$, where $v_i^{*}\in[-1,1]$. Note that $i\in\mathcal{S}$ means a key-value pair $\langle i,\cdot\rangle\in\mathcal{S}$. Denote the sampled key-value pair by Padding-and-Sampling in Algorithm \[alg:PS\] as $x=\langle k,v\rangle$, where $v\in\{1,-1\}$ (the discretized value). According to Line-5 in Algorithm \[alg:PS\], we have $v_k^{*}=0$ for $k\in\{d+1,\cdots,d^{\prime}\}$, where $d^{\prime}=d+\ell$. For vector $\mathbf{x}$ in PCKV-UE, only the $k$-th element is $v$ ($1$ or $-1$) while others are 0s. Then, the probability of outputting a vector $\mathbf{y}$ is $$\begin{aligned} \Pr(\mathbf{y}|\mathcal{S},k) &=\Pr(\mathbf{y}[k]|v_k^{*})\prod_{i\in\mathcal{K}^{\prime}\backslash k}\Pr(\mathbf{y}[i]|\mathbf{x}[i]=0)\\ &=\frac{\Pr(\mathbf{y}[k]|v_k^{*})}{\Pr(\mathbf{y}[k]|\mathbf{x}[k]=0)}\cdot\prod_{i\in\mathcal{K}^{\prime}}\Pr(\mathbf{y}[i]|\mathbf{x}[i]=0)\end{aligned}$$ According to Figure \[fig:perturbation\], the perturbation probabilities of the $k$-th element from the raw value can be represented as $$\begin{aligned} \Pr(\mathbf{y}[k]|v_k^{*})= \begin{cases} \frac{1+(2p-1)v_k^{*}}{2}\cdot a,& \text{if }\mathbf{y}[k]=1\\ \frac{1-(2p-1)v_k^{*}}{2}\cdot a,& \text{if }\mathbf{y}[k]=-1\\ 1-a,& \text{if }\mathbf{y}[k]=0\\ \end{cases}\end{aligned}$$ where $v_k^{*}\in[-1,1]$. For convenience, denote $$\begin{aligned} \Psi(\mathbf{y},k) =\frac{\Pr(\mathbf{y}[k]|v_k^{*})}{\Pr(\mathbf{y}[k]|\mathbf{x}[k]=0)},~~ \Phi(\mathbf{y}) = \prod_{i\in\mathcal{K}^{\prime}}\Pr(\mathbf{y}[i]|\mathbf{x}[i]=0)\end{aligned}$$ then we have $\Pr(\mathbf{y}|\mathcal{S},k)=\Psi(\mathbf{y},k)\cdot\Phi(\mathbf{y})$ and $$\begin{aligned} \Psi(\mathbf{y},k)= \begin{cases} (1+(2p-1)v_k^{*})\cdot\frac{a}{b},& \text{if }\mathbf{y}[k]=1\\ (1-(2p-1)v_k^{*})\cdot\frac{a}{b},& \text{if }\mathbf{y}[k]=-1\\ \frac{1-a}{1-b},& \text{if }\mathbf{y}[k]=0\\ \end{cases}\end{aligned}$$ where $a,p\in[\frac{1}{2},1)$ and $b\in(0,\frac{1}{2}]$ (in Algorithm \[alg:PCKV-UE\]). **Case 1.** For $k\in\{1,2,\cdots,d\}$, we have $v_k^{*}\in[-1,1]$ and $$\begin{aligned} \frac{1-a}{1-b}\leqslant\frac{2pa}{b},\quad \frac{2(1-p)a}{b}\leqslant (1\pm(2p-1)v_k^{*})\cdot\frac{a}{b}\leqslant \frac{2pa}{b}\end{aligned}$$ then the upper bound and lower bound of $\Psi(\mathbf{y},k)$ are $$\begin{aligned} \Psi_\text{upper}=\frac{2pa}{b},\quad \Psi_\text{lower}=\min\left\{\frac{1-a}{1-b},\frac{2(1-p)a}{b}\right\}\end{aligned}$$ **Case 2.** For $k\in\{d+1,\cdots,d^{\prime}\}$, we have $v_k^{*}=0$, then the upper bound and lower bound of $\Psi(\mathbf{y},k)$ are $$\begin{aligned} \Psi_\text{upper}^{\prime}=\frac{a}{b},\quad \Psi_\text{lower}^{\prime}=\frac{1-a}{1-b}\end{aligned}$$ Note that $\Psi_\text{lower}\leqslant\Psi_\text{lower}^{\prime}\leqslant\Psi_\text{upper}^{\prime}\leqslant\Psi_\text{upper}$. Then, the probability of perturbing $\mathcal{S}$ into $\mathbf{y}$ is bounded by $$\begin{aligned} &\Pr(\mathbf{y}|\mathcal{S}) = \eta\sum_{k\in\mathcal{S}}\frac{\Pr(\mathbf{y}|\mathcal{S},k)}{|\mathcal{S}|} + (1-\eta)\sum_{k=d+1}^{d^{\prime}}\frac{\Pr(\mathbf{y}|\mathcal{S},k)}{\ell}\\ &=\Phi(\mathbf{y})\left[\frac{\eta}{|\mathcal{S}|}\sum_{k\in\mathcal{S}}\Psi(\mathbf{y},k)+\frac{1-\eta}{\ell}\sum_{k=d+1}^{d^{\prime}}\Psi(\mathbf{y},k)\right]\\ &\leqslant \Phi(\mathbf{y})\left[\frac{\eta}{|\mathcal{S}|}\cdot |\mathcal{S}|\Psi_\text{upper} +\frac{1-\eta}{\ell}\cdot \ell\Psi_\text{upper}^{\prime}\right] \leqslant\Phi(\mathbf{y})\cdot\Psi_\text{upper}\end{aligned}$$ where the last inequality holds since $\eta=\frac{|\mathcal{S}|}{\max\{|\mathcal{S}|,\ell\}}\in(0,1]$ and $\Psi_\text{upper}^{\prime}\leqslant\Psi_\text{upper}$. Similarly, $\Pr(\mathbf{y}|\mathcal{S}) \geqslant \Phi(\mathbf{y})\cdot\Psi_\text{lower}$ holds. Then, for two different key-value sets $\mathcal{S}_1$ and $\mathcal{S}_2$, we have $$\begin{aligned} &\frac{\Pr(\mathbf{y}|\mathcal{S}_1)}{\Pr(\mathbf{y}|\mathcal{S}_2)} \leqslant\frac{\Phi(\mathbf{y})\cdot\Psi_\text{upper}}{\Phi(\mathbf{y})\cdot\Psi_\text{lower}} =\frac{\Psi_\text{upper}}{\Psi_\text{lower}} =\frac{2pa/b}{\min\left\{\frac{1-a}{1-b},\frac{2(1-p)a}{b}\right\}}\\ &=\max\left\{2p\cdot\frac{a(1-b)}{b(1-a)},\frac{p}{1-p}\right\} = \max\left\{\frac{2e^{\epsilon_1}}{1+e^{-\epsilon_2}},e^{\epsilon_2}\right\}=e^{\epsilon}\end{aligned}$$ where $\epsilon$ is defined in (\[equ:epsilon\_UE\]). Proof of Theorem \[thm:LDP\_GRR\] {#apx:thm:LDP_GRR} ================================= In PCKV-GRR, for an input $\mathcal{S}$ with pairs $\langle i,v_i^{*}\rangle$ for all $i\in\mathcal{S}$ and an output $y^{\prime}=\langle k^{\prime},v^{\prime}\rangle$ , denote the sampled pair as $x=\langle k,v\rangle$. When the sampled key is $k$, the probability of outputting a pair $y^{\prime}=\langle k^{\prime},v^{\prime}\rangle$ is $$\begin{aligned} \Pr(y^{\prime}|\mathcal{S},k)= \begin{cases} \frac{1+(2p-1)v_k^{*}}{2}\cdot a,& \text{if } k^{\prime}=k, v^{\prime}=1\\ \frac{1-(2p-1)v_k^{*}}{2}\cdot a,& \text{if } k^{\prime}=k, v^{\prime}=-1\\ b/2,& \text{if } k^{\prime}\neq k \end{cases}\end{aligned}$$ where $v_k^{*}=0$ for $k\in\{d+1,\cdots,d^{\prime}\}$. **Case 1.** If $k^{\prime}\in\mathcal{S}$, then $$\begin{aligned} & \Pr(y^{\prime}|\mathcal{S})=\eta\sum_{k\in\mathcal{S}}\frac{\Pr(y^{\prime}|\mathcal{S},k)}{|\mathcal{S}|} + (1-\eta)\sum_{k=d+1}^{d^{\prime}}\frac{\Pr(y^{\prime}|\mathcal{S},k)}{\ell}\\ &= \frac{\eta}{|\mathcal{S}|}\left[a\cdot\frac{1+(2p-1)v_{k^{\prime}}^{*}v^{\prime}}{2}+(|\mathcal{S}|-1)\frac{b}{2}\right]+(1-\eta)\frac{b}{2}\end{aligned}$$ Considering $v_{k^{\prime}}^{*}\in[-1,1]$ and $v^{\prime}\in\{1,-1\}$, we have $$\begin{aligned} \label{equ:case1_upper} \Pr(y^{\prime}|\mathcal{S})\leqslant\frac{\eta}{|\mathcal{S}|}ap+(1-\frac{\eta}{|\mathcal{S}|})\frac{b}{2}\leqslant \frac{1}{\ell}ap+(1-\frac{1}{\ell})\frac{b}{2}\end{aligned}$$ where $\frac{\eta}{|\mathcal{S}|}=\frac{1}{\max\{|\mathcal{S}|,\ell\}}\in[\frac{1}{d},\frac{1}{\ell}]$ and $ap>\frac{1}{4}>\frac{b}{2}$. Also, $$\begin{aligned} \label{equ:case1_lower} \Pr(y^{\prime}|\mathcal{S})\geqslant\frac{\eta}{|\mathcal{S}|}a(1-p)+(1-\frac{\eta}{|\mathcal{S}|})\frac{b}{2}\end{aligned}$$ **Case 2.** If $k^{\prime}\notin\mathcal{S}$, i.e., $k^{\prime}\in\{d+1,\cdots,d^{\prime}\}$, then $$\begin{aligned} \label{equ:case2_upper} &\quad\Pr(y^{\prime}|\mathcal{S})=\eta\cdot\frac{b}{2}+\frac{1-\eta}{\ell}\left[\frac{a}{2}+(\ell-1)\frac{b}{2}\right] \notag\\ &<\frac{1}{\ell}\left[\frac{a}{2}+(\ell-1)\frac{b}{2}\right] <\frac{1}{\ell}ap+(1-\frac{1}{\ell})\frac{b}{2}\end{aligned}$$ where $\eta=\frac{|\mathcal{S}|}{\max\{|\mathcal{S}|,\ell\}}\in[\frac{1}{\ell},1]$, and $a,p>\frac{1}{2}>b$. Also, $$\begin{aligned} \label{equ:case2_lower} \Pr(y^{\prime}|\mathcal{S})=\eta\cdot\frac{b}{2}+\frac{1-\eta}{\ell}\left[\frac{a}{2}+(\ell-1)\frac{b}{2}\right] \geqslant\frac{b}{2}\end{aligned}$$ **Bound of Probability Ratio.** Denote $\Phi=\Pr(y^{\prime}|\mathcal{S})$. By combining (\[equ:case1\_upper\]) and (\[equ:case2\_upper\]), the upper bound is $$\begin{aligned} \Phi_{\text{upper}} =\frac{1}{\ell}ap+(1-\frac{1}{\ell})\frac{b}{2}\end{aligned}$$ According to (\[equ:case1\_lower\]) and (\[equ:case2\_lower\]), the lower bound can be discussed by the following two cases. **Case 1.** If $a(1-p)<\frac{b}{2}$, i.e., $e^{\epsilon_1}<\frac{e^{\epsilon_2}+1}{2}$, we have $$\begin{aligned} \Phi_{\text{lower}} &= \frac{\eta}{|\mathcal{S}|}a(1-p)+(1-\frac{\eta}{|\mathcal{S}|})\frac{b}{2}\bigg|_{\frac{\eta}{|\mathcal{S}|}=\frac{1}{\ell}}\\ &= \frac{1}{\ell}a(1-p)+(1-\frac{1}{\ell})\frac{b}{2}\end{aligned}$$ where $\Phi_{\text{lower}}<\frac{b}{2}$. Then, for any two different inputs $\mathcal{S}_1$ and $\mathcal{S}_2$, the probability ratio is bounded by $$\begin{aligned} \label{equ:bound1} \frac{\Pr(y^{\prime}|\mathcal{S}_1)}{\Pr(y^{\prime}|\mathcal{S}_2)} &\leqslant\frac{\Phi_{\text{upper}}}{\Phi_{\text{lower}}} =\frac{\frac{1}{\ell}ap+(1-\frac{1}{\ell})\frac{b}{2}}{\frac{1}{\ell}a(1-p)+(1-\frac{1}{\ell})\frac{b}{2}}\notag\\ &=\frac{\frac{ap}{b}+\frac{\ell-1}{2}}{\frac{a(1-p)}{b}+\frac{\ell-1}{2}} =\frac{e^{\epsilon_1+\epsilon_2}+(\ell-1)\frac{e^{\epsilon_2}+1}{2}}{e^{\epsilon_1}+(\ell-1)\frac{e^{\epsilon_2}+1}{2}}\end{aligned}$$ **Case 2.** If $a(1-p)\geqslant\frac{b}{2}$, i.e., $e^{\epsilon_1}\geqslant\frac{e^{\epsilon_2}+1}{2}$, then $\Phi_\text{lower}=\frac{b}{2}$ $$\begin{aligned} \label{equ:bound2} &\quad\frac{\Pr(y^{\prime}|\mathcal{S}_1)}{\Pr(y^{\prime}|\mathcal{S}_2)} \leqslant\frac{\Phi_{\text{upper}}}{\Phi_{\text{lower}}} =\frac{\frac{1}{\ell}ap+(1-\frac{1}{\ell})\frac{b}{2}}{\frac{b}{2}}\notag\\ &=\frac{2e^{\epsilon_1+\epsilon_2}}{\ell(e^{\epsilon_2}+1)}+1-\frac{1}{\ell} =\frac{e^{\epsilon_1+\epsilon_2}+(\ell-1)\frac{e^{\epsilon_2}+1}{2}}{\ell\cdot\frac{e^{\epsilon_2}+1}{2}}\end{aligned}$$ By combining the results in (\[equ:bound1\]) and (\[equ:bound2\]), we have $$\begin{aligned} \frac{\Pr(y^{\prime}|\mathcal{S}_1)}{\Pr(y^{\prime}|\mathcal{S}_2)}\leqslant\frac{e^{\epsilon_1+\epsilon_2}+(\ell-1)(e^{\epsilon_2}+1)/2}{\min\{e^{\epsilon_1},(e^{\epsilon_2}+1)/2\}+(\ell-1)(e^{\epsilon_2}+1)/2}\end{aligned}$$ Proof of Theorem \[thm:estimation\] {#apx:thm:estimation_UE} =================================== **Step 1. calculate the expectation and variance of $n_1$ and $n_2$.** Denote $$\begin{aligned} q_1=a\cdot[1+(2p-1)m_k^{*}]/2,\quad q_2=a\cdot[1-(2p-1)m_k^{*}]/2\end{aligned}$$ where $m_k^{*}$ is the true mean of key $k$. For a user $u\in\mathcal{U}_k$ (the set of users who possess key $k\in\mathcal{K}$), denote the expected contribution of supporting $1$ and $-1$ as $q_{u1}^{*}$ and $q_{u2}^{*}$ respectively. According to the perturbation steps of PCKV-UE in Figure \[fig:perturbation\] (note that PCKV-GRR has the similar perturbation), $q_{u1}^{*}$ and $q_{u2}^{*}$ are computed by $$\begin{aligned} q_{u1}^{*}=a\cdot[1+(2p-1)v^{*}_u]/2,\quad q_{u2}^{*}=a\cdot[1-(2p-1)v^{*}_u]/2\end{aligned}$$ where $\frac{\sum_{u\in\mathcal{U}_k}q_{u1}^{*}}{|\mathcal{U}_k|}=q_1$ and $\frac{\sum_{u\in\mathcal{U}_k}q_{u2}^{*}}{|\mathcal{U}_k|}=q_2$. Then the expected contribution of supporting $1$ of a group of users $\mathcal{U}_k$ is $$\begin{aligned} \mathbb{E}_{\mathcal{U}_k}[n_1]=\frac{1}{\ell}\sum\nolimits_{u\in\mathcal{U}_k}q_{u1}^{*} =\frac{1}{\ell}|\mathcal{U}_k|q_1 =n\frac{f_k^{*}}{\ell}q_1\end{aligned}$$ where $|\mathcal{U}_k|=nf_k^{*}$. And the corresponding variance is $$\begin{aligned} &\text{Var}_{\mathcal{U}_k}[n_1]=\frac{1}{\ell}\sum_{u\in\mathcal{U}_k}q_{u1}^{*}(1-q_{u1}^{*}) =\frac{1}{\ell}\left[\sum_{u\in\mathcal{U}_k}q_{u1}^{*}-\sum_{u\in\mathcal{U}_k}{q_{u1}^{*}}^2\right]\\ &\leqslant\frac{1}{\ell}\left[\sum_{u\in\mathcal{U}_k}q_{u1}^{*}-\frac{1}{|\mathcal{U}_k|}(\sum_{u\in\mathcal{U}_k}{q_{u1}^{*}})^2\right] =n\frac{f_k^{*}}{\ell}q_1(1-q_1)\end{aligned}$$ where $\sum_{u\in\mathcal{U}_k}{q_{u1}^{*}}^2\geqslant\frac{1}{|\mathcal{U}_k|}(\sum_{u\in\mathcal{U}_k}{q_{u1}^{*}})^2$ from Cauchy-Schwarz inequality. Similarly, we can compute $\mathbb{E}_{\mathcal{U}_k}[n_2]$ and the upper bound of $\text{Var}_{\mathcal{U}_k}[n_2]$. Then, for all users, the expectation and the upper bound of variance are ($t=1$ or $2$) $$\begin{aligned} &\mathbb{E}[n_t]=\mathbb{E}_{\mathcal{U}_k}[n_t]+\mathbb{E}_{\mathcal{U}\backslash\mathcal{U}_k}[n_t] =n\frac{f_k^{*}}{\ell}q_t+n(1-\frac{f_k^{*}}{\ell})\frac{b}{2}\\ &\text{Var}[n_t]\leqslant n\frac{f_k^{*}}{\ell}q_t(1-q_t)+n(1-\frac{f_k^{*}}{\ell})\frac{b}{2}(1-\frac{b}{2})\end{aligned}$$ where $\mathcal{U}\backslash\mathcal{U}_k$ denotes the set of users not in $\mathcal{U}_k$. Note that $$\begin{aligned} &\text{Var}_{\mathcal{U}_k}[n_1]-\text{Var}_{\mathcal{U}_k}[n_2] =\frac{1}{\ell}\sum_{u\in\mathcal{U}_k}(q_{u1}^{*}-q_{u2}^{*})(1-q_{u1}^{*}-q_{u2}^{*})\\ &=n\frac{f_k^{*}}{\ell}(q_1-q_2)(1-a) =n\frac{f_k^{*}}{\ell}(1-a)a(2p-1)m_k^{*}\end{aligned}$$ because of $q_{u1}^{*}+q_{u2}^{*}=a$ and $\sum_{u}(q_{u1}^{*}-q_{u2}^{*})=nf_k^{*}(q_1-q_2)$, where $q_1-q_2=a(2p-1)m_k^{*}$. Then, for all users $u\in\mathcal{U}$, $$\begin{aligned} &\text{Var}[n_1]-\text{Var}[n_2]=n\frac{f_k^{*}}{\ell}(1-a)a(2p-1)m_k^{*}\\ &\text{Var}[n_1+n_2]=n\frac{f_k^{*}}{\ell}a(1-a)+n(1-\frac{f_k^{*}}{\ell})b(1-b)\end{aligned}$$ Note that $n_1$ and $n_2$ are correlated variables. **Step 2. calculate the expectation and variance of frequency estimation.** According to the frequency estimator in (\[equ:hat\_f\]), we have $$\begin{aligned} \mathbb{E}[\hat{f}_k]&=\frac{\mathbb{E}[n_1+n_2]/n-b}{a-b}\ell=\frac{\frac{f_k^{*}}{\ell}a+(1-\frac{f_k^{*}}{\ell})b-b}{a-b}\ell=f_k^{*}\\ \text{Var}[\hat{f}_k]&=\frac{\ell^2 \text{Var}[n_1+n_2]}{n^2(a-b)^2} = \frac{\ell^2 b(1-b)}{n(a-b)^2} + \frac{\ell\cdot f_k^{*}(1-a-b)}{n(a-b)}\end{aligned}$$ which are equivalent to the results for itemset data in [@wang2018locally] (note that [@wang2018locally] focuses on the count $c_k=nf_k^{*}$ while we consider the proportion $f_k^{*}$). **Step 3. calculate the expectation and variance of mean estimation.** From the multivariate Taylor Expansions of functions of random variables [@casella2002statistical], the expectation of quotient of two random variables $X$ and $Y$ can be approximated by $$\begin{aligned} \label{equ:E[X/Y]} \mathbb{E}\left[\frac{X}{Y}\right]&\approx \frac{\mathbb{E}[X]}{\mathbb{E}[Y]}-\frac{\text{Cov}_{X,Y}}{\mathbb{E}[Y]^2}+ \frac{\mathbb{E}[X]}{\mathbb{E}[Y]^3}\cdot\text{Var}[Y]\\ \label{equ:Var[X/Y]} \text{Var}\left[\frac{X}{Y}\right]&\approx \frac{\text{Var}[X]}{\mathbb{E}[Y]^2}-\frac{2\mathbb{E}[X]\text{Cov}_{X,Y}}{\mathbb{E}[Y]^3}+ \frac{\mathbb{E}[X]^2}{\mathbb{E}[Y]^4}\text{Var}[Y]\end{aligned}$$ For convenience, denote $X=n_1-n_2,Y=n_1+n_2-nb$, then $$\begin{aligned} \mathbb{E}[X]=n\frac{f_k^{*}}{\ell}a(2p-1)m_k^{*},\quad \mathbb{E}[Y]=n\frac{f_k^{*}}{\ell}(a-b)\end{aligned}$$ The variances are $$\begin{aligned} \text{Var}[X]&=\text{Var}[n_1-n_2]=2(\text{Var}[n_1]+\text{Var}[n_2])-\text{Var}[n_1+n_2]\\ &\leqslant nb+n\frac{f_k^{*}}{\ell}[(a-b)-a^2(2p-1)^2{m_k^{*}}^2]\\ \text{Var}[Y]&=\text{Var}[n_1+n_2] =n\frac{f_k^{*}}{\ell}a(1-a)+n(1-\frac{f_k^{*}}{\ell})b(1-b)\end{aligned}$$ The covariance is $$\begin{aligned} &\quad\text{Cov}_{X,Y}=\text{Cov}[n_1-n_2,n_1+n_2]\\ &=\mathbb{E}[(n_1-n_2)(n_1+n_2)]-\mathbb{E}[n_1-n_2]\mathbb{E}[n_1+n_2]\\ &=\mathbb{E}[n_1^2-n_2^2]-\left(\mathbb{E}[n_1]^2-\mathbb{E}[n_2]^2\right)=\text{Var}[n_1]-\text{Var}[n_2]\\ &=n\frac{f_k^{*}}{\ell}a(1-a)(2p-1)m_k^{*}=(1-a)\cdot\mathbb{E}[X]\end{aligned}$$ Note that only $\text{Var}[X]$ is computed by its upper bound, while $\mathbb{E}[X]$, $\mathbb{E}[Y]$, $\text{Var}[Y]$ and $\text{Cov}_{X,Y}$ are computed by their exact values. For convenience, denote $\delta=\frac{f_k^{*}}{\ell}(a-b)$ and $\gamma=\frac{f_k^{*}}{\ell}a(2p-1)$. According to (\[equ:hat\_m\]) and (\[equ:E\[X/Y\]\]), we have $$\begin{aligned} \mathbb{E}[\hat{m}_k]&=\frac{(a-b)\mathbb{E}\left[\frac{X}{Y}\right]}{a(2p-1)} \approx\frac{(a-b)\mathbb{E}[X]}{a(2p-1)\mathbb{E}[Y]}\left[1-\frac{1-a}{\mathbb{E}[Y]}+\frac{\text{Var}[Y]}{\mathbb{E}[Y]^2}\right]\\ &=m_k^{*}\left[1+\frac{ (1-b-\delta)b }{n\delta^2}\right]\end{aligned}$$ Similarly, according to (\[equ:hat\_m\]) and (\[equ:Var\[X/Y\]\]), we have $$\begin{aligned} \text{Var}[\hat{m}_k]=\frac{(a-b)^2\text{Var}\left[\frac{X}{Y}\right]}{a^2(2p-1)^2} \lesssim\frac{b+\delta}{n\gamma^2} + \frac{b(1-b)-\delta}{n\delta^2} \cdot{m_k^{*}}^2\end{aligned}$$ Proof of Lemma \[lem:hat\_n\] {#apx:lem:hat_n} ============================= According to the perturbation mechanism, we have $$\begin{aligned} \mathbb{E}[n_1]&=n_1^{*}ap+n_2^{*}a(1-p)+(n-n_1^{*}-n_2^{*})b/2\\ \mathbb{E}[n_2]&=n_1^{*}a(1-p)+n_2^{*}ap+(n-n_1^{*}-n_2^{*})b/2\end{aligned}$$ which can be rewritten as $$\begin{aligned} \begin{bmatrix} \mathbb{E}[n_1]\\ \mathbb{E}[n_2] \end{bmatrix}= A \begin{bmatrix} n_1^{*}\\ n_2^{*} \end{bmatrix}+ \begin{bmatrix} nb/2\\ nb/2 \end{bmatrix}\end{aligned}$$ where $$\begin{aligned} A= \begin{bmatrix} ap-\frac{b}{2} & a(1-p)-\frac{b}{2}\\ a(1-p)-\frac{b}{2} & ap-\frac{b}{2} \end{bmatrix}\end{aligned}$$ According to the linear property, the expectation of $\hat{n}_1$ and $\hat{n}_2$ in (\[equ:hat\_n\]) are $$\begin{aligned} \begin{bmatrix} \mathbb{E}[\hat{n}_1]\\ \mathbb{E}[\hat{n}_2] \end{bmatrix}= A^{-1} \begin{bmatrix} \mathbb{E}[n_1]-nb/2\\ \mathbb{E}[n_2]-nb/2 \end{bmatrix} =A^{-1}A\begin{bmatrix} n_1^{*}\\ n_2^{*} \end{bmatrix}= \begin{bmatrix} n_1^{*}\\ n_2^{*} \end{bmatrix}\end{aligned}$$ Note that $$\begin{aligned} \det(A)&=(ap-b/2)^2-(a(1-p)-b/2)^2\\ &=a(a-b)(2p-1)>0\end{aligned}$$ thus $A^{-1}$ exists. Therefore, $(\hat{n}_1,\hat{n}_2)$ are unbiased estimators of $(n_1^{*},n_2^{*})$. Proof of Lemma \[lem:budget\_UE\] {#apx:lem:budget_UE} ================================= According to budget allocation in (\[equ:epsilon\_1&2\]) and perturbation probabilities setting of OUE, we can rewrite $a,b,p$ with respect to $\theta$ $$\begin{aligned} a=\frac{1}{2},\quad b=\frac{1}{e^{\epsilon_1}+1}=\frac{1}{\theta+1},\quad p = \frac{1}{1+{e^{-\epsilon_2}}} =\frac{e^{\epsilon}}{2\theta}\end{aligned}$$ where $\frac{e^{\epsilon}+1}{2}\leqslant\theta <e^{\epsilon}$. Then, $g$ and $h$ in (\[equ:gh\]) can be rewritten as the function of $\theta$ $$\begin{aligned} g(\theta)=\frac{4}{(\theta+1)(e^{\epsilon}/\theta-1)^2},\quad h(\theta) =\frac{4\theta}{(\theta-1)^2}\end{aligned}$$ and their derivative functions are $$\begin{aligned} g^{\prime}(\theta)=\frac{4\theta[\theta^2+(\theta+2)e^{\epsilon}]}{(\theta+1)^2(e^{\epsilon}-\theta)^3}>0,\quad h^{\prime}(\theta)=-\frac{4(\theta+1)}{(\theta-1)^3}<0\end{aligned}$$ For convenience, denote $$\begin{aligned} \label{equ:Phi_theta} \Phi(\theta)=\text{MSE}_{\hat{m}_k}/\mu=g(\theta)+h(\theta)\cdot{m_k^{*}}^2\end{aligned}$$ and $\theta_0=\frac{e^{\epsilon}+1}{2}$, which is the minimum value of $\theta$. In the following, we show that $\Phi(\epsilon_1)$ is an approximately increasing function of $\epsilon_1$. Considering both $g^{\prime}(\theta)$ and $h^{\prime}(\theta)$ are increasing functions of $\theta$, we have $$\begin{aligned} &\quad\Phi^{\prime}(\theta)=g^{\prime}(\theta)+h^{\prime}(\theta)\cdot{m_k^{*}}^2 \geqslant g^{\prime}(\theta_0)+h^{\prime}(\theta_0)\cdot{m_k^{*}}^2\\ &= \frac{16(e^\epsilon+3)}{(e^\epsilon-1)^3}\cdot\left[\frac{(e^\epsilon+1)(3e^{2\epsilon}+12e^\epsilon+1)}{(e^\epsilon+3)^3}-{m_k^{*}}^2\right]\end{aligned}$$ where $-1\leqslant m_k^{*}\leqslant 1$. Denote $$\begin{aligned} \label{equ:Psi_epsilon} \Psi(\epsilon) = \frac{(e^\epsilon+1)(3e^{2\epsilon}+12e^\epsilon+1)}{(e^\epsilon+3)^3}\end{aligned}$$ whose value is plotted in Figure \[fig:proof\] (a), where $0.5<\Psi(\epsilon)<3$ for all $\epsilon>0$, and $\Psi(0.85)\approx 1$. Therefore, we have $\quad\Phi^{\prime}(\theta)\geqslant0$ for all $\epsilon_1\in[\ln\frac{e^\epsilon+1}{2},\epsilon)$ when $\Psi(\epsilon)\geqslant {m_k^{*}}^2$, which always holds if ${m_k^{*}}^2\leqslant 0.5$ or $\epsilon\geqslant 0.85$. Moreover, with different $\epsilon$, the value of $\Phi(\theta)$ in (\[equ:Phi\_theta\]) when ${m_k^{*}}^2=1$ (the worst case) is shown in Figure \[fig:proof\] (b), which validates that $\Phi(\theta)$ is approximately increasing function of $\theta$ for all possible $\epsilon$ and $m_k^{*}$. Therefore, $\theta_0=\frac{e^{\epsilon}+1}{2}$ is the optimal solution of minimizing $\text{MSE}[\hat{m}_k]=\mu\cdot\Phi(\theta)$. By substituting $\theta=\frac{e^{\epsilon}+1}{2}$ into (\[equ:epsilon\_1&2\]), we finally obtain the budgets as in . \[The value of $\Phi(\theta)$ in (\[equ:Phi\_theta\]) when ${m_k^{*}}^2=1$ (the worst case), where $\Phi(\theta)=\text{MSE}_{\hat{m}_k}/\mu$ ($\mu$ is a constant) and $\theta\in[\frac{e^\epsilon+1}{2},e^{\epsilon})$ according to (\[equ:epsilon\_1&2\]).\][![Illustrations in Appendix \[apx:lem:budget\_UE\].[]{data-label="fig:proof"}](fig_proof2.pdf "fig:"){width="1.6in"}]{}\ Proof of Lemma \[lem:budget\_GRR\] {#apx:lem:budget_GRR} ================================== According to (\[equ:abp\_GRR\]) and , we have $$\begin{aligned} \label{equ:proof_g} g&=\frac{b}{a^2(2p-1)^2}= \frac{e^{-\epsilon_1}+(d^{\prime}-1)e^{-2\epsilon_1}}{(\frac{2}{1+e^{-\epsilon_2}}-1)^2}\\ \label{equ:proof_h} h&= \frac{(1-b)b}{(a-b)^2}= \frac{e^{\epsilon_1}+d^{\prime}-2}{(e^{\epsilon_1}-1)^2}\end{aligned}$$ where $d^{\prime}=d+\ell$ and $h^{\prime}(\epsilon_1)<0$. In the following, we discuss the optimal $\epsilon_1$ in two cases. **Case 1.** If $\ell=1$, then (\[equ:epsilon\_GRR\]) reduces to (\[equ:epsilon\_UE\]) because of $\lambda=0$, thus we can obtain the same result as in PCKV-UE $$\begin{aligned} \epsilon_1=\ln \theta,\quad \epsilon_2=\ln\frac{1}{2\theta e^{-\epsilon}-1},\quad \text{for }\frac{e^{\epsilon}+1}{2}\leqslant \theta <e^{\epsilon}\end{aligned}$$ then we have $$\begin{aligned} g(\theta) =\frac{\theta+(d^{\prime}-1)}{(e^{\epsilon}-\theta)^2},\quad h(\theta)= \frac{\theta+d^{\prime}-2}{(\theta-1)^2}\end{aligned}$$ where $g^{\prime}(\theta)>0$ and $h^{\prime}(\theta)<0$. Similar to the proof in Appendix \[apx:lem:budget\_UE\], the optimal solution of minimizing $g(\theta)+h(\theta)\cdot{m_k^{*}}^2$ can be approximated at $\theta=\frac{e^{\epsilon}+1}{2}$, then $\epsilon_1=\ln\frac{e^{\epsilon}+1}{2}$ and $\epsilon_2=\epsilon$. **Case 2.** If $\ell>1$, denote $\theta=e^{\epsilon_1}$ and let $\epsilon_2=0$ in (\[equ:epsilon\_GRR\]), then $$\begin{aligned} e^{\epsilon}=\frac{\theta+(\ell-1)}{\ell} ~~\Rightarrow~~ \theta=\ell\cdot(e^{\epsilon}-1)+1\end{aligned}$$ Thus, to guarantee $\epsilon_1,\epsilon_2>0$ under a given $\epsilon$, variable $\theta$ should in the following range $$\begin{aligned} \label{equ:range_theta} 1<\theta<\ell\cdot(e^{\epsilon}-1)+1\end{aligned}$$ On the other hand, let $\theta=e^{\epsilon_1}=(e^{\epsilon_2}+1)/2$ in (\[equ:epsilon\_GRR\]), i.e., the two values in the min operation equal with each other, then $\theta=\ell\cdot(e^\epsilon-1)/2+1$. For the parameter $g$ calculated in , we discuss its derivative function $g^{\prime}(\theta)$ in the two ranges - For $1<\theta\leqslant\ell\cdot(e^\epsilon-1)/2+1$, we have $$\begin{aligned} \min\{e^{\epsilon_1},(e^{\epsilon_2}+1)/2\}=e^{\epsilon_1} ~~\Rightarrow~~ e^\epsilon=\frac{e^{\epsilon_1+\epsilon_2}+\lambda}{e^{\epsilon_1}+\lambda} \end{aligned}$$ where $\lambda=(\ell-1)(e^{\epsilon_2}+1)/2$. Then, $$\begin{aligned} &\frac{2}{1+e^{-\epsilon_2}}-1 = \frac{e^{\epsilon}-1}{e^{\epsilon}+1}\cdot[1+(\ell-1)/\theta]\\ \Rightarrow~~ &g(\theta)=\left(\frac{e^{\epsilon}+1}{e^{\epsilon}-1}\right)^2\cdot\frac{\theta+d^{\prime}-1}{(\theta+\ell-1)^2} \end{aligned}$$ where $g^{\prime}(\theta)<0$ and $d^{\prime}=d+\ell$. - For $\ell\cdot(e^\epsilon-1)/2+1\leqslant\theta<\ell\cdot(e^\epsilon-1)+1$, we have $$\begin{aligned} e^\epsilon=\frac{e^{\epsilon_1+\epsilon_2}+\lambda}{(e^{\epsilon_2}+1)/2+\lambda} \end{aligned}$$ then $$\begin{aligned} &\frac{2}{1+e^{-\epsilon_2}}-1 = [\ell(e^{\epsilon}-1)-(\theta-1)]/\theta\\ \Rightarrow~~ &g(\theta)=\frac{\theta+d^{\prime}-1}{[\ell(e^{\epsilon}-1)-(\theta-1)]^2} \end{aligned}$$ where $g^{\prime}(\theta)>0$. Therefore, $g(\theta)$ approaches to the minimum value at $\theta=\ell\cdot(e^\epsilon-1)/2+1$. Note that $g(\theta)\rightarrow+\infty$ when $\theta\rightarrow\ell\cdot(e^{\epsilon}-1)+1$ (the upper bound in ), and $h(\theta)\rightarrow+\infty$ when $\theta\rightarrow1$ (the lower bound in ). Similar to the proof in Appendix \[apx:lem:budget\_UE\], the optimal solution of minimizing $g(\theta)+h(\theta)\cdot{m_k^{*}}^2$ can be approximated at $\theta=\ell\cdot(e^\epsilon-1)/2+1$. Then, we have $$\begin{aligned} \epsilon_1=\ln[\ell\cdot(e^\epsilon-1)/2+1],\quad \epsilon_2=\ln\left[\ell\cdot(e^\epsilon-1)+1\right]\end{aligned}$$ By combining the results in Case 1 (when $\ell=1$) and Case 2 (when $\ell>1$), we obtain (\[equ:budget\_GRR\]).

--- abstract: 'We present theoretical studies of above threshold ionization (ATI) produced by spatially inhomogeneous fields. This kind of field appears as a result of the illumination of plasmonic nanostructures and metal nanoparticles with a short laser pulse. We use the time-dependent Schrödinger equation (TDSE) in reduced dimensions to understand and characterize the ATI features in these fields. It is demonstrated that the inhomogeneity of the laser electric field plays an important role in the ATI process and it produces appreciable modifications to the energy-resolved photoelectron spectra. In fact, our numerical simulations reveal that high energy electrons can be generated. Specifically, using a linear approximation for the spatial dependence of the enhanced plasmonic field and with a near infrared laser with intensities in the mid- $10^{14}$ W/cm$^{2}$ range, we show it is possible to drive electrons with energies in the near-keV regime. Furthermore, we study how the carrier envelope phase influences the emission of ATI photoelectrons for few-cycle pulses. Our quantum mechanical calculations are supported by their classical counterparts.' author: - 'M. F. Ciappina$^{1}$' - 'J. A. Pérez-Hernández$^{2}$' - 'T. Shaaran$^{1}$' - 'J. Biegert$^{1,3}$' - 'R. Quidant$^{1,3}$' - 'M. Lewenstein$^{1,3}$' title: 'Above threshold ionization by few-cycle spatially inhomogeneous fields' --- Introduction ============ In the field of the interaction of laser fields with matter, above-threshold ionization (ATI) has been a particularly interesting subject in both experimental and theoretical physics. ATI, which was experimentally observed more than 30 years ago [@Agostini1979], occurs when an atom or molecule absorbs more photons than the minimum number required to ionize it, with the leftover energy being converted to the kinetic energy of the released electron. With recent advances in laser technology, it has become possible to generate few-cycle pulses, which find a wide range of applications in science, such as controlling chemical reactions and molecular motion [@schnurer2000; @vdHoff2009], and generating high-order harmonics and even the creation of isolated extreme ultraviolet (XUV) pulses [@ferrari2010; @schultze2007]. These allow even more control on an attosecond temporal scale. The electric field in a few-cycle pulse can be characterized by its duration and by the so-called carrier-envelope phase (CEP). In comparison to a multicycle pulse, the electric field of few-cycle pulses is greatly affected by the CEP [@Wittmann2009; @kling2008]. The influence of CEP has been experimentally observed in high-harmonic generation (HHG) [@nisoli2003], the emission direction of electrons from atoms [@paulus2001] and in the yield of nonsequential double ionization [@liu2004]. In order to have a better control of the system on an attosecond temporal scale it is, therefore, important to find reliable schemes to measure the absolute phase of few-cycle pulses. Recently, the investigation of ATI generated by few-cycle driving laser pulses has attracted so much interest due to the sensitivity of the energy and angle-resolved photoelectron spectra to the absolute value of the CEP [@paulus_cleo; @sayler]. Consequently, this feature renders the ATI phenomenon a very valuable tool for laser pulse characterization. In order to characterize the CEP of a few-cycle laser pulse, the so-called backward-forward asymmetry of the ATI spectrum is measured and from the information collected the absolute CEP can be obtained [@paulus]. Furthermore, nothing but the high energy region of the photoelectron spectra appears to be sensitive to the absolute CEP and consequently electrons with kinetic energy are needed in order to characterize it [@milosevic_rev; @paulus2003]. New experiments have demonstrated that the harmonic cutoff and electron spectra of ATI could be extended further by using plasmon field enhancement [@kim; @kling]. This field appears when a metal nanostructure or nanoparticle is illuminated by a short laser pulse and it is not spatially homogeneous, due to the strong confinement of the plasmonics spots and the distortion of the electric field by the surface plasmons induced in the nanosystem. One should note, however, that a recent controversy about the outcome of the experiments of Ref. [@kim] has arisen [@sivis; @Kimreply; @corkum_priv]. Consequently, alternative systems to the metal bow-tie shaped nanostructures have appeared [@Kimnew]. A related process employing solid state targets instead of atoms and molecules in gas phase is the so called Above Threshold Photoemission (ATP). This laser driven phenomenon has received special attention recently due to its novelty and considering new physics could be involved. In ATP electrons are emitted from metallic surfaces or metal nanotips and they present distinct characteristics, namely higher energies, far beyond the usual cutoff for noble gases and consequently the possibility to reach similar electron energies with smaller laser intensities (see e.g. [@peterprl2006; @peterprl2010; @peternature; @peterjpbreview; @jensdombi; @ropers]). Furthermore, the photoelectrons emitted from these nanosources are sensitive to the CEP and consequently it plays an important role in the angle and energy resolved photoelectron spectra [@apolonski; @dombi; @kling; @peternature]. Despite new developments, all numerical and semiclassical approaches to model the ATI phenomenon are based on the assumption that the external field is spatially homogeneous in the region where the electron dynamics take place [@keitel; @krausz]. For an inhomogeneous field, however, important changes will occur to the features of strong field phenomena [@kim; @kling] since the laser-driven electric field, and consequently the force applied to the electron, will also depend on position. Up to now, there have been very few studies to investigate the strong field phenomena in such kind of fields [@husakou; @ciappi2012; @yavuz; @ciappi_prl]. From a theoretical viewpoint, the ATI process can be tackled using different approaches (for a summary see e.g. [@milosevic_rev; @schafer1993; @telnov2009; @bauer2006; @Blaga2009; @Quan2009] and references therein). In this article, we concentrate our effort in extending one of the most and widely used approaches: the numerical solution of time-dependent Schrödinger Equation (TDSE) in reduced dimensions. We have developed our numerical tool in such a way to allow the treatment of a very general set of nonhomogeneous fields. Furthermore, based on our model, we examine the influence of the CEP on photoelectron spectra of ATI. The kinetic energy for the rescattered electron is classically calculated and compared to our quantum mechanical approach. This article is organized as follows. In Sec. II, we present our theoretical approach to model ATI produced by nonhomogeneous fields. Subsequently, in Sec. III, we employ this method to compute the ATI energy-resolved photoelectron spectra using few-cycle laser pulses for both homogeneous and inhomogeneous fields. In addition, we perform classical simulations to support our quantum mechanical method. Finally, in Sec. IV, we conclude with a short summary and outlook. Theoretical approach ==================== In order to calculate the energy resolved photoelectron spectra, we use the one-dimensional time-dependent Schrödinger equation (1D-TDSE) $$\begin{aligned} \label{tdse} {\mathrm{i}}\frac{\partial \Psi(x,t)}{\partial t}&=&\mathcal{H}(t)\Psi(x,t) \\ &=&\left[-\frac{1}{2}\frac{\partial^{2}}{\partial x^{2}}+V_{atom}(x)+V_{laser}(x,t)\right]\Psi(x,t) \nonumber\end{aligned}$$ where $V_{laser}(x,t)$ represents the laser-atom interaction. For the atomic potential, we use the quasi-Coulomb or soft core potential $$\begin{aligned} \label{atom} V_{atom}(x)&=&-\frac{1}{\sqrt{x^2+a^2}}\end{aligned}$$ which was introduced in [@eberly] and has been widely used in the study of laser-matter processes in atoms. The parameter $a$ in Eq. (\[atom\]) allows us to match the ionization potential of the atom under consideration. We consider the field to be linearly polarized along the $x$-axis and modify the interaction term $V_{laser}(x,t)$ in order to treat spatially nonhomogeneous fields, although maintaining the dipole character. Consequently we write $$\begin{aligned} \label{vlaser} V_{laser}(x,t)&=&-E(x,t)\,x\end{aligned}$$ where $E(x,t)$ is the laser electric field defined as $$\label{electric} E(x,t)=E_0\,f(t)\, (1+\varepsilon h(x))\,\sin(\omega t+\phi).$$ In Eq. (\[electric\]), $E_0$, $\omega$ and $\phi$ are the peak amplitude, the frequency of the laser pulse and the CEP, respectively. We refer to sin(cos)-like laser pulses where $\phi=0$ ($\phi=\pi/2$). The pulse envelope is given by $f(t)$ and $\varepsilon$ is a small parameter that characterizes the inhomogeneity strength. The function $h(x)$ represents the functional form of the nonhomogeneous field and, in principle, could take any form and be supported by the numerical algorithm [@ciappi_prl]. In this work, however, we concentrate our efforts on the simplest form for $h(x)$, i.e. the linear term: $h(x)=x$. This choice is motivated by previous investigations in high-order harmonic generation [@husakou; @ciappi2012; @yavuz; @ciappi_prl; @tahirsfa].[^1] In the linear model we are using in this work, the units of $\varepsilon$ are inverse length (see also [@husakou; @yavuz; @ciappi2012]). We have written $V_{laser}$ in Eq. (\[vlaser\]) in such a way to emphasize the fact we are working within the dipole approximation and any deviation of it is considered small, i.e. higher electric multipole terms and magnetic effects are neglected [@reiss]. To model short laser pulses, we use a sin-squared envelope $f(t)$ of the form $$f(t)=\sin^{2}\left(\frac{\omega t}{2 n_p}\right)$$ where $n_p$ is the total number of optical cycles. The total duration of the laser pulse will then be $T_p=n_p \tau$ where $\tau=2\pi/\omega$ is the laser period. We assume the target atom is in the ground state ($1s$) before we turn on the laser ($t=-\infty$). This state can be found by solving an eigenvector and eigenvalue problem once the spatial coordinate $x$ has been discretized. We chose $a^2=1.412$ to match the atomic ionization potential of our target, which is an hydrogen atom ($I_p=-0.5$ a.u.). Eq.(\[tdse\]) is solved numerically by using the Crank-Nicolson scheme with an adequate spatial grid [@keitel]. We employ boundary reflections mask functions  [@mask] in order to avoid spurious contributions. For calculating the energy-resolved photoelectron spectra $P(E)$ we use the window function technique developed by Schafer [@schaferwop1; @schaferwop]. This tool has been widely used, both to calculate angle-resolved and energy-resolved photoelectron spectra [@schaferwop2] and it represents a step forward with respect to the usual projection methods. Results ======= In this section, we will determine the energy-resolved photoelectron spectra $P(E)$ using Eq. (\[tdse\]), in order to investigate the role of the inhomogeneities of the field. Furthermore, we demonstrate how the CEP $\phi$ will effect the the energy-resolved photoelectron spectra of ATI. We employ a four-cycle (total duration 10 fs) sin-squared laser pulse with an intensity $I=3\times10^{14}$ W/cm$^{2}$ and wavelength $\lambda=800$ nm. We chose three different values for the parameter that characterizes the inhomogeneity strength, namely $\varepsilon =0$ (homogeneous case), $0.003$ and $0.005$. Figures 1 and 2 show the cases with $\phi =0$ (a sin-like laser pulse) and $\phi =\pi/2$ (a cos-like laser pulse), respectively. Panels (a) of both Figures represent the homogeneous case, i.e. $\varepsilon =0$, and panels (b) and (c) show the nonhomogeneous case with $\varepsilon =0.003$ and $\varepsilon =0.005$, respectively. ![Energy-resolved photoelectron spectra $P(E)$ calculated using the 1D-TDSE for a model atom with $I_{p}=-0.5$. The laser parameters are $I=3\times10^{14}$ W/cm$^{2}$ and $\lambda=800$ nm. We have used a sin-squared shaped pulse with a total duration of 4 cycles (10 fs) and $\phi=0$ (a sin-like pulse). The arrows indicate the $2 U_p$ and $10 U_p$ cutoffs predicted by the classical model [@milosevic_rev]. Panel (a) $\varepsilon=0$ (homogeneous case), (b) $\varepsilon=0.003$ and (c) $\varepsilon=0.005$. []{data-label="fig:figure1"}](figure1.eps){width="42.00000%"} For the homogeneous case, the spectra exhibits the usual distinct behavior, namely the $2U_{p}$ cutoff ($\approx 36$ eV for our case) and the $10U_{p}$ cutoff ($\approx 180$ eV), where $U_{p}=E_{0}^{2}/4\omega^{2}$ is the ponderomotive potential. The former cutoff corresponds to those electrons that, once ionized, never return to the atomic core, while the latter one corresponds to the electrons that, once ionized, return to the core and elastically rescatter. It is well established using classical arguments that the maximum kinetic energies of the *direct* and the *rescattered* electrons are $E_{max}^{d}=2U_{p}$ and $E_{max}^{r}=10U_{p}$, respectively. In a quantum mechanical approach, however, it is possible to find electrons with energies beyond the 10$U_p$, although their yield of them drops several orders of magnitude [@milosevic_rev]. Experimentally, both mechanisms contribute to the energy-resolved photoelectron spectra and consequently the theoretical approach to tackle the problem should to include them. In that sense the TDSE, which can be considered as an exact approach to the problem, is able to predict the $P(E)$ in the whole range of electron energies. In addition, the most energetic electrons, i.e. those with $E_{k}\gg 2U_{p}$, are used to characterize the CEP of few-cycle pulses. As a result, a correct description of the rescattering mechanism is needed. ![Idem Fig. 1 but $\phi=\pi/2$ (a cos-like pulse).[]{data-label="fig:figure2"}](figure2.eps){width="42.00000%"} For the inhomogeneous case, the cutoff positions of the *direct* and the *rescattered* electrons are extended towards larger energies. For the *rescattered* electrons, this extension is very prominent. In fact, for $\varepsilon =0.003$ and $\varepsilon =0.005,$ it reaches $\approx 260$ eV and $\approx 420$ eV (panels b and c of Fig. 1, respectively). Furthermore, it appears that the high energy region of $P(E)$, for instance, the region between $200-400$ eV for $\varepsilon =0.005$ (see panels (c) of Figs. 1 and 2), is strongly sensitive to the CEP. This feature indicates that the high energy region of the photoelectron spectra could resemble a new and better CEP characterization tool. It should be, however, complemented by other well known and established CEP characterization tools, as, for instance, the forward-backward asymmetry (see [@milosevic_rev]). Furthermore, the utilization of nonhomogeneous fields would open the avenue for the production of high energy electrons, reaching the keV regime, if a reliable control of the spatial and temporal shape of the laser electric field is attained. {width="80.00000%"} We now concentrate our efforts in order to explain the extension of the energy-resolved photoelectron spectra using classical arguments. From the simple-man’s model [@corkum] we can describe the physical origin of the ATI process as follows: an atomic electron at a position $x=0$, is released or *born* at a given time, that we call *ionization* time $t_{i}$, with zero velocity, i.e. $\dot{x}(t_{i})=0$. This electron now moves only under the influence of the oscillating laser electric field (the residual Coulomb interaction is neglected in this model) and will reach the detector either directly or through the rescattering process. By using the classical equation of motion, it is possible to calculate the maximum energy of the electron for both direct and rescattered processes. The Newton equation of motion for the electron in the laser field can be written as (\[vlaser\]): $$\begin{aligned} \ddot{x}(t) &=&-\nabla _{x}V_{laser}(x,t) \notag \label{newton} \\ &=&E(x,t)+\left[ \nabla _{x}E(x,t)\right] x \notag \\ &=&E(t)(1+2\varepsilon x(t)),\end{aligned}$$ where we have collected the time dependent part of the electric field in $E(t)$, i.e. $E(t)=E_{0}f(t)\sin (\omega t+\phi )$ and we have specialized to the case $h(x)=x$. In the limit where $\varepsilon =0$ in Eq. (\[newton\]), we recover the homogeneous case. For the direct ionization, the kinetic energy of an electron released or born at time $t_{i}$ is $$\label{direct} E_{d}=\frac{\left[ \dot{x}(t_{i})-\dot{x}(t_{f})\right] ^{2}}{2},$$ where $t_{f}$ is the end time of the laser pulse. For the rescattered ionization, in which the electron returns to the core at a time $t_{r}$ and reverses its direction, the kinetic energy of the electron yields $$\label{rescattered} E_{r}=\frac{\left[ \dot{x}(t_{i})+\dot{x}(t_{f})-2\dot{x}(t_{r})\right] ^{2}}{2}.$$ For homogeneous fields, Eqs. (\[direct\]) and (\[rescattered\]) become as $E_{d}=\frac{\left[ A(t_{i})-A(t_{f})\right] ^{2}}{2}$ and $E_{r}=\frac{\left[ A(t_{i})+A(t_{f})-2A(t_{r})\right] ^{2}}{2}$, with $A(t)$ being the laser vector potential $A(t)=-\int^{t} E(t')dt'$. For the case with $\varepsilon=0$, it can be shown that the maximum value for $E_{d}$ is $2U_{p}$ while for $E_{r}$ it is $10U_{p}$ [@milosevic_rev]. These two values appear as cutoffs in the energy resolved photoelectron spectrum as can be observed in panels (a) of Figs. 1 and 2 (see the respective arrows). In Fig. 3, we present the numerical solutions of Eq. (\[newton\]), which is plotted in terms of the kinetic energy of the direct and rescattered electrons. We employ the same laser parameters as in Figs. 1 and 2. Panels (a), (b) and (c) correspond to the case of $\phi=0$ (sin-like pulses) and for $\varepsilon=0$ (homogeneous case), $\varepsilon=0.003$ and $\varepsilon=0.005$, respectively. Meanwhile, panels (d), (e) and (f) correspond to the case of $\phi=\pi/2$ (cos-like pulses) and for $\varepsilon=0$ (homogeneous case), $\varepsilon=0.003$ and $\varepsilon=0.005$, respectively. From the panels (b), (c), (e) and (f) we can observe the strong modifications that the nonhomogeneous character of the laser electric field produces in the electron kinetic energy. These are related to the changes in the electron trajectories (for details see e.g. [@yavuz; @ciappi2012; @ciappi_prl]). In short, the electron trajectories are modified in such a way that now the electron ionizes at an earlier time and recombines later, and in this way it spends more time in the continuum acquiring energy from the laser electric field. Consequently, higher values of the kinetic energy are attained. A similar behavior with the photoelectrons was observed recently in ATP using metal nanotips. According to the model presented in Ref. [@ropers] the localized fields modify the electron motion in such a way to allow sub-cycle dynamics. In our studies, however, we consider both direct and rescattered electrons (in Ref. [@ropers] only direct electrons are modeled) and the characterization of the dynamics of the photoelectrons is more complex. Nevertheless, the higher kinetic energy of the rescattered electrons is a clear consequence of the strong modifications of the laser electric field in the region where the electron dynamics takes place, as in the above mentioned case of ATP. Conclusions and Outlook ======================= We have extended previous studies of high-order harmonic generation produced by nonhomogeneous fields to above threshold ionization (ATI). An example is the field generated in a vicinity of a metal nanostructure or nanoparticle when it is irradiated by a short laser pulse. We have modified the time dependent Schrödinger equation to model the ATI phenomenon driven by nonhomogeneous fields. We predict an extension in the cutoff position and an increase of the yield of the energy-resolved photoelectron spectra in certain regions. These features are reasonable well reproduced by classical simulations. Our predictions would pave the way to the production of high energy photoelectrons, reaching the keV regime, using plasmon enhanced fields. Application of our model to a broader range of laser parameters, including an exhaustive study of CEP effects, and a systematic survey over different atomic species using a full dimensional scheme will be subject of further investigations. Acknowledgments {#acknowledgments .unnumbered} =============== We acknowledge the financial support of the MINCIN projects (FIS2008-00784 TOQATA and Consolider Ingenio 2010 QOIT) (M. F. C. and M.L.); ERC Advanced Grant QUAGATUA, Alexander von Humboldt Foundation and Hamburg Theory Prize (M. L.); Spanish MINECO (FIS2009-09522) (J. A. P-H); Spanish Ministry of Education and Science through its Consolider Program Science (SAUUL CSD 2007-00013), Plan Nacional (FIS2008-06368-C02-01), LASERLAB-EUROPE (grant agreement n° 228334, EC’s Seventh Framework Programme) (J. B.); this research has been partially supported by Fundació Privada Cellex. We thank Dane Austin for valuable comments and suggestions. [48]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , , , ****, (). , , , , , , , ****, (). , , , , ****, (). , , , , , , , ****, (). , , , , , , , , ****, (). , , , , , , , , ****, (). , , , , , , ****, (). , , , , , , , , , ****, (). , , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , p.  (). , , , , , , ****, (). , ****, (). , , , , ****, (). , , , , , , , ****, (). , , , , , , ****, (). , ****, (). , , , , ****, (). , , , , , , ****, (). , (). , , , , , , , ****, (). , , , , ****, (). , , , ****, (). , , , ****, (). , , , , ****, (). , , , , , , , , pp. (). , , , , ****, (). , ****, (). , , , ****, (). , , , ****, (). , ****, (). , , , ****, (). , , , , ****, (). , , , , ****, (). , , , , , , p. (). , , , , ****, (). , ****, (). , ****, (). , , , , , , , ****, (). , , , , , , , , , , , ****, (). , ****, (). , , , p. (). , ****, (). , , , ****, (). , ****, (). , ****, (). , ** (, ). , ****, (). [^1]: The actual spatial dependence of the enhanced near-field in the surrounding of a metal nanostructure can be obtained by solving the Maxwell equations incorporating both the geometry and material properties of the nanosystem under study and the input laser pulse characteristics (see e.g. [@ciappi_prl]). The electric field retrieved numerically is then approximated using a power series $h(x) =\sum_{i=1}^{N}b_{i}x^{i}$, where the coefficients $b_i$ are obtained by fitting the real electric field that results from a finite element simulation. Furthermore, in the region relevant for the strong field physics and electron dynamics and in the range of the parameters we are considering, the electric field can be indeed approximated by its linear dependence.

--- abstract: 'We investigate three kinds of heat produced in a system and a bath strongly coupled via an interaction Hamiltonian. By studying the energy flows between the system, the bath, and their interaction, we provide rigorous definitions of two types of heat, $Q_{\rm S}$ and $Q_{\rm B}$ from the energy loss of the system and the energy gain of the bath, respectively. This is in contrast to the equivalence of $Q_{\rm S}$ and $Q_{\rm B}$, which is commonly assumed to hold in the weak coupling regime. The bath we consider is equipped with a thermostat which enables it to reach an equilibrium. We identify another kind of heat $Q_{\rm SB}$ from the energy dissipation of the bath into the super bath that provides the thermostat. We derive the fluctuation theorems (FT’s) with the system variables and various heats, which are discussed in comparison with the FT for the total entropy production. We take an example of a sliding harmonic potential of a single Brownian particle in a fluid and calculate the three heats in a simplified model. These heats are found to equal on average in the steady state of energy, but show different fluctuations at all times.' author: - Chulan Kwon - Jaegon Um - Joonhyun Yeo - Hyunggyu Park title: Three heats in strongly coupled system and bath --- The nonequilibrium fluctuation theorem (FT) has been proven originally for deterministic systems [@evans93; @evans94; @gallavotti], later for stochastic systems [@jarzynski1; @jarzynski2; @crooks; @kurchan; @lebowitz; @seifert; @esposito], and recently for quantum systems [@campisi; @campisi_RMP; @hanggi; @talkner]. It takes into account thermodynamic quantities such as heat and work which are continuously produced even in the steady state. Such quantities accumulated for a long time exhibit huge fluctuations around their means, which is especially prominent in small systems. Compared to work, heat is intriguing because it is interpreted as an energy exchange with practically unrecognizable bath. By assuming the master equation or the Langevin equation, heat is found as a function of stochastic trajectories [@schnakenberg; @sekimoto]. Recent studies, mostly quantum mechanical, have more concentrated on a system strongly coupled with a bath [@nieuwenhuizen; @hoerhammer; @ilki_kim; @esposito_qm1; @pucci; @horowitz1; @hekking; @horowitz2; @silaev; @gallego; @ankerhold; @carrega; @esposito_qm2; @seifert_strong_coupling; @esposito_strong; @iyoda_sagawa; @funo1]. In spite of extensive efforts, however, it is pointed out in Ref. [@hanggi] that a consistent definition of heat for the strong coupling regime is currently not known and most of the studies are restricted to the assumption of an initial product state. A proper means to treat the interaction energy or Hamiltonian between the system and the bath is still missing. This limitation is also present in classical approaches. In this study, we develop a theoretical framework to deal with the interaction rigorously for strongly coupled classical systems, which is expected to extend to quantum systems. As the interaction energy changes in time, it accompanies energy changes in both system and bath, hence we expect two different forms of heat. We consider the bath to be equipped with a thermostat provided by another external system, which we call a super bath, so that the total system and bath is able to reach an equilibrium in the absence of a nonequilibrium source. We then find another form of heat in the bath, which is dissipated into the super bath and plays a crucial role to prevent the bath from heating up indefinitely. In this paper, we present detailed mathematical definitions for the three heats based on the rigorous treatment of the interaction Hamiltonian. We show that there are many different versions of the FT for entropy production due to the three forms of heat. We find that three heats exhibit different fluctuations even in the steady state. We explicitly confirm these properties from a specific example. First, we consider a general particle Hamiltonian system and bath coupled. The system variables are given by a collection of momentums ${\vec p}=({\vec p}_1, {\vec p}_2, \ldots,)^{\rm t}$ and positions ${\vec x}=({\vec x}_1, {\vec x}_2, \ldots)^{\rm t}$, where the superscript ${\rm t}$ denotes the transposition of vector or matrix. Similarly, the bath variables are given by ${\vec p}_{\rm B}$ and ${\vec x}_{\rm B}$. The Hamiltonian of the total system is composed of three parts: $H_{\rm S}={\vec p}^2/(2\mu)+U({\vec x},\lambda(t))$ for the system, $H_{\rm B}={\vec p}_{\rm B}^2/(2m)+U_{\rm B}({\vec x}_{\rm B})$ for the bath, and $H_{\rm I}= V({\vec x}, {\vec x}_{\rm B})$ for the interaction, where the time-dependent protocol $\lambda(t)$ is prescribed only in the system potential $U$ and the interaction Hamiltonian $V$ is a pairwise potential between the system and bath particles. We take the same mass $\mu$ for all system particles and $m$ for all bath particles, just for notational convenience. We assume that the bath is equipped with a Langevin thermostat provided by super bath. Then, equations of motion read as $\dot{\vec x}=\partial H_{\rm S}/\partial {\vec p}$, $\dot{\vec p}=-\partial(H_{\rm S}+ H_{\rm I})/\partial {\vec x}$, $\dot{\vec x}_{\rm B}=\partial H_{\rm B}/\partial {\vec p}_{\rm B}$, and $\dot{\vec p}_{\rm B}=-\partial(H_{\rm B}+ H_{\rm I})/\partial {\vec x}_{\rm B}-\gamma {\vec p}_{\rm B}/m+{\vec \xi}(t)~$ with the white noise ${\vec \xi}(t)$ satisfying $\langle \xi_i(t)\xi_j(t')\rangle=2\gamma\beta^{-1}\delta_{ij}\delta(t-t')$ for the inverse temperature $\beta$ and the viscosity coefficient $\gamma$. Equations of motion lead to energy relations: $$\frac{{\rm d}H_{\rm S}}{{\rm d}t}=\dot{W}-\dot{Q}_{\rm S}, ~ \frac{{\rm d}(H_{\rm S}+H_{\rm I})}{{\rm d}t}=\dot{W}-\dot{Q}_{\rm B}, ~\frac{{\rm d}H}{{\rm d}t}=\dot{W}-\dot{Q}_{\rm SB}~. \label{H_tot}$$ where $\dot{W}=\partial U/\partial t$ is the rate of the work produced by the time-dependent protocol $\lambda(t)$, $\dot{Q}_{\rm S}$ ($\dot{Q}_{\rm B}$) the rate of heat loss (gain) of the system (bath). $\dot{Q}_{\rm SB}$ is the rate of heat loss of the bath flowing into the super bath (SB) surrounding it, which was also considered in a recent study [@esposito_strong]. The rates of the three heats are defined as $$\dot{Q}_{\rm S}=\frac{\partial V}{\partial {\vec x}}\cdot \frac{\vec p}{\mu},~ \dot{Q}_{\rm B}=-\frac{\partial V}{\partial {\vec x}_{\rm B}}\cdot \frac{{\vec p}_{\rm B}}{m},~ \dot{Q}_{\rm SB}=\left( \frac{\gamma {\vec p}_{\rm B}}{m}-{\vec \xi}\right)\cdot \frac{{\vec p}_{\rm B}}{m}.\label{3heats}$$ Note that ${\rm d}H_{\rm I}/{\rm dt}=\dot{Q}_{\rm S}- \dot{Q}_{\rm B}\neq 0$, which is contrary to the usual expectation about the heat exchange between system and bath. From the bath point of view, we get ${\rm d}H_{\rm B}/{\rm dt}=\dot{Q}_{\rm B}-\dot{Q}_{\rm SB}$, where the thermostat slows down the increase of the bath energy. For equilibrium bath, the driving on the system by the time-dependent protocol should be mild enough to maintain the bath energy saturated in the long-time limit. Even in this case, the $\dot{Q}_{\rm B}$ and $\dot{Q}_{\rm SB}$ may show different fluctuations. Now, we examine the FT for our model. Though the Langevin thermostat is connected partially only to the bath, the total system plus bath are governed by the Langevin dynamics for which various forms of FT are already known to hold [@kurchan; @lebowitz]. For example, the integral FT holds for the total entropy production $\Delta S$ accumulated during a finite time interval as $$\langle e^{-\Delta S}\rangle =1 \quad {\rm with} \quad \Delta S=-\Delta \ln \rho+\beta Q_{\rm SB}~, \label{TFT}$$ where $-\Delta \ln \rho$ represents the Shannon entropy change with $\rho$ the probability distribution function (PDF) of the total system and $Q_{\rm SB}$ the accumulated heat flowing into the super bath. Here and throughout our paper, we set the Boltzmann constant $k_{\rm B}=1$. These trivial FT’s are, however, not very informative from the system point of view. Let $\mathbf{q}=({\vec x}, {\vec p}, {\vec x}_{\rm B}, {\vec p}_{\rm B})^{\rm t}$ be a state vector of the total system with $\mathbf{q}_{\rm S}=({\vec x}, {\vec p})^{\rm t}$ for system and $\mathbf{q}_{\rm B}=({\vec x}_{\rm B}, {\vec p}_{\rm B})^{\rm t}$ for bath, respectively. The reduced system PDF $\rho_{\rm S}(\mathbf{q}_{\rm S})$ defined as ${\rm Tr}_{\rm B} \rho(\mathbf{q})$ is obtained by tracing out the bath variable $\mathbf{q}_{\rm B}$ for the total system PDF $\rho(\mathbf{q})$. Then, the Bayes’ rule leads to $\rho(\mathbf{q})=\rho_{\rm S}(\mathbf{q}_{\rm S}) {\rho}(\mathbf{q}_{\rm B}|\mathbf{q}_{\rm S})$ with ${\rho}(\mathbf{q}_{\rm B}|\mathbf{q}_{\rm S})$ the conditional PDF. In deriving the FT for the system variables, it is useful to introduce a reference state for the total system. In this paper, we consider two typical reference states in the form of $\tilde{\rho}(\mathbf{q})=\rho_{\rm S}(\mathbf{q}_{\rm S})\tilde{\rho}(\mathbf{q}_{\rm B}|\mathbf{q}_{\rm S})$, where the conditional PDF’s for the two cases are given by $$\begin{aligned} \textrm{(a)}&&~~\tilde{\rho}(\mathbf{q}_{\rm B}|\mathbf{q}_{\rm S})= Z_{\rm B}^{-1}e^{\beta\tilde{H}_{\rm S}}e^{-\beta (H_{\rm B}+H_{\rm I})}, \nonumber\\ \textrm{(b)}&&~~\tilde{\rho}(\mathbf{q}_{\rm B}|\mathbf{q}_{\rm S})=Z_{\rm B}^{-1}e^{-\beta H_{\rm B}}, \label{Ref}\end{aligned}$$ where the equilibrium bath partition function $Z_{\rm B}={\rm Tr}_{\rm B} e^{-\beta H_{\rm B}}$ and the [*additional*]{} system Hamiltonian $\tilde{H}_{\rm S}(\mathbf{q}_{\rm S})$ originates from the normalization as $e^{-\beta \tilde{H}_{\rm S}}=Z_{\rm B}^{-1} {\rm Tr}_{\rm B} e^{-\beta (H_{\rm B}+H_{\rm I})}$. One can see easily that $\tilde{H}_{\rm S}$ vanishes in the limit $H_{\rm I} \approx 0$. The case (a) is a special type recently considered by Seifert [@seifert_strong_coupling]. If the total system is in equilibrium, (a) is exact and the reduced system PDF becomes $\rho_{\rm S} (\mathbf{q}_{\rm S})\sim e^{-\beta H_{\rm S}^{\rm eff}}$ with $H_{\rm S}^{\rm eff}=H_{\rm S} + \tilde{H}_{\rm S}$, indicating that the strong coupling induces an additional term in the system Hamiltonian. The case (b) corresponds to the usual assumption of the product state of system and bath. Difference between the true and reference states can be measured by the relative entropy as $D(\rho||\tilde{\rho})=\ln[\rho/\tilde{\rho}]$. Then, the total entropy production $\Delta S$ in Eq.  can be rewritten in terms of the system PDF and other energy variables along with the relative entropy such as $$\begin{aligned} \textrm{(a)} &&~\Delta S = -\Delta \ln\rho_{\rm S}+\beta (Q_{\rm S}-\Delta \tilde{H}_{\rm S})-\Delta D_{\rm a}, \nonumber\\ \textrm{(b)} &&~\Delta S =-\Delta \ln\rho_{\rm S} +\beta Q_{\rm B}-\Delta D_{\rm b}, \label{TFT1}\end{aligned}$$ where $\Delta D_{\rm a, b}$ are the relative entropy changes for the types (a) and (b), respectively. In this derivation, we utilized energy relations as $Q_{\rm SB}-Q_{\rm B}=-\Delta H_{\rm B}$ and $Q_{\rm SB}-Q_{\rm S}=-\Delta (H_{\rm B}+H_{\rm I})$. Then, the thermodynamic second laws yield the inequalities, $$\begin{aligned} \textrm{(a)}&&~R_a=\langle-\Delta \ln\rho_{\rm S}+\beta (Q_{\rm S}-\Delta \tilde{H}_{\rm S})-\Delta D_{\rm a}\rangle \ge 0 ,\label{ineq_a}\nonumber\\ \textrm{(b)}&&~R_b=\langle -\Delta \ln\rho_{\rm S}+\beta Q_{\rm B}-\Delta D_{\rm b}\rangle \ge 0 ,\label{ineq1}\end{aligned}$$ where the equality holds for non-thermostatted bath ($Q_{\rm SB}=0$), due to $\Delta \ln\rho=0$ for the Louiville dynamics. Nevertheless, the FT’s and second laws in the above forms still require the knowledge of the relative entropy change which cannot be accessible without knowing the true PDF ${\rho}(\mathbf{q}_{\rm B}|\mathbf{q}_{\rm S})$ of the bath. One can get around this when the initial state is not arbitrary but of our reference state (a) or (b) in Eq. . For example, consider a quantity $\Delta A\equiv -\Delta \ln\rho_{\rm S}+\beta (Q_{\rm S}-\Delta \tilde{H}_{\rm S})$, appeared in Eq. , which does not require the knowledge of the bath PDF. With the initial condition prepared with the reference state (a), we get, for a finite time interval $t=[0,\tau]$, $$\begin{aligned} &&\langle e^{-\Delta A} \rangle_{\rm a} =\int\!\!{\rm D}\mathbf{q}(t)~ e^{-\Delta A} \Pi[\mathbf{q}(t);\lambda(t)] \frac{\rho_{\rm S}(0) e^{-\beta (H_{\rm B}(0)+H_{\rm I}(0))}}{Z_{\rm B}e^{-\beta\tilde{H}_{\rm S}(0)}}\nonumber\\ &&=\int\!\!{\rm D}\mathbf{q}_{\rm R}(t)~ \Pi[\mathbf{q}_{\rm R}(t);\lambda_{\rm R}(t)] \frac{ \rho_{\rm S}(\tau) e^{-\beta (H_{\rm B}(\tau)+H_{\rm I}(\tau))}}{Z_{\rm B}e^{-\beta\tilde{H}_{\rm S}(\tau)}} = 1, \label{FTA}\end{aligned}$$ where $ \Pi[\mathbf{q}(t);\lambda(t)]$ ($ \Pi[\mathbf{q}_{\rm R}(t);\lambda_{\rm R}(t)]$) is the standard conditional probability for the path (reverse path) $\mathbf{q}(t)$ ($\mathbf{q}_R(t))$ and the protocol (time-reversed protocol) is $\lambda(t)$ ($\lambda^R(t)$), and the Schnakenberg relation $\Pi[\mathbf{q}(t);\lambda(t)]/\Pi[\mathbf{q}_R(t);\lambda^R(t)]=e^{\beta Q_{\rm SB}}$ is used. Note that $\langle\cdots\rangle_{\rm a}$ is the average with the initial state of reference type (a). The final equality comes from the probability normalization because the second integral represents the sum of all possible paths in the reverse process with its initial state of the same reference type (a). Similarly, we get for $\Delta B\equiv -\Delta \ln\rho_{\rm S}+\beta Q_{\rm B}$ as $\langle e^{-\Delta B}\rangle_{\rm b} =1$, when the initial condition is prepared with the reference state (b). The FT for $\Delta A$ in Eq.  has been recently found by Seifert [@seifert_strong_coupling] in the case without the super bath, and the FT for $\Delta B$ has been known for quantum systems [@iyoda_sagawa]. The corresponding inequalities are given as $$\begin{aligned} \textrm{(A)}&&~R_A=\langle-\Delta \ln\rho_{\rm S}+\beta (Q_{\rm S}-\Delta \tilde{H}_{\rm S})\rangle_{\rm a} \ge 0 ,\label{ineq_a}\nonumber\\ \textrm{(B)}&&~R_B=\langle -\Delta \ln\rho_{\rm S}+\beta Q_{\rm B}\rangle_{\rm b} \ge 0 . \label{ineq2}\end{aligned}$$ One should notice that $R_A$ or $R_{B}$ do not necessarily increase with interval time $\tau$ ($dR_A/d\tau$ and $dR_B/d\tau$ can be negative), because the total system PDF does not maintain its form of reference states as soon as the evolution starts. In contrast, $R_a$ or $R_b$ should increase always with $\tau$. These properties will be shown explicitly from rigorous calculations for a simple example later shown in Fig. \[fig1\]. We remark that, with reference initial states, $R_A \ge \langle D_a(\tau)\rangle_a \ge 0$ from Eq. , which implies the inequality for the total entropy production provides a tighter bound by the amount of the relative entropy at the final time. A similar result was found in Ref. [@esposito_strong]. Now, we take a concrete example for explicit calculation of three heats in average and also their PDF’s. Consider a Brownian colloidal particle submerged in a fluid bath. This colloid interacts with bath particles nearby through a finite-range interaction. These perturbed bath particles relax fast into equilibrium and new bath particles begin to interact as the colloid moves through the bath. For an analytic approach, we mimic this situation by considering only a small number $N$ of bath particles moving along with the colloid through strong harmonic interactions [@exp1]. All other non-interacting bath particles are in equilibrium. For simplicity, we only consider the one-dimensional model and take the bath potential $U_{\rm B}=0$. The total system state is given by $\mathbf{q}=(x, p, x_1, p_1, \cdots, x_N, p_N)^{\rm t}$ with the system state $\mathbf{q}_{\rm S}=(x,p)^{\rm t}$ and the bath state $\mathbf{q}_{\rm B}=(x_1, p_1, \cdots, x_N, p_N)^{\rm t}$. We use a different ordering of the components of the state vectors from the previous one. Note that we dropped state variables of all other bath particles which do not interact with the colloid. The interaction Hamiltonian is written as $H_I=\sum_iV_i$ where the interaction potential between the colloid and the $i$-th bath particle is chosen as $V_i=\kappa(x-x_i)^2/2$, which is long-ranged enough to keep interacting bath particles near the colloid. In order to study non-equilibrium motion, we introduce a sliding harmonic potential with a constant velocity $u$ given by $U(x,\lambda(t))=k(x-\lambda(t))^2/2$ with $\lambda(t)=ut$. This protocol for a Brownian particle has been extensively studied experimentally [@wang; @hummer; @wang05] and theoretically [@vanzon1; @vanzon2; @kwangmoo] for a single-particle Langevin system. ![(Color online) $R_A$ (red, upper) and $R_a$ (blue, lower) versus time $t$ for $\beta'=1.5$, $u=0.02$. We use $N=2$, $\gamma=30$, $\mu=m=1$, $k=\kappa=1$, and $\beta=1$. $R_A>R_a$ and $R_A$ is not monotonous for this weak nonequilibrium case, as expected.[]{data-label="fig1"}](fig1){width="\columnwidth"} We define $\mathbf{q}^*=\mathbf{q}-\mathbf{u}t$ for $\mathbf{u}=(u,0,u,0,\ldots)^{\rm t}$. Then, the total Hamiltonian at time $t$ can be expressed as a function of $\mathbf{q}^*$ with no explicit time dependence, given as $$H(\mathbf{q}^*)=\frac{1}{2}\mathbf{q}^*\cdot\mathsf{A}_{\rm eq}\cdot\mathbf{q}^*=\frac{1}{2}\mathbf{q}^*\cdot(\mathsf{A}_{\rm S}+\mathsf{A}_{\rm I}+\mathsf{A}_{\rm B})\cdot\mathbf{q}^*$$ where various matrices $\mathsf{A}$ are obtained from the corresponding Hamiltonians $H$, $H_{\rm S}$, $H_{\rm I}$, and $H_{\rm B}$ which are quadratic in $\mathbf{q}^*$. We decompose $\mathbf{q}^*$ into a stochastic part $\mathbf{z}$ and a deterministic part $\mathbf{d}$, which are governed by $$\begin{aligned} \dot{\mathbf{d}}=-\mathsf{F}\cdot\mathbf{d}-\mathbf{u}~,~~ \dot{\mathbf{z}}=-\mathsf{F}\cdot\mathbf{z}+\boldsymbol{\xi}(t)~.\label{separation}\end{aligned}$$ Here, $\mathsf{F}$ is a $d\times d$ positive-definite matrix with $d=2(N+1)$. The white noise $\boldsymbol{\xi}(t)$ acts exclusively on the momenta of bath particles. See Sec. I in the Supplementary Material (SM) for the explicit forms of matrices [@SM]. We get $\mathbf{d}(t)=-\mathsf{F}^{-1}(\mathbf{I}-e^{-\mathsf{F}t})\cdot\mathbf{u}$ for the initial condition $\mathbf{d}(0)=\mathbf{0}$. The PDF for $\mathbf{z}$ at time $t$ is given [@kwon-ao-thouless; @kwon] as $\sigma(\mathbf{z},t)=\sqrt{\frac{|\beta\mathsf{A}_t|}{(2\pi)^d}} \exp\left[-\frac{\beta}{2}\mathbf{z}^{\rm t}\cdot\mathsf{A}_t\cdot\mathbf{z}~\right]$ where $\mathsf{A}_t^{-1}=\mathsf{A}_{\rm eq}^{-1}-\mathsf{U}_{t,0}(\mathsf{A}_{\rm eq} -\mathsf{A}_0^{-1}){\mathsf{U}}_{t,0}^{\rm t}$ for $\mathsf{U}_{t,t'}=e^{-\mathsf{F}(t-t')}$. Then, the PDF for $\mathbf{q}$ at $t$ is given by $$\rho(\mathbf{q},t)=\sqrt{\frac{|\beta\mathsf{A}_t|}{(2\pi)^d}} e^{-\frac{\beta}{2}\left[\mathbf{q}-\mathbf{u}t-\mathbf{d}(t)\right]^{\rm t}\cdot\mathsf{A}_t\cdot\left[\mathbf{q}-\mathbf{u}t-\mathbf{d}(t)\right]}. \label{pdf}$$ The nonequilibrium nature of the system is characterized by a nonzero value of $\langle\mathbf{q}\rangle=\mathbf{u}t+\mathbf{d}(t)$. We write the three heats and work accumulated for $0<t<\tau$ using Eq. (\[H\_tot\]) as $$\begin{aligned} W&=&\int_0^\tau dt\frac{\partial U(x,t)}{\partial t}=-ku\int_0^\tau\!\!\!\! {\rm d}t[\mathbf{q}^*(t)]_x \label{work}\\ Q_{\alpha}&=&W-\Delta\left[\frac{1}{2}\mathbf{q}^*\cdot \mathsf{B}_\alpha\cdot\mathbf{q}^*\right] \label{3heats}\end{aligned}$$ where the subscript $x$ denotes the first (system position) component of the vector and $\mathsf{B}_{\alpha}=\mathsf{A}_{\rm S}$, $\mathsf{A}_{\rm S}+\mathsf{A}_{\rm I}$, and $\mathsf{A}_{\rm eq}$, respectively for $\alpha={\rm S},~{\rm B},~{\rm SB}$. We can choose an initial condition according to type (a) such as $\mathsf{A}_0=(\beta'/\beta)\mathsf{A}_{\rm S}+\mathsf{A}_{\rm B}+\mathsf{A}_{\rm I}$ and find $\rho(\mathbf{q},t)$ from Eq. (\[pdf\]). Then, we find $R_{a}$ and $R_{A}$ for Fig. \[fig1\]; see Sec. II in SM [@SM]. The behavior of the two quantities in time is presented in Fig. \[fig1\], which is consistent with the expectation. ![(Color online) Plots for ${\cal P}_{\alpha}^{\rm eq}(r)$ for $r=\beta Q$. The distributions become broader for larger $N$. For the same $N$, ${\cal P}_{\rm SB}^{\rm eq}$ is the broadest and ${\cal P}_{\rm S}^{\rm eq}$ is the sharpest. ${\cal P}_{\rm S}^{\rm eq}(r)=e^{-|r|/2}$ is independent of $N$. []{data-label="fig2"}](fig2){width="\columnwidth"} The generating function for the heat distribution is defined as ${\cal G}_\alpha(\lambda)=\langle e^{-\beta \lambda Q_{\alpha}}\rangle$ where $\langle\cdots\rangle$ denotes the average over all trajectories $\mathbf{z}(t)$ for $0<t<\tau$, and the initial and final states, $\mathbf{z}_0$ and $\mathbf{z}_\tau$, for an initial PDF and the trajectory probability. For convenience, we only consider the $\beta^\prime=\beta$ case for the initial condition, implying that the total system is in equilibrium at the beginning. We have $$\begin{aligned} {\cal G}_{\alpha}(\lambda)&=&\left\langle e^{\frac{\beta\lambda}{2}{\mathbf{q}_\tau^*}^{\rm t}\cdot\mathsf{B}_{\alpha} \cdot\mathbf{q}_\tau^*} e^{\beta\lambda ku \int_0^\tau{\rm d}t [\mathbf{q}^*(t)]_x} e^{-\frac{\beta\lambda}{2}{\mathbf{q}_0^*}^{\rm t}\cdot\mathsf{B}_{\alpha} \cdot\mathbf{q}_0^*}\right\rangle \nonumber\\ &=&c_\alpha N_\alpha\left\langle e^{\beta\lambda\mathbf{d}_\tau^{\rm t}\cdot\mathsf{B}_{\alpha} \cdot\mathbf{z}_\tau+\beta\lambda ku \int_0^\tau{\rm d}t[\mathbf{z}(t)]_x}\right\rangle_{\rm ren}~. \label{generating}\end{aligned}$$ Here $\mathbf{q}_\tau^*=\mathbf{z}_\tau+\mathbf{d}_\tau$ and $\mathbf{q}_0^*=\mathbf{z}_0$ are used to get the second line. $c_\alpha$ is the multiplicative factor independent of integration. $N_\alpha$ is the normalization factor for the renormalized integral due to the alteration of the initial and final PDF’s by $\mathsf{B}_{\alpha}$. The renormalized integral $\langle\cdots\rangle_{\rm ren}$ in Eq. (\[generating\]) can be performed by using the cumulant expansion in terms of renormalized correlation functions $\langle\mathbf{z}(t)^{\rm t}\mathbf{z}(t')\rangle_{\rm ren}$ [@kwangmoo]; see Sec. III in SM [@SM]. In the following, we consider the long-time limit, neglecting terms with $e^{-\mathsf{F}\tau}$ and $e^{-\mathsf{F}^{\rm t}\tau}$. For the equilibrium case ($u=0$), the generating function is given by $N_\alpha$ only, given in the large $\tau$ as $${\cal G}_{\alpha}^{\rm eq}(\lambda)=\sqrt{\frac{|\mathsf{A}_{\rm eq}|^2}{|\mathsf{A}_{\rm eq}+\lambda \mathsf{B}_{\alpha}||\mathsf{A}_{\rm eq}-\lambda \mathsf{B}_{\alpha}|}}=\frac{1}{(1-\lambda^2)^{\nu}}~,$$ where $\nu=1,~1+N/2,~N+1$ for $\alpha={\rm S},~{\rm B},~{\rm SB}$, respectively. Using the Fourier transformation, we evaluate the equilibrium heat distributions for dimensionless heat $r=\beta Q$, given as $${\cal P}_{\rm \alpha}^{\rm eq}(r)= \frac{(|r|/2)^{\nu-1/2}}{\sqrt{\pi}\Gamma(\nu)}K_{\nu-1/2}(|r|)~,$$ where $K_{\nu}(z)$ is the second-kind modified Bessel function of order $\nu$. Figure \[fig2\] shows a clear difference in three heat distributions depending on $N$, but their averages vanish as expected in equilibrium. It is interesting to note that ${\cal P}_{\rm S}^{\rm eq}(r)=e^{-|r|/2}$ is independent of $N$ and has been found to be consistent with the equilibrium heat distribution for the single-particle Langevin system [@kwon_langevin; @CM]. For the nonequilibrium case with $u\neq 0$, we find $\langle W\rangle=ku\tau[ \mathsf{F}^{-1}\cdot\mathbf{u}]_{x}\to N\gamma u^2 \tau$ for large $\tau$. This is exactly $N$ times larger than the corresponding value for the single-particle Langevin system [@kwon_langevin; @CM], which implies that the dissipation coefficient for the colloid particle increases linearly with the number of interacting bath particles. This is consistent with the usual Stokes’ formula [@stokes; @donghwan] and indicates that our rather oversimplified model still describes the colloidal particle dynamics reasonably well. We compute Eq. (\[generating\]) for the large $\tau$ limit and find $${\cal G}_{\alpha}(\lambda)\simeq\frac{e^{-N\tau w\lambda(1-\lambda)-Nw b_{\alpha}\lambda^3/[2(1+\lambda)]}}{(1-\lambda^2)^\nu}~,$$ where $w=\beta\langle W\rangle/(N\tau)=\gamma u^2$ and $b_{\alpha}$’s in unit of time differ for three heats $Q_{\alpha}$; see Sec. IV in SM [@SM]. The heat distribution function for $\beta Q=N\tau w q$ can be obtained by the Fourier integral as $${\cal P}_{\alpha}(q)=\int_{-i\infty}^{i\infty}\frac{{\rm d}\lambda}{2\pi i}\frac{e^{-N\tau w[\lambda(1-\lambda)-q\lambda]-Nw b_{\alpha}\lambda^3/[2(1+\lambda)]}}{(1-\lambda^2)^\nu}~, \label{integral_NEQ}$$ which can be evaluated by using the saddle-point approximation due to singularities [@jslee1; @jslee2; @kwangmoo]. The saddle point $\lambda^*$ occurs in the range $-1<\lambda^*<1$. We consider three piecewise regions: (1) far from $\lambda^*=\pm 1$ corresponding to $-1<q<3$ (center); (2) $\lambda^*\simeq 1$ corresponding to $q<-1$ (left wing); (3) $\lambda^*\simeq -1$ corresponding to $q>3$ (right wing). After some algebra (see Sec. V in SM [@SM]), we find $${\cal P}_{\alpha}(q)=\left\{ \begin{array}{ll} \exp\left[-\frac{N\tau}{4} w(1-q)^2-\frac{1}{2}\ln(w\tau)\right]& \textrm{; $-1< q<3$}\\ \exp\left[N\tau w q+(\nu-1)\ln(w\tau) \right]&\textrm{; $q<-1,~|q+1|\gg \left(\frac{Nw\tau}{8\nu}\right)^{-1/2}$}\\ \exp\left[-N\tau w (q-2)+N\sqrt{2w^2b_\alpha\tau(q-3)}+\frac{1}{2}\left(\nu -\frac{3}{2}\right)\ln\left[w^2b_\alpha\tau(q-3)\right]\right]&\textrm{; $q-3\gg \left(\frac{2 \tau}{b_\alpha}\right)^{-2/3}$} \end{array}\right.~.$$ The most significant corrections to the large deviation function, proportional to $\tau$ in the exponent of the above equation, appear in the right wings for $q>3$. Difference arises from the initial memory effect of different Hamiltonians in Eq. (\[3heats\]) [@farago; @puglisi; @jslee1; @jslee2; @kwangmoo]. For a general strongly coupled system and bath, we find the appearance of the three different heats and various forms of FT involving different heats. 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--- abstract: | Somewhat outside the topic of this conference, some preliminary results on the ongoing spectroscopic study of the six brightest Orion Trapezium stars is presented here. The main purpose of this work is to better understand the multiplicity and stability of each of these subsystems and the dynamical future of the group. So far the most interesting results reached are: 1) The orbit of the secondary star of the eclipsing Component A (V1016 Ori) is highly inclined with respect to the equatorial plane of its primary star. 2) The also eclipsing binary BMOri (Trapezium Component B) does have a tertiary member with period about 3.5 years, as proposed by [@VitriKloch04], and is the same as the companion recently found by the [@GRAVITY+18]. 3) Component D is indeed a spectroscopic and interferometric double star with a relatively high-mass companion ($q=M_2/M_1=0.5$) and period $52.90\pm0.05\,d$. 4) Component F, is a CP star (B7.5pSi); its radial velocity, $23.2\pm4.2\,km\,s^{-1}$, is smaller than that of all other Trapezium members and, possibly, the evolutionary stage of the star is more advanced than that of members with similar mass. Consequently, Component F is probably not physically related to the Trapezium. Several evidences point to the extreme youth of this stellar group; its further study, most likely, will shed light on the formation processes of massive stars. **Key words:** binaries: spectroscopic — stars: early-type — stars: pre-main-sequence — stars: chemically peculiar — stars: individual: $\theta^1$Ori author: - 'R. Costero$^1$' bibliography: - 'Costero.bib' title: Multiplicity of the Orion Trapezium stars --- 1.0cm Introduction ============ It has been a long time since I last participated in research on Gaseous Nebulae, so I apologize for my momentarily switching the subject matter of this meeting down to stellar affairs. In the last years I have been working on the spectral analysis of the Orion Trapezium stars, based on Échelle spectra ($R\simeq$ 15000) obtained with the 2.1-m telescope at the Observatorio Astronómico Nacional in San Pedro Mártir, Baja California. Many colleagues have contributed to obtain the spectra, reduce the data and calculate orbital and eclipsing parameters, among which Juan Echevarría, Yilen Gómez-Maqueo, Lester Fox and Alan Watson stand out. The aim of this work is to clarify some observational parameters of this famous and complex stellar group. The Orion Trapezium ($\theta^{1}$Ori = ADS4186) is the center of the massive star forming region closest to the Sun and the main ionization source of the Orion Nebula. It consists of six bright stars (their $V$ magnitudes spanning from 5.1 to 10.3) that fit inside a circle $22\arcsec$ in diameter (see Figure \[Trap1m\]) and are immersed in the brightest part of the nebula. These circumstances have undoubtedly contributed to the fact that there are many important observational properties associated with this notorious stellar system that are either uncertain or remained unknown until recently. Indeed, the Trapezium stars are difficult targets for small telescopes because of their mutual proximity, whereas they are too bright for most stellar surveys undertaken with bigger telescopes. In particular, the radial velocity of most of its components is poorly known. This is understandable since, in addition to the already mentioned problems, nearly all of these stars are at least spectroscopic binaries. In fact, as we will show here, all but one of the six Trapezium stars are double-lined spectroscopic binaries, three of which are also known eclipsing binaries. It is important to obtain reliable velocity curves of these binaries, because the precise knowledge of their orbital parameters will allow a better calibration of the physical properties of high-mass young stars, in addition to understand their formation process and the evolution of massive multiple systems. For instance, to illustrate the latter point, [@Allen+17] find that realistic N-body simulations of the Orion Trapezium predict that the system will probably become dynamically unstable in a very short time ($< 30\,000\,y$) unless the mass of the components (as presently known) is substantially increased. ![ The Orion Trapezium in a two-second exposure image obtained with the $1.0\,m$ telescope at the Observatorio Astronómico Nacional in Tonantzintla, Puebla, using a Johnson $V$ filter. The primary mirror was covered except for two equal circular openings $20\,cm$ in diameter. Components A and D are separated by about $21\arcsec$. Notice that Component A was near minimum light during primary eclipse.[]{data-label="Trap1m"}](fig1){height="5.0cm"} A few months before this meeting, a very high-resolution interferometric study of the Orion Trapezium Cluster was published [@GRAVITY+18], where some of the spectroscopic binaries observed were spatially resolved and the existence of suspected companions were confirmed. These results highlight the importance of securing precise orbital parameters, derived from radial velocity curves, for the double and multiple members of this very young, compact and interesting stellar system. Here, a short review of what is known about the multiplicity and the orbital elements of the Orion Trapezium members is presented. In addition, some relevant (though preliminary) results of our ongoing research are given, including a brief discussion about the membership to this very young system of Component F, apparently the only single star in this very young and compact stellar group. Component A {#a} =========== $\theta^1$OriA=HD37020=V1016Ori is at least a triple stellar system, with one B0.5 Main Sequence (MS) star and two intermediate-mass, pre-MS companions. Considering that the Orion Nebula Cluster (ONC) has been well studied for stellar variability since the XIX Century, it is surprising that the eclipsing [@Lohsen75] and spectroscopic [@Lohsen76; @Bossi+89] binary nature of this $V= 6.73$mag star was discovered not long ago, rising speculations on a possible recent perturbation or capture in this system. In addition, $\theta^1$OriA is a strong and variable radio [GMR12 in @Garay+87] and X-ray ([*COUP*]{} 745) source. At its maximum, this object becomes the brightest radio source in de ONC. It is important to note that the radio source [**is not**]{} the spectroscopic binary, as originally believed, but the third component in this system [@PetrMassi08 and references therein]. The light curve of V1016Ori during primary eclipse has been rather well established by [@Bondar+00] and by [@LloydStich99]. It is about one magnitude deep, only slightly color dependent in the $UBVI$ photometric range, and approximately 21 hours long. Though wide, its bottom is not flat and corresponds to a partial eclipse produced by a relatively opaque object passing in front of a similar-sized but much brighter star. The secondary eclipse has never been observed and is expected to be very shallow (just a few hundredths of magnitud in V). The period of the eclipsing binary, $P=65.433\,d$, obtained by both the above mentioned groups from the timing of primary minima, is frequently adopted when calculating the spectroscopic orbital parameters, as is the case of those determined by [@Vitri+98] and by [@StickLloyd00] using nearly equal archival data and a few additional measurements of their own. Not surprisingly, both groups reach nearly equal orbital parameters: $e = 0.65(3)$ $K_1 = 33(2)\,km\,s^{-1}$, $\gamma = 28(1)\,km\,s^{-1}$, $\omega = 180^{\circ}(4) $ (numbers in parenthesis are representative of the error in the last digit, as given by those authors). The physical properties of the secondary star of the eclipsing binary have eluded convincing identification, though the initial (and correct) educated guess by [@Lohsen75], based on the eclipse light curve, has almost always been confirmed: it is a pre-MS star. [@VitriPlachinda01], in a very high signal-to noise ratio (S/N) spectrum obtained during the descending branch of a primary eclipse, clearly detected low-excitation lines that they interpret to arise from the secondary and tertiary stars, the former with $T_{eff}\,\approx\,8000\,K$ and the latter with $T_{eff}\,\approx\,3500\,K$. Indeed, the average heliocentric radial velocity of the 13 lines attributed to the secondary (eclipsing) component is $128.8\pm5.5\,km\,s^{-1}$ and that of that of the 7 lines associated to the tertiary star is $33.4\pm11.6\,km\,s^{-1}$ (errors are the standard deviation from the mean). From the former radial velocity and the orbital elements obtained by [@Vitri+98], the first reliable value of $q=M_2/M_1=0.19\pm0.01$ was obtained by [@VitriPlachinda01]. The radial velocity of the tertiary is, within errors, similar to the systemic velocity of the eclipsing binary. Although these results are very important, the spectral range these authors analyzed (5300-5365Å) is small and the fit of the combined synthetic spectra with that of $\theta^1$OriA is rather poor; however, their work is indicative of the excellent possibilities that high-dispersion spectroscopy may bring to the study of this stellar system. Our early attempts (2004-2006) to improve the results for $\theta^1$OriA obtained by [@VitriPlachinda01] were fruitful. In the high S/N spectra obtained inside and around three primary eclipses (at minimum light in two of them), we realized that the spectrum arising from the secondary star was detected, even outside eclipse, when doing a cross-correlation with them and that of an early G-type standard star, and that the width of the spectral lines from the primary star was definitively smaller when inside eclipse as compared with their width well outside eclipse. This change in line width during primary eclipse is due to the Rossiter-McLaughlin (RM) effect [@Rossiter24; @McLaughlin24], that consists of an anomaly in the observed radial velocity of the occulted component, normally (when the the ecuatorial and orbital planes are coplanar) first rising above the orbital radial velocity, back to normal at mid-eclipse, and then below the expected radial velocity until the eclipse ends. However, in $\theta^1$OriA, during the 21-hour long primary eclipse, the radial velocity of the primary star is always noticeably bellow the predicted orbital value, except perhaps in the first few hours. This abnormality is due to the highly inclined orbit of the secondary star with respect to the projected ecuatorial plane of the primary star (large orbital obliquity). Consequently, during the (partial) eclipse, the secondary occults mostly one side of the rotating primary, in this case the hemisphere receding from us. The observed velocity curve inside and around primary eclipse, displaying the RM effect in $\theta^1$OriA, is shown in Figure \[RMenA\]. In this figure, the horizontal axis is the photometric phase calculated using $P = 65.4330\,d$ and $HJD_\circ = 2\,453\,744.7585$. The latter date is that of the observed minimum light in the primary eclipse of 2006 Jan 6, calculated here from the photometric data obtained by Raul Michel Murillo and kindly made available to us. Note in this figure that the minimum radial velocity during the eclipse does not occur at phase zero (minimum light), but slightly later, and that there is a small anomalous rise in the velocity curve in the very inicial part of ingress. These features are indicative that the spin-orbit inclination is close to, but smaller than $90^{\circ}$. Other orbital parameters we derive for $\theta^1$OriA (still subject to revision), using the above mentioned photometric period, are: $e=67\pm0.01$, $\omega=180^\circ\pm2^\circ$ and $q=M_2/M_1=0.20\pm0.1\,M_\odot$. They are in excellent agreement with those by [@Vitri+98] and by [@StickLloyd00]. From the mass ratio and assuming the mass of the primary to be $M_1=15\,M_\odot$ (that of a B0.5V star), the mass of the pre-MS secondary star is derived to be $M_2=3.0\,M_\odot$. ![Velocity curve of $\theta^1$OriA obtained during and around seven primary eclipses, showing the abnormal Rossiter-McLaughlin effect. The horizontal axis is the photometric phase (see text for details). Colors and shapes of data points correspond to different observing seasons.[]{data-label="RMenA"}](fig2){height="5.0cm"} Preliminary analysis of the Échelle spectra obtained in 2004 and 2006, during and around three primary eclipses of $\theta^1$OriA, was performed by [@Valle2011] for his BA thesis. All together, the spectra covered the complete ingress of the eclipse and well passed the minimum light. Adopting the projected rotational velocity, $v\,sin\,i = 55\,km\,s^{-1}$, obtained by [@SimonDiaz+06] of the primary component, and by means of a very simple numerical simulations (no limb darkening in the primary component; opaque, non emitting secondary) [@Valle2011] reproduced the observed RM effect by adjusting the ratio of the stellar radii ($r=R_2/R_1$) and the spin-orbit (obliquity) angle, $l$. In every simulation, the impact parameter (the minimum projected distance between the two stars, in units of the primary star radius $R_1$, was fixed in order to fulfill one magnitude depth of the eclipse. The best fit found with this simple model occurs around $l=70^{\circ}\pm10^{\circ}$ and $r=0.8\pm0.1$. Such high orbital obliquity (spin-orbit angle) is totally abnormal in (eclipsing) binary stars. The only exception I know of is that of DI Her [@Albrecht+09; @Albrecht+11], where the equatorial plane of both members of the binary are strongly tilted with respect to their orbital plane; though no definitive explanation has been given to such misalignment, it could be the consequence of a relatively recent and strong orbital perturbation in a triple system, an extreme case of the Lidov-Kozai mechanism [@Lidov62; @Kozai62], or due to the capture of the secondary star by the primary. In any case, a third component in the system is required, either in a very eccentric orbit or ejected from the system as a consequence of the dynamical perturbation. Indeed, in $\theta^1$OriA there is a third component (see below). [@Valle2011] also derived the effective temperature of the secondary star to be $T_{eff}=5850\pm250\,K$, much lower than any of the other previous estimates. This was done by measuring nine close pair line ratios sensitive to temperature, in two very high S/N spectra of the secondary star (obtained during minimum light in two eclipses), and comparing each of the same pair ratios in synthetic spectra created in the 5000-9000$K$ temperature range, with log g = 3.5 and solar abundances. The third (hierarchical) companion in $\theta^1$OriA, located about 0.2north of the binary, was discovered by [@Petr+98] using holographic speckle interferometry in the $H$ and $K$ bands. In these bands, this star is about 1.4 magnitudes weaker than the out-of-eclipse, combined brightness of the binary components. As mentioned above, it is the highly variable (by at least a factor of 30) radio source that is frequently misidentified with the eclipsing binary; the physical process responsible for the radio emission and its large fluctuations has not been well established [for details, see @PetrMassi08]. The possibility that this third component is an interloper was ruled out very recently, when the [@GRAVITY+18] proved that this interferometric companion is gravitationally bound to the eclipsing binary. When in cross-correlation of some of our high S/N spectra taken inside the primary eclipse, with a G5V standard, there is a hint of a late-type star at about the heliocentric systemic radial velocity of the eclipsing binary. This and other (currently preliminary) results await further analysis. Component B {#b} =========== The weakest of the four stars that originally gave its name to the Orion Trapezium, $\theta^1$OriB=HD37021=BMOri, is at least a sextuple stellar system, so itself constitutes a subtrapezium or mini-cluster. Its brightest member is, as in the case of Component A, a hierarchical triple star also consisting of an eclipsing and spectroscopic binary (BM Ori, in a nearly circular orbit with period $P=6.4705\,d$) and a tertiary component, whose existence was recently, but inadvertently, confirmed beyond doubt by [@GRAVITY+18]. The reminding three components of the sextet are a resolved binary (separated from each other by about $0.12\arcsec$ and at nearly $0.97\arcsec$ from BM Ori), and a much fainter star located $0.6\arcsec$ northwest from the main component. More information about these companions may be found in [@GRAVITY+18] and references therein. The precarious dynamical stability of this subtrapezium has been studied by [@Allen+15]. In what follows, only the close and massive triple system will be analyzed. The eclipsing binary, BM Ori, is a very peculiar Algol-type system: 1) The light curve of the primary eclipse seams to be a total one, with a wide flat bottom (though with small fluctuations) but, during this part of the eclipse, the spectrum of the star is almost equal to that outside eclipse; 2) the secondary eclipse is very shallow, only reasonably well observed in the R and I filters; 3) some time between 1990 and 2010, the duration of the flat bottom of the primary eclipse changed from about 8 hours to less than 4 hours [@Windemuth+13]. Several models have been proposed to explain the first two peculiarities, including the presence of opaque circumsecondary material, a strongly oblate Pre-Main-Sequence secondary, and a compact third star in the system [see @PopperPlavec76; @VasileiskiiVitri00 and references therein]. The change in the duration of the eclipse was interpreted by [@Windemuth+13] as probably due to the very young secondary star actively accreting its circumstellar disk. The main star in Component B is a fast rotator [$v\,sin\,i = 240\,km\,s^{-1}$ according to @Abt+02] so it is a challenging spectroscopic target. On the visible range and at low dispersion, only neutral Hydrogen and Helium lines from the primary component are detectable, all of them contaminated by their nebular counterparts to a greater or lesser degree. The only exception is the Mg II $\lambda 4481 A$ line, that has a strongly variable, non-Gaussian profile. Consequently, it is not surprising that its spectrum has been assigned diverse classifications, from B0 to B4, and that some radial velocity data points and certain parameters of the binary orbit, derived from the primary component spectral lines, differ strongly between authors and epochs. The spectrum of the secondary star of this binary was first weakly detected by [@PopperPlavec76] in high dispersion photographic spectra, mainly during minimum light, as well as in the D, Na I doublet near quadrature. From the latter lines, these authors obtained orbital parameters for the secondary star and, from a group of HeI lines in the photographic range, those of the primary component. These authors conclude that the secondary must be a late A or early F Pre-MS star with mass ratio $q = M_2/M_1 = 0.31$, where $M_1$ and $M_2$ are the masses of the primary and secondary stars, respectively. It took some time for observers to realize that large (up to 30$\,km\,s^{-1}$) discrepancies in some radial velocity data, obtained at different epochs, were not only due to instrumental and measurement errors. [@VitriKloch04] first proposed that a third star was needed in order to explain the discordant orbital parameters and radial velocity outliers. They conclude that the center of mass of the eclipsing binary moves around the center of mass of the putative triple system in a very eccentric orbit (e=0.92) with $P=1302\,d$ and $K_{1,2}=20\,km\,s^{-1}$. The problem with this proposed orbit is that, at periastron (assuming both orbits are coplanar), the tertiary star lays inside the orbit of the eclipsing binary, as shown by [@Vitri+06], who, additionally, cross-correlated a single high S/N spectrum of $\theta^1$OriB with that of synthetic spectra calculated in the 5100-5500Å interval; in doing so, the latter mentioned authors clearly detect the secondary star with a $T_{eff} = 7000\,K$ synthetic spectrum, and claim to have detected the tertiary (with a much lower correlation height) with a $T_{eff} = 4000\,K$ template. Using their previously published orbital parameters [@VitriKloch04] and the spectroscopic mass ($M_1=6.3\pm0.3\,M_\odot$) for the primary star, they estimate of the masses of the two [*satellite stars*]{} to be $M_2=2.5\pm0.1\,M_\odot$ and $M_3=1.8\pm0.2\,$ where subscript 2 and 3 refer to the eclipsing companion and the tertiary star, respectively. The work by [@Vitri+06] show that high resolution and S/N spectra of $\theta^1$OriB, when cross-correlated with that of an early F-type narrow line star or with a $T_{eff} \approx 7000\,K$ synthetic spectrum, may yield precise radial velocities of the secondary component and, hence, accurate orbital parameters of the eclipsing binary, possibly with better precision than those obtained from the rapidly rotating primary star. In fact, the orbital eccentricity of the binary has been assumed to be zero in recent parameter determinations, even though in the first published orbital parameters, those by [@StruveTitus44] and [@Doremus70], it was calculated to be 0.14 and 0.095, respectively. According to [@VitriKloch04], the systemic velocity of the eclipsing binary, $\gamma_{1,2}$, is expected to vary with a period of about $1302\,d=3.56\,y$ due to the putative tertiary star; consequently, the orbital parameters of the close binary (derived from the secondary star) should be obtained from data secured during a single cicle or, at most, during very few consecutive orbital cicles. We did that in January 2010, when we acquired (at least) one high S/N spectrum of the star every night during nine consecutive nights. Setting $P=6.470524\,d$ [the photometric period of BM Ori revised by @Vitri08], we derive the following orbital parameters of the secondary star: $e=0.05$, $K_2=170\,km\,s^{-1}$, $\omega=82^\circ$ and $\gamma_{1,2}=5.9\,km\,s^{-1}$. Notice the very low value of the systemic velocity, as compared to $24\pm3\,km\,s^{-1}$ obtained in previous publications for this binary. This result alone is in agreement with [@VitriKloch04] proposal of a considerably massive tertiary. This result encouraged us to pursue further observations, ideally covering one cycle in each run, by exchanging observing time with other programs. With these data we derive $\gamma_{1,2}$ at different epochs from the velocity curve of the secondary star and by fixing all the other orbital parameters to those calculated in the January 2010 observing run (by far the best sampled one). In Figure \[B12Vrad\] the preliminary results of this work are shown; they correspond to the first nearly five years of observations (from 2010 Jan to 2014 Dec, spanning 1795 days). In this figure, the five available data points are folded with the 1302 d period proposed by [@VitriKloch04]; the vertical axis is the systemic velocity of the eclipsing binary system $\gamma_{1,2}$, obtained from the secondary star as described above. Care should be taken when interpreting this result since we are adjusting four orbital parameters of the triple system ($e, \omega, K_{1,2}$ and $\gamma$) with only five data points and, of course, there are other possible solutions, specially if the period is set free. What is clearly seen is that the systemic velocity of the eclipsing binary, $\gamma_{1,2}$, is indeed variable, with a semi-amplitude of about $20\,km\,s^{-1}$, in excellent agreement with [@VitriKloch04], but with a much smaller eccentricity ($e \approx 0.3$ in the solution shown in Figure \[B12Vrad\]), surely a more stable configuration than that proposed by those authors. It is important to point out taht the stellar object detected by the [@GRAVITY+18] around the eclipsing binary in $\theta^1$OriB is the same tertiary star [@VitriKloch04] postulate, and that its gravitational effects on the close binary are those shown here in Figure \[B12Vrad\]) ; at least, the one-year orbital segment obtained through the interferometric observations by the [@GRAVITY+18] is fully consistent with the 1302-day period. ![Velocity curve of the center of mass of the eclipsing binary in $\theta^1$OriB (BM Ori) folded by the orbital period, $P=1302\,d$, proposed by [@VitriKloch04] for the tertiary star. Error bars are estimates, mostly due to zero-point shifts yet to be determined.[]{data-label="B12Vrad"}](fig3){height="5.0cm"} Components C and D {#other} ================== Nothing can be added here to what is mentioned by the [@GRAVITY+18] about Component C. Summarizing, the brightest star in this subsystem, the intermediate O-type star that is the main source of ionizing photons in the Orion Nebula, is an oblique magnetic rotator, a rare characteristic that has been interpreted as evidence of a collision process in the formation of the star [@ZinneckerYorke07]. In addition, it is an interferometric binary, with a relatively massive companion (also detected in high S/N spectra), in an eccentric orbit ($e=0.69$) with period of 11.3 years. Additionally, it is suspected to be a spectroscopic binary, with a one solar mass secondary star in a 61.5.day period. The systemic radial velocity is, understandably, poorly known; probably its best estimate comes from the spectroscopic orbital parameters of the 11.3-year binary obtained by [@Balega+15], derived from a few usable primary and secondary lines, from which $\gamma(C_{1,2})=29.4\pm0.6\,km\,s^{-1}$ was obtained. Component D ($\theta^1$OriD = HD 37023), a $V=6.7\,mag$, spaectral type B0.5V and $T_{eff}=32\,000\,K$ according to [@SimonDiaz+06], has been suspected to be a spectroscopic binary ever since its first spectral series was observed. However, efforts to find a period for it had been scarce, to put it mildly. [@Vitri02] gathered published radial velocities and measured available IUE spectra for this star; he derived two possible solutions for the orbit with periods $20.27 d$ and $40.53 d$. In some of our high dispersion and S/N spectra we clearly noticed that certain lines, namely the SiIII triplet around $\lambda\,4560$ and CII $\lambda\,4267$, appear double when the weaker component is significantly blue-shifted. These lines reach their maximum intensity at around $T_{eff}=20\,000\,K$ (about type B2 in the MS), so it is reasonable to estimate that the secondary star is about two magnitudes weaker than the primary. Considering $\theta^1$OriD as a double-lined spectroscopic binary, we have measured radial velocities of this star in several high quality spectra obtained, randomly and whenever posible, during the last 14 years. From these we derive $P=52.90\pm0.05\,d$ and, from the primary component lines, the orbital parameters $e=0.42$, $K_1=36\,km\,s^{-1}$, $\omega=9.8^\circ$ and $\gamma=38\,km\,s^{-1}$. All these parameters are being updated and revised, specially the systemic velocity, that is surprisingly large when compared with the radial velocities of other Trapezium members. From those spectra in which the secondary star lines could be deblended from those of the primary, an average mass ratio $q=M_1/M_2=0.47\pm0.05$ has been derived, which is consistent with that expected for a binary made up of a B0V primary and a B2V secondary. The period and the eccentricity we obtain for $\theta^1$OriD are equal, within errors, to those derived by the [@GRAVITY+18], $P=53.0\pm0.7\,d$ and $e=0.43\pm0.7$, an amazing and wonderful result of modern interferometry for a spectroscopic binary at $400 pc$ from the Sun! A very weak additional visual companion, at $1.4\arcsec$ from the bright binary, has not been proved to be physically related to the binary [for details see @GRAVITY+18]. Components E and F {#other} ================== Components E and F are about 3.5 and 5.0 magnitudes weaker than, and located less than $5\arcsec$ to, their closer Trapezium members (Component A and Component C, respectively; see Figure \[Trap1m\]). This explains why their basic observational parameters were (and some, still are) poorly known. $\theta^1$OriE was discovered to be a a double-lined spectroscopic binary by Costero et al. (2006). Its nearly identical members are pre-MS stars with approximate spectral type G2IV; they are in circular orbit with $P=9.8952\,d$ and $K_1=K_2=84.4\,km\,s^{-1}$ [@Costero+08]. This binary is a variable radio [e.g. see @Felli+93] and X-ray source source ([*COUP*]{} 732); it also varies by several tenths of magnitud in the visual range [@Wolf94] and by hundredths of magnitud in the $3,5\,\mu$ and $4.5\,\mu$ bands [@Morales+12]. However, the scarcity of published absolute photometry data on this star is amazing, except for its K-magnitude for which five measures are found in the literature, averaging about 6.8$\,mag$. In a [*Spitzer*]{} survey dedicated to the search and characterization of variability of the ONC stars, [@Morales+12] found that Component E is a grazing eclipsing binary and derived the mass of its members to be $M_1=M_2=2.80\,M_{\odot}$ with 2% accuracy. Together with its measured high- precision proper motion [@Dzib+17] and parallax [@Menten+07; @Kounkel+17], these results make of $\theta^1$OriE the highest mass pre-MS star with well known physical parameters. Component F has received very little attention and practically nothing was known about it. [@Herbig50], in the last sentence of his early paper on the spectroscopy of variable stars in the ONC, just before the acknowledgements and after discussing the spectral type of Component E, writes: “[*Star F can be classified, with considerably more confidence as of type B8*]{}”. This short entry was noticed and registered by [@Parenago54] in his catalogue of stars in the ONC, perpetuating the only spectral classification I know of, published for this star. We have obtained Échelle spectra of $\theta^1$OriF in six nights spanning six years, in order to verify this classification and, of course, to find out its radial velocity and possible multiplicity. To our surprise, the metallic spectral lines in this star are very narrow and numerous. The spectral type we estimate from them is in excellent agreement with that given by [@Herbig50], though in our spectra we register obvious peculiarities, like abnormally strong SiII, SiIII and PII lines, indicating overabundance of these elements and the chemically peculiar (CP) character of the star (also called Ap stars). The HeI lines, expected to be quite strong at $13\,000\,K$ (the temperature we estimate for it) are undetected, probably because they are both weak and filled-in by their nebular counterparts. Hence, we classify $\theta^1$OriF as a CP B7.5p Si star. The mean heliocentric radial velocity we obtain for $\theta^1$OriF from our six spectra is $23.2\pm4.2\,km\,s^{-1}$; the error is the standard deviation from the mean. The internal error in our spectra is expected to be about $2\,km\,s^{-1}$, so this result is not conclusive upon the star being a spectroscopic binary or not. Hence, together with the non detection of a companion to this star by the [@GRAVITY+18], there is no convincing evidence of multiplicity in Component F. Any way, this radial velocity is smaller than that of the three brighter Trapezium members (in particular, those of Components C and D). It is interesting to note that the spectroscopic mass of this B7.5 star should be about $3.7 M_\odot$ if it is in the Main Sequence, slightly larger than that of the identical members of Component E ($2.8 M_\odot$) and of the secondaries of components A ($3.0 M_\odot$) and B ($2.5\,M_{\odot}$), all of them definitively pre-MS stars. So, if Component F is coeval with the other Trapezium members, it must be at or near the turn-on of the zero-age MS of the Trapezium Cluster. Alternatively, Component F is not a member af the Orion Trapezium, but an evolved star located by chance in front of theTrapezium, as proposed by [@Olivares+13]. Conclusions {#discussion} =========== In agreement with the short-term dynamical instability that [@Allen+17] found in $\theta^1$Ori, the Orion Trapezium ($\theta^1$Ori) stellar system, we have shown here that there is clear evidence of extreme youth of the components of this nearby Trapezium: 1\) The large orbital obliquity (spin-orbit angle) we find in the eclipsing binary of Component A (part of a hierarchical triple system) is almost unique among binary stars and probably the consequence of tidal, secular friction with the tertiary member. 2\) The change of the eclipse duration in the close binary, also hierarchical triple Component B [itself part of a six-member unstable mini-cluster, @Allen+17], is probably the result of a sudden change in the circumstellar disk around the very young secondary star [@Windemuth+13], possibly induced by the tertiary, for which we obtain a plausible, medium-eccentricity orbit. 3\) The oblique magnetic rotator nature of the hottest and most massive member of Component C, that possibly originated in a collision during its formation [@ZinneckerYorke07], a process that could still be in progress through gravitational interactions between the massive interferometric companion and the relatively close, $1M_\odot$ spectroscopic companion proposed by [@Vitri02b] and [@Lehmann10+]; 4\) Component E —a variable radio, infrared, optical and X-ray source— is the highest-mass ($2.80 M_\odot$) pre-MS binary known, with most of the physical characteristics of its (practically identical) components determined with high precision. Modern observational techniques will enable the precise dynamical study of the Orion Trapezium system that is probably in flagrant disintegration. They will also be important to better understand the physical characteristics that are still uncertain in several of its members, and to follow the evolution of those observables that are clearly varying. In doing so, I am sure, our knowledge of massive star formation will be greatly benefited. I am deeply grateful to the meeting organizers for this affectionate celebration, and for their including me in it. I am particularly grateful to Oli Dors for obtaining the generous support that made possible my participation. Finally, I sincerely thank Yilen Gómez Maqueo Chew for her valuable suggestions that substantially improved this paper.

--- author: - Shaull Almagor - Udi Boker - Piotr Hofman - Patrick Totzke bibliography: - 'autocleaned.bib' title: 'Parametrized Universality Problems for One-Counter Nets' --- Introduction {#sec:intro} ============ One-Counter Nets (OCNs) are finite-state machines equipped with an integer counter that cannot decrease below zero and which cannot be explicitly tested for zero. They are the same as 1-dimensional Vector Addition Systems (or Petri nets with exactly one unbounded place). In order to use them as formal language acceptors we assume that transitions are labelled with letters from a finite alphabet and that some states are marked as accepting. OCNs are a syntactic restriction of One-Counter Automata – Minsky Machines with only one counter, which can have zero-tests, i.e., transitions that depend on the counter value being exactly zero. If counter updates are restricted to $\pm 1$, the model corresponds to Pushdown automata with a single-letter stack alphabet. OCNs are one of the simplest types of discrete infinite-state systems, which makes them suitable for exploring the decidability border of classical decision problems from automata and formal-language theory. ##### Universality Problems. {#universality-problems. .unnumbered} The universality problem for a class of automata asks if a given automaton accepts all words over its input alphabet. Due to their lack of an explicit zero-test, OCNs are monotone with respect to counter values: if it is possible to make an $a$-labelled step from a configuration with state $p$ and counter $n$ to state $q$ with counter $n+d$, written as $(p,n) \step{a} (q,n+d)$ here, then the same holds for any larger counter value $m\ge n$: $(p,m) \step{a}(q,m+d)$. Consequently, if we define the language via acceptance by reaching a final control state, then for all states $s$ and $n\le m\in\N$, the language $\Lang{s,n}$ of the initial configuration $(s,n)$ is included in that of $(s,m)$. This motivates our first variation of the universality problem. The *Initial-Value Universality* problem asks if there exists a sufficiently large initial counter to make the resulting language universal. The second question we consider is the *Bounded Universality* problem, which asks if there exists a large enough upper bound on the counter so that every word can be accepted via a run that remains within this bound. Writing $\bLang{\initialstate,c_0}{b}\subseteq \alphabet^*$ for the $b$-bounded language from configuration $(\initialstate,c_0)$, the decision problem is as follows. The motivation for studying these parameterized problems comes from the observation that the “vanilla” universality problem, without existentially quantifying over parameters, is decidable, but Ackermann-complete [@HT2017], and the lower bound depends strongly on the assumption that we start with a fixed initial counter (and that its value is not bounded). The two new variants of the universality problem relax these assumptions in an attempt to allow efficient decision procedures via simple cycle analysis or similar. ##### Our Results. {#our-results. .unnumbered} We show that both initial-value universality and bounded universality are undecidable (\[sec:nondet\]). The proofs use techniques from weighted automata [@DDGRT10; @ABK11], reducing the halting problem of two-counter machines to our setting. In light of these negative results, we proceed to study restricted classes of OCNs, for which the problems become decidable, as we elaborate below. In most cases, the complexity crucially depends on how transition updates are encoded: we consider both the case of “succinct”, binary-encoded updates, and the case of unary-encoded updates, which corresponds to systems where transitions can only update the counter by $\pm 1$. The most intricate and interesting case is that of OCNs over a single-letter alphabet (\[sec:unary\]). In order to analyze this model, we split universality to criteria on “short” words, and on longer words that admit a cyclic behavior. In particular, we devise a canonical representation of “pumpable” paths, akin to the so-called linear-path schemes [@LS2004; @BFGHM15]. We show that the complexity of some of the problems is $\coNP$ complete, where others range between $\coNP$ and $\SigmaTwo$ (see \[tbl:Unary,tbl:Binary\]). We then consider deterministic, and unambiguous OCNs (\[sec:deterministic,sec:unambiguous\], respectively). For such systems, deciding (bounded) universality problems mostly reduces to checking simple conditions on the cyclic structure of the control automaton underlying the OCN. Based on known (but in some cases very recent) results on unambiguous finite automata and vector-addition systems, we derive relatively low complexity upper bounds, in polynomial time (assuming unary encoding) and space (assuming binary encoding). summarize the status quo, following our results. ##### Related work. {#related-work. .unnumbered} The undecidability of language universality for pushdown automata is textbook. In his 1973 PhD thesis [@Val1973], Valiant showed that the problem remains undecidable for the strictly weaker model of one-counter automata (OCA, with zero tests) by recognizing the complement of all accepting runs of a two-counter machine. Language inclusion is undecidable for the further restricted model of OCNs [@HLMT2016]. If one considers $\omega$-regular languages defined by OCNs with Büchi acceptance condition then the resulting universality problem is undecidable [@BGHH2017]. On the positive side, universality is decidable for vector addition systems [@JEM1999] and Ackermann-complete for the special case of OCNs [@HT2017]. One-counter systems have received some attention in regards to checking bisimulation and simulation relations, which under-approximate language equivalence (and inclusion, respectively) and are computationally simpler. For OCAs/OCNs, bisimulation is -complete [@BGJ2010], while weak bisimulation is undecidable for OCNs [@Mayr2003]. Both strong and weak simulation are -complete for OCNs, and checking if an OCN simulates an OCA is decidable [@AAHMKT2014]. Universality problems for OCNs over single-letter alphabets are related to the termination problem for VASS, which asks if there exists an infinite run. Non-termination naturally corresponds to the property that $a^n \in \Lang{\initialstate,\vec{v_0}}$, i.e., all finite words are accepted, assuming that all states are accepting. Termination reduces to boundedness (finiteness of the reachability set) which is -complete [@Rac1978; @Dem2013] in general and -complete for systems with fixed dimensions [@RY1985]. In contrast, the *structural* termination problem (there exists no infinite run, regardless of the initial configuration) is equivalent to finding an executable cycle that is non-decreasing on all dimensions, and can be solved in polynomial time [@KS1988]. Finally, the idea to existentially quantify over some initial resource is commonplace in the formal verification literature. Examples include unknown initial-credit problems for energy games [@BFLMS2008; @AAHMKT2014] and R-Automata [@AKY2008], timed Petri nets [@AACMT2018], and inclusion problems for weighted automata [@DDGRT10; @ABK11]. We defer most proofs to the Appendix. Preliminaries {#sec:preliminaries} ============= ##### One-Counter Nets. {#one-counter-nets. .unnumbered} A *one-counter net* (OCN) is a finite directed graph where edges carry both an integer weight and a letter from a finite alphabet. We write $\sys{A}=\ocntuple$ for the net $\sys{A}$ where $\states$ is a finite set of *states*, $\alphabet$ is a finite set of *letters*, $\initialstate\in \states$ is an *initial state*, $\transitions\subseteq \states\x\alphabet\x\Z\x\states$ is the *transition* relation, and $\fstates\subseteq \states$ are the *accepting* states. For a transition $t=(s,a,e,s')\in\transitions$ we write $\effect{t}\eqdef e$ for its (counter) *effect*, and write $\norm{\delta}$ for the largest absolute effect among all transitions. By the *underlying automaton* of an OCN we mean the NFA obtained from the OCN by disregarding the transition effects. A path in the OCN is a sequence $\pi=(s_1,a_1,e_1,s_2)(s_2,a_2,e_2,s_3)\dots (s_{k},a_k,e_k,s_{k+1})\in\transitions^*$. Such a path $\pi$ is a *cycle* if $s_1=s_{k+1}$, and is a *simple cycle* if no other cycle is a proper infix of it. We say that the path above *reads* word $a_1a_2\dots a_k\in\alphabet^*$ and is accepting if $s_{k+1}\in\fstates$. Its $\effect{\pi}\eqdef \sum_{i=1}^k e_i$ is the sum of its transition effects . Its *height* is the maximal effect of any prefix and, similarly, its *depth* is the inverse of the minimal effect of any prefix. An OCN naturally induces an infinite-state labelled transition system in which each *configuration* is a pair $(s,c)\in \states\x\N$ comprising a state and a non-negative integer. We call such a configuration *final*, or *accepting*, if $s\in F$. Every letter $a\in\alphabet$ induces a step relation $\step{a}~\subseteq (Q\x\N)^2$ between configurations where, for every two configurations $(s,c)$ and $(s',c')$, $$(s,c)\step{a}(s',c') \iff (s,a,d,s')\in\transitions \quad\text{and } c'=c+d. $$ A *run* on a word $w=a_1a_2\ldots a_k\in\alphabet^*$ is a path in this induced infinite system; that is, a sequence $\rho=(\initialstate,c_0),(s_1,c_1), (s_2,c_2),\ldots (s_k,c_k)$ such that $(s_{i-1},c_{i-1})\step{a_i}(s_i,c_i)$ holds for all $1\le i\le k$. Naturally, a run uniquely describes a path in the underlying finite OCN. Conversely, for every such path and initial counter value $c_0\in\N$, there is at most one corresponding run: A path $\pi$ is *executable from $c_0$* if its depth is at most $c_0$ (that is, we do not allow the counter to become negative). A run as above is called a (simple) *cycle* if its underlying path is a (simple) cycle. It is *accepting* if it ends in an accepting configuration. We call a run *bounded* by $b\in\N$ if $c_i\le b$ for all $0\le i\le k$. For any fixed initial configuration $(s,c)$, we define its *language* $\Lang[\?A]{s,c}\subseteq\alphabet^*$ to contain exactly all words on which an accepting run starting in $(s,c)$ exists. (We omit the subscript $\?A$ if the OCN is clear from context.) Similarly, the *$b$-bounded language* $\bLang{s,c}{b}$ is the set of those words on which there is a $b$-bounded run starting in $(s,c)$. The OCN is *deterministic* if for every pair $(s,a) \in Q\x\alphabet$ there is at most one pair $(d,q)\in\N\x Q$ with $(s,a,d,s')\in\delta$. A net together with an initial configuration $(\initialstate,c_0)$ is *unambiguous* if for every word $w\in\alphabet^*$ there is at most one accepting run starting in $(\initialstate,c_0)$. ##### Two-Counter Machines. {#two-counter-machines. .unnumbered} A two-counter machine (Minsky Machine) $\M$ is a sequence $(l_1,\ldots,l_n)$ of commands involving two counters $x$ and $y$. We refer to $\set{1,\ldots,n}$ as the [*locations*]{} of the machine. There are five possible forms of commands: `inc(c)`, `dec(c)`, `goto l_i`, `halt`, `if c=0 goto l_i else goto l_j`, where $c\in \set{x,y}$ is a counter and $1\le i,j\le n$ are locations. The counters are initially set to $0$. Since we can always check whether $c=0$ before a $\texttt{dec(c)}$ command, we assume that the machine never reaches $\texttt{dec(c)}$ with $c=0$. That is, the counters never have negative values. Undecidability {#sec:nondet} ============== We show that both initial-value universality and bounded universality are undecidable by reduction from the undecidable halting problem of two-counter machines (2CM) [@Min67]. The idea underlying both reductions is that the initial counter value, or the bound on the allowed counter, prescribes a bound on the number of steps until the OCN must make a decision weather the input word, which encodes a prefix of the run of the 2CM, either halts or cheats. After this decision the OCN is reset and continues to read the remaining word within an adjusted bound. If the decision was correct then the bound remains the same and otherwise, it is strictly reduced. The existence of a halting run of the 2CM now implies that its length corresponds to a sufficient initial bound for this simulating OCN to be universal. Conversely, if the run of the machine does not halt then for every bound $n$, there exists a non-cheating, and non-terminating prefix of length $n$. Repeating this prefix $n$ times witnesses non-universality for the simulating OCN with initial counter $n$. Initial-Value Universality {#sec:NondetInitial} -------------------------- Given a two-counter machine $\M$, we construct a one-counter net $\A$ as follows (see Figure \[fig:A\]). Intuitively, an input word $w$ to $\A$ is a sequence of segments separated by $\#$, where each segment is a sequence of commands from $\M$. Accordingly, the alphabet of $\A$ consists of $\#$ and all possible commands of $\M$. We build $\A$ to accept $w$, once starting with a big enough initial counter value, if one of the following conditions holds: i) one of $w$’s segments is shorter than the length of the (legal halting) run of $\M$; or ii) one of $w$’s segments does not respect the control structure underlying $\M$, which is called a “non-counting cheat” here; or iii) all of $w$’s segments do not describe a prefix of the run of $\M$, making “counting cheats”. The OCN reads every segment in between two $\#$’s starting in, and returning to, a central state $q_0$. Non-counting cheats are easy to verify—for every line $l$ of $\M$, there is a corresponding state $q$ in $\A$, and when $\A$ is at state $q$ and reads a letter $a$, $\A$ checks if $a$ matches the command in $l$. For example, if $l=$ and $a=$ , the transition from $q$ goes to a forever accepting state ($heaven$), and if $a=$, it goes to the state of $\A$ that corresponds to the line $l_i$. This is the “command-checker gadget” of $\A$. Counting cheats are more challenging to verify, as OCNs cannot branch according to a counter value. We consider separately “positive cheats” and “negative cheats”. The former stands for the case that the input letter is (or ) while the value of $x$ (or $y$) in the legal run of $\M$ should be positive. The latter stands for the case that the input letter is (or ) while the value of $x$ (or $y$) in the legal run of $\M$ should be $0$. Positive cheats can be verified by directly simulating the respective counter of $\M$ using the counter in $\A$ (states $q_3$ and $q_5$ in \[fig:A\]). Once the cheat occurs, $\A$ can return to $q_0$ with a penalty of $-1$, and since the counter in $\M$ is positive, we are guaranteed that the counter in $\A$ did not decrease since leaving $q_0$, allowing $\A$ to continue the run. For verifying a negative cheat, we simulate the counting of $\M$ by an “opposite-counting” in $\A$ (states $q_4$ and $q_6$ in \[fig:A\]), whereby an increment of the counter in $\M$ results in a decrement of the counter in $\A$, and vice versa—once the cheat occurs, $\A$ can return to $q_0$ with no penalty, and since the counter in $\M$ is $0$, we are guaranteed that the counter in $\A$ did not decrease since leaving $q_0$, allowing $\A$ to continue the run. Formally, we construct $\A$ from $\M$ as follows. - The alphabet $\Sigma$ of $\A$ consists of $\#$ and the descriptive commands for the counter machine $\M$ : , , , , , and for every line $i$ of $\M$, the commands , , , , and . - The initial state $q_0$ is accepting, it has a self transition over $\Sigma\setminus\{\#\}$ and nondeterministic transitions to the states $q_1\ldots q_6$ over $\#$, all with weight $0$. - There is a $heaven$ state, which is accepting, and has a self loop over $\Sigma$ with weight $0$. - The state $q_1$ is accepting and intuitively allows to accept short segments between consecutive $\#$’s: It has a self transition over $\Sigma\setminus\{\#\}$ and a transition to $heaven$ over $\#$, all with weight $-1$. - The state $q_2$ starts the command-checker gadget, which looks for a non-counting violation of $\M$’s commands (which is a simple regular check). Once reaching a violation it goes to $heaven$. All of its transitions are with weight $0$. If it does not find a violation, it cannot continue the run. - The state $q_3$ is a positive-cheat checker for $\M$’s counter $x$. It has a self loop over with weight $+1$ and over with weight $-1$. Over it can nondeterministically choose between a self loop with weight $0$ and a transition to $q_0$ with weight $-1$. Over the rest of the alphabet lettres, except for and $\#$, it has a self loop with weight $0$. (Over and $\#$ it cannot continue the run.) - The state $q_4$ is a negative-cheat checker for $\M$’s counter $x$. It has a self loop over with weight $-1$ and over with weight $+1$. Over it can nondeterministically choose between a self loop with weight $0$ and a transition to $q_0$ with weight $0$. Over the rest of the alphabet lettres, except for and $\#$, it has a self loop with weight $0$. - The states $q_5$ and $q_6$ provide positive-cheat checker and negative-cheat checker for $\M$’s counter $y$, respectively, analogously to states $q_3$ and $q_4$. \[thm:Undecidable1\] The initial-value universality problem for one-counter nets is undecidable. We show that a given two-counter machine $\M$ halts if and only if the corresponding one-counter net $\A$, as constructed in \[sec:NondetInitial\], is initial-value universal. $\Rightarrow:$ *If $\M$ halts*, its (legal) run has some length $n-1$. We claim that $\A$ is universal with the initial value $n$. Consider some word $w$ over the alphabet of $\A$. We shall describe an accepting run $\rho$ of $\A$ on $w$. Until the first occurrence of $\#$, the run $\rho$ is deterministically in $q_0$, which is accepting. We show that for every segment between two consecutive $\#$’s, as well as the segment after the last $\#$, the run $\rho$ may either reach $heaven$ or reach $q_0$ with counter value at least $n$ (and remains there until the next $\#$ or the end of the word), from which it follows that $\rho$ is accepting. If the segment is shorter than $n$, $q_0$ can choose to go to $q_1$ over $\#$, and from there it will reach heaven. If the segment is longer than $n$, it cannot describe the legal run of $\M$. Then, it must cheat within up to $n$ steps. We show that each of the 5 possible cheats fulfills the claim. 1. : If it makes a non-counting cheat, $q_0$ will go to $q_2$ over $\#$, and will reach $heaven$. (This is also the case if it has additional letters different from $\#$ after the letter.) 2. : If it makes a positive cheat on $x$, $q_0$ will go to $q_3$ upon reading the next $\#$. When the cheat occurs, the value of $x$ is positive, while reading the letter . Notice that the value of $\A$’s counter is accordingly bigger than its value when entering $q_3$ (and by the inductive assumption bigger than $n$). Then, $q_3$ goes to $q_0$ with weight $-1$, guaranteeing that $\A$’s counter value is at least $n$. Notice that the counter value cannot go below $n$ at any point, since $\M$ cannot make the value of $x$ negative without a counting cheat. (We equipped $\M$ with a counter check before every decrement.) 3. : If it makes a negative cheat on $x$, $q_0$ will go to $q_4$. Then, when the cheat occurs, the value of $x$ is $0$, while there is the letter . Notice that the value of $\A$’s counter is accordingly exactly its value when entering $q_3$ (and by the inductive assumption at least $n$). Then, $q_4$ goes to $q_0$ with weight $0$, guaranteeing that $\A$’s counter value is at least $n$. Notice that the counter might go below $n$ between getting to $q_4$ and returning to $q_0$. Yet, since the violation must occur within up to $n$ steps, and the value of the counter when entering $q_4$ is at least $n$, we are guaranteed to be able to properly continue with the run, as the counter need not go below $0$. 4-5. : Analogously, if it makes a positive or negative cheat over $y$, the choice of $q_0$ will be $q_5$ or $q_6$, respectively. $\Leftarrow:$ *If $\M$ does not halt*, for every positive integer $n$, we build the word $w_n$ and show that it is not accepted by $\A$ with an initial counter value $n$. The word $w_n$ consists of $n+1$ segments between $\#$’s, where each segment is the prefix of length $n+1$ of the (legal) run of $\M$. Consider the possible runs of $\A$ on $w_n$. It cannot go from $q_0$ to $q_1$, because it will stop after $n$ steps. It also cannot go to $q_2$, because there is no cheating. We show that if it goes to $q_3..q_6$, it must return to $q_0$ before the next $\#$, while decreasing the value of $\A$’s counter, which can be done only $n$ times until the run stops. If it goes to $q_3$, it must return to $q_0$ upon some , as it cannot continue the run on $\#$. Yet, as there is no cheating, it returns to $q_0$ when $x=0$, which implies that $\A$’s counter has the same value as when entering $q_3$, and due to the $-1$ weight of the transition to $q_0$, it returns to $q_0$ while decreasing the value of $\A$’s counter by $1$. An analogous argument follows if it goes to $q_5$. If it goes to $q_4$, it must return to $q_0$ upon some , as it cannot continue the run on $\#$. Yet, as there is no cheating, it returns to $q_0$ while the value of $x$ is indeed strictly positive, which implies that the value of $\A$’s counter is smaller than the value it had when entering $q_4$, and therefore due to the $0$-weight transition to $q_0$, it returns to $q_0$ with a smaller value of $\A$’s counter. An analogous argument follows if it goes to $q_6$. Bounded Universality {#sec:NondetBounded} -------------------- We show that the problem is undecidable by making some changes to the undecidability proof of the initial-value universality problem. Given a two-counter machine $\M$, we construct a one-counter net $\A'$ that is similar to $\A$, as constructed above, except for the following changes (see Figure \[fig:A’\]): - There is an additional state $q'_0$ that is accepting, it is the new initial state, and it has a nondeterministic choice over $\Sigma$ of either taking a self loop with weight $+1$ or going to $q_0$ with weight $0$. - The state $q_0$ is no longer initial, and it has an additional transition over $\#$ to a new state $q_7$ with weight $0$. - The state $q_7$ is accepting, and it has nondeterministic choice over $\Sigma$ of either taking a self loop with weight $-1$ or going to $q_0$ with weight $-1$. Now $\M$ halts if and only if $\A'$ is bounded universal for an initial counter value $0$. A detailed proof can be found in \[apx:thm:Undecidable1\]. [theorem]{}[undecBU]{} \[thm:Undecidable2\] The bounded universality problem for one-counter nets is undecidable. Singleton Alphabet {#sec:unary} ================== In this section we study universality problems on OCN over singleton alphabets. The universality problem for NFA over singleton alphabets is already -hard [@MS1973], a lower bound which trivially carries over to all problems considered here[^1]. For simplicity, we identify languages $L\subseteq\{a\}^*$ with their Parikh image, so that the universality problems ask if the (bounded) language of a given OCN equals $\N$. Throughout this section, fix an OCN $\?A=\ocntuple$. We start by sketching our approach. Observe that the language of an OCN is not universal iff the OCN does not accept some word $w$. To show that such $w$ exists, we distinguish between two cases: either $w$ is “relatively short”, in which case we use a guess-and-check approach to find it, or it is long, in which case we deduce its existence by analyzing some cyclic behaviour of the OCN. The details of both the guess-and-check elements and the cyclic behaviour depend on the encoding of the weights and the variant of universality. Universality {#subsec: singleton universality} ------------ We start by describing a procedure to decide the ordinary universality problem for OCN over singleton alphabets – with fixed initial configuration and no bounds on the counter. Consider a cycle $\gamma=s_1,s_2,\ldots,s_k$ (with $s_1=s_k$). Recall that $\effect{\gamma}$ is the sum of weights along $\gamma$ and $\depth{\gamma}$ is the inverse of the lowest effect along the prefixes of $\gamma$. We call $1\le d\le k$ a *nadir* of $\gamma$ if it is the index of a prefix that attains the depth of $\gamma$. That is, $\effect{s_1,\ldots, s_d}=-\depth{\gamma}$. We say that $\gamma$ is *positive* if $\effect{\gamma}$ is positive (and similarly for negative, non-negative, zero, etc.). We call $\gamma$ *good* if it a simple, non-negative cycle, and $\depth{\gamma}=0$. \[obs:shifted cycle\] If $\gamma$ is non-negative and it has a nadir $d$, then the *shifted cycle* $\gamma^{\leftarrow d}\eqdef s_d\, s_{d+1},\cdots , s_k,s_2,\cdots ,s_d$ is good. Similarly, if $\gamma$ is negative, then $\effect{\gamma^{\leftarrow d}}=-\depth{\gamma^{\leftarrow d}}$. For a state $r\in\states$ and an initial configuration $\initialstate,c_0$, let $\LangVia{\initialstate,c_0}{r} \subseteq \Lang{\initialstate,c_0}$ be the language of words accepted by a run that visits $r$. The first tool we use in studying the universality problem is a canonical form for accepting runs, akin to *linear path schemes* of [@LS2004; @BFGHM15]. \[def:linear path scheme\] A path $\pi$ is in *linear form* if there exist simple cycles $\gamma_1,\ldots,\gamma_k$ and paths $\tau_0,\ldots,\tau_{k}$ such that $\pi=\tau_0\gamma_1^{e_1}\tau_1\cdots \tau_{k-1}\gamma_k^{e_k}\tau_k$ for some numbers $e_1,\ldots,e_k\in\N$, and such that every non-negative cycle $\gamma_i$, is taken from a nadir, and so is executable with any counter value. We call $e_i$ the *exponent* of $\gamma_i$, and we refer to $\tau_0\gamma_1\tau_1\ldots\gamma_k\tau_k$ as the *underlying path* of $\pi$. The *length* of the linear form is the length of the underlying path. A linear form is described by the components above, where the exponents are given in binary. In the following, we show that every path can be transformed to a path in linear form with a small description size. [lemma]{}[unaryLPF]{} \[lem:linear form of paths\] Let $\pi$ be an executable path of length $n$ from $(p,c)$ to $(q,c')$. Then there exists an executable path $\pi'$ of length $n$ in linear form whose length is at most $2|\states|^2$, from $(p,c)$ to $(q,c'')$ with $c''\ge c'$. $\pi'$ is obtained from $\pi$ in two steps, namely rearranging simple cycles, and then choosing a small set of “representative” simple cycles to replace others. The crux of the proof is the first step, where instead of simply moving a cycle, we also shift it so that it is taken from its nadir. Then, for every set of simple cycles of the same length and on the same state, we take the one with maximal effect as a representative. We now turn to identify states that have a special significance in analyzing universality. \[def:pump states\] Let $\Pump\subseteq\states$ be the set of states that admit good cycles. For each such state $r$ fix a shortest good cycle $\gamma_r$. Intuitively, a state $r$ is in if it has a cycle that can be taken with any counter value, any number of times. That is, it can be used to “pump” the length of the word. Another important property is that if a path never visits a state in then *all* its simple cycles must be negative. Indeed, any non-negative cycle must contain a non-negative simple cycle and any state at a nadir of such cycle must be in . If however, a state in occurs along an accepting run, we can accept the same word using a run in a short linear form, as we now show. [lemma]{}[unaryLPFpump]{} \[lem:linear form with pump state\] There exists a bound $\bound{1}\in\poly(|\states|,\norm{\transitions})$ such that, for every $n\in\N$, if $n$ is accepted by a run that visits a state $r\in \Pump$, then $n$ has an accepting run of the form $\eta_1\gamma_r^t\eta_2$ for paths $\eta_1,\eta_2$ of length at most $\bound{1}$. Using \[lem:linear form of paths\], we split an accepting run on $n$ that visits $r$ to the form $\pi_1,r,\pi_2$ where $\pi_1$ and $\pi_2$ are in linear form. Then, we successively shorten $\pi_1$ and $\pi_2$ by eliminating simple cycles along them, and instead pumping the non-negative cycle $\gamma_r$. Some careful accounting is needed so that the length of the path is maintained, and so that it remains executable. We now characterize the regular language $\LangVia{\initialstate,c_0}{r}$ using a DFA of bounded size. [lemma]{}[unaryPumpDFA]{} \[lem:pump state DFA\] There exists a bound $\bound{2}\in\poly(\norm{\transitions}\cdot\card{\states})$ such that, for every $r\in \Pump$, there exists a DFA that accepts $\LangVia{\initialstate,c_0}{r}$ and is of size at most $\bound{2}$. Define $\PS\eqdef\bigcup_{r\in \Pump} \LangVia{\initialstate,c_0}{r}$. Notice that $\PS\subseteq \Lang{\initialstate,c_0}$ and that $\Lang{\initialstate,c_0}\setminus \PS$ must be finite. Indeed, if $w\in \Lang{\initialstate,c_0}\setminus \PS$ then it can only be accepted by runs with *only* negative cycles, of which there are finitely many. In particular, if $\N\setminus \PS$ is infinite, then $\Lang{\initialstate,c_0}\neq \N$. Using the bounds from \[lem:pump state DFA\], we have the following. [lemma]{}[unaryWitnessBound]{} \[lem:singleton alphabet witness bound\] There exists $\bound{3}\in\poly(\norm{\transitions},\card{\states})$ such that $\Lang{\initialstate,c_0}\neq \N$ if, and only if, there exists $n\in \N$ such that either $n<\bound{2}$ and $n\notin \Lang{\initialstate,c_0}$, or $\bound{3}^{|\states|}\le n\le 2\bound{3}^{|\states|}$ and $n\notin \PS$. suggests the following algorithmic scheme for deciding non-universality: non-deterministically either (1) guess $n<\bound{3}$, and check that $n\notin \Lang{\initialstate,c_0}$, or (2) guess $\bound{3}^{|\states|}\le n\le 2\bound{3}^{|\states|}$ and check that $n\notin \LangVia{\initialstate,c_0}{r}$ for all $r\in \Pump$, which implies that $n\notin\?P$. Note that even if the transitions are encoded in unary, $n$ still needs to be guessed in binary for part (2) (and also for part (1) if the encoding is binary). The complexity of the checks involved in both parts of the algorithm depend on the encoding of the transitions, and are handled separately in the following. ##### Unary Encoding. {#unary-encoding. .unnumbered} If the transitions are encoded in unary, then $\bound{3}$ is polynomial in the size of the OCN. Consequently, we can check for $n<\bound{3}$ whether $n\in \Lang{\initialstate,c_0}$ by simulating the OCN for $n$ steps, while keeping track of the maximal run to each state. Indeed, due to the monotonicity of executability of OCN paths it suffices to remember, for each state $s$, the maximal possible counter-value $c$ so that $(s,c)$ is reachable via the current prefix, which must be a number $\le c_0 + n\cdot \norm{\transitions}$ or $-\infty$ (to represent that no configuration $(s,c)$ can be reached). Next, in order to check whether $n\notin \LangVia{\initialstate,c_0}{r}$ for all $r\in \Pump$ for $\bound{3}^{|\states|}\le n\le 2\bound{3}^{|\states|}$ written in binary, we notice that since $\bound{3}$ is polynomial in the description of the OCN, then the size of each DFA for $\LangVia{\initialstate,c_0}{r}$ constructed as per \[lem:pump state DFA\] is polynomial in the OCN. Since the proof in \[lem:pump state DFA\] is constructive, we can obtain an explicit representation of these DFAs. Finally, given a DFA (or indeed, and NFA) over a singleton alphabet and $n$ written in binary, we can check whether $n$ is accepted in time $O(\log n)$ by repeated squaring of the transition matrix for the DFA [@MS1973]. We conclude with the following. \[thm:singleton alphabet unary coNP\] The universality problem for singleton-alphabet one-counter nets with transitions encoded in unary is in , and is thus -complete. ##### Binary Encoding. {#binary-encoding. .unnumbered} When the transitions are encoded in binary, $\bound{3}$ is potentially exponential in the encoding of the OCN. Thus, naively adapting the methods taken in the unary case (with basic optimization) will lead to a $\PSPACE$ algorithm for universality (using Savitch’s Theorem). As we now show, by taking a different approach, we can obtain an upper bound of $\SigmaTwo$, placing the problem in the second level of the polynomial hierarchy. In order to obtain this bound, we essentially show that given $n$ encoded in binary, checking whether $n$ is accepted by the OCN can be done in . This is based on the linear form of \[lem:linear form of paths\]. [lemma]{}[unaryCheckLPS]{} \[lem:checking linear path scheme\] Let $\pi=\tau_0\gamma_1^{e_1}\tau_1\cdots \tau_{k-1}\gamma_k^{e_k}\tau_k$ be a run in linear form, then we can check whether $\pi$ is executable from counter value $c$ in time polynomial in the description of $\pi$. shows that, given $n$ in binary, we can check whether $n\in \Lang{\initialstate,c_0}$ in $\NP$. Indeed, we guess the structure of an accepting run in linear form (including the exponents of the cycles), and check in polynomial time whether this run is executable, and whether it is accepting. In order to complete our algorithmic scheme for universality, it remains to show how we can check in $\NP$, given $n$ in binary, whether $n\notin \LangVia{\initialstate,c_0}{r}$ for every $r$. In contrast to the case of unary encoding, this is fairly simple. Given $r$, we can construct an OCN $\?A^r$ such that $\Lang[\?A^r]{\initialstate,c_0}=\LangVia[\?A]{\initialstate,c_0}{r}$ by taking two copies of $\?A$, and allowing a transition to the second copy only once $r$ is reached. The accepting states are then those of the second copy. Thus, checking whether $n\notin \LangVia{\initialstate,c_0}{r}$ amounts to checking whether $n\notin \Lang[\?A^r]{\initialstate,c_0}$. We can now complete the algorithmic scheme. [theorem]{}[unaryUnivBinary]{} \[thm:singleton alphabet binary Sigma2\] The universality problem for singleton-alphabet one-counter nets with transitions encoded in binary is in $\SigmaTwo$. Initial-Value Universality {#initial-value-universality} -------------------------- The characterization of universality given in \[lem:singleton alphabet witness bound\] can be simplified in the case of initial-value universality, in the sense that the freedom in choosing an initial value allows us to work with the underlying automaton of the OCN, disregarding the transition effects. This also allows us to obtain the same complexity results under unary and binary encodings. Recall that $\Pump$ is the set of states that admit good cycles (see \[def:pump states\]). Let $\?N$ be the underlying NFA of $\?A$. For a state $r\in \Pump$, define $\LangVia[\?N]{\initialstate}{r}$ to be the set of words accepted by $\?N$ via a run that visits $r$. Overloading the notation of \[subsec: singleton universality\], we define $\PS\eqdef\bigcup_{r\in \Pump}\LangVia[\?N]{\initialstate}{r}$. [lemma]{}[unaryIVchar]{} \[lem:singleton alphabet init-univ characterization\] There exists $c_0$ such that $\Lang[\?A]{\initialstate,c_0}=\N$ iff $\Lang[\?N]{\initialstate}=\N$ and $\N\setminus \PS$ is finite. Following similar arguments to those in \[lem:linear form with pump state,lem:pump state DFA\], and using the fact that we work with the underlying NFA, we can show the following. [lemma]{}[unaryIVdfa]{} \[lem:singleton alphabet init-univ DFA\] There exists a bound $\bound{4}\in \poly(|\states|)$ such that, for every $r\in \Pump$ there exists a DFA that accepts $\LangVia{\initialstate}{r}$ and which is of size at most $\bound{4}$. We can now solve the initial-value universality problem. \[thm:Singleton-iv-universality\] The initial-value universality problem for one-counter nets (in unary or binary encoding) is -complete. First, observe that the problem is -hard by reduction from the universality problem for NFAs. We now turn to show the upper bound. By \[lem:singleton alphabet init-univ characterization\], it is enough to decide whether $\Lang[\?N]{\initialstate}=\N$ and $\N\setminus \PS$ is finite. Checking whether $\Lang[\?N]{\initialstate}=\N$, i.e., deciding the universality problem for NFA over a single-letter alphabet, can be done in  [@MS1973]. By \[lem:singleton alphabet init-univ DFA\], there exists a DFA $\?D$ for $\N\setminus \PS$ of size at most $M=\bound{4}^{\card{\states}}$, by taking the intersection of the respective DFAs over every $r\in\Pump$. Thus, $\N\setminus \PS$ is infinite iff $\?D$ accepts a word of length $M< n\le 2M$ (as such a word induces infinitely many other words). Thus, we can decide in $\NP$ whether $\N\setminus \PS$ is infinite, by guessing $M<n\le 2M$, and checking that it is in $\LangVia{\initialstate}{r}$ for every $r\in \Pump$ (using repeated squaring on the respective DFAs). We conclude that both checking whether $\Lang[\?N]{\initialstate}=\N$ and whether $\N\setminus \PS$ is finite can be done in $\coNP$, and so the initial value universality problem is also in $\coNP$. Bounded Universality {#bounded-universality} -------------------- For bounded universality, the states in are not restrictive enough: in order to keep the counter bounded, a state must admit a $0$-effect cycle. However, these cycles need not be simple. Thus, we need to adjust our definitions somewhat. Fortunately, however, once the correct definitions are in place, most of the proofs carry out similarly to those of \[subsec: singleton universality\]. \[def:stable states\] A state $q\in \states$ is *stable* if either: 1. it is at the nadir of a simple positive cycle, and admits a negative cycle, or 2. it is at the nadir of a simple zero cycle. We denote by $\Stable$ the set of stable states. Identifying stable states can be done in polynomial time (see e.g. \[lem:DOCN-conditions\]). The motivation behind this definition is to identify states that admit a zero-effect (not necessarily simple) cycle. [lemma]{}[unaryBUstable]{} \[lem:singleton alphabet bounded-univ stable zero cycles\] There exists a bound $\bound{5}\in \poly(|\states|,\norm{\transitions})$ such that, every stable state $q$ admits a zero cycle of length and depth at most $\bound{5}$. By \[lem:singleton alphabet bounded-univ stable zero cycles\] we can fix, for each $q\in \Stable$, some zero-cycle $\zeta_q$ with effect and depth bounded by $\bound{5}$. Recall that $\LangVia{\initialstate,c_0}{r}$ is the set of words that are accepted with a path that passes through $r$. Let $\?S\eqdef\bigcup_{r\in \Stable}\LangVia{\initialstate,c_0}{r}$. We prove an analogue of \[lem:linear form with pump state\]. [lemma]{}[unaryBUlinearForm]{} \[lem:singleton alphabet bounded-univ linear form stable state\] There exists a bound $\bound{6}\in\poly(|Q|,\norm{\transitions})$ such that every $n\in \LangVia{\initialstate,c_0}{r}$ has an accepting run of the form $\eta_1\zeta_r^t\eta_2$ for paths $\eta_1,\eta_2$ of length at most $\bound{6}$. The proof follows *mutatis-mutandis* that of \[lem:linear form with pump state\], with one important difference: before replacing cycles with iterations of the zero cycle $\zeta_r$, we replace a bounded number of cycles with the positive cycle on $r$, on which $r$ is at a nadir,[^2] so that the counter value goes above $\depth{\zeta_r}$, enabling us to take $\zeta_r$ arbitrarily many times. Note that this lengthens the prefix $\eta_1$ at most polynomially in $(|\states|\cdot \norm{\transitions})$. implies that every word $n\in\?S$ can be accepted by a run whose counter values are bounded because there must by an accepting run that, except for some bounded prefix and suffix, only iterates some zero-cycle $\zeta_r$. More precisely, we have the following. \[cor:singleton alphabet bounded-univ bounded langvia\] There exists $\bound{6}\in \poly(|\states|,\norm{\transitions})$ such that every word $n\in \?S$ is accepted by a run whose counter value remains below $2\bound{6}+c_0$. In addition, \[lem:singleton alphabet bounded-univ linear form stable state\] immediately gives us (with an identical proof) an analogue of \[lem:pump state DFA\]. [lemma]{}[unaryBUdfa]{} \[lem:singleton alphabet bounded-univ DFA\] There exists a bound $\bound{7}\in\poly(|Q|,\norm{\transitions})$ such that, for every $r\in \Stable$ there exists a DFA that accepts $\LangVia{\initialstate,c_0}{r}$ and is of size at most $\bound{7}$. We can now characterize bounded universality in terms of $\?S$, the set of stable states. [lemma]{}[unaryBUchar]{} \[lem:singleton alphabet bounded-univ characterization\] $\Lang{\initialstate,c_0}$ is bounded-universal if, and only if, the underlying automaton $\?N$ is universal ($\Lang[\?N]{\initialstate}=\N$) and $\N\setminus \?S$ is finite. Finally, checking whether $\N\setminus \?S$ is finite can be done similarly to \[subsec: singleton universality\] (and the complexity depends on the transition encoding), by checking that a candidate word $n$ of bounded length is not in $\LangVia{\initialstate,c_0}{r}$ for all stable states $r$. We conclude with the following. \[thm:singleton alphabet bounded-univ complexity\] Bounded universality of one-counter nets is $\coNP$-complete assuming unary encoding, and in $\SigmaTwo$ assuming binary encoding. Deterministic Systems {#sec:deterministic} ===================== We turn to deterministic one-counter nets (DOCNs) for which the underlying finite automaton is a DFA. We assume without loss of generality that the graphs underlying the DOCNs are connected, i.e., that all states are reachable from the initial state. For such systems, (bounded) universality problems can be decided by checking a suitable combination of simple conditions on cycles and short words. In order to prevent tedious repetition, we list these conditions first and prove (in \[apx:lem:DOCN-conditions\]) upper bounds for checking each of them (\[lem:DOCN-conditions\]). We then show which combination allows to solve each decision problem (\[lem:DOCN-universalities-char\]). All mentioned upper bounds follow either easily from first principles, or from the result that the state reachability problem (a.k.a., coverability) for OCN is in  [@almagor2019coverability Theorem 15]. We will also use the following fact, which follows from [@IntroductionToCircuitComplexity] (see \[apx:lem:binary-addition\]). [lemma]{}[detBinaryAddition]{} \[lem:binary-addition\] \[lem:addition of n numbers in binary\] Given a set $S=\{\alpha_1, \alpha_2\ldots \alpha_n\}$ of integers written in binary, the question whether the sum of all elements in $S$ is non-negative is in [^2^]{}. [lemma]{}[detConditions]{} \[lem:DOCN-conditions\] Consider the following conditions on a deterministic one-counter net $\sys{A}=\ocntuple$, initial value $c_0\in\N$, and bound $b\in\N$. : The underlying automaton is universal. : Every word $w$ of length $\len{w}\le \card{\states}$ is in $\Lang{\initialstate,c_0}$ : Every word $w$ of length $\len{w}\le \card{\states}$ is in $\bLang{\initialstate,c_0}{b}$ : All simple cycles have non-negative effect. : All simple cycles have $0$-effect. Condition can be checked in non-deterministic logspace (), independently of the encoding of numbers. All other conditions can be verified in  assuming unary encoding, and in  (conditions and even in [^2^]{}) assuming binary encoding. [lemma]{}[detUs]{} \[lem:DOCN-universalities-char\] Consider a deterministic one-counter net with initial state $\initialstate$. 1. For any $c_0\in\N$, the language $\Lang{\initialstate,c_0}$ is universal if, and only if, all simple cycles are non-negative , and all words shorter than the number of states are accepting . 2. There exists an initial counter value $c_0\in\N$ such that $\Lang{\initialstate,c_0}$ is universal if, and only if, all simple cycles are non-negative , and the underlying automaton is universal . 3. For any $c_0\in\N$, there exists a bound $b\in\N$ such that the bounded language $\bLang{\initialstate,c_0}{b}$ is universal if, and only if,  the effect of all simple cycles is $0$ and  all words shorter than the number of states are in $\bLang{\initialstate,c_0}{b'}$ for $b'\eqdef\card{\states}\cdot\norm{\transitions}$. The following is a direct consequence of \[lem:DOCN-conditions,lem:DOCN-universalities-char\]. \[thm:DOCN-complexities\] The universality, initial-value universality, and bounded universality problems for deterministic one-counter nets are in  assuming unary encoding, and in [^^]{} assuming binary encoding. For the special case of DOCN over single letter alphabets, it is possible to derive even better upper bounds, based on the particular shape of the underlying automaton. Recall that a deterministic automaton over a singleton alphabet is in the shape of a lasso: it consists of an acyclic path that ends in a cycle. [lemma]{}[detUnaryConditions]{} \[lem:DOCN-conditions\_singleton\] For any given deterministic one-counter net $\sys{A}=\ocntuple$ with $\card{\alphabet}=1$ and $c_0,b\in\N$, one can verify in deterministic logspace (Ł) that the underlying DFA is universal. Moreover, conditions , , , and as defined in \[lem:DOCN-conditions\] can be verified in Ł assuming unary encodings and in [^2^]{} assuming binary encodings. Using \[lem:DOCN-conditions\_singleton\] and the characterisation of the three universality problems by \[lem:DOCN-universalities-char\], we get the desired complexity upper bounds. \[lem:DOCN-singleton-complexities\] The universality, initial-value universality, and bounded universality problems of deterministic one-counter nets over a singleton alphabet are in Ł assuming unary encoding and in [^2^]{} assuming binary encoding. Unambiguous Systems {#sec:unambiguous} =================== In line with the usual definition of unambiguous finite automata, we call an OCN with a given initial configuration *unambiguous* iff for every word in its language there exists exactly one accepting run. Since the language of an OCN depends in a monotone fashion on the initial counter value, there is also a related, but different, notion of unambiguity. We call an OCN (which has a fixed initial state $\initialstate$) *structurally unambiguous* if the unambiguity condition holds for every initial counter $c_0$. Notice that every OCN that has an unambiguous underlying automaton is necessarily structurally unambiguous. We will show (\[lem:UOCA-structural-unambiguity\]) that these conditions are in fact equivalent. In [@czerwiski:hal-02483495], the complexity of the universality problem for unambiguous vector addition systems with states (VASSs) was studied. In particular, for unambiguous OCNs, it is shown that checking universality is in [^2^]{} and -hard, assuming unary encoded inputs, and in  and -hard, assuming binary encoding. The special case of unambiguous OCN over a single letter alphabet is not considered there, nor are the initial-counter – and bounded universality problems. We discuss these problems in the remainder of this section. We assume w.l.o.g, that for any given OCN, all states in the underlying automaton are reachable from the initial state, and that from every state it is possible to reach an accepting state. States that do not satisfy these properties can be removed in . Moreover, all algorithms we propose need to check universality for the underlying automaton, and hence rely on the following computability result (see [@WENGUEY199643] for a proof for general alphabet, and \[apx:lem:universality\_of\_UFA\] for singleton alphabet). [lemma]{}[UFAUniv]{} \[lem:universality\_of\_UFA\] Universality of an unambiguous finite automaton over single letter alphabet is in , and over general alphabet is in [^2^]{}. We will start by considering the universality problem for unambiguous OCNs over a single letter alphabet. Here, unambiguity implies a strong restriction on accepting runs: if a run is accepting then it contains at most one positive cycle (which may be iterated multiple times). [lemma]{}[unambOneLoop]{} \[lem:UOCA-single-loop\] Let $\pi=\pi_1\pi_2\pi_3$ be an accepting run where $\pi_2$ is a positive simple cycle. Then $\pi_3=\pi_2^k\pi_4$ for some $k\in\N$ and acyclic path $\pi_4$. Assume towards contradiction that there is an accepting run $\pi=\pi_1\pi_2\pi_3\pi_4\pi_5$, where $\pi_2$ is a positive simple cycle and $\pi_4$ is a simple cycle. Based on this we show that the system cannot be unambiguous. Let $c=\card{\states}\cdot\norm{\transitions}$ and denote by $\len{\pi}$ the length of path $\pi$. Since $\pi_2$ has a positive effect, it follows that $\pi'=\pi_1\pi_2^{\len{\pi_4}+c\cdot \len{\pi_2}}\pi_3\pi_4\pi_5$ is an accepting run. But there is a second run that reads the same word, namely $\pi''=\pi_1\pi_2^{c\cdot \len{\pi_2}}\pi_3\pi_4^{\len{\pi_2}}\pi_5$. The second run is indeed a run as the increment along $ \pi_2^{c\cdot\len{\pi_2}}$ is bigger than any possible negative effect of $\pi_4^{\len{\pi_2}}$. Moreover the lengths of both runs are the same as $\pi_2^{\len{\pi_4}}=\pi_4^{\len{\pi_2}}$. A consequence of \[lem:UOCA-single-loop\] is that if along any accepting run the value of the counter exceeds $\bound{0}=\card{\states}\cdot\norm{\transitions}$ then it cannot drop to zero afterwards, as it would require at least one negative cycle to do so. One can therefore encode all counter values up to $\bound{0}$ into the finite-state control and solve universality for the resulting UFA. thus yields the following. [theorem]{}[unambU]{} \[thm:UOCA-universality-unary-single\] The universality problem of unary encoded unambiguous one-counter nets over a singleton alphabet is in . We consider next the initial-value universality problem for unambiguous OCNs. Since whether an OCN is unambiguous depends on the initial counter value, the initial-value universality problem is only meaningful for structurally unambiguous systems, those which are unambiguous regardless of the initial counter. We first observe a simple fact about these definitions. [lemma]{}[unambSU]{} \[lem:UOCA-structural-unambiguity\] An OCN is structurally unambiguous if and only if its underlying automaton is unambiguous. [lemma]{}[unambSUstruct]{} \[lem:SUOBA-structure\] Consider a structurally unambiguous OCN with initial state $\initialstate$. There exists an initial counter $c_0$ so that $\Lang{\initialstate,c_0}=\alphabet^*$ if, and only if, the underlying automaton is universal and has no negative cycles. The following is a direct consequence of \[lem:SUOBA-structure\] and the complexity bounds provided by \[lem:universality\_of\_UFA,lem:DOCN-conditions\], for the cycle condition . \[thm:SUOCN-iv-universality\] The initial-value universality problem of structurally unambiguous one-counter nets is in [^2^]{} assuming binary encoding, and in  assuming unary encoding and single-letter alphabets. Finally, we turn our attention to the bounded universality problem for unambiguous OCNs. This turns out to be quite easy, due to the following observation. [lemma]{}[unambBUnoLoops]{} \[lem:UOCN-no-positive-loops\] If an unambiguous OCN is bounded universal then no accepting run contains a positive cycle. [theorem]{}[unambBU]{} \[thm:UOCN-bu\] \[thm:UOCN-bu-bin\] The bounded universality problem of unambiguous one-counter nets with unary-encoded transition weights is in [^2^]{}, and in  if the alphabet has only one letter, and for binary-encoded transition weights it is in . Proofs of \[sec:nondet\] {#apx:thm:Undecidable1} ======================== \[apx:thm:Undecidable2\] We show that a given two-counter machine $\M$ halts if and only if the corresponding one-counter net $\A'$, as constructed in \[sec:NondetBounded\], is bounded universal for an initial counter value $0$. $\Rightarrow:$*If $\M$ halts*, its (legal) run has some length $n-1$. We claim that $\A'$ is universal with the counter bound $2n$. Consider some word $w'$ over the alphabet of $\A'$. We shall describe an accepting run $\rho'$ of $\A'$ on $w'$. In the first $n$ steps, $\rho'$ remains in $q'_0$, increasing the counter to $n$. Then, it moves to $q_0$. In the rest of the run, $\rho'$ continues as the accepting run $\rho$ of $\A$ on the word $w$ that is the suffix of $w'$ from the $n+1$ position (as described in the proof of \[thm:Undecidable1\]), except for the following changes: whenever it is in $q_0$ and the counter is bigger than $n$, it goes to $q_7$ on $\#$. In $q_7$, it uses the self loop until the counter’s value becomes $n$ and then goes to $q_0$. If the length of $w'$ is up to $n$, then $\rho'$ is obviously accepting, as it remains in the accepting states $q'_0$ and $q_0$, and the counter need not exceed $2n$ nor go below $0$. If the length of $w'$ is more than $n$, we prove that for every segment between two consequent $\#$’s, as well as the segment after the last $\#$, the run $\rho'$ may either reach $heaven$ or reach $q_0$ with counter value at least $n$, and proceed from $q_0$ to $q_1..q_6$ with counter value exactly $n$. This will immediately imply that $\rho'$ is accepting. The challenge is to show that the counter of $\A'$ never needs to exceed $2n$. (It does not go below $0$, since we go from $q_0$ to $q_1..q_6$ with a counter value of at least $n$ (in this case exactly $n$), which satisfies the assumptions in the proof of \[thm:Undecidable1\].) Now, in states $q_1, q_2, q_4, q_6$, and $q_7$ there is no problem, as the counter never gets above its value when entering these states. Yet, in states $q_3$ and $q_5$ there is a potential problem, since $\A'$’s counter increases when $\M$’s counters increase. However, since the (legal) run of $\M$ is of length $n-1$, a violation must occur within up to $n$ steps. Hence, getting to states $q_3$ and $q_5$ with counter value of exactly $n$, the run $\rho'$ may return to $q_0$ over the first violation, and thus need not increase the counter’s value to more than $2n$. Observe that when returning to $q_0$ the counter’s value might be bigger than $n$, in which case $\rho'$ will later decrease it to exactly $n$ by going to $q_7$. $\Leftarrow:$ *If $\M$ does not halt*, for every positive integer $n$, we build the word $w'_n$ and show that it is not accepted by $\A'$ for an initial counter value $0$ and a bound $n$ on the counter. The word $w'_n$ consists of $n+2$ segments between $\#$’s, where each segment is the prefix of length $n$ of the (legal) run of $\M$. Consider the possible runs of $\A'$ on $w'_n$. In $q'_0$ it can stay up to $n$ steps, entering $q_0$ with a counter value of up to $n$. Then it should accept from $q_0$ the suffix of $w'_n$, which contains $n+1$ segments as described above. However, as shown in the proof of \[thm:Undecidable1\], using all states except for $q_7$, it must decrease the counter value in each segment, and so is the case if using $q_7$. Hence, the run must stop after at most $n$ segments and cannot be accepting. Proofs of \[sec:unary\] ======================= Let $n\in \Lang{\initialstate,c_0}$, and let $\pi=\initialstate,s_1,\ldots,s_n$ be an accepting run of the OCN on $n$. For each state $q$ visited by $\pi$, let $\first(q)$ and $\last(q)$ denote the first and last indices where $q$ occurs in $\pi$, respectively. Let $\Marks\eqdef\{\first(q),\last(q) : q\mbox{ occurs in }\pi\}$ be the set of all markings in $\pi$. Observe that $|\Marks|\le 2|\states|$. We reshape $\pi$ into linear form in two phases. In the first phase, we move cycles around such that in the obtained path, any infix between two marked positions consists of a simple path, and a collection of simple cycles. In the second phase, we replace most of the simple cycles, such that any infix between two marked positions consists of a relatively short path, and a single repeating cycle (which completes the linear form). Crucially, in both phases we must take care that the path remains executable. The crux of the proof is that instead of simply shifting cycles, we also change their starting point, such that they always start from a nadir, thus making them executable with any counter value. For the first phase, consider an interval $[i,i+|\states|]$ in $\pi$ that does not intersect $\Marks$ (if no such interval exists, we proceed to the second phase). Since this interval has $|\states|+1$ states, it contains some simple cycle $\gamma=x_1,x_2,\ldots,x_k$. Let $d$ be a nadir of $\gamma$, and observe that necessarily $\first(x_d)<i$ and $\last(x_d)>i+|\states|$, since the interval $[i,i+|\states|]$ does not contain any marks. We now split into two cases. - If $\effect{\gamma}\ge 0$, we modify $\pi$ by removing the cycle $\gamma$ from the interval $[i,i+|\states|]$, and instead adding the shifted cycle $\gamma^{\leftarrow d}$ at index $\first(x_d)$. Observe that the modified path is still executable, since by \[obs:shifted cycle\] the cycle $\gamma^{\leftarrow d}$ is good, and can be executed with any counter value, and following its execution, the remaining path either has higher counters (up to where $\gamma$ occurred) or the same values as in $\pi$ (after where $\gamma$ occurred). - If $\effect{\gamma}< 0$, we modify $\pi$ by removing the cycle $\gamma$ from the interval $[i,i+|\states|]$, and instead adding the shifted cycle $\gamma^{\leftarrow d}$ at index $\last(x_d)$. Observe that the modified path is still executable. Indeed, by \[obs:shifted cycle\] $\effect{\gamma^{\leftarrow d}}=-\depth{\gamma^{\leftarrow d}}$, and so $\gamma^{\leftarrow d}$ can be executed as long as the counter is at least $\effect{\gamma^{\leftarrow d}}$. Moreover, removing this negative cycle results in a run in which, all counter-values from the index of removal are increased by $-\effect{\gamma}$. In particular, at index $\last(x_d)$ it is at least $0+\effect{\gamma^{\leftarrow d}}$, so $\gamma^{\leftarrow d}$ can be executed. Notice that moving a negative cycle like this results in a path that is executable an has the same effect as $\pi$. This completes the first phase. We remark that conceptually, this cycle modification takes place in a single “shot” for all cycles, so that the indices in $\Marks$ do not change after every cycle is moved, but are rather the same for all cycles being moved (otherwise intervals may “expand”, and $\Marks$ becomes ill-defined). We now proceed to the second phase. Let $\pi'$ be the path obtained after the first phase. We refer to any cycle that was moved in $\pi$ as a *dangling cycle*. Thus, $\pi'$ consists of at most $2|\states|$ intervals[^3] that contain no non-dangling cycles, and at most $2|\states|$ indices on which there are dangling cycles (namely the indices in $\Marks$). Furthermore, the dangling cycles always start at their respective nadirs. We now proceed to eliminate most dangling cycles at each state. Consider some mark $\first(q)$ or $\last(q)$ in $\Marks$. For each $1\le t\le |\states|$, consider all simple cycles of length $t$ where $q$ is a nadir, and let $\mu_{q,t}$ be such a cycle of maximal effect. We now replace every dangling cycle of length $t$ in $\first(q)$ with $\mu_{q,t}$. Clearly the effect of the cycles does not decrease, so the path remains executable. Furthermore, we maintain the length of the paths, so the path still represents a run on $n$. Finally, within each mark, we can bunch the cycles by length, so that all cycles of the same length are executed consecutively. Thus, the obtained path consists of at most $2|\states|$ simple paths and $2|\states|\cdot |\states|=2|\states|^2$ simple cycles, which is a linear form as required. Let $\gamma_r$ be a shortest good cycle on $r$, and let $\rho$ be a an accepting run that passes through $r$. We write $\rho=\pi_1,r,\pi_2$, where $\pi_r$ is a prefix of the run before it visits $r$ and $\pi_2$ is the suffix after visiting $r$ (note that $r$ may occur in $\pi_2$). Furthermore, by \[lem:linear form of paths\] we can assume $\pi_1$ and $\pi_2$ are in linear form of length at most $2|Q|^2$. Thus, we can write $\pi_1=\tau_0\gamma_1^{e_1}\tau_1\cdots \tau_{k-1}{\gamma_k}^{e_k}\tau_k$ with $k\le 2|Q|^2$, and similarly for $\pi_2$. We now start by replacing negative cycles in $\pi_1$ and in $\pi_2$ by repetitions of $\gamma_r$ (the good cycle on $r$). This is done as follows. For every subset of cycles whose combined length equals $m|\gamma_r|$ for some $m\in \N$, we remove those cycles and replace them by $m$ iterations of the good cycle $\gamma_r$. Since we only remove negative cycles, and since $\gamma_r$ has non-negative effect and depth $0$, the run remains executable. Recall that the $\gamma_i$ cycles are simple, and are therefore of length at most $|Q|$. Thus, after removing cycles in this manner, we are left with at most $|\gamma_r|-1\le |\states|$ negative cycles of every length. We now aim to remove non-negative cycles in the same fashion. This, however, requires some caution, as some cycles might have effect greater than that of $\gamma_r$, or appear before the run visits state $r$ for the first time, and therefore replacing them with $\gamma_r$ may cause the path to become non-executable. Recall that by \[def:linear path scheme\] (and indeed, by the construction in the proof of \[lem:linear form of paths\]) all the non-negative $\gamma_i$ cycles start from their nadir, and therefore have depth $0$. In addition, after removing the negative cycles as done above, the path length (excluding the non-negative cycles) is at most $2|\states|^2+|\states|^2=3|\states|^2$ in each of $\pi_1$ and $\pi_2$. Thus, the maximal depth possible along the entire path is $6|\states|^2\norm{\transitions}$. Thus, as long as a (strictly) positive cycle (or a combination thereof) is taken enough times to maintain the counter above $6|\states|^2\norm{\transitions}$, the path remains executable. We can now proceed to replace non-negative cycles with $\gamma_r$ in the same manner done for negative cycles, while maintaining executability. We thus end up with a modified run of the form $\eta_1 \gamma_r^{t} \eta_2$ where $\eta_1$ and $\eta_2$ are of length $\poly(|\states|,\norm{\transitions})$, which implies the claim. From \[lem:linear form with pump state\] it follows that there exists a bound $\bound{1}\in \poly(|\states|,\norm{\transitions})$ such that every word accepted with a run that goes through $r$ is of the form $x+y|\gamma_r|$ where $x,y\in \+N$ and $x\le \bound{0}$. Thus, we can construct a DFA of size $\bound{2}\eqdef\bound{1}+|\gamma_r|$ whose form is an initial prefix of length $\bound{1}$, followed by a cycle of length $|\gamma_r|$, and whose accepting states correspond to all the $x$ above, with corresponding accepting states on the cycle. Let $\bound{2}$ be as per \[lem:pump state DFA\], and define $\bound{3}\eqdef\bound{2}^{|\Pump|}\le \bound{2}^{|\states|}$. Observe that by taking the product of the DFAs obtained in \[lem:pump state DFA\], we can construct a DFA $\?D$ of size at most $\bound{3}$ for $\N\setminus \PS$. Then, $\N\setminus \PS$ is infinite iff there exists a word of length $\bound{3}\le n\le 2\bound{3}$ that is accepted by $\?D$ (as such a word is necessarily accepted by a run that contains a cycle in $\?D$). Towards the claim, if $\N\setminus \PS$ is infinite, then $\Lang{\initialstate,c_0}\neq \N$, and clearly if there exists $n<\bound{2}$ such that $n\notin \Lang{\initialstate,c_0}$ then again, $\Lang{\initialstate,c_0}\neq \N$. Conversely, assume $\Lang{\initialstate,c_0}\neq \N$. We claim that either there exists $n<\bound{2}$ with $n\notin \Lang{\initialstate,c_0}$, or $\N\setminus \PS$ is infinite. Indeed, observe that since $\?D$ is obtained as the product of singleton-alphabet DFAs, then it has a “lasso” shape: a finite prefix of states, followed by a cycle. Moreover, the size of the prefix is at most $\bound{2}$, namely the maximal size of the prefix in each of the DFAs in the product. Thus, if there exists $n<\bound{2}$ with $n\notin \Lang{\initialstate,c_0}$ then we are done, and otherwise there is some $n>\bound{2}$ with $n\notin \Lang{\initialstate,c_0}$, and in particular $n\notin \PS$, so $\?D$ accepts some word along its cycle, and so accepts infinitely many words, and in particular some word $\bound{3}\le n\le 2\bound{3}$. Checking that the transitions follow those of the OCN can be done in polynomial time, since we only need to check the underlying path, regardless of the exponents. In order to check that the counter value remains non-negative, we observe that for any cycle $\gamma_i$, if $\effect{\gamma_i}\ge 0$, then $\gamma_i$ is taken from a nadir (by \[def:linear path scheme\]), and hence can be taken with any counter value. If that is the case, then we can compute directly $\effect{\gamma_i^{e_i}}=e_i \cdot\effect{\gamma_i}$. Otherwise, if $\effect{\gamma_i}< 0$, then in order to check if $\gamma_i^{e_i}$ is executable from counter value $c$, it suffices to check that $(e_i-1)\cdot\effect{\gamma_i}-\depth{\gamma_i}\le c$. Indeed, for negative cycles, the last iteration is the “hardest”. Again, we can now compute $\effect{\gamma_i^{e_i}}=e_i\cdot \effect{\gamma_i}$. Thus, we can keep track of the counter value along the underlying path, and update it directly for every cycle. This takes polynomial time overall. For the first direction, assume $\Lang[\?A]{\initialstate,c_0}=\N$ for some $c_0$. Clearly $\Lang[\?N]{\initialstate}=\N$ as otherwise some word is not accepted in the underlying NFA, let alone the OCN. Assume by way of contradiction that $\N\setminus \PS$ is infinite, and recall that in every accepting run on a word $n\in \N\setminus \PS$, all cycles must be negative. Thus, for long enough words, the counter value, starting at $c_0$, must become negative, which is a contradiction. Conversely, if $\N\setminus \PS$ is finite and $\Lang[\?N]{\initialstate}=\N$, we can take an initial counter value large enough so that all words not in $\PS$ have accepting runs. Then, similarly to \[lem:linear form with pump state\], we can show that every word in $\PS$ has an accepting run of the form $\tau_1\gamma_r^{t}\tau_2$ with $\tau_1$ and $\tau_2$ of length $\poly(|\states|)$ and where $\gamma_r$ is the canonical good cycle from state $r\in\Pump$ with maximal effect. Notice here that the bound on the lengths of paths $\tau_1$ and $\tau_2$ is polynomial only in the number of states and not, as in \[lem:linear form with pump state\], also in $\norm{\transitions}$. This is because we can safely remove any combination of simple cycles in these sub-paths without preserving the executability of the resulting path in the net. A large enough counter value ensures that the prefix and suffix are executable, so all words in $\PS$ are accepted as well. If $q$ is at the nadir of a simple zero cycle, then $\card{\states}$ bounds its length and we are done. Otherwise, since $q$ admits a negative cycle, then there is a state $x\in Q$ that admits a simple negative cycle $\gamma$ such that $x$ and $q$ are reachable from each other. Let $\tau_1$ and $\tau_2$ be simple paths from $q$ to $x$ and from $x$ to $q$, respectively. Let $s=\effect{\tau_1\tau_2}+1$, then $\chi=\tau_1\gamma^s\tau_2$ is a negative cycle of length at most $3|\states|\cdot \norm{\transitions}$. Let $\eta$ be a simple positive cycle that has a nadir at $q$. Then $q$ admits the zero cycle $\zeta_q=\eta^{-\effect{\chi}}\cdot \chi^{\effect{\eta}}$ and $\bound{5}\eqdef \card{\states} \cdot (\card{\states}\cdot\norm{\transitions}) + (3\card{\states}\cdot\norm{\transitions}) \cdot \card{\states} $ satisfies the claim. By \[cor:singleton alphabet bounded-univ bounded langvia\], there exists a bound $\bound{7}$ such that all words in $\?S$ are accepted with paths whose counter values remains below $\bound{7}$. Hence, if there are only finitely many words that are outside $\?S$, and $\Lang[\?N]{c_0}=\N$, then the counter values among the runs on the remaining finite set of words are clearly bounded. Hence, $\Lang{\initialstate,c_0}$ is bounded-universal. Conversely, assume $\N\setminus \?S$ is infinite, we show that $\Lang{\initialstate,c_0}$ is not bounded-universal. First, if $\Lang[\?N]{\initialstate}\neq \N$ the OCN cannot be universal, and in particular it is not bounded-universal. Observe that by \[def:stable states\], words outside $\?S$ can be accepted only with paths on which the number of alternations between positive and negative cycles is at most $|\states|$, and that do not contain zero cycles. Since only finitely many words can be accepted using a bounded number of positive cycles, it follows that if $\N\setminus\?S$ is infinite, then for every $M\in\N$ there exists a word that is only accepted by runs that have a positive cycle taken at least $M$ times, and hence have effect at least $M$. It follows that $\Lang{\initialstate,c_0}$ is not bounded-universal. Following our algorithmic scheme, an $\NP^\NP$ algorithm for non-universality proceeds as follows. non-deterministically either (1) guess $n<\bound{3}$, and check (using an $\NP$ oracle as per \[lem:checking linear path scheme\]) that $n\notin \Lang{\initialstate,c_0}$, or (2) guess $\bound{3}^{|\states|}\le n\le 2\bound{3}^{|\states|}$ and check that $n\notin \Lang[\?A^r]{\initialstate,c_0}$ for all $r\in \Pump$, using $|\states|$ calls to an $\NP$ oracle as per \[lem:checking linear path scheme\]. Proofs of \[sec:deterministic\] {#apx:lem:binary-addition} =============================== Addition of two integers written in binary can be done in ${\AC\textsuperscript{0}}$ [@IntroductionToCircuitComplexity], and therefore in ${\NC\textsuperscript{1}}$. As the summation of $n$ numbers can be done in $\log n$ iterations (whereby each iteration reduces the number of elements by a factor of $2$ by adding up $\alpha_{2i}$ and $\alpha_{2i+1}$, for every index $i$ up to half the number of elements), and each iteration is in ${\NC\textsuperscript{1}}$ (by performing in parallel all of these additions), we get that the overall problem is in [^2^]{}. \[apx:lem:DOCN-conditions\] **Unary encoding**. All conditions can be shown to be in   using the theorems of Savitch (reachability in finite directed graphs is in ) and Immerman–Szelepcsényi ($\NL=\coNL$). Indeed, holds iff no non-accepting state is reachable in the underlying automaton. For the remaining conditions, just notice that the assumption that inputs are given in unary means that all relevant numbers are bounded polynomially in the input. For instance, to show that does not hold, one simply guesses the offending simple cycle and stepwise computes its effect in binary representation. **Binary encoding**. Let’s first consider condition . This fails iff there is a short word whose run in $\sys{A}$ either ends in a non-accepting state or reduces the counter below zero. The first case is again a simple reachability condition in the underlying DFA. The second case reduces to a coverability problem as follows. For $k\in\N$, let $\sys{A}\x k \eqdef (\states\x\{0,1,\ldots,k\},\alphabet,\transitions',\fstates',\initialstate')$ be the OCN that results from $\sys{A}$ by adding a step-counter up to $k$ into the states. That is, $\transitions'\eqdef \{((p,i),\alpha, e, (q,i+1)): (p,\alpha,e,q)\in \transitions, i\le k\}$, $\fstates'\eqdef \fstates\x\{0\ldots k\}$, and $\initialstate'\eqdef (\initialstate,0)$. Further, let $\sys{B}$ denote the OCN $\sys{A}\x\card{\states}$, in which all transition effects are inverted. Notice that for every word $w$ of length $\len{w}\le \card{\states}$, the effect of its induced run in $\sys{A}$ (and $\sys{B}$) is between $-\bound{}$ and $\bound{}$, for $\bound{}\eqdef \card{\states}\cdot\norm{\transitions}$. Such a word cannot be accepted by $\sys{A}$ from $(\initialstate,c_0)$ iff the run it induces in $\sys{B}$ starting from $(\initialstate', \bound{})$ leads to some configuration $((q,\len{w}),(\bound{}+c_0+1))$. This reachability question about $\sys{B}$ can be answered in [^^]{} [@almagor2019coverability Lemma 1 and Theorem 15 ], and since $\sys{A}$ and $\sys{B}$ are of polynomially the same size, also in [^^]{} with respect to $\sys{A}$. An  upper bound for condition is completely analogous and differs only in that an additional reachability check should be taken, in which the weights in $\sys{B}$ are not inverted and the target configuration is $((q,\len{w}),(\bound{}+b-c_0+1))$. Conditions and on the effect of simple cycles can be verified in by a similar reduction to coverability. For example, to check if a simple cycle with negative effect exists it suffices to check that it is possible in $\sys{B}$ to start in a configuration $((q,0),\bound{})$ and cover a configuration $((q,k),(\bound{}+1))$ for some $0< k < |Q|$. We can do slightly better than that and check these conditions in [^2^]{}, as follows. Let $Q=\{p_1,p_2, \ldots, p_{|Q|}\}$, and for every $0< k < \card{\states}$, let $M_k$ denote the $\card{\states}\times \card{\states}$ matrix of elements in $\Z\cup\infty$, where the entry for $i,j$ equals the minimal effect of a path of length $k$ from state $p_i$ to $p_j$. Then, $M_k$ can be computed in [^2^]{} using standard repeated-squaring in the min-plus semiring [@aho1974design] To check condition , that all simple cycles have non-negative effect, we just need to check (in parallel) that all entries in the main diagonal of all the $M_k$ matrices are non-negative. The same procedure, applied to an OCN that is derived from $\sys{A}$ by inverting all transition weights, allows to check for the presence of positive simple cycles, and hence for an [^2^]{} algorithm to check condition . 1. (Normal Universality): \[apx:lem:DOCN-universality\] Clearly both conditions are necessary for the system to be universal. To see why they are sufficient for universality, assume that holds and consider shortest word $w\not\in\Lang{\initialstate,c_0}$. Then the run on $w$ cannot contain any non-negative cycle because this would contradict the minimality assumption. Since we assume , that all cycles are non-negative, the run on $w$ must have no cycles. Thus, $\len{w}\le \card{\states}$ which is impossible due to . 2. (Initial-Value Universality): \[apx:lem:DOCN-iv-universality\] If both conditions hold then any cycle on any run must have non-negative effect. So if one picks $c_0\eqdef \card{\states}\cdot\norm{\transitions}$ then the counter cannot become negative on any run and the language $\Lang{\initialstate,c_0}$ equals that of the underlying automaton, namely $\alphabet^*$ by condition . Conversely, since $\Lang{\initialstate,c_0}$ is always included in the language of the underlying automaton, condition  is clearly necessary. If  fails then, because the system is deterministic, for every number $c_0$ there must be a word $w(c_0)\in\alphabet^*$ whose run has an effect strictly below $-c_0$. Then $w\notin\Lang{\initialstate,c_0}$. Therefore both conditions are necessary. 3. (Bounded Universality): \[apx:lem:DOCN-bounded-universality\] Trivially, both conditions are necessary. For the opposite direction, assume that the conditions hold. We contradict the assumption that $\bLang{\initialstate,c_0}{b'}\neq\alphabet^*$. If that was the case, we can pick a shortest word $w$ not in that language. The run of this word cannot contain a cycle, because by condition all cycles have zero effect on the counter and therefore the presence of a cycle on the run would contradict the assumed minimality of $\len{w}$. This implies that $w$ is no longer than the number of states, and by condition  it must be in $\bLang{\initialstate,c_0}{b'}$. Contradiction. \[apx:lem:DOCN-conditions\_singleton\] Condition is equivalent to checking that all states are accepting ($\states=\fstates$). For the other conditions, notice that if all numbers are encoded in unary then one only needs to compute numbers bounded polynomially in $\card{Q}$ and $\norm{\transitions}$. This can be done in deterministic logspace by representing them in binary. If numbers are already encoded in binary then the [^2^]{} bounds follow from \[lem:binary-addition\]. Proofs of \[sec:unambiguous\] ============================= \[apx:lem:universality\_of\_UFA\] The lemma was proven in [@WENGUEY199643], for the general alphabet. For the single letter alphabet we have that if the language is not universal then the shortest not accepted word is bounded by $|Q|$ [@DBLP:conf/dcfs/Colcombet15] (Lemma 2). Thus to verify universality, we need to test if for every $0\leq i\leq |Q|$ there is an accepting run of length $i$, which can be tested in $\NL$. \[apx:thm:UOCA-universality-unary-single\] By \[lem:UOCA-single-loop\] it is possible to construct an unambiguous finite automaton (UFA) of polynomial size, which is universal if and only if the net is universal. This can be done by bounding the counter from above by $\bound{0}$, remembering its value in the states, and switching to a copy of the underlying automaton once the counter is observed to exceed this bound. It is easy to see that every run in the net induces a run in the automaton and vice-versa. The number of states of this new finite automaton is $\card{\states}\cdot (1+\bound{0}) + \card{\states}$. Since the constructed UFA is still over a single letter alphabet, we can check if it is universal  by \[lem:universality\_of\_UFA\]. \[apx:lem:UOCA-structural-unambiguity\] If the underlying automaton is unambiguous then the net is as well, as every run of the net is also a run of the automaton. In the opposite direction, suppose that the underlying automaton is not unambiguous, then there is a word $w$ read by two accepting runs $\pi_1$ and $\pi_2$. If we start with the counter value bigger than $(\len{\pi_1}+\len{\pi_2}) \cdot \norm{\transitions}$ then the both runs in the underlying automaton will describe two different accepting runs in the OCN. \[apx:lem:SUOBA-structure\] *“If”*. If all cycles have non-negative effect then an initial value of $c_0\eqdef\bound{0}$ suffices to ensure that no run can drop the counter below zero. Consequently, the system behaves just like its underlying automaton, which is universal by assumption. *“Only if”*. The language $\Lang{\initialstate}$ of the underlying automaton clearly includes $\Lang{\initialstate,c}$ for any value $c\in\N$. By assumption that there is $c_0$ with $\Lang{\initialstate,c_0}=\alphabet^*$, the underlying automaton must be universal. It remains to show that it cannot contain any (reachable) simple cycles with negative effect. Towards a contradiction, suppose that $\pi_1\pi_2\pi_3$ is an accepting run from a configuration $(\initialstate, c_0)$ and that $\effect{\pi_2}<0$. Then there is must exist $k\in \N$ such that $\pi_1\pi_2^k\pi_3$ is not a run from the configuration $(\initialstate,c_0)$, as the counter runs out. By assumption, that the language of the net with initial configuration $(\initialstate,c_0)$ is universal, there must be another run $\pi_4$ on the same word, and which is accepting. But now both runs, $\pi_4$ and $\pi_1\pi_2^k\pi_3$, are accepting from the configuration $(\initialstate,c_0+\norm{\transitions}\cdot \len{\pi_2}\cdot k)$ as the effect of $\pi_2^k$ is larger than $\norm{\transitions}\cdot \len{\pi_2}\cdot k$. This means that the net is not structurally unambiguous, which contradicts our assumptions. Suppose otherwise, then for any bound $k$ there will be an accepting run which is going through configurations with counter value bigger than $k$, and from unambiguity, there is no other run that stays below the bound. \[apx:thm:UOCN-bu\] **Unary encoded transitions:** By \[lem:UOCN-no-positive-loops\], if the OCN is bounded universal then every accepting run will only visit counter values below $\bound{1}\eqdef c_0+\bound{0}=c_0+\card{\states}\cdot\norm{\transitions}$. This means that the OCN is bounded universal if, and only if, $\bLang{\initialstate,c_0}{\bound{1}}=\alphabet^*$. This can be verified by checking universality for the UFA that results by remembering all bounded counter values in the finite state space. The claim now follows by \[lem:universality\_of\_UFA\]. **Binary encoded transitions:** By \[lem:UOCN-no-positive-loops\], if the OCN is bounded universal then every accepting run will only visit counter values below $\bound{1}\eqdef c_0+\bound{0}=c_0+\card{\states}\cdot\norm{\transitions}$. This means that the OCN is bounded universal if, and only if, $\bLang{s_0,c_0}{\bound{1}}=\alphabet^*$. This can be verified by checking universality for the UFA that results by remembering all bounded counter values in the finite state space. The claim now follows by \[lem:universality\_of\_UFA\] and the following fact [^^]{}$=PolyLog$ applied to the UFA which is of exponential size. [^1]: The proof in [@MS1973 Theorem 6.1] in fact shows -completeness of the problem of whether two regular expressions over $\{0\}$ define different languages. Hardness is shown by reduction from Boolean satisfiability to non-universality of expressions using prime-cycles, and it is straightforward to rephrase it in terms of DFAs. [^2]: That is, unless $r$ is the nadir of a zero cycle, in which case the proof requires no changes. [^3]: The first and last indices of $\pi$ must be marked and so there are in fact at most $2|\states|-1$ intervals.

--- abstract: 'Unsupervised domain adaption aims to learn a powerful classifier for the target domain given a labeled source dataset and an unlabeled target dataset. The key challenge lies in aligning the distribution of the two domains to alleviate the effect of the domain shift. Recent research indicates that generative adversarial network (GAN) based adversarial learning can help to learn domain-invariant representations. In this paper, we propose a very simple model for unsupervised domain adaption. Specially, we share a single encoder between the source domain and target domain which is expected to extract domain-invariant representations with the help of a discriminator. By sharing the encoder, the model could receive images from both source and target domains and does not discriminate the source of images during testing. Besides, the distributions will be aligned in an online way rather than aligning the target features with pre-prepared source features from a pretrained model as done in previous studies. In addition, we highlight the importance of learning discriminative features for unsupervised domain adaption which is ignored by previous studies and integrate the classification loss with the center loss. Besides the marginal distribution, we also align the conditional distributions during the adversarial learning. We evaluate the proposed method on several unsupervised domain adaption benchmarks and achieve better performance than state-of-the-art methods.' author: - | First Author\ Institution1\ Institution1 address\ [firstauthor@i1.org]{} - | Second Author\ Institution2\ First line of institution2 address\ [secondauthor@i2.org]{} bibliography: - 'egbib.bib' title: 'Domain-invariant Discriminative Feature Learning for Unsupervised Domain Adaption' --- Introduction ============ Deep neural networks have drawn broad attention in many fields due to its impressive performance on a variety of tasks. Training a deep neural network usually requires a large labeled dataset. However, collecting and annotating a dataset for each new task is time-consuming and expensive. Fortunately, the big data era makes a large amount of data available for other domains and tasks and using the auxiliary data related to the current task may be helpful to alleviate the necessity of annotating large scale labeled data. However, due to some factors such as image condition, illumination and pose, the distributions of datasets from two domains are usually different. When the model trained on one dataset is directly tested on another one, the performance will be greatly discounted due to the existence of “domain shift". Domain adaption, as a sub-line of transfer learning, is proposed to solve the “domain shift" problem. For unsupervised domain adaption where all samples in the target domain are unlabeled, many studies try to align the statistical distributions of the source domain and target domain using various mechanisms. The most commonly used methods for comparing and reducing the distribution discrepancy include maximum mean discrepancy (MMD) [@long2015learning; @long2016unsupervised; @pan2011domain], correlation alignment (CORAL) [@sun2016return; @sun2016deep] and Kullback-Leibler (KL) divergence [@zhuang2015supervised]. Recently, some studies adopted the adversarial domain adaption method to align the feature distribution by extracting features which are indistinguishable by the discriminator about the domain information [@tzeng2017adversarial]. However, they usually train two separate encoders for source domain and target domain and this will require the source of samples to be known during testing. And the source encoder is usually pretrained first and then fixed when training the target encoder and the discriminator. In addition, only aligning the distribution of samples may be not enough for learning a powerful classifier for target domain. Although the distributions are aligned, there may be many samples falling into the margin of clusters which will be likely misclassified. Therefore, it is important to learn discriminative features for both source domain and target domain. In this paper, we propose a very simple but effective model for unsupervised domain adaption. Inspired by the phenomenon that people can receive massive images of one object under various environment and then correctly recognize it, we design our model which could receive images from both source domain and target domain without distinction about image source, and could extract domain-invariant features. In detail, instead of using two separate feature extractors for source domain and target domain, we share a single feature extractor between these two domains. Samples from both the source and target domain will pass through the feature extractor and then the features of source and target samples will be distinguished by the discriminator. The feature extractor and the discriminator will play the min-max game with the goal that the features are distinguished by the discriminator about the domain information. By this way, the encoder will learn to extract content information shared by the two domains and ignore private domain information. In addition, the distributions of source domain and target domain are online aligned, which is different from previous studies where the source features are prepared with a pretrained encoder on the source domain. Furthermore, we highlight the importance of extracting discriminative features for unsupervised domain adaption and augment the classification loss with the center loss. Besides the marginal distribution, we also align the conditional distribution of source and target domain, which has been ignored by most adversarial-based domain adaption methods. Since aligning conditional distribution $P(Y|X)$ is quite challenging due to the absence of labels in target domain, we resort to align the class-conditional distribution $P(X|Y)$. Synthesizing the above sections, the learned features will be domain-invariant and discriminative for both source domain and target domain. The main contributions of this paper are summarized as following. - A very simple model is proposed for unsupervised domain adaption, which does not distinct the source of images and can extract domain-invariant and discriminative features for both source and target domains. - The model shares one feature extractor between two domains and the data distributions of two domains will be online aligned to extract domain-invariant features. - We highlight the importance of extracting discriminative features for unsupervised domain adaption and augment the classification loss with the center loss. - Besides the marginal distribution, we also align the conditional distributions of source and target domain. Related Work ============ For unsupervised domain adaption, the main approach is to guide the feature learning by minimizing the difference between the distributions of source domain and target domain. Several methods have used the MMD to measure the difference of distributions. [@pan2011domain] proposed the Transfer Component Analysis (TCA) to minimize the discrepancy of two domains in a Reproducing Kernel Hilbert Space (RKHS) using MMD. Then, [@long2015learning; @tzeng2014deep] extended the MMD to deep neural networks and achieved great success. Rather than using a single adaption layer and linear MMD, Long et al. [@long2015learning] proposed the deep adaptation network (DAN) which matches the shift in marginal distributions across domains by adding multiple adaptation layers and exploring multiple kernels. Further, Long et al. proposed a joint adaptation network (JAN) [@long2016deep] which aligns the shift in the joint distributions of input images and output labels. Different than MMD, CORAL [@sun2016return] learns a linear transformation that aligns convariance of the source and target domains. Then, Sun et al. [@sun2016deep] extended CORAL to deep neural networks. Another commonly used metric to measure the discrepancy between domains is central moment discrepancy (CMD) [@zellinger2017central], which restrains the domain discrepancy by matching the higher-order moments of the domain distributions. Inspired by the generative adversarial networks (GANs) [@goodfellow2014generative], adversarial learning is introduced to restrain the domain discrepancy by learning representations which is simultaneously discriminative in source labels and indistinguishable in domains. Tzeng et al. proposed the adversarial discriminative domain adaption (ADDA) [@tzeng2017adversarial], which uses GANs to train an encoder for target samples, by making the features extracted with this encoder indistinguishable from the ones extracted through an encoder trained with source samples. Then, Volpi et al. [@volpi2017adversarial] extended the ADDA framework by forcing the learned feature extractor to be domain-invariant and training it through data augmentation in the feature space. {height="1.9in" width="6in"} Other methods have chosen generative methods to minimize the domain discrepancy. In [@ghifary2016deep], a deep reconstruction-classification network (DRCN) is introduced to learn common representations for both domains through the joint optimization of supervised classification of labeled source data and unsupervised reconstruction of unlabeled target data. In [@bousmalis2016domain], Bousmalis et al. proposed the domain separation network (DSN) which explicitly learns to extract image representations that are partitioned into two components, one for the private information of each domain (domain feature) and the other for the shared representation across domains (content feature), to reconstruct the images and features from both domains. Recently, several image-to-image translation based methods are proposed for domain adaption by transferring images into target domain and then directly training classifiers on them. Taigman et al. proposed the Domain Transfer Network (DTN) [@taigman2016unsupervised] which is optimized by a compound loss function including a multi-class GAN loss, an f-constancy component, and a regularizing component that encourages the transfer network to map samples from target domain to themselves. This network can transfer one image from the source domain to the target domain. In [@liu2016coupled], Liu and Tuzel introduced the coupled GANs (CoGAN) which can learn a joint distribution across multiple domains without requirement for paired images. CoGAN consists of a pair of GANs and each has a generative model for synthesizing realistic images in one domain and a discriminative model for classifying whether an image is real or synthesized. It can be applied for domain adaption by attaching a softmax layer to the last hidden layer of the discriminator. As an extension of CoGAN, Liu et al. proposed the unsupervised image-to-image translation (UNIT) [@liu2017unsupervised] network, which combined the GANs with variational auto-encoders (VAEs) and achieved unsupervised image-to-image translation based on the shared-latent space assumption. Lu et al proposed the duplex GAN (DupGAN) [@hu2018duplex] to achieve domain invariant feature extraction and domain transformation. Cycle-Consistent Adversarial Domain Adaptation (CyCADA) proposed in [@hoffman2017cycada] adapts representations at both the pixel-level and feature-level while enforcing semantic consistency, which achieved satisfying performance on both digit classification and semantic segmentation. Model ===== In this section, we introduce the proposed model in detail, whose architecture is displayed in Figure \[fig:network\]. The whole network consists of an encoder, a classifier and a discriminator, which is elegant. Our goal is to learn both domain-invariant and discriminative features which will benefit the domain adaption. In the following, we will introduce how we achieve this. Domain-invariant Feature Extraction ----------------------------------- For unsupervised domain adaption, extracting domain-invariant features is critical to alleviate the effect of the domain shift. In previous studies [@rozantsev2018beyond; @tzeng2017adversarial], totally separate or partially tied feature extractors are usually used for source domain and target domain. However, in this study, we claim that sharing a single feature extractor seems more effective to learn domain-invariant features for unsupervised domain adaption. In Figure \[fig:network\], the images from the source domain and target domain are passed through one shared encoder $E$ and we aim to extract features which only contain the information about the content of the image, namely domain-invariant features. Here, we adopt the adversarial learning and add a discriminator to distinguish which domain the extracted feature is from and simultaneously, the encoder tries to extract features which are indistinguishable for the discriminator. The following adversarial loss is applied: $$\begin{aligned} \min_{\theta_E} \max_{\theta_D} \mathcal{L}_{GAN} & = \textrm{E}_{x_i\in X_s}logD(E(x_i)) \\ \nonumber & + \textrm{E}_{x_i\in X_t}log(1-D(E(x_i)))\end{aligned}$$ where $D(\cdot)$ is the probability of being source features predicted by the discriminator $D$, $\theta_E$ and $\theta_D$ are parameters of encoder $E$ and discriminator $D$, $X_s$ and $X_t$ are distributions of samples in source domain and target domain, respectively. Although the inputs of the encoder are from two different domains, the extracted features cannot be distinguished by the discriminator about domains. By this limited condition, we expect the encoder to only extract the content information which is shared between these two domains and ignore the private domain information. Besides, with the shared encoder, the model can receive images from both source domain and target domain and we do not need to know the source of images during testing. Discriminative Feature Extraction --------------------------------- Since we have labels for samples in the source domain, the features of the source domain will be classified by the classifier $C$, which is a fully connected softmax layer with the size dependent on the task. The optimization function for the classification of the labeled data in source domain is defined as: $$\min_{\theta_E,\theta_C} \mathcal{L}_{s} = E_{(x_i,y_i)\in (X_s,Y_s)} H(C(E(x_i)),y_i)$$ where $H(\cdot)$ is the cross entropy loss used in the softmax layer, ($X_s$, $Y_s$) is the distribution of samples and labels in the source domain and $\theta_C$ are parameters of the classifier. Only aligning the distribution of samples may be not enough for domain adaption, because there will be some samples falling into inter-class gaps, which proposes the requirement for learning more discriminative features. In the learning literature, there exists several methods for learning discriminative features, such as the triplet loss [@schroff2015facenet], the contrastive loss [@sun2014deep] and the center loss [@wen2016discriminative]. Both the triplet loss and the contrastive loss need to construct a lot of image pairs and compute the distance between images of each pair, which is computationally complicated. Therefore, in this study, we define our objective function based on the center loss, which can be flexibly combined with the above classification loss. For samples in source domain which have labels, we adopt the following loss to cluster the features belonging to the same class: $$\mathcal{L}_{cs} = \sum_{(x_i, y_i)\in (X_s, Y_s)} ||E(x_i) - c_{y_i} ||_2^2$$ where $c_{y_i}$ is a $d$-dimensional vector representing the center of the $y_i$-th class. Ideally, each class center should be calculated using the features of all samples belonging to that class. But due to we optimize the model with mini-batch samples, it is difficult to compute the average of all samples. Therefore, we first initialize the class center by the batch in the first iteration, then update the centers by the following strategy: $$c_k^{t+1} = c_k^t - \gamma \Delta c_k^t, \quad k=1,2,\ldots,K$$ where $c_k^t$ is the center for the $k$-th class in iteration $t$, $\gamma$ is the learning rate for updating the centers, $K$ is the total number of classes and $$\Delta c_k^t = \frac{\sum_{(x_i,y_i)\in \mathcal{B}^t} \mathbb{I}(y_i=k) (c_k^t-E(x_i))}{1+N_k}$$ where $\mathcal{B}^t$ represents the mini-batch in iteration $t$, $\mathbb{I}(\cdot)$ is an indicator function and $N_k=\sum_{(x_i,y_i)\in \mathcal{B}^t}$ $\mathbb{I}(y_i=k)$ is the number of samples in batch $\mathcal{B}^t$ which belong to class $k$. Conditional Distribution Adaption --------------------------------- In most adversarial-based domain adaption methods, only marginal distribution adaption is concerned by aligning the distribution $P(X)$. However, as verified in some previous research [@long2013transfer], the conditional distribution $P(Y|X)$ of two domains may also be different. Since we have no labels for target domain, directly aligning the $P(Y|X)$ is challenging. Inspired by [@long2013transfer; @volpi2017adversarial], we explore the pseudo labels of target samples and resort to explore the sufficient statistics of class-conditional distributions $P(X|Y)$ instead as done in [@long2013transfer]. Therefore, for unlabeled samples in target domain, we assign each sample with the pesudo label predicted by the source classifier and define the following loss: $$\mathcal{L}_{ct} = \sum_{x_i\in \Phi(X_t)} ||E(x_i) - c_{\hat{y}_i}||_2^2 \label{eq:mag}$$ where $\hat{y}_i$ is the label of $x_i$ predicted by the classifier $C$. Since not all the predicted labels are accurate, we calculate the above $\mathcal{L}_{ct}$ only on $\Phi(X_t)$ which is a subset of $X_t$ and the samples in it satisfy: $$\Phi(X_t) = \{ x_i| x_i\in X_t \quad \textrm{and} \quad \max(p(x_i)) \geq T\}$$ where $p(x_i)$ is a $K$-dimensional vector with the $i$-th dimension being the predicted probability of belonging to $i$-th class, $\max(p(x_i))$ is the probability of sample $x_i$ belonging to the predicted class, and $T$ is the threshold we set. By this way, samples in target domain will also be close to corresponding clusters. Therefore, the total objective function of the model can be formulated as: $$\min_{\theta_E, \theta_C} \max_{\theta_D} \mathcal{L}_{GAN} + \alpha \mathcal{L}_s + \beta_1 \mathcal{L}_{cs} + \beta_2 \mathcal{L}_{ct}$$ where $\alpha$, $\beta_1$ and $\beta_2$ are weighted parameters. layer1 CONV-(N64,K5,S1), ReLU, MAX-POOL-(K3,S2) -------- ------------------------------------------ layer2 CONV-(N64,K5,S1), ReLU, MAX-POOL-(K3,S2) layer3 CONV-(N128,K5,S1), ReLU layer4 FC-(N500), ReLU layer5 FC-(N128), ReLU layer6 FC-(N10) : The modified LeNet model architecture used for digital dataset.[]{data-label="tab:lenet"} Method MNIST$\rightarrow$USPS(P1) USPS$\rightarrow$MNIST(P1) MNIST$\rightarrow$USPS(P2) USPS$\rightarrow$MNIST(P2) SVHN$\rightarrow$MNIST ------------------------------- ---------------------------- ---------------------------- ---------------------------- ---------------------------- ------------------------ -- -- Source 74.59$\pm$1.30 60.54$\pm$1.33 86.33$\pm$0.47 67.49$\pm$1.34 67.95$\pm$0.89 DANN [@ganin2014unsupervised] 77.1$\pm$1.8 73.0$\pm$2.0 - - 73.9 DDC [@tzeng2014deep] 79.1$\pm$0.5 66.5$\pm$3.3 - - 68.1$\pm$0.3 ADDA [@tzeng2017adversarial] 89.4$\pm$0.2 90.1$\pm$0.8 - - 76.0$\pm$1.8 UNIT [@liu2017unsupervised] - 95.97 93.58 90.53 CoGAN [@liu2016coupled] 91.2$\pm$0.8 89.1$\pm$0.8 95.65 93.15 - DI [@volpi2017adversarial] 91.4$\pm$0.0 87.9$\pm$0.5 95.4$\pm$0.2 - 85.1$\pm$2.6 DIFA [@volpi2017adversarial] 92.3$\pm$0.1 89.7$\pm$0.5 96.2$\pm$0.2 - 89.2$\pm$2.0 CyCADA [@hoffman2017cycada] - - 95.6$\pm$0.2 96.5$\pm$0.1 90.4$\pm$0.4 DupGAN [@hu2018duplex] - - 96.0 98.75 92.46 Ours **95.02$\pm$0.22** **97.28$\pm$0.25** **97.06$\pm$0.20** **99.12$\pm$0.06** **95.85$\pm$0.81** Target 95.65$\pm$0.14 97.83$\pm$0.49 96.71$\pm$0.18 99.36$\pm$0.04 99.36$\pm$0.04 Experiments =========== In this section, we evaluate the proposed method by comparing it with several state-of-the-art methods for unsupervised domain adaption. We first introduce the datasets we used. Then, we introduce the implementation details and finally, the experimental results are analyzed in detail. Datasets -------- We evaluate our proposed method on two image classification adaptation datasets. One is a large-scale digital recognition dataset which is commonly used in domain adaption research and the other is the Office-31 dataset [@saenko2010adapting], which is a standard domain adaptation benchmark. For digit classification, the datasets of MNIST [@lecun1998gradient], USPS [@denker1989neural] and SVHN [@netzer2011reading] are used for evaluating all of the methods. All three datasets contain images of digits 0-9 but with different styles. MNIST is composed of 60,000 training and 10,000 testing images. USPS consists of 7,291 training and 2,007 testing images. SVHN contains 73,257 training, 26,032 testing and 531,131 extra training images. Following previous work [@volpi2017adversarial], we evaluate the adaption between MNIST and USPS by setting two protocols. For the first protocol (P1), we follow the training protocol established in [@long2013transfer] and randomly sampled 2,000 MNIST images and 1,800 USPS images. The second protocol (P2) uses the whole MNIST training set and USPS training set. For both P1 and P2, we evaluate the two directions of the split (MNIST$\rightarrow$USPS and USPS$\rightarrow$MNIST). For the adaption of SVHN and MNIST, we only evaluate on SVHN$\rightarrow$MNIST following previous studies [@tzeng2017adversarial; @volpi2017adversarial]. We trained the model using the whole training set of both datasets and then tested on the test set of MNIST. All images of MNIST and USPS are transformed to be RGB with size 32$\times$32 which is the size of images in SVHN. The Office-31 dataset consists of 4,110 images spread across 31 classes in 3 domains: amazon (2,817 images), webcam (795 images), and dslr (498 images). We compare the methods on all of the six combination pairs of the three domains: A$\leftrightarrow$W, A$\leftrightarrow$D and W$\leftrightarrow$D. Following previous work [@tzeng2017adversarial], we train the model on all of the labeled data in the source domain and all unlabeled data in the target domain, and then test it on the data in target domain. Implementation Details ---------------------- For experiments on digital classification, we use the simple modified LeNet architecture which is displayed in Table \[tab:lenet\]. The discriminator is composed of 3 fully connected layers: two layers with 500 hidden units followed by the final discriminator output. Each of the 500-unit layers uses a ReLU activation function. The whole network is optimized by the RMSPropOptimizer with batch size 64 for MNIST$\leftrightarrow$USPS in P1 and 256 for other cases. We set the initial learning rate being 0.001 and decaying 0.5 times for very 60 epochs. The threshold $T$ is set 0.99 for all cases. During adaption between MNIST and USPS, we first train the model with $\alpha$=10, $\beta_1$=0.001 and $\beta_2$=0 for 30 epochs and then we change $\beta_1$=$\beta_2$=0.002 for another 30 epochs and then train the model with $\beta_1$=$\beta_2$=0.02 until convergence. For domain adaption of SVHN$\rightarrow$MNIST, we first train the model with $\alpha$=10, $\beta_1$=0.001, $\beta_2$=0 for 30 epochs and then change $\beta_1$ and $\beta_2$ to be 0.1 and 0.2 for the rest epochs. For the experiments on Office-31 dataset, we adopt the ResNet-50 [@he2016deep] as the base model which is pretrained on the ImageNet, and the activations of the last layer pool5 are used as the image representations. Since the dataset is small, we only fine-tune the last block and the fully-connected layer of ResNet-50. We optimize the network by the Stochastic Gradient Descent (SGD) optimizer with the momentum of 0.9 and batch size 64, namely total 128 images from the source and target domain. We set the initial learning rate as 0.001 and decay 0.5 times for every 50 epochs to avoid over-fitting. We first train the model with $\alpha$=10, $\beta_1$=0.001 and $\beta_2$=0 for 50 epochs and $\beta_1$=$\beta_2$=0.002 for another 50 epochs and then train the model with $\beta_1$=$\beta_2$=0.01 until convergence. For fair comparison, we conduct each experiment for several times with random initialization and show the mean$\pm$std as the result. Method A$\rightarrow$W D$\rightarrow$W W$\rightarrow$D A$\rightarrow$D D$\rightarrow$A W$\rightarrow$A Average ----------------------------------- -------------------- ------------------ ------------------- -------------------- -------------------- -------------------- ----------- AlexNet [@krizhevsky2012imagenet] 61.6$\pm$0.5 95.4$\pm$0.3 99.0$\pm$0.2 63.8$\pm$0.5 51.1$\pm$0.6 49.8$\pm$0.4 70.1 DDC [@tzeng2014deep] 61.8$\pm$0.4 95.0$\pm$0.5 98.5$\pm$0.4 64.4$\pm$0.3 52.1$\pm$0.6 52.2$\pm$0.4 70.6 DeepCORAL [@sun2016deep] 66.4$\pm$0.4 95.7$\pm$0.3 99.2$\pm$0.1 66.8$\pm$0.6 52.8$\pm$0.2 51.5$\pm$0.3 72.1 DAN [@long2015learning] 68.5$\pm$0.5 96.0$\pm$0.3 99.0$\pm$0.3 67.0$\pm$0.4 54.0$\pm$0.5 53.1$\pm$0.5 72.9 DANN [@ganin2014unsupervised] 73.0$\pm$0.5 96.4$\pm$0.3 99.2$\pm$0.3 72.3$\pm$0.3 53.4$\pm$0.4 51.2$\pm$0.5 74.3 JAN [@long2016deep] 75.2$\pm$0.4 96.6$\pm$0.2 99.6$\pm$0.1 72.8$\pm$0.3 57.5$\pm$0.2 56.3$\pm$0.2 76.3 VGG-16 [@simonyan2014very] 67.6$\pm$0.6 96.1$\pm$0.3 99.2$\pm$0.2 73.9$\pm$0.9 58.2$\pm$0.5 57.8$\pm$0.4 75.5 CMD [@zellinger2017central] 77.0$\pm$0.6 96.3$\pm$0.4 99.2$\pm$0.2 79.6$\pm$0.6 63.8$\pm$0.7 63.3$\pm$0.6 79.9 DRCN [@ghifary2016deep] 68.7$\pm$0.3 96.4$\pm$0.3 99.0$\pm$0.2 66.8$\pm$0.5 56.0$\pm$0.5 54.9$\pm$0.5 73.6 DupGAN [@hu2018duplex] 73.2$\pm$0.2 - - 74.1$\pm$0.6 61.5$\pm$0.5 59.1$\pm$0.5 - ResNet-50 [@he2016deep] 68.4$\pm$0.2 96.7$\pm$0.1 99.3$\pm$0.1 68.9$\pm$0.2 62.5$\pm$0.3 60.7$\pm$0.3 76.1 DDC [@tzeng2014deep] 75.6$\pm$0.2 96.0$\pm$0.2 98.2$\pm$0.1 76.5$\pm$0.3 62.2$\pm$0.4 61.5$\pm$0.5 78.3 ADDA [@tzeng2017adversarial] 75.1 97.0 99.6 - - - - DAN [@long2015learning] 80.5$\pm$0.4 **97.1$\pm$0.2** 99.6$\pm$0.1 78.6$\pm$0.2 63.6$\pm$0.3 62.8$\pm$0.2 80.4 DANN [@ganin2014unsupervised] 82.0$\pm$0.4 96.9$\pm$0.2 99.1$\pm$0.1 79.7$\pm$0.4 68.2$\pm$0.4 67.4$\pm$0.5 82.2 JAN [@long2016deep] 86.0$\pm$0.4 96.7$\pm$0.3 99.7$\pm$0.1 85.1$\pm$0.4 69.2$\pm$0.4 70.7$\pm$0.5 84.6 Ours **91.13$\pm$0.66** 95.94$\pm$0.28 **99.8$\pm$0.00** **86.85$\pm$0.74** **70.66$\pm$0.77** **71.35$\pm$0.24** **85.96** Results ------- In this section, we will first display and analyze the experimental results on both digital classification and object recognition tasks. And then we visualize the features extracted by the encoder to further verify the proposed model. ### Results on Digital Classification For experiments on digital classification, we compare the proposed method with state-of-the-art methods to verify that the proposed method is effective when the domain size is large. The results are displayed in Table \[tab:digital\]. The row of “Source" reports the accuracies on target data achieved by non-adapted classifiers trained on the source data. And the row of “Target" reports the results on target data achieved by classifiers trained on target data. Since we follow the same experimental settings with most compared methods, we directly copy the results from corresponding papers. As observed in Table \[tab:digital\], deep transfer learning models perform better than non-adapted classifiers trained on the source data, indicating that integrating domain adaption modules into deep networks will help reduce the domain discrepancy. Among the deep transfer learning models, our proposed method outperforms all baselines on all tasks. In particular, our method improves the accuracy for a large marge even on difficult transfer tasks, e.g. SVHN$\rightarrow$MNIST, where the SVHN dataset contains significant variations in scale, background, rotation and so on, and there is only slightly variation in the digits shapes, which makes it substantially different from MNIST dataset. These experimental results demonstrate that the proposed method is effective for unsupervised domain adaption with large scale datasets. We owe the improvements to three points. First, we share one encoder between the source domain and target domain to extract domain-invariant representations and the sample distributions are online aligned. Secondly, we extract discriminative features by integrating the classification loss with the center loss, which makes samples in the same class more compact. Thirdly, rather than only aligning the marginal distributions, we also consider aligning the conditional distributions of the source domain and target domain, by which samples in target domain are more close to corresponding class clusters. Method MNIST$\rightarrow$USPS(P1) USPS$\rightarrow$MNIST(P1) MNIST$\rightarrow$USPS(P2) USPS$\rightarrow$MNIST(P2) SVHN$\rightarrow$MNIST ------------------------------------------------------------------------- ---------------------------- ---------------------------- ---------------------------- ---------------------------- ------------------------ -- -- $M_{\mathcal{L}_s+\mathcal{L}_{GAN}}$ 91.56$\pm$0.91 93.81$\pm$1.36 96.29$\pm$0.39 98.89$\pm$0.11 88.34$\pm$0.89 $M_{\mathcal{L}_s+\mathcal{L}_{GAN}+\mathcal{L}_{cs}}$ 93.52$\pm$0.81 96.89$\pm$0.99 96.06$\pm$0.30 99.06$\pm$0.05 89.27$\pm$1.76 $M_{\mathcal{L}_s+\mathcal{L}_{GAN}+\mathcal{L}_{cs}+\mathcal{L}_{ct}}$ **95.02$\pm$0.22** **97.28$\pm$0.25** **97.06$\pm$0.20** **99.12$\pm$0.06** **95.85$\pm$0.81** Method A$\rightarrow$W D$\rightarrow$W W$\rightarrow$D A$\rightarrow$D D$\rightarrow$A W$\rightarrow$A Average ------------------------------------------------------------------------- -------------------- ----------------- ------------------- -------------------- -------------------- -------------------- ----------- $M_{\mathcal{L}_s+\mathcal{L}_{GAN}}$ 81.33$\pm$0.76 96.49$\pm$0.32 96.99$\pm$0.49 78.35$\pm$0.75 61.90$\pm$0.30 66.30$\pm$0.36 80.31 $M_{\mathcal{L}_s+\mathcal{L}_{GAN}+\mathcal{L}_{cs}}$ 90.00$\pm$0.96 96.18$\pm$0.23 99.68$\pm$0.1 84.48$\pm$1.18 68.64$\pm$0.82 69.08$\pm$0.96 84.68 $M_{\mathcal{L}_s+\mathcal{L}_{GAN}+\mathcal{L}_{cs}+\mathcal{L}_{ct}}$ **91.13$\pm$0.66** 95.94$\pm$0.28 **99.8$\pm$0.00** **86.85$\pm$0.74** **70.66$\pm$0.77** **71.35$\pm$0.24** **85.96** ### Results on Object Recognition In contrast to digital classification which has a large domain size, the object recognition on office-31 is a task where both source domain and target domain only have a small number of samples. The comparison results of the proposed method and some state-of-the-art methods are presented in Table \[tab:office\]. For fair comparison with identical evaluation setting, the AlexNet-based results of DeepCORAL [@sun2016deep], DAN [@long2015learning], DANN [@ganin2014unsupervised] and JAN [@long2016deep], the VGG-16-based results of CMD [@zellinger2017central] and the ResNet-based results of ADDA [@tzeng2017adversarial] are directly reported from their published papers. The results of DRCN [@ghifary2016deep] and DupGAN [@volpi2017adversarial] are also copied from the published papers. The other results are reported in JAN [@long2016deep]. The rows of “AlexNet", “VGG-16" and “ResNet50" display the results on target domain obtained by fine-tuning the AlexNet, VGG-16 and ResNet-50 with the source data respectively, which are pretrained on ImageNet. As shown in Table \[tab:office\], our proposed approach performs better than all baselines for most tasks and achieves the highest average score, demonstrating that our method is also effective for unsupervised domain adaption when source domain and target domain only have a small number of samples. It is worth noting that our method significantly improves the accuracies on difficult tasks, such as A$\rightarrow$W and A$\rightarrow$D where samples in source domain and target domain are very different, and W$\rightarrow$A and D$\rightarrow$A where the size of the source domain is very small. Besides, our method achieved comparable results with state-of-the-art methods on relatively easy tasks such as W$\rightarrow$D and D$\rightarrow$W, where the source domain and target domain is similar. In addition, similar results can be observed that deep transfer learning models perform better than base deep learning models such as AlexNet, indicating that integrating domain adaption modules into deep networks will help reduce the domain shift. Further, models based on VGG-16 and ResNet-50 perform better than AlexNet-based models which implies that very deep models like VGG-16 and ResNet-50 not only learn better representations for general learning tasks but also learn more generalizable features for domain adaption. \ ### Ablation Study To verify the effectiveness of each component in our model, we conduct ablation study on both digital classification and object recognition. We trained another two models: one model is trained only using the softmax loss and the GAN loss which is denoted as “$M_{\mathcal{L}_s+\mathcal{L}_{GAN}}$"; and the other model is trained using the softmax loss, the GAN loss and the center loss for the source domain $\mathcal{L}_{cs}$, which is denoted as “$M_{\mathcal{L}_s+\mathcal{L}_{GAN}+\mathcal{L}_{cs}}$". We compare the two models with the model trained with all losses and the results are shown in Table \[tab:ablation1\] and Table \[tab:ablation2\]. As shown in these tables, after removing one or more parts, the performance degrades in most cases. The more parts are removed, the worse the performance is. This indicates that all parts are designed reasonably and they work harmoniously forming an effective solution for unsupervised domain adaption. Comparing Table \[tab:digital\] and Table \[tab:ablation1\], we can observe that “$M_{\mathcal{L}_s+\mathcal{L}_{GAN}}$" performs better than models which uses two different encoders for source domain and target domain, such as ADDA, indicating that sharing one feature extractor between two domains is better for unsupervised domain adaption to extract domain-invariant features. In addition, “$M_{\mathcal{L}_s+\mathcal{L}_{GAN}}$" performs better than DI model which also shares one encoder for source and target domain. However, in DI model, the features extracted by the encoder will be compared with features from a fixed pretrained source feature generator. This is different from our model where the features distinguished by the discriminator are from the same being trained encoder, namely the distributions are online aligned. Experimental results demonstrate that directly building the GAN loss on the features extracted by the shared encoder is better. In Table \[tab:ablation1\] and Table \[tab:ablation2\], “$M_{\mathcal{L}_s+\mathcal{L}_{GAN}+\mathcal{L}_{cs}}$" achieved better performance than “$M_{\mathcal{L}_s+\mathcal{L}_{GAN}}$" for most tasks, showing that learning discriminative representations in source domain is helpful for learning a powerful classifier. Furthermore, after adding the loss $\mathcal{L}_{ct}$, the accuracies are improved further, which demonstrates that aligning the class-conditional distribution $P(X|Y)$ is helpful for unsupervised domain adaption. Besides, we test models $M_{\mathcal{L}_s}$ and $M_{\mathcal{L}_s+\mathcal{L}_{cs}}$ and their corresponding models $M_{\mathcal{L}_s+\mathcal{L}_{GAN}}$ and $M_{\mathcal{L}_s+\mathcal{L}_{GAN}+\mathcal{L}_{cs}}$ which are adapted with data in target domain, on test set in source domain, and the results are shown in Table \[tab:com\_source\]. As observed, the accuracies on source domain have little difference before (96.39 and 96.75) and after (96.29 and 96.64) adaption, implying that our model can still keep good performance on source domain and simultaneously improve the performance on target domain, and that our model can be used to test images from both source and target domains. Source $M_{\mathcal{L}_s}$ $M_{\mathcal{L}_s+\mathcal{L}_{GAN}}$ $M_{\mathcal{L}_s+\mathcal{L}_{cs}}$ $M_{\mathcal{L}_s+\mathcal{L}_{GAN}+\mathcal{L}_{cs}}$ -------- --------------------- --------------------------------------- -------------------------------------- -------------------------------------------------------- M(P1) 97.83 97.56 98.56 98.34 M(P2) 99.36 99.43 98.34 99.52 U(P1) 95.65 95.28 96.51 96.62 U(P2) 96.71 96.98 97.02 97.54 S 92.44 92.18 93.34 91.16 Ave 96.39 96.29 96.75 96.64 : Results on test dataset in source domain classified with model $M_{\mathcal{L}_s}$ and $M_{\mathcal{L}_s+\mathcal{L}_{cs}}$ and the domain adapted models $M_{\mathcal{L}_s+\mathcal{L}_{GAN}}$ and $M_{\mathcal{L}_s+\mathcal{L}_{GAN}+\mathcal{L}_{cs}}$. M: MNIST, U: USPS, S: SVHN. “Ave" represents the average accuracy.[]{data-label="tab:com_source"} ### Feature Visualization To further evaluate the effectiveness of the proposed method, we visualize the t-SNE embeddings of features extracted by four models trained with different loss functions in Figure \[fig:visualize\]. We use different colors to represent different classes in Figure \[fig:visualize\] (a)-(d) and we use two colors to denote the source domain (red) and target domain (blue) in Figure \[fig:visualize\] (e)-(h). As shown in Figure \[fig:visualize\]-(a)(e), the features of these two domains are separately distributed and there exists an obvious separation between features in two domains. In Figure \[fig:visualize\]-(b)(f), the features from two domains are merged in distribution and cannot be distinguished, which benefits from the domain-invariant feature extraction. However, there are many points scattered in the inter-class gap, whose labels may be misclassified with large possibility. After adding the center loss on source samples, the features are more compacted and form clear clusters as observed in Figure \[fig:visualize\]-(c)(g), indicating that the features learned with center loss are much more discriminative. Furthermore, since the model tries to extract domain-invariant features, adding the center loss for the source samples also makes the features of target samples look like clusters, but they may fall into incorrect clusters (seeing the center cluster in Figure \[fig:visualize\]-(c) containing points of many colors). As shown in Figure \[fig:visualize\]-(d)(h), after adding the $\mathcal{L}_{ct}$, the number of misclustered points decreased and more points fall into correct area, which indicates that aligning the conditional distribution will guide more target samples to correct clusters. Through the feature visualization, we further validate the effectiveness of each component in our method. Conclusion ========== In this paper, we propose a very simple model for unsupervised domain adaption which could learn both domain-invariant and discriminative representations. The proposed model consists of an encoder, a classifier and a discriminator. The encoder is totally shared between the source domain and target domain, which is expected to extract domain-invariant features with the help of the discriminator. By sharing the encoder, our model could receive images from both source and target domains without distinction of the source of images. Besides, the data distributions will be aligned in an online way. In addition, we highlight the importance of learning discriminative representations for unsupervised domain adaption and augment the classification loss with the center loss. Besides the marginal distribution, we also align the conditional distributions of the two domains. Finally, we evaluate the proposed method on several benchmarks of unsupervised domain adaption and achieve better performance than state-of-the-art methods.

--- abstract: 'In this note, we prove that the kernel of the linearized equation around a positive energy solution in $\rn$, $n\geq 3$, to $-\Delta W-\gamma|x|^{-2}V=|x|^{-s}W^{\crits-1}$ is one-dimensional when $s+\gamma>0$. Here, $s\in [0,2)$, $0\leq\gamma<(n-2)^2/4$ and $\crits=2(n-s)/(n-2)$.' address: 'Frédéric Robert, Institut Élie Cartan, Université de Lorraine, BP 70239, F-54506 Vand[œ]{}uvre-lès-Nancy, France' author: - Frédéric Robert date: December 29th 2016 title: 'Nondegeneracy of positive solutions to nonlinear Hardy-Sobolev equations' --- We fix $n\geq 3$, $s\in [0,2)$ and $\gamma<\frac{(n-2)^2}{4}$. We define $\crits=2(n-s)/(n-2)$. We consider a nonnegative solution $W\in C^2(\rnp)\setminus\{0\}$ to $$\label{eq:V} -\Delta W-\frac{\gamma}{|x|^{2}}W=\frac{W^{\crits-1}}{|x|^{s}}\hbox{ in }\rnp.$$ Due to the abundance of solutions to , we require in addition that $W$ is an energy solution, that is $W\in \dundeux$, where $\dundeux$ is the completion of $C^\infty_c(\rn)$ for the norm $u\mapsto \Vert\nabla u\Vert_2$. Linearizing yields to consider $$\label{def:KV} K:=\left\{\varphi\in \dundeux/\, -\Delta\varphi-\frac{\gamma}{|x|^2}\varphi=(\crits-1)\frac{W^{\crits-2}}{|x|^s}\varphi\hbox{ in }\dundeux\right\}$$ Equation is conformally invariant in the following sense: for any $r>0$, define $$W_r(x):=r^{\frac{n-2}{2}}W(rx)\hbox{ for all }x\in\rnp,$$ then, as one checks, $W_r\in C^2(\rnp)$ is also a solution to , and, differentiating with respect to $r$ at $r=1$, we get that $$-\Delta Z-\frac{\gamma}{|x|^2}Z=(\crits-1)\frac{W^{\crits-2}}{|x|^s}Z\hbox{ in }\rnp,$$ where $$Z:=\frac{d}{dr}{W_r}_{|r=1}= \sum_ix^i\partial_i W+\frac{n-2}{2}W\in \dundeux.$$ Therefore, $Z\in K$. We prove that this is essentially the only element: \[th:main\] We assume that $\gamma\geq 0$ and that $\gamma+s>0$. Then $K=\rr Z$. In other words, $K$ is one-dimensional. Such a result is useful when performing Liapunov-Schmidt’s finite dimensional reduction. When $\gamma=s=0$, the equation is also invariant under the translations $x\mapsto W(x-x_0)$ for any $x_0\in\rn$, and the kernel $K$ is of dimension $n+1$ (see Rey [@Rey] and also Bianchi-Egnell [@BE]). After this note was completed, we learnt that Dancer-Gladiali-Grossi [@dgg] proved Theorem \[th:main\] in the case $s=0$, and that their proof can be extended to our case, see also Gladiali-Grossi-Neves [@ggn]. This note is devoted to the proof of Theorem \[th:main\]. Since $\gamma+s>0$, it follows from Chou-Chu [@ChouChu], that there exists $r>0$ such that $W=\lambda^{\frac{1}{\crits-2}}U_r$, where $$U(x):=\left(|x|^{\frac{2-s}{n-2}\am}+|x|^{\frac{2-s}{n-2}\ap} \right)^{-\frac{n-2}{2-s}}.$$ with $$\eps:=\sqrt{\frac{(n-2)^2}{4}-\gamma}\hbox{ and }\alpha_{\pm}(\gamma):=\frac{n-2}{2}\pm\sqrt{\frac{(n-2)^2}{4}-\gamma}.$$ As one checks, $U\in \dundeux\cap C^\infty(\rnp)$ and $$\label{eq:U} -\Delta U-\frac{\gamma}{|x|^2}U=\lambda\frac{U^{\crits-1}}{|x|^s}\hbox{ in }\rnp,\hbox{ with }\lambda:=4\frac{n-s}{n-2}\eps^2.$$ Therefore, proving Theorem \[th:main\] reduces to prove that $\tilde{K}$ is one-dimensional, where $$\label{def:tK} \tilde{K}:=\left\{\varphi\in \dundeux/\, -\Delta\varphi-\frac{\gamma}{|x|^2}\varphi=(\crits-1)\lambda\frac{U^{\crits-2}}{|x|^s}\varphi\hbox{ in }\dundeux\right\}$$ [**I. Conformal transformation.**]{} We let $\sn:=\{x\in\rn/\, \sum x_i^2=1\}$ be the standard $(n-1)-$dimensional sphere of $\rn$. We endow it with its canonical metric $\can$. We define $$\left\{\begin{array}{cccc} \Phi: & \rr\times\sn &\mapsto &\rnp\\ &(t,\sigma) & \mapsto & e^{-t}\sigma \end{array}\right.$$ The map $\Phi$ is a smooth conformal diffeomorphism and $\Phi^\star\eucl=e^{-2t}(dt^2+\can)$. On any Riemannian manifold $(M,g)$, we define the conformal Laplacian as $L_g:=-\Delta_g+\frac{n-2}{4(n-1)}R_g$ where $\Delta_g:=\hbox{div}_g(\nabla)$ and $R_g$ is the scalar curvature. The conformal invariance of the Laplacian reads as follows: for a metric $g'=e^{2\omega}g$ conformal to $g$ ($\omega\in C^\infty(M)$), we have that $L_{g'}u=e^{-\frac{n+2}{2}\omega}L_g(e^{\frac{n-2}{2}\omega}u)$ for all $u\in C^\infty(M)$. It follows from this invariance that for any $u\in C^\infty_c(\rnp)$, we have that $$\label{transfo:delta} (-\Delta u)\circ \Phi(t,\sigma)=e^{\frac{n+2}{2}t}\left(-\partial_{tt}\hat{u}-\Delta_{\can}\hat{u}+\frac{(n-2)^2}{4}\hat{u}\right)(t,\sigma)$$ for all $(t,\sigma)\in\rr\times \sn$, where $\hat{u}(t,\sigma):=e^{-\frac{n-2}{2}t}u(e^{-t}\sigma)$ for all $(t,\sigma)\in \rr\times\sn$. In addition, as one checks, for any $u,v\in C^\infty_c(\rnp)$, we have that $$\begin{aligned} \int_{\rn}(\nabla u,\nabla v)\, dx&=& \int_{\rr\times\sn}\left(\partial_t\hat{u}\partial_t\hat{v}+\left(\nabla^\prime\hat{u},\nabla^\prime\hat{v}\right)_{\can}+\frac{(n-2)^2}{4}\hat{u}\hat{v}\right)\, dt\, d\sigma\nonumber\\ &:=&B(\hat{u},\hat{v})\label{def:B}\end{aligned}$$ where we have denoted $\nabla^\prime\hat{u}$ as the gradient on $\sn$ with respect to the $\sigma$ coordinate. We define the space $H$ as the completion of $C_c^\infty(\rr\times\sn)$ for the norm $\Vert\cdot\Vert_H:=\sqrt{B(\cdot,\cdot)}$. As one checks, $u\mapsto \hat{u}$ extends to a bijective isometry $\dundeux\to H$. The Hardy-Sobolev inequality asserts the existence of $K(n,s,\gamma)>0$ such that $\left(\int_{\rn}\frac{|u|^{\crits}}{|x|^s}\, dx\right)^{\frac{2}{\crits}}\leq K(n,s,\gamma)\int_{\rn}\left(|\nabla u|^2-\frac{\gamma}{|x|^2}u^2\right)\, dx$ for all $u\in C^\infty_c(\rnp)$. Via the isometry $\dundeux\simeq H$, this inequality rewrites $$\left(\int_{\rr\times \sn}|v|^{\crits}\, dt d\sigma\right)^{\frac{2}{\crits}}\leq K(n,s,\gamma)\int_{\rr\times\sn}\left((\partial_t v)^2+|\nabla^\prime v|_{\can}^2+\eps^2v^2\right)\, dtd\sigma,$$ for all $v\in H$. In particular, $v\in L^{\crits}(\rr\times\sn)$ for all $v\in H$. We define $H_1^2(\rr)$ (resp. $H_1^2(\sn)$) as the completion of $C^\infty_c(\rr)$ (resp. $C^\infty(\sn)$) for the norm $$u\mapsto \sqrt{\int_{\rr}(\dot{u}^2+u^2)\, dx}\; \left(\hbox{resp. }u\mapsto \sqrt{\int_{\sn}(|\nabla^\prime u|^2_{\can}+u^2)\, d\sigma}\right).$$ Each norm arises from a Hilbert inner product. For any $(\varphi,Y)\in C^\infty_c(\rr)\times C^\infty(\sn)$, define $\varphi\star Y\in C^\infty_c(\rr\times\sn)$ by $(\varphi\star Y)(t,\sigma):=\varphi(t)Y(\sigma)$ for all $(t,\sigma)\in\rr\times\sn$. As one checks, there exists $C>0$ such that $$\label{eq:star} \Vert \varphi\star Y\Vert_H\leq C\Vert \varphi\Vert_{H_1^2(\rr)}\Vert Y\Vert_{H_1^2(\sn)}$$ for all $(\varphi,Y)\in C^\infty_c(\rr)\times C^\infty(\sn)$. Therefore, the operator extends continuously from $H_1^2(\rr)\times H_1^2(\sn)$ to $H$, such that holds for all $(\varphi,Y)\in H_1^2(\rr)\times H_1^2(\sn)$. \[lem:2\] We fix $u\in C^\infty_c(\rr\times\sn)$ and $Y\in H_1^2(\sn)$. We define $$u_Y(t):=\int_{\sn}u(t,\sigma)Y(\sigma)\, d\sigma=\langle u(t,\cdot),Y\rangle_{L^2(\sn)}\hbox{ for all }t\in\rr.$$ Then $u_Y\in H_1^2(\rr)$. Moreover, this definition extends continuously to $u\in H$ and there exists $C>0$ such that $$\Vert u_Y\Vert_{H_1^2(\rr)}\leq C\Vert u\Vert_H\Vert Y\Vert_{H_1^2(\sn)}\hbox{ for all }(u,Y)\in H\times H_1^2(\sn).$$ [*Proof of Lemma \[lem:2\]:*]{} We let $u\in C^\infty_c(\rr\times\sn)$, $Y\in H_1^2(\sn)$ and $\varphi\in C^\infty_c(\rr)$. Fubini’s theorem yields: $$\int_{\rr}\left(\partial_t u_Y\partial_t\varphi+u_Y\varphi\right)\, dt=\int_{\rr\times\sn}\left(\partial_t u\partial_t(\varphi\star Y)+u\cdot (\varphi\star Y)\right)\, dtd\sigma$$ Taking $\varphi:=u_Y$, the Cauchy-Schwartz inequality yields $$\begin{aligned} &&\Vert u_Y\Vert_{H_1^2(\rr)}^2\\ &&\leq \sqrt{\int_{\rr\times\sn}\left((\partial_t u)^2+u^2\right)dtd\sigma} \times \sqrt{\int_{\rr\times\sn}\left((\partial_t (u_Y\star Y))^2+ (u_Y\star Y)^2\right) dtd\sigma}\\ &&\leq C\Vert u\Vert_H\Vert u_Y\star Y\Vert_H\leq C\Vert u\Vert_H\Vert u_Y\Vert_{H_1^2(\rr)}\Vert Y\Vert_{H_1^2(\sn)},\end{aligned}$$ and then $\Vert u_Y\Vert_{H_1^2(\rr)}\leq C\Vert u\Vert_H\Vert Y\Vert_{H_1^2(\sn)}$. The extension follows from density. [**II. Transformation of the problem.**]{} We let $\varphi\in \tilde{K}$, that is $$-\Delta\varphi-\frac{\gamma}{|x|^2}\varphi=(\crits-1)\lambda\frac{U^{\crits-2}}{|x|^s}\varphi\hbox{ weakly in }\dundeux.$$ Since $U\in C^\infty(\rnp)$, elliptic regularity yields $\varphi\in C^\infty(\rnp)$. Moreover, the correspondance yields $$\label{eq:hphi} -\partial_{tt}\hphi-\Delta_{\can}\hphi+\eps^2\hphi=(\crits-1)\lambda \hU^{\crits-2}\hphi$$ weakly in $H$. Note that since $\hphi,\hU\in H$ and $H$ is continuously embedded in $L^{\crits}(\rr\times\sn)$, this formulation makes sense. Since $\varphi\in C^\infty(\rnp)$, we get that $\hphi\in C^\infty(\rr\times\sn)\cap H$ and equation makes sense strongly in $\rr\times\sn$. As one checks, we have that $$\hU(t,\sigma)=\left(e^{\frac{2-s}{n-2}\eps t}+e^{-\frac{2-s}{n-2}\eps t}\right)^{-\frac{n-2}{2-s}}\hbox{ for all }(t,\sigma)\in \rr\times\sn.$$ In the sequel, we will write $\hU(t)$ for $\hU(t,\sigma)$ for $(t,\sigma)\in \rr\times\sn$. The eigenvalues of $-\Delta_{\can}$ on $\sn$ are $$0=\mu_0<n-1=\mu_1<\mu_2<....$$ We let $\mu\geq 0$ be an eigenvalue for $-\Delta_{\can}$ and we let $Y=Y_\mu\in C^\infty(\sn)$ be a corresponding eigenfunction, that is $$-\Delta_{\can}Y=\mu Y\hbox{ in }\sn.$$ We fix $\psi\in C^\infty_c(\rr)$ so that $\psi\star Y\in C^\infty_c(\rr\times\sn)$. Multiplying by $\psi\star Y$, integrating by parts and using Fubini’s theorem yields $$\int_{\rr}\left(\partial_{t}\hphi_Y\partial_t\psi+(\mu+\eps^2)\hphi_Y\psi\right)\, dt=\int_{\rr}(\crits-1)\lambda \hU^{\crits-2}\hphi_Y\psi\, dt,$$ where $\hphi_Y\in H_1^2(\rr)\cap C^\infty(\rr)$. Then $$A_\mu \hphi_Y=0\hbox{ with }A_\mu:=-\partial_{tt}+(\mu+\eps^2-(\crits-1)\lambda \hU^{\crits-2})$$ where this identity holds both in the classical sense and in the weak $H_1^2(\rr)$ sense. We claim that $$\label{eq:phi:0} \hphi_Y\equiv 0\hbox{ for all eigenfunction }Y\hbox{ of }\mu\geq n-1.$$ We prove the claim by taking inspiration from Chang-Gustafson-Nakanishi ([@gustaf], Lemma 2.1). Differentiating with respect to $i=1,...,n$, we get that $$-\Delta\partial_i U-\frac{\gamma}{|x|^2}\partial_i U-(\crits-1)\lambda\frac{U^{\crits-2}}{|x|^s}\partial_i U=-\left(\frac{2\gamma}{|x|^{4}}U+\frac{s\lambda}{|x|^{s+2}}U^{\crits-1}\right)x_i$$ On $\rr\times\sn$, this equation reads $$-\partial_{tt}\hat{\partial_i U}-\Delta_{\can}\hat{\partial_i U}+\left(\eps^2-(\crits-1)\lambda \hU^{\crits-2}\right)\hat{\partial_i U}=-\sigma_i e^t \left(2\gamma\hU+s\lambda \hU^{\crits-1}\right)$$ Note that $\hat{\partial_i U}=-V\star \sigma_i$, where $\sigma_i:\sn\to \rr$ is the projection on the $x_i$’s and $$V(t):=-e^{-\frac{n-2}{2}t}U^\prime(e^{-t})=e^{(1+\eps)t}\left(\ap +\am e^{2\frac{2-s}{n-2}\eps t}\right)\left(1+e^{2\frac{2-s}{n-2}\eps t}\right)^{-\frac{n-s}{2-s}}>0$$ for all $t\in\rr$. Since $-\Delta_{\can}\sigma_i=(n-1)\sigma_i$ (the $\sigma_i$’s form a basis of the second eigenspace of $-\Delta_{\can}$), we then get that $$A_\mu V\geq A_{n-1}V= e^t\left(2\gamma\hU+s\lambda \hU^{\crits-1}\right)>0\hbox{ for all }\mu\geq n-1\hbox{ and }V>0.$$ Note that for $\gamma>0$, we have that $\am>0$, and that for $\gamma=0$, we have that $\am=0$. As one checks, we have that $$\begin{aligned} (i)\;\left\{\left(\gamma>0\hbox{ and }\eps>1\right)\hbox{ or }\left(\gamma=0\hbox{ and }s<\frac{n}{2}\right)\right\}&\Rightarrow & V\in H_1^2(\rr)\\ (ii)\; \left\{\left(\gamma>0\hbox{ and }\eps\leq1\right)\hbox{ or }\left(\gamma=0\hbox{ and }s\geq \frac{n}{2}\right)\right\}&\Rightarrow & V\notin L^2((0,+\infty))\end{aligned}$$ [*Assume that case (i) holds:*]{} in this case, $V\in H_1^2(\rr)$ is a distributional solution to $A_\mu V>0$ in $H_1^2(\rr)$. We define $m:=\inf \{\int_{\rr}\varphi A_\mu \varphi\, dt\}$, where the infimum is taken on $\varphi\in H_1^2(\rr)$ such that $\Vert\varphi\Vert_2=1$. We claim that $m>0$. Otherwise, it follows from Lemma \[lem:3\] below that the infimum is achieved, say by $\varphi_0\in H_1^2(\rr)\setminus \{0\}$ that is a weak solution to $A_\mu\varphi_0=m\varphi_0$ in $\rr$. Since $|\varphi_0|$ is also a minimizer, and due to the comparison principle, we can assume that $\varphi_0>0$. Using the self-adjointness of $A_\mu$, we get that $0\geq m\int_{\rr}\varphi_0V\, dt=\int_{\rr}(A_\mu \varphi_0)V\, dt=\int_{\rr}(A_\mu V)\varphi_0\, dt>0$, which is a contradiction. Then $m>0$. Since $A_\mu\varphi_Y=0$, we then get that $\varphi_Y\equiv 0$ as soon as $\mu\geq n-1$. This ends case (i). [*Assume that case (ii) holds:*]{} we assume that $\varphi_Y\not\equiv 0$. It follows from Lemma \[lem:4\] that $V(t)=o(e^{-\alpha |t|})$ as $t\to -\infty$ for all $0<\alpha<\sqrt{\eps^2+n-1}$. As one checks with the explicit expression of $V$, this is a contradiction when $\eps<\frac{n-2}{2}$, that is when $\gamma>0$. Then we have that $\gamma=0$ and $\eps=\frac{n-2}{2}$. Since $\frac{n}{2}\leq s<2$, we have that $n=3$. As one checks, $(\mu+\eps^2-(\crits-1)\lambda \hU^{\crits-2})>0$ for $\mu\geq n-1$ as soon as $n=3$ and $s\geq 3/2$. Lemma \[lem:4\] yields $\varphi_Y\equiv 0$, a contradiction. So $\varphi_Y\equiv 0$, this ends case (ii). These steps above prove . Then, for all $t\in\rr$, $\hphi(t,\cdot)$ is orthogonal to the eigenspaces of $\mu_i$, $i\geq 1$, so it is in the eigenspace of $\mu_0=0$ spanned by $1$, and therefore $\hphi=\hphi(t)$ is independent of $\sigma\in\sn$. Then $$-\hphi^{\prime\prime}+(\eps^2-(\crits-1)\lambda \hU^{\crits-2})\hphi=0\hbox{ in }\rr\hbox{ and }\hphi\in H_1^2(\rr).$$ It follows from Lemma \[lem:5\] that the space of such functions is a most one-dimensional. Going back to $\varphi$, we get that $\tilde{K}$ is of dimension at most one, and then so is $K$. Since $Z\in K$, then $K$ is one dimensional and $K=\rr Z$. This proves Theorem \[th:main\]. [**III. Auxiliary lemmas.**]{} \[lem:5\] Let $q\in C^0(\rr)$. Then $$\hbox{dim}_{\rr}\{\varphi\in C^2(\rr)\cap H_1^2(\rr)\hbox{ such that }-\ddot{\varphi}+q\varphi=0\}\leq 1.$$ [*Proof of Lemma \[lem:5\]:*]{} Let $F$ be this space. Fix $\varphi,\psi\in F\setminus\{0\}$: we prove that they are linearly dependent. Define the Wronskian $W:=\varphi \dot{\psi}-\dot{\varphi}\psi$. As one checks, $\dot{W}=0$, so $W$ is constant. Since $\varphi,\dot{\varphi},\psi,\dot{\psi}\in L^2(\rr)$, then $W\in L^1(\rr)$ and then $W\equiv 0$. Therefore, there exists $\lambda\in\rr$ such that $(\psi(0),\dot{\psi}(0))=\lambda (\varphi(0),\dot{\varphi}(0))$, and then, classical ODE theory yields $\psi=\lambda\varphi$. Then $F$ is of dimension at most one. \[lem:3\] Let $q\in C^0(\rr)$ be such that there exists $A>0$ such that $\lim_{t\to\pm\infty}q(t)=A$, and define $$m:=\inf_{\varphi\in H_1^2(\rr)\setminus\{0\}}\frac{\int_{\rr}\left(\dot{\varphi}^2+q\varphi^2\right)\, dt}{\int_{\rr}\varphi^2\, dt}.$$ Then either $m>0$, or the infimum is achieved. Note that in the case $q(t)\equiv A$, $m=A$ and the infimum is not achieved. [*Proof of Lemma \[lem:3\]:*]{} As one checks, $m\in\rr$ is well-defined. We let $(\varphi_i)_i\in H_1^2(\rr)$ be a minimizing sequence such that $\int_{\rr}\varphi_i^2\, dt=1$ for all $i$, that is $\int_{\rr}\left(\dot{\varphi}_i^2+q\varphi_i^2\right)\, dt=m+o(1)$ as $i\to +\infty$. Then $(\varphi_i)_i$ is bounded in $H_1^2(\rr)$, and, up to a subsequence, there exists $\varphi\in H_1^2(\rr)$ such that $\varphi_i\rightharpoonup \varphi$ weakly in $H_1^2(\rr)$ and $\varphi_i\to \varphi$ strongly in $L^2_{loc}(\rr)$ as $i\to +\infty$. We define $\theta_i:=\varphi_i-\varphi$. Since $\lim_{t\to \pm\infty}(q(t)-A)=0$ and $(\theta_i)_i$ goes to $0$ strongly in $L^2_{loc}$, we get that $\lim_{i\to +\infty}\int_{\rr}(q(t)-A)\theta_i^2\, dt=0$. Using the weak convergence to $0$ and that $(\varphi_i)_i$ is minimizing, we get that $$\int_{\rr}\left(\dot{\varphi}^2+q\varphi^2\right)\, dt+\int_{\rr}\left(\dot{\theta}_i^2+A\theta_i^2\right)\, dt=m+o(1)\hbox{ as }i\to +\infty.$$ Since $1-\Vert\varphi\Vert_2^2=\Vert\theta_i\Vert_2^2+o(1)$ as $i\to +\infty$ and $\int_{\rr}\left(\dot{\varphi}^2+q\varphi^2\right)\, dt\geq m\Vert\varphi\Vert_2^2$, we get $$m\Vert\theta_i\Vert_2^2\geq \int_{\rr}\left(\dot{\theta}_i^2+A\theta_i^2\right)\, dt+o(1)\hbox{ as }i\to +\infty.$$ If $m\leq 0$, then $\theta_i\to 0$ strongly in $H_1^2(\rr)$, and then $(\varphi_i)_i$ goes strongly to $\varphi\not\equiv 0$ in $H_1^2$, and $\varphi$ is a minimizer for $m$. This proves the lemma. \[lem:4\] Let $q\in C^0(\rr)$ be such that there exists $A>0$ such that $\lim_{t\to\pm\infty}q(t)=A$ and $q$ is even. We let $\varphi\in C^2(\rr)$ be such that $-\ddot{\varphi}+q\varphi=0$ in $\rr$ and $\varphi\in H_1^2(\rr)$. - If $q\geq 0$, then $\varphi\equiv 0$. - We assume that there exists $V\in C^2(\rr)$ such that $$-\ddot{V}+qV>0\; ,\; V>0\hbox{ and }V\not\in L^2((0,+\infty)).$$ Then either $\varphi\equiv 0$ or $V(t)=o(e^{-\alpha |t|})$ as $t\to -\infty$ for all $0<\alpha<\sqrt{A}$. [*Proof of Lemma \[lem:4\]:*]{} We assume that $\varphi\not\equiv 0$. We first assume that $q\geq 0$. By studying the monotonicity of $\varphi$ between two consecutive zeros, we get that $\varphi$ has at most one zero, and then $\ddot{\varphi}$ has constant sign around $\pm\infty$. Therefore, $\varphi$ is monoton around $\pm\infty$ and then has a limit, which is $0$ since $\varphi\in L^2(\rr)$. The contradiction follows from studying the sign of $\ddot{\varphi}$, $\varphi$. Then $\varphi\equiv 0$ and the first part of Lemma \[lem:4\] is proved. We now deal with the second part and we let $V\in C^2(\rr)$ be as in the statement. We define $\psi:=V^{-1}\varphi$. Then, $-\ddot{\psi}+h \dot{\psi}+Q \psi=0$ in $\rr$ with $h,Q\in C^0(\rr)$ and $Q>0$. Therefore, by studying the zeros, $\dot{\psi}$ vanishes at most once, and then $\psi(t)$ has limits as $t\to\pm\infty$. Since $\varphi=\psi V$, $\varphi\in L^2(\rr)$ and $V\not\in L^2(0,+\infty)$, then $\lim_{t\to +\infty}\psi(t)=0$. We claim that $\lim_{t\to-\infty}\psi(t)\neq 0$. Otherwise, the limit would be $0$. Then $\psi$ would be of constant sign, say $\psi>0$. At the maximum point $t_0$ of $\psi$, the equation would yield $\ddot{\psi}(t_0)>0$, which contradicts the maximum. So the limit of $\psi$ at $-\infty$ is nonzero, and then $V(t)=O(\varphi(t))$ as $t\to-\infty$. We claim that $\varphi$ is even or odd and $\varphi$ has constant sign around $+\infty$. Since $t\mapsto \varphi(-t)$ is also a solution to the ODE, it follows from Lemma \[lem:5\] that it is a multiple of $\varphi$, and then $\varphi$ is even or odd. Since $\dot{\psi}$ changes sign at most once, then $\psi$ changes sign at most twice. Therefore $\varphi=\psi V$ has constant sign around $+\infty$. We fix $0<A'<A$ and we let $R_0>0$ such that $q(t)>A'$ for all $t\geq R_0$. Without loss of generality, we also assume that $\varphi(t)>0$ for $t\geq R_0$. We define $b(t):=C_0e^{-\sqrt{A'}t}-\varphi(t)$ for all $t\in\rr$ with $C_0:=2\varphi(R_0)e^{\sqrt{A'}R_0}$. We claim that $b(t)\geq 0$ for all $t\geq R_0$. Otherwise $\inf_{t\geq R_0}b(t)<0$, and since $\lim_{t\to +\infty}b(t)=0$ and $b(R_0)>0$, then there exists $t_1>R_0$ such that $\ddot{b}(t_1)\geq 0$ and $b(t_1)<0$. However, as one checks, the equation yields $\ddot{b}(t_1)<0$, which is a contradiction. Therefore $b(t)\geq 0$ for all $t\geq R_0$, and then $0<\varphi(t)\leq C_0e^{-\sqrt{A'}t}$ for $t\to +\infty$. Lemma \[lem:4\] follows from this inequality, $\varphi$ even or odd, and $V(t)=O(\varphi(t))$ as $t\to-\infty$. [12]{}

--- author: - Arianna Carbone - Arianna Carbone bibliography: - 'biblio.bib' title: | Self-consistent\ Green’s functions\ with three-body forces ---

--- abstract: 'Two-dimensional transition metal dichalcogenides (TMDs) offer the possibility to address the electron’s valley degree of freedom making them interesting for valley-based electronics, so-called valleytronics[@NatPhys.10.343; @NatureReviewsMaterials.1.16055; @Nat.Nanotechnol.13.11]. The device performance of potential valleytronic applications largely depends on the valley lifetimes of free charge carriers. Here, we report on nanosecond long, gate-dependent valley lifetimes of free charge carriers in WSe$_2$, unambiguously identified by the combination of time-resolved Kerr rotation (TRKR) and electrical transport measurements. While the valley polarization increases when tuning the Fermi level into the conduction or valence band, there is a strong decrease of the respective valley lifetime consistent with both electron-phonon and spin-orbit scattering. The longest lifetimes are seen for spin-polarized bound excitons in the band gap region. We explain our findings via two distinct, Fermi level-dependent scattering channels of optically excited, valley polarized bright trions either via dark or bound states. By electrostatic gating we demonstrate that WSe$_2$ can be tuned to be either an ideal host for long-lived localized spin states or allow for nanosecond valley lifetimes of free charge carriers ($> 10$ ns) explaining the huge variation in previously reported lifetimes extracted from TRKR[@NatPhys.11.830; @PhysRevLett.119.137401; @ScienceAdvances.5.eaau4899; @NanoLetters.17.4549; @NanoLetters.19.4083; @NatureComm.6.896; @NanoLett.16.5010; @APL.111.082404; @2DMaterials.5.011010; @PhysRevB.90.161302].' author: - Manfred Ersfeld - Frank Volmer - Lars Rathmann - Luca Kotewitz - Maximilian Heithoff - Mark Lohmann - Bowen Yang - Kenji Watanabe - Takashi Taniguchi - Ludwig Bartels - Jing Shi - Christoph Stampfer - Bernd Beschoten bibliography: - 'Literature.bib' title: 'Unveiling valley lifetimes of free charge carriers in monolayer WSe$_2$' --- With band gaps in the optically visible energy range[@AdvMater.9.1399; @NatureReviewsMaterials.2.17033; @NaturePhotonic.10.216] and spin-split valleys which allow the creation of valley- and spin-polarized excitons by circularly polarized light[@NatPhys.10.343; @RevModPhys.90.021001], monolayer (ML) TMDs are a very promising family of materials in the field of both spin- and valleytronics[@NatPhys.10.343; @NatureReviewsMaterials.1.16055; @Science.360.893]. However, the full potential of TMD-based devices for valleytronic applications critically depends on their valley lifetime of free charge carriers. So far, there is a huge variation in experimentally reported valley lifetimes, which range from the picosecond up to the microsecond timescale[@NatPhys.11.830; @PhysRevLett.119.137401; @ScienceAdvances.5.eaau4899; @NanoLetters.17.4549; @NanoLetters.19.4083; @NatureComm.6.896; @NanoLett.16.5010; @APL.111.082404; @2DMaterials.5.011010; @PhysRevB.90.161302]. Especially the longer lifetimes are not consistent with ab-initio studies that predict valley lifetimes in the picosecond range limited by electron-phonon coupling[@NanoLetters.17.4549; @NanoLetters.19.4083]. This discrepancy is partly due to the ambiguous use of the term valley polarization, which can be explained in respect to two measurement techniques typically used for exploring spin and valley dynamics in ML TMDs: Time-resolved photoluminescence (TRPL) and time-resolved Kerr rotation (TRKR). Both techniques rely on the valley-selective optical excitation of excitons by circularly polarized laser pulses (see Fig. 1a, circles represent the photo-excited electron-hole pair, shaded areas represent the filling of the bands with free charge carriers). The situation in Fig. 1a is sometimes called a valley polarization as, e.g., the numbers of holes in the valence band differ between the K and K’ valleys. In this article we identify how such an exciton valley polarization can create a net valley polarization of free conduction and valence band states after the photo-excited charge carriers have recombined (see Fig. 1b) and determine the valley lifetimes of the respective free charge carrier valley polarization. Here, TRPL reaches its limitations as it is restricted to exciton lifetimes[@RevModPhys.90.021001] and is therefore not capable to determine a valley polarization of free charge carriers after exciton recombination. However, the probe pulse in TRKR can detect the temporal decay of the valley polarization from the Kerr rotation angle when tuned to the trion energy, as the creation of the trion depends on the availability of free charge carriers within each valley. {width="\linewidth"} \[fig1\] To identify the existence of a valley polarization of free charge carriers and determine its valley lifetimes, we combine TRKR, electrical transport measurements, and photoluminescence (PL) spectroscopy on ML WSe$_2$ protected by hexagonal boron nitride and contacted via graphite electrodes (see Method section and Supporting Information (SI) for details on device fabrication). The combination of electrical transport and PL measurements (Fig. 1c) allows to assign the position of the Fermi level to the corresponding back-gate voltage ($V_\text{BG}$)[@NatureNanotechnology.12.144]: Between and the Fermi-level is pinned within the band gap of the TMD due to mid-gap states, resulting in a very low conductance with a current of some tens of pA due to residual hopping transport[@ACSNano.5.7707; @Nat.Commun.4.2642] and the dominance of bound exciton states (X$_\text{bound}$) in the PL spectrum[@PhysRevLett.121.057403; @SciRep.3.2657; @RevModPhys.90.021001]. Between and the current undergoes an exponential increase as the Fermi level moves through the tail states of the conduction band[@NatureCommunications.5.3087; @IEEEElectronDeviceLetters.39.761]. Above $V_\text{BG}=\unit{20}{V}$ the Fermi level is tuned into the conduction band, resulting in a linear gate voltage dependence of the current in accordance to diffusive charge transport of free electrons and the complete disappearance of the neutral exciton (X$_0$) as enough free charge carriers are present for the formation of the energetically favourable charged excitons (trions, X$_{+/-}$)[@RevModPhys.90.021001]. At even higher electron densities we observe the appearance of the X$'_-$ feature, which can either be attributed to the interaction of an exciton with the Fermi sea of free electrons or to the onset of filling the energetically higher, spin-split conduction band[@PhysRevLett.120.066402; @PhysRevB.95.035417; @NaturePhysics.13.255]. Fig. 1d shows TRKR traces for representative gate voltages measured with pump and probe energies in the trion regime at a temperature of $T=\unit{40}{K}$, showing two exponentially decaying signals. Within the band gap ($V_\text{BG}=\unit{-40}{V}$, orange data points, dashed line represents the axis with the Kerr rotation angle $\theta_\text{K}$ being zero) a long-lived polarization with a lifetime exceeding the laser repetition interval of can be observed together with a short-lived polarization of around . In the transition regime between tail states and conduction band ($V_\text{BG}=\unit{20}{V}$, green curve) the long-lived polarization starts to vanish, whereas the short-lived polarization first undergoes a sign reversal and then shows an increasing amplitude towards higher charge carrier densities at larger $V_\text{BG}$ values (compare green, blue, and violet curves within the first two ns). We fitted the data over the whole gate-voltage range by the sum of two exponential decays (solid lines) and plotted the extracted amplitudes and the lifetimes in Figs. 1e and 1f, respectively. The Kerr rotation amplitude $\theta_\text{K}$ of the long-lived polarization is almost constant over the whole gap regime and starts to disappear as soon as the Fermi level reaches the tail regime (red data points in Fig. 1e), demonstrating its connection to band gap states. On the other hand, the amplitude of the short-lived polarization (black data points) shows a linear increase with increasing charge carrier density between $V_\text{BG}=\unit{10}{V}$ and $\unit{50}{V}$ (see green line in Fig. 1e as a guide to the eye). As this increase goes hand in hand with the linear increase in current in this gate voltage range, it is clearly connected to the increasing number of free charge carriers in the conduction band and, hence, can be attributed to a valley polarization of these free charge carriers. This assignment is backed up by the fact that the amplitude reaches a maximum at around the same gate voltages where the X$'_-$ feature appears in the PL map of Fig. 1c. The decrease of $\theta_\text{K}$ is expected at even larger $V_\text{BG}$ values as the filling of the upper, spin-inverted conduction band will reduce the overall net valley polarization of free charge carriers. Finally, the strong decrease in lifetime towards higher gate-voltages (see black data points in Fig. 1f) also supports our assignment of the measured valley polarization to the free band carriers, as such a decay is expected from both wave vector dependent electron-phonon and spin-orbit scattering mechanisms[@PhysRevB.87.245421; @PhysRevB.90.235429; @PhysRevB.93.075415; @PhysRevB.90.035414; @PhysRevB.93.035414; @NanoLetters.17.4549]. {width="\linewidth"} \[fig2\] The free carrier valley polarization and the band gap polarization show distinctively different temperature dependencies of their respective lifetimes as depicted in Fig. 1g, where for $V_\text{BG}<\unit{0}{V}$ only the band gap polarization and for $V_\text{BG}>\unit{0}{V}$ only the valley polarization is plotted for simplicity (see dashed lines in Fig. 1f). The valley lifetime of the conduction band polarization strongly decreases from at to below at . Such a strong temperature dependence is expected if electron-phonon scattering limits the valley lifetimes[@PhysRevB.93.035414; @NanoLetters.17.4549; @NanoLetters.19.4083]. In contrast, the long-lived polarization in the band gap is far more robust against temperature and retains lifetimes of at and can be well observed up to temperatures of around . This is the temperature at which the bound exciton emission in PL typically disappears[@PhysRevB.90.161302; @APL.105.101901; @ScientificRep.6.22414], highlighting a possible connection between this long-lived polarization and bound excitons. This notion is backed up by comparing other publications showing either long-lived TRKR signals or bound exciton features in PL, respectively, showing that both signals often have similar temperature dependencies[@PhysRevB.90.161302; @APL.105.101901; @ScientificRep.6.22414; @NanoLett.16.5010; @NatureComm.6.896; @PhysRevLett.119.137401; @PhysRevB.95.235408]. The mechanisms which explain both the formation of the free carrier valley polarization and the long-lived polarization in the gap region are depicted in Fig. 2. We start with the Fermi level tuned into the conduction band ($V_\text{BG}>\unit{10}{V}$) at equilibrium conditions, i.e. an equal number of free charge carriers in both valleys (indicated by the shaded areas in the band structure). A circularly polarized laser pump pulse now excites valley-selectively electron-hole pairs (circles in Fig. 2a)[@RevModPhys.90.021001]. By the interaction with the free conduction band carriers, the electron-hole pairs first create bright trions which, however, cannot directly be responsible for the TRKR signals in the ns range as their recombination times are typically in the ps range[@RevModPhys.90.021001; @PhysRevB.93.205423; @PhysRevLett.123.067401]. The recombination of the photo-exited electron-hole pairs within the same valley can also not lead to a net valley polarization of resident charge carriers. In contrast, a net valley polarization can be formed only if one charge of the electron-hole pair scatters into the other valley. In case of WSe$_2$, this can easily happen by the intervalley transition of the electron into the energetically lower conduction band of the same spin-orientation in the K’-valley under the emission of an optical phonon (Fig. 2b)[@RevModPhys.90.021001; @PhysicalReviewMaterials.2.014002]. The subsequent recombination of the photo-excited holes with free electrons of the K-valley on the time-scale $\tau_\text{rec}^\text{dark}$ (this process has a finite possibility as explained in the SI) will lead to a situation where the photo-exited electrons have increased the electron number in the K’-valley. At the same time the K-valley is missing the same number of free electrons which recombined with the photo-excited holes (see Figs. 2c and 2d). This leads to a net valley polarization of conduction band electrons. We assign the Fermi level dependent valley lifetimes $\tau_\text{v}^\text{CB}$ of the valley polarization created by this process to the black data points in Fig. 1f for $V_\text{BG}>\unit{10}{V}$. Our model also explains the linear increase in Kerr rotation amplitude between $V_\text{BG}=\unit{10}{V}$ and $\unit{50}{V}$ in Fig. 1e, as with increasing free charge carrier density it becomes more likely that photo-excited holes can recombine with free charge carriers of the K-valley’s conduction band. Next to the scattering via dark states, bright trions can also bind to localized states within the band gap caused e.g. by vacancies or dopant atoms. We identify these bound states as the origin of the long-lived polarization seen as the red data points in Fig. 1e and 1f. We note that this polarization is also measured with a laser probe pulse at the bright trion energy. When probing at the energetically lower bound exciton energies no Kerr rotation signal is observed. Therefore, we do not directly probe the polarization of the bound exciton states but rather have to consider a charge transfer process between bound excitons and the optically accessible conduction band states. This leads to a complication: On the one hand, the Fermi level has to be in the conduction band to have free carriers available both for the formation of the initially excited bright trions by the pump pulse and also for the creation of the final valley polarization. But at the same time, having the Fermi level in the conduction band also means that all band gap states are occupied by unpolarized charge carriers, preventing an interaction with bright trions. In this context, it is important to consider the existence of charge puddles in the transition regions between gap and bands. Scanning tunneling spectroscopy studies revealed that monolayer TMDs can show conduction band minima (CBM) and valence band maxima (VBM) energies which can spatially vary by hundreds of meV over length-scales as small as due to strain variations (see schematic in Fig. 2e)[@AdvancedMaterials.28.9378]. In this situation, charge puddles are formed as depicted by the shaded areas in Fig. 2e. While bright trions are easily formed with the charges in these puddles, their formation is diminished in nearby regions where the Fermi level is in the band gap. This picture accounts for the coexistence of both the exciton (X$_0$) and the trion feature (X$_{+/-}$) over a part of the gate voltage range in PL measurements as seen both in Fig. 1c and previous work[@RevModPhys.90.021001]. Our model for the long-lived polarization is now based on the assumption that a photo-excited electron-hole pair can create a bound trion at a band gap state with an additional charge carrier taken from the conduction band (i.e. charges from the puddles) (Fig. 2f). The resulting imbalance of free charge carriers between K and K’ valley will relax to equilibrium conditions within the time-scale of the valley lifetime $\tau_\text{v}^\text{CB}$ (Fig. 2g). This explains why we also see the valley polarization within the gap region (compare Figs. 2d with 2g) and that the lifetime of the valley polarization (black data points in Fig. 1f) shows a smooth transition over the whole gate voltage range. The bound exciton lifetimes can reach up to the $\mu$s range[@PhysRevLett.121.057403; @PhysRevB.96.121404], which is comparable to the longest reported lifetimes in TRKR measurements[@PhysRevLett.119.137401; @NanoLetters.19.4083; @ScienceAdvances.5.eaau4899]. Furthermore, it was demonstrated that these bound excitons show a certain degree of polarization in PL[@PhysRevLett.121.057403], indicating that a spin polarization of these bound excitons is not completely relaxed at the time of their recombination. Therefore, during the bound exciton’s recombination (Fig. 2h), we argue that the initially caught charge carrier will predominately transfer back into the valley where it came from (the one with the same spin orientation) and, hence, will again create a valley polarization (Fig. 2i). As it is discussed in the SI in more detail, the measured long-lived TRKR signal mainly reflects the bound trion recombination time as $\tau_\text{v}^\text{CB} \ll \tau_\text{rec}^\text{bound}$. The decrease in its amplitude (red data points in Fig. 1e) towards the conduction band can be well understood, as an increasing Fermi level will decrease the number of unoccupied localized states in the band gap at which a bright trion can be bound[@PhysRevLett.121.057403]. An important aspect of our model is that it predicts a sign reversal of the net valley polarization for the two scattering mechanisms via dark and bound excitons (compare Figs. 2d and 2i where the net valley polarization is carried by K’ and by K states, respectively), which is indeed seen in the measurements (see black data points for positive and red data points for negative gate voltages in Fig. 1e). {width="1\linewidth"} \[fig3\] To confirm that our model indeed describes the underlying valley- and spin-dynamics consistently, we study a device made from ML MoSe$_2$. In contrast to WSe$_2$, in MoSe$_2$ the formation of dark excitons is energetically not favorable due to its inverted conduction band spin ordering (see inset in Fig. 3a)[@RevModPhys.90.021001; @PhysicalReviewMaterials.2.014002]. Hence, the scattering mechanism via dark excitons (Figs. 2b to 2d), which is responsible for the linear increase of the valley polarization with increasing charge carrier density, is not expected. In contrast, the scattering mechanism via bound excitons is still feasible in MoSe$_2$ (Figs. 2f to 2i), but in that case the amplitude of the valley polarization is not limited by the amount of free charge carriers but rather by the number of bound excitons which can be created (see transition from Fig. 2f to 2g). And in fact, the TRKR data from the MoSe$_2$ sample shows both a long-lived and a short-lived polarization (see Figs. 3b and 3c) consistent to the scattering mechanism via bound states, but no increase of the Kerr rotation amplitude towards higher charge carrier densities, although the increase of the measured current by more than four orders of magnitude (Fig. 3a) clearly demonstrates that the Fermi level can be tuned into the conduction band of MoSe$_2$. {width="1\linewidth"} \[fig4\] Finally, to confirm that our model also holds for the valence band we studied another WSe$_2$ sample in which we were able to tune the Fermi level all the way from the valence band into the conduction band. Figs. 4a and 4b show the Kerr rotation amplitude and the lifetime of the respective valley polarizations of both bands which got polarized by the scattering channel via dark trion states (Figs. 2b to 2d). Next to the conduction band, also the valence band shows the expected increase in the valley polarization and decrease in valley lifetimes with higher charge carrier densities. Interestingly, the valley lifetimes of the valence band states are a factor of 30 longer than the corresponding valley lifetimes of the conduction band states (see different y-axes). Remarkably, the valley lifetimes in our samples are found to vary between and tens of ns at , which is much longer than expected from ab-initio studies of pristine TMD monolayers that predict electron-phonon-limited valley lifetimes in the lower ps-range[@NanoLetters.17.4549; @NanoLetters.19.4083]. Accordingly, to achieve such long-lived valley lifetimes the intervalley electron-phonon scattering has to be suppressed, which is most likely accomplished by fabrication-induced local strain variations in our samples. In this respect, it was shown that strain indeed has a significant impact on the band structure of TMDs[@AdvancedMaterials.28.9378; @NanoLetters.13.3626], being able to suppress the electron-phonon coupling[@PhysRevB.90.035414; @NanoLetters.18.1751] and, hence, can e.g. lead to an increase in charge carrier mobilities[@PhysRevB.90.035414; @NatureNanotechnology.14.223]. Together with our model in Fig. 2 this possible strain-induced decrease in electron-phonon coupling can explain how an exciton valley polarization can create a nanosecond long valley polarization of free charge carriers. This valley polarization of free charge carriers can be clearly identified in our experiments, as 1.) its amplitude increases linearly with the gate-induced charge carrier density which goes hand in hand with a simultaneous increase in electrical conductance, 2.) its amplitude decreases as soon as the upper, spin-inverted conduction band starts to be filled, and 3.) its lifetime decreases towards higher charge carrier densities in accordance to theory about wave vector dependent electron-phonon and spin-orbit scattering mechanisms[@PhysRevB.87.245421; @PhysRevB.90.235429; @PhysRevB.93.075415; @PhysRevB.90.035414; @PhysRevB.93.035414; @NanoLetters.17.4549]. Overall, we have shown that by changing the Fermi level via gating, TMDs can be tuned to be either ideal hosts for long-lived localized spin states or allow valley lifetimes of conduction and valence band states exceeding at , which are adequate timescales for both spin manipulation and valley transport. [**Acknowledgements:**]{} The authors thank P.M.M.C. de Melo, M.J. Verstraete and Z. Zanolli for helpful discussions. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 785219 (Graphene Flagship), by US National Science Foundation (NSF) under ECCS Award \# 1610447 and under grant DMR-1609918, and by the Helmholtz Nanoelectronic Facility (HNF)[@HNF] at the Forschungszentrum Jülich. Growth of hexagonal boron nitride crystals was supported by the Elemental Strategy Initiative conducted by the MEXT, Japan, A3 Foresight by JSPS and the CREST (Grant No. JPMJCR15F3), JST. [**Author information:**]{} Manfred Ersfeld, Frank Volmer, and Lars Rathmann are equally contributing authors. [**Methods:**]{} [*Measurement scheme to minimize photo-induced screening of the gate electric field –*]{} For optical measurements on 2-dimensional materials it is important to account for the screening of the gate electric field by photo-excited charged defects in the dielectric layer[@NatureNanotechnology.9.348; @Nanoscale.11.7358]. We note that the true gate-dependent spin and valley dynamics can be masked when using a conventional measurement scheme, in which the time delay between pump and probe pulses is swept at a fixed gate voltage. Within this conventional measurement scheme a photo-induced temporal shift of the Fermi level on a laboratory time-scale after setting a new gate voltage will lead to the fact that Kerr rotation signals within one time delay trace get recorded at effectively different charge carrier densities. We therefore developed a new measurement scheme in which we continuously sweep the gate voltage for fixed time delays between pump and probe pulses. In our new method each data point of a specific TRKR curve is recorded exactly the same amount of time after setting the corresponding gate voltage. All data points are thus recorded at the same charge carrier density. See the Supporting Information for a more detailed discussion on this matter. [*Sample fabrication –*]{} Hexagonal boron nitride (hBN)/ML TMD half-stacks were transferred via a dry transfer method onto prepatterned electrodes fabricated on Si$^{++}$/SiO$_2$ () wafers used as back gates. The fabrication of CVD WSe$_2$ is described in Ref.. See SI for more details. [*TRKR –*]{} We use two mode-locked Ti:sapphire lasers to independently tune the energies of both pump and probe laser pulses. The pump energy was kept above the probe energy. Both energies were kept near the peak position of the trion feature in PL. An electronic delay between both pulses covers the full laser repetition interval of with a jitter of less than . The pulse widths are and the FWHM spot sizes are $\unit{8}{\mu m}$. The power was kept between $\unit{200-600}{\mu W}$. A detailed scheme of the experimental setup can be found in the SI. [*PL –*]{} For excitation a diode cw-laser with an energy of and a power of $\unit{1}{\mu W}$ was used. A lens with a numerical aperture of 0.66 focused the laser beam to a FWHM spot size of $\unit{6}{\mu m}$ and also collects the emitted PL light.

--- abstract: | We provide some new estimates for Bellman type functions for the dyadic maximal opeator $\mathbb{R}^{n}$ and of maximal operators on martingales related to weighted spaces. Using a type of symmetrization principle, introduced for the dyadic maximal operator in earlier works of the authors we introduce certain conditions on the weight that imply estimate for the maximal operator on the corresponding weighted space. Also using a well known estimate for the maximal operator by a double maximal operators on different measures related to the weight we give new estimates for the above Bellman type functions. This research has been co-financed by the European Union and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF). ARISTEIA I, MAXBELLMAN 2760, research number 70/3/11913. address: 'Department of Mathematics, University of Athens, Panepistimiopolis 15784, Athens, Greece' author: - 'Antonios D. Melas' - 'Eleftherios N. Nikolidakis' - Dimitrios Cheliotis date: 'September 25, 2015' title: 'Estimates for Bellman functions related to dyadic-like maximal operators on weighted spaces ' --- Introduction ============ The dyadic maximal operator on $\mathbb{R}^{n}$ is defined by $$M{}\,_{d}\phi{}(x)=\sup\left\{ \frac{1}{\left\vert S\right\vert }\int _{S}\left\vert \phi(u)\right\vert du:x\in S\text{, }S\subseteq\mathbb{R}^{n}\text{ is a dyadic cube}\right\} \label{i1}$$ for every $\phi\in L_{\text{loc}}^{1}(\mathbb{R}^{n})$ where the dyadic cubes are the cubes formed by the grids $2^{-N}\mathbb{Z}^{n}$ for $N=0,1,2,...$. As it is well known it satisfies the following weak $L^{p}$ inequality (for martingales known as Doob’s inequality) $$\left\Vert M_{d}\phi\right\Vert _{p}\leq\dfrac{p}{p-1}\left\Vert \phi\right\Vert _{p}\label{i3}$$ for every $p>1$ and every $\phi\in L^{p}(\mathbb{R}^{n})$ which is best possible (see [@Burk1], [@Burk2] for the general martingales and [@Wang] for dyadic ones). An approach for studying more in depth the behavior of this maximal operator is the introduction of the so called Bellman functions (see [@Naz]) related to them which reflect certain deeper properties of them by localizing. Such functions related to the $L^{p}$ inequality (\[i3\]) have been precisely evaluated in [@Mel1]. Actually defining for any $p>1$ $$\mathcal{B}_{p}(F,f)=\sup\left\{ \dfrac{1}{\left\vert Q\right\vert }\int _{Q}(M_{d}^{\prime}\phi)^{p}:\operatorname{Av}_{Q}(\phi^{p})=F,\operatorname{Av}_{Q}(\phi)=f\right\} \label{i4}$$ where $Q$ is a fixed dyadic cube, $R$ runs over all dyadic cubes containing $Q$, $\phi$ is nonnegative in $L^{p}(Q)$ and the variables $F,f$ satisfy $0\leq f,f^{p}\leq F$ which is independent of the choice of $Q$ (so we may take $Q=[0,1]^{n}$) and where the localized maximal operator $M_{d}^{\prime }\phi$ is defined as in (\[i1\]) with the dyadic cubes $S$ being assumed to be contained in the ambient dyadic cube $Q$. It has been shown in [@Mel1] that $$\mathcal{B}_{p}(F,f)=F\omega_{p}\left( \dfrac{f^{p}}{F}\right) ^{p}\label{i5}$$ where $\omega_{p}:$ $[0,1]\rightarrow\lbrack1,\frac{p}{p-1}]$ is the inverse function of $H_{p}(z)=-(p-1)z^{p}+pz^{p-1}$. Actually (see [@Mel1]) the more general approach of defining Bellman functions with respect to the maximal operator on a nonatomic probability space $(X,\mu)$ equipped with a tree $\mathcal{T}$ (see Section 2) can be taken and the corresponding Bellman function is always the same. The fact that the range of $\omega_{p}$ is $[1,\frac{p}{p-1}]$ shows in a sense the extend that the constant in Doob’s inequality can be approached only by functions whose integral is very small compared to its $p$-norm. For example for $p=2$ we get the following sharp improvement of Doob’s inequality $$\left\Vert M_{\mathcal{T}}\phi\right\Vert _{2}\leq\left\Vert \phi\right\Vert _{2}+(\left\Vert \phi\right\Vert _{2}^{2}-\left\Vert \phi\right\Vert _{1}^{2})^{1/2}<2\left\Vert \phi\right\Vert _{2}\label{i12}$$ which aside from the $L^{2}$ norm of $\phi$ involves also in a sharp way the *variance* of $\phi$. Here we will be concerned with the behavior of these maximal operators on weighted spaces. As it is well known for any positive locally integrable function $w$ on $Q$ the estimate$$\int_{Q}(M_{d}^{\prime}\phi)^{p}w\leq C\int_{Q}\phi^{p}w\label{a}$$ holds for all $\phi$ if and only if $w$ is a dyadic $A_{p}$ weight in the sense that $$\sup\{\left\vert I\right\vert ^{-p}(\int_{I}w)(\int_{I}w^{-\frac{1}{p-1}})^{p-1}:I\text{ dyadic subcube of }Q\}=[w]_{p}<+\infty\text{.}$$ Also it is known that the best possible $C$ is of the order of $[w]_{p}^{p/(p-1)}$, the exponent being best possible. Related to this one may define the following Bellman function given a weight $w$$$\mathcal{B}_{p,w}(F,f)=\sup{\huge \{}\dfrac{1}{\left\vert Q\right\vert }\int_{Q}(M_{d}^{\prime}\phi)^{p}w:\operatorname{Av}_{Q}(\phi^{p}w)=F,\operatorname{Av}_{Q}(\phi)=f{\huge \}}\label{i6a}$$ which is finite only if $w$ is in $A_{p}$ and seek estimates for this in order to improve the above estimate (\[a\]). One may add more variables to the above Bellman function as the integrals of $w$ and of $w^{-1/(p-1)}$ over $Q$ but we will not treat those here. The estimates here will be proved in the general setting of tree like families on probability spaces and its related maximal operator, as will be described in the next section. ‘We will derive two types of estimates related to the above problems. In the first we will use a related condition on some symmetrization of the weight to find the exact form of a related to weights Bellman function and this is done in section 2. Then in section 3 we obtain certain new estimates for the above Bellman function related to $A_{p}$ with respect to a tree, and to the corresponding maximal operator, by using an estimate of the maximal operator via two applications of maximal operators on the same tree but with different measures, and this is described in section 3. There are several other problems in Harmonic Analysis where Bellman functions naturally arise. Such problems (including the dyadic Carleson imbedding and weighted inequalities) are described in [@Naz2] (see also [@Naz], [@Naz1]) and also connections to Stochastic Optimal Control are provided, from which it follows that the corresponding Bellman functions satisfy certain nonlinear second order PDE. The exact computation of a Bellman function is a difficult task which is connected with the deeper structure of the corresponding Harmonic Analysis problem. Thus far several Bellman functions have been computed (see [@Burk1], [@Burk2], [@Mel1], [@Sla], [@Sla1], [@Vas], [@Vas1], [@Vas2]). L.Slavin and A.Stokolos [@SlSt] linked the Bellman function computation to solving certain PDE’s of the Monge Ampere type, and in this way they obtained an alternative proof of the Bellman functions relate to the dyadic maximal operator in [@Mel1]. Also in [@Vas2] using the Monge-Ampere equation approach a more general Bellman function than the one related to the dyadic Carleson imbedding Theorem has be precisely evaluated thus generalizing the corresponding result in [@Mel1]. Trees, maximal operators and symmetrization =========================================== As in [@Mel1] we let $(X,\mu)$ be a nonatomic probability space (i.e. $\mu(X)=1$). Two measurable subsets $A$, $B$ of $X$ will be called almost disjoint if $\mu(A\cap B)=0$. Then we give the following. A set $\mathcal{T}$ of measurable subsets of $X$ will be called a tree if the following conditions are satisfied: \(i) $X\in\mathcal{T}$  and for every $I\in\mathcal{T}$  we have $\mu(I)>0$. \(ii) For every $I\in\mathcal{T}$  there corresponds a finite subset $\mathcal{C}(I)\subseteq\mathcal{T}$  containing at least two elements such that: \(a) the elements of $\mathcal{C}(I)$ are pairwise almost disjoint subsets of $I$, \(b) $I=\bigcup\mathcal{C}(I)$. \(iii) $\mathcal{T}=\bigcup_{m\geq0}\mathcal{T}_{(m)}$ where $\mathcal{T}_{(0)}=\{X\}$ and $\mathcal{T}_{(m+1)}=\bigcup_{I\in\mathcal{T}_{(m)}}\mathcal{C}(I)$. \(iv) We have $\lim\limits_{m\rightarrow\infty}\sup\limits_{I\in\mathcal{T}_{(m)}}\mu(I)=0$ and $\mathcal{T}$ differentiates $L^{1}$. By removing the measure zero exceptional set $E(\mathcal{T})=\bigcup _{I\in\mathcal{T}}\bigcup_{\substack{J_{1},J_{2}\in\mathcal{C}(I)\\J_{1}\neq J_{2}}}(J_{1}\cap J_{2})$ we may replace the almost disjointness above by disjointness. Now given any tree $\mathcal{T}$ we define the maximal operator associated to it as follows $$M_{\mathcal{T}}\phi(x)=\sup\left\{ \frac{1}{\mu(I)}\int_{I}\left\vert \phi\right\vert d\mu:x\in I\in\mathcal{T}\right\} \label{t1}$$ for every $\phi\in L^{1}(X,\mu)$. The above setting can be used not only for the dyadic maximal operator but also for the maximal operator on martingales, hence many of the results here can be viewed as generalizations and refinements of the classical Doob’s inequality. Also for any locally integrable positive function $w$ on $X$, which will be called weight, we denote $\sigma=w^{-\frac{1}{p-1}}$, and for any $I\in\mathcal{T}$ we write $w(I)=\int_{I}wd\mu$, $\sigma(I)=\int_{I}\sigma d\mu$. Now we give the following. A weight $w$ on $X$ will be called $A_{p}$ with respect to $\mathcal{T}$ if the following expression$$\lbrack w]_{\mathcal{T}\text{,}p}=[w]_{p}=\sup_{I\in\mathcal{T}}\frac{w(I)\sigma(I)^{p-1}}{\mu(I)^{p}}$$ is finite. A way to study estimates for the above maximal operator is through the symmetrization of $\phi$ as has been introduced in [@Mel2] and [@Nik] and used in [@Mel5] to evaluate Bellman functions related to Lorentz norms. In order to apply this in the context of weights we introduce the following condition on a weight $w$ on $X$. A weight $w$ on $X$ will be called $A_{p}^{\ast}$ if for some equimeasurable rearrrangement $w^{\ast\ast}$ of $w$ on $(0,1)$ (not necessarily decreasing) there exist two constants $c,a>0$ such that for every $t$ in $(0,1]$ the following estimate holds$$\int_{t}^{1}\frac{w^{\ast\ast}(s)}{s^{p}}ds+c\leq a\frac{w^{\ast\ast}(t)}{t^{p-1}}\label{Ap1}$$ and also$$\lim_{t\rightarrow0^{+}}t^{p}\int_{t}^{1}\frac{w^{\ast\ast}(s)}{s^{p}}ds=0\label{Ap1a}$$ Note that by writing $r(t)=\frac{w^{\ast\ast}(t)}{t^{p-1}}$ the first condition implies that $r(t)>\frac{c}{a}>0$ for all $t$ hence $\lim _{t\rightarrow0^{+}}\int_{t}^{1}\frac{w^{\ast\ast}(s)}{s^{p}}ds=+\infty$ and so $\lim_{t\rightarrow0^{+}}r(t)=0$. Hence we conclude that there is a best possible pair $(a,c)$ for each such weight, namely $a=\sup_{t}r(t)^{-1}\int_{t}^{1}\frac{r(t)}{t}dt$ and $c=\sup_{t}(ar(t)-\int_{t}^{1}\frac{r(t)}{t}dt)$. We will refer to this pair as the *constants of the corresponding* $A_{p}^{\ast}$ weight $w$. **Example.** Suppose that $w^{\ast\ast}(t)=kt^{b}$ with $k,b\in\mathbf{R}$. Then the above conditions hold if and only if $-1<b<p-1$ which is exactly the range making $w^{\ast\ast}$ an $A_{p}$ weight on $(0,1)$. Moreover the corresponding constants $c,a$ can be easily seen to be $a=\frac{1}{p-1-b},~c=\frac{k}{p-1-b}$. Now we take into consideration the following theorem proved in [@Nik] and [@Mel2]. Let $G:[0,+\infty)\rightarrow\lbrack0,+\infty)$ be non-decrasing, $h:(0,1]\rightarrow$ $\mathbb{R}^{+}$ be any locally integrable function. Then for any nonatomic probability space $(X,\mu)$, equipped with any tree-like family $\mathcal{T}$ , for any non-increasing right continuous integrable function $g:(0,1]\rightarrow$ $\mathbb{R}^{+}$ and any $k\in(0,1]$, the following equality holds (where by $\psi^{\ast}$ we denote the decreasing equimeasurable rearrangement of $\psi$):en $$\begin{gathered} \sup\left\{ \int_{0}^{k}G[(M_{\mathcal{T}}\phi)^{\ast}(t)]h(t)dt:\phi\text{ measurable on }X\text{ with }\phi^{\ast}=g\right\} =\\ =\int_{0}^{k}G\left( \frac{1}{t}\int_{0}^{t}g(u)du\right) h(t)dt.\end{gathered}$$ After this given an $A_{p}^{\ast}$ weight $w$, we define the following variant of the Bellman function (\[i6a\]).$$\mathcal{B}_{p,w}^{\ast}(F,f)=\sup{\huge \{}\int_{0}^{1}((M_{\mathcal{T}}\phi)^{\ast})^{p}w^{\ast\ast}:\int_{0}^{1}(\phi^{\ast})^{p}w^{\ast\ast}=F,\int_{X}\phi=f{\huge \}}$$ where here by $\phi^{\ast},~(M_{\mathcal{T}}\phi)^{\ast}$ we denote the equimeasurable **decreasing** rearrangement of $\phi$ and $M_{\mathcal{T}}\phi$ whereas by $w^{\ast\ast}$ we denote the equimeasurable rearrangement of $w$ that appears in the above definition. Note that in case $w^{\ast\ast}$ is decreasing $\int_{X}((M_{\mathcal{T}}\phi)^{\ast})^{p}w^{\ast\ast}$ is greater than or equal to $\int_{X}(M_{\mathcal{T}}\phi)^{p}w$ and $\int_{0}^{1}(\phi^{\ast})^{p}w^{\ast\ast}$ is greater than or equal to $\int_{X}\phi^{p}w$ and when $w^{\ast\ast}$ is increasing then the opposite relations hold. Then we can prove the following. For the above function we have$$\mathcal{B}_{p,w}^{\ast}(F,f)=(p-1)^{p}a^{p}F\omega_{p}\left( \frac{cf^{p}}{(p-1)^{p-1}a^{p}F}\right) ^{p}.$$ where $c,a$ are the constants of the $A_{p}^{\ast}$ weight $w$, the domain of this function being all $(F,f)$ such that $cf^{p}\leq(p-1)^{p-1}a^{p}F$. In view of the above mentioned result it suffices to consider the expression $\Delta_{w}(g)=\int_{0}^{1}(t^{-1}\int_{0}^{t}g(u)du)^{p}w^{\ast\ast}(t)dt$ when $g$ runs over all nonnegative decreasing right continuous functions on $(0,1]$ satisfying $\int_{0}^{1}g(t)dt=f$ and $\int_{0}^{1}g(t)^{p}w^{\ast \ast}(t)dt=F$. We next define the following function on $(0,1)$$$u(t)=\int_{t}^{1}\frac{w^{\ast\ast}(s)}{s^{p}}ds+c$$ so that $u^{\prime}(t)=t^{-p}w^{\ast\ast}(t)$. Considering first any bounded such function $g$ we compute by integration by parts$$\begin{gathered} \int_{0}^{1}u(t)(\int_{0}^{t}g(u)du)^{p-1}g(t)dt=\frac{1}{p}\int_{0}^{1}u(t)[(\int_{0}^{t}g(u)du)^{p}]^{\prime}dt=\\ =\frac{1}{p}(\int_{0}^{1}g(u)du)^{p}u(1)+\frac{1}{p}\int_{0}^{1}(t^{-1}\int_{0}^{t}g(u)du)^{p}w^{\ast\ast}(t)=c\frac{f^{q}}{p}+\frac{1}{p}\Delta(g)\end{gathered}$$ the integration by parts term $\lim_{t\rightarrow0+}u(t)(\int_{0}^{t}g(u)du)^{p}$ being zero because of condition (\[Ap1a\]) since $g$ is assumed bounded. Now using Young’s inequality $xy\leq\frac{xp}{p}+\frac{y^{p^{\prime}}}{p^{\prime}}$ (where $p^{\prime}=p/(p-1)$) in the first integral as follows, ($\lambda>0$ to be determined later) combined with the condition $\frac{u(t)t^{p-1}}{w^{\ast\ast}(t)}\leq a$ from the above definition we get$$\begin{gathered} \int_{0}^{1}u(t)(\int_{0}^{t}g(u)du)^{p-1}g(t)dt=\\ =\int_{0}^{1}(\lambda g(t)w^{\ast\ast}(t)^{1/p})(\frac{w^{\ast\ast}(t)^{1/p}}{\lambda^{1/(p-1)}t}\int_{0}^{t}g(u)du)^{p-1}\frac{u(t)t^{p-1}}{w^{\ast\ast }(t)}dt\leq\\ \leq\frac{a}{p}\int_{0}^{1}\lambda^{p}g(t)^{p}w^{\ast\ast}(t)dt+\frac {a}{p^{\prime}}\int_{0}^{1}\lambda^{-p^{\prime}}(\frac{1}{t}\int_{0}^{t}g(u)du)^{p}w^{\ast\ast}(t)dt=\\ =\frac{a\lambda^{p}}{p}\int_{0}^{1}g(t)^{p}w^{\ast\ast}(t)dt+a\frac {\lambda^{-p^{\prime}}}{p^{\prime}}\int_{0}^{1}(\frac{1}{t}\int_{0}^{t}g(u)du)^{p}w^{\ast\ast}(t)dt=a\frac{\lambda^{p}}{p}F+a\frac{\lambda ^{-p^{\prime}}}{p^{\prime}}\Delta_{w}(g)\text{.}$$ Therefore we have by writing $\lambda^{p^{\prime}}=(p-1)a(\beta+1),~\beta>0$ and using the above inequalities we get that$$\Delta_{w}(g)\leq(1+\frac{1}{\beta})\frac{(\beta+1)^{p-1}(p-1)^{p}a^{p}F-(p-1)cf^{p}}{(p-1)}.\label{Ap2}$$ Next, given an arbitrary $g$, the above estimate can be used for the truncations $g_{M}=\min(g,M)$ and $F,f$ replaced by the corresponding quantities for $g_{M}$ and then take $M\rightarrow+\infty$ and use monotone convergence to infer that (\[Ap2\]) holds for the general nonnegative decreasing right continuous function on $(0,1]$ satisfying $\int_{0}^{1}g(t)dt=f$ and $\int_{0}^{1}g(t)^{p}w^{\ast\ast}(t)dt=F$. Moreover since $\Delta_{w}(g)>0$ the inequality (\[Ap2\]) implies that $(\beta +1)^{p-1}(p-1)^{p}a^{p}F-(p-1)cf^{p}>0$ for every $\beta>0$ and so letting $\beta\rightarrow0^{+}$ we conclude that $(F,f)$ must satisfy the inequality $cf^{p}\leq(p-1)^{p-1}a^{p}F$ given in the statement of the Theorem. Writing $A=(p-1)^{p}a^{p}F$ and $B=(p-1)cf^{p}$ it is easy to compute (see for example [@Mel1] pg. 326) that the minimum possible value of the right hand side of (\[Ap2\]) is equal to $A\omega_{p}\left( \frac{B}{A}\right) ^{p}$. This proves the inequality $$\mathcal{B}_{p,w}^{\ast}(F,f)\leq(p-1)^{p}a^{p}F\omega_{p}\left( \frac {cf^{p}}{(p-1)^{p-1}a^{p}F}\right) ^{p}.\label{Ap2a}$$ Now we consider the continuous positive decreasing function$$g_{\alpha}(t)=f(1-\alpha)t^{-\alpha}\label{Ap3}$$ where $0\leq\alpha<1$, and any $A_{p}^{\ast}$ weight $w$ that is equimeasurable to$$w^{\ast\ast}(t)=kt^{b}\text{, }k>0,-1<b<p-1\label{Ap4}$$ Clearly $\int_{0}^{1}g_{\alpha}(t)dt=f$ and $\int_{0}^{1}g_{\alpha}(t)^{p}w^{\ast\ast}(t)dt=\frac{kf^{p}(1-\alpha)^{p}}{1+b-\alpha p}$ assuming that $\alpha<\frac{1+b}{p}$. Next note that $\frac{1}{t}\int_{0}^{t}g_{\alpha }(u)du=\frac{g_{\alpha}(t)}{1-\alpha}$ for all $t\in(0,1]$ and so we have $\Delta_{w}(g_{\alpha})=\left( \frac{1}{1-\alpha}\right) ^{p}\int_{0}^{1}g_{\alpha}(t)^{p}w^{\ast\ast}(t)dt$. The condition $\int_{0}^{1}g_{\alpha }(t)^{p}w^{\ast\ast}(t)dt=F$ is then equivalent to the following equation in $\alpha$$$\frac{(1-\alpha)^{p}}{1+b-\alpha p}=\frac{F}{kf^{q}}\text{.}\label{Ap5}$$ To study this equation we write$$z=\frac{p-1-b}{p-1}\frac{1}{1-\alpha}\label{Ap6}$$ and note that (\[Ap5\]) is then equivalent to$$-(p-1)z^{p}+pz^{p-1}=\frac{kf^{p}}{(\frac{p-1}{p-1-b})^{p-1}F}$$ thus$$z=\omega_{p}(\frac{kf^{p}}{(\frac{p-1}{p-1-b})^{p-1}F})\label{Ap7}$$ and so using (\[Ap6\])$$\Delta_{w}(g_{\alpha})=(\frac{p-1}{p-1-b})^{p}F\omega_{p}(\frac{kf^{p}}{(\frac{p-1}{p-1-b})^{p-1}F}).$$ But now note that the constants $c,a$ of the weight $w$ are $a=\frac{1}{p-1-b},~c=\frac{k}{p-1-b}$ and so$$\Delta_{w}(g_{\alpha})=(p-1)^{p}a^{p}F\omega_{p}\left( \frac{cf^{p}}{(p-1)^{p-1}a^{p}F}\right) ^{p}$$ and moreover by varying $k,b$ with $-1<b<p-1$ we can achieve all possible pairs of constants $c,a$. This completes the proof. Estimation via double maximal operators ======================================= Here we will use an inequality introduced by A. Lerner, see [@Ler], for the nondyadic case. We fix $p>1$, let $w$ be an $A_{p}$ weight with respect to the tree $\mathcal{T}$ and we denote for any $I$ in $\mathcal{T}$, $w(I)=\int_{I}wd\mu$, $\sigma=w^{-\frac{1}{p-1}}$, $\sigma(I)=\int_{I}\sigma d\mu$. Also by $M_{\mathcal{T},w}$ we denote the maximal operator with respect to the tree $\mathcal{T}$  but when $X$ is equipped by the measure $w\mu$ instead of $\mu$, and similarly for $M_{\mathcal{T}\text{,}\sigma}$. Then the following holds. Let $w$ be an $A_{p}$ weight with respect to the tree $\mathcal{T}$ and $\mathcal{T}$-constant $[w]_{p}=\sup_{I\in\mathcal{T}}\frac{w(I)\sigma (I)^{p-1}}{\mu(I)^{p}}$ Then for any $\phi$ we have the following pointwise estimate$$(M_{\mathcal{T}}\phi)^{p-1}\leq\lbrack w]_{p}M_{\mathcal{T},w}[(M_{\mathcal{T}\text{,}\sigma}(\phi\sigma^{-1}))^{p-1}w^{-1}].$$ The proof follows from the following inequalities valid for any $I\in \mathcal{T}$.$$\begin{aligned} \left( \frac{1}{\mu(I)}\int_{I}\phi d\mu\right) ^{p-1} & =\frac {w(I)\sigma(I)^{p-1}}{\mu(I)^{p}}\left( \frac{\mu(I)}{w(I)}\left( \frac {1}{\sigma(I)}\int_{I}\phi\sigma^{-1}\sigma d\mu\right) ^{p-1}\right) \leq\\ & \leq[w]_{p}\frac{1}{w(I)}\int_{I}M_{\mathcal{T}\text{,}\sigma}(\phi \sigma^{-1})^{p-1}w^{-1}wd\mu\end{aligned}$$ since $M_{\mathcal{T}\text{,}\sigma}(\phi\sigma^{-1})(x)\geq\frac{1}{\sigma(I)}\int_{I}\phi\sigma^{-1}\sigma d\mu$ for every $x$ in $I$. As a first application of this fixing a tree $\mathcal{T}$ on a probability space $(X,\mu)$ and given an $A_{p}$ weight $w$ in the sense of Definition 2, we define the following generalization of the Bellman function (\[i6a\]), where $p>1$$$\mathcal{B}_{p,w}^{\mathcal{T}}(F,f)=\sup{\huge \{}\int_{X}(M_{\mathcal{T}}\phi)^{p}wd\mu:\int_{X}\phi^{p}wd\mu=F,\int_{X}\phi d\mu=f{\huge \}}$$ and we have the following estimates For any tree $\mathcal{T}$ on a probability space $(X,\mu)$ and any $A_{p}$ weight $w$ and any $\phi$ with $\int_{X}\phi^{p}wd\mu=F,\int_{X}\phi d\mu=f$ we have$$\begin{gathered} \int_{X}(M_{\mathcal{T}}\phi)^{p}wd\mu\leq\nonumber\\ \leq\lbrack w]_{p}^{1/(p-1)}F\omega_{p}\left( \frac{f^{p}}{\sigma(X)^{p-1}F}\right) ^{p}\omega_{p^{\prime}}\left( \frac{(\int_{X}\phi^{p-1}wd\mu)^{p^{\prime}}}{w(X)^{p^{\prime}-1}F\omega_{p}\left( \frac{f^{p}}{\sigma(X)^{p-1}F}\right) ^{p}}\right) ^{p^{\prime}}\label{W1}$$ In particular$$\mathcal{B}_{p,w}^{\mathcal{T}}(F,f)\leq p^{p^{\prime}}[w]_{p}^{1/(p-1)}F\omega_{p}\left( \frac{f^{p}}{\sigma(X)^{p-1}F}\right) ^{p}\label{W2}$$ By applying estimate (1.12) after Theorem 1 in [@Mel1] for the exponent $p^{\prime}=\frac{p}{p-1}$ to the function $\rho=(M_{\mathcal{T}\text{,}\sigma}(\phi\sigma^{-1}))^{p-1}w^{-1}$ and with respect to the tree $\mathcal{T}$ but on the probability space $(X,\frac{1}{w(X)}wd\mu)$ (where as usual $w(X)=\int_{X}wd\mu$) we get$$\begin{gathered} \frac{1}{[w]_{p}^{1/(p-1)}}\int_{X}(M_{\mathcal{T}}\phi)^{p}wd\mu\leq w(X)\int_{X}(M_{\mathcal{T},w}\rho)^{p^{\prime}}w\frac{d\mu}{w(X)}\leq\nonumber\\ \leq w(X)\int_{X}\rho^{p^{\prime}}w\tfrac{d\mu}{w(X)}.\omega_{p^{\prime}}\left( \frac{(\int_{X}\rho w\tfrac{d\mu}{w(X)})^{p^{\prime}}}{\int_{X}\rho^{p^{\prime}}w\tfrac{d\mu}{w(X)}}\right) ^{p^{\prime}}.\end{gathered}$$ Note that (as proved in [@Mel1]) the function $x\omega_{p^{\prime}}\left( \dfrac{y^{p^{\prime}}}{x}\right) ^{p^{\prime}}$ is increasing in $x$ and decreasing in $y$. Now we have$$\int_{X}\rho w\tfrac{d\mu}{w(X)}=\int_{X}(M_{\mathcal{T}\text{,}\sigma}(\phi\sigma^{-1}))^{p-1}\tfrac{d\mu}{w(X)}\geq\int_{X}(\phi\sigma^{-1})^{p-1}\tfrac{d\mu}{w(X)}=\int_{X}\phi^{p-1}w\tfrac{d\mu}{w(X)}$$ and using estimate (1.12) after Theorem 1 in [@Mel1] for the exponent $p$ to the function $\rho=\phi\sigma^{-1}$ and with respect to the tree $\mathcal{T}$ but on the probability space $(X,\frac{1}{\sigma(X)}\sigma d\mu)$ we get (since $\sigma^{-(p-1)}=w$)$$\begin{gathered} \int_{X}\rho^{p^{\prime}}wd\mu=\int_{X}(M_{\mathcal{T}\text{,}\sigma}(\phi\sigma^{-1}))^{p}w^{-p^{\prime}}.wd\mu=\int_{X}(M_{\mathcal{T}\text{,}\sigma}(\phi\sigma^{-1}))^{p}\sigma d\mu\leq\nonumber\\ \sigma(X)\int_{X}(\phi\sigma^{-1})^{p}\sigma\tfrac{d\mu}{\sigma(X)}.\omega _{p}\left( \frac{(\int_{X}\phi\sigma^{-1}\sigma\tfrac{d\mu}{\sigma(X)})^{p}}{\int_{X}(\phi\sigma^{-1})^{p}\sigma\tfrac{d\mu}{\sigma(X)}}\right) ^{p}=\nonumber\\ =\int_{X}\phi^{p}wd\mu.\omega_{p}\left( \frac{(\int_{X}\phi d\mu)^{p}}{\sigma(X)^{p-1}\int_{X}\phi^{p}wd\mu}\right) ^{p}=F\omega_{p}\left( \frac{f^{p}}{\sigma(X)^{p-1}F}\right) ^{p}\text{.}$$ Now combining the above estimates we get$$\begin{gathered} \frac{1}{[w]_{p}^{1/(p-1)}}\int_{X}(M_{\mathcal{T}}\phi)^{p}wd\mu \leq\nonumber\\ \leq F\omega_{p}\left( \frac{f^{p}}{\sigma(X)^{p-1}F}\right) ^{p}\omega_{p^{\prime}}\left( \frac{(\int_{X}\phi^{p-1}wd\mu)^{p^{\prime}}}{w(X)^{p^{\prime}-1}F\omega_{p}\left( \frac{f^{p}}{\sigma(X)^{p-1}F}\right) ^{p}}\right) ^{p^{\prime}}$$ which proves (\[W1\]). Since $\omega_{p^{\prime}}(x)\leq\frac{p^{\prime}}{p^{\prime}-1}=p$ the estimate (\[W2\]) follows also. To get lower bounds for the Bellman function we invoke the following construction. Fixing $\alpha$ with $0<\alpha<1$ and using Lemma 1 in [@Mel1], we fix now a tree $\mathcal{T}$, for example the dyadic subintervals of $[0,1]$, and choose for every $I\in\mathcal{T}$ a family $\mathcal{F}(I)\subseteq \mathcal{T}$ of pairwise almost disjoint subsets of $I$ such that $$\sum_{J\in\mathcal{F}(I)}\mu(J)=(1-\alpha)\mu(I)\text{.}\label{e13}$$ Then we define $\mathcal{S}=\mathcal{S}_{\alpha}$ to be the smallest subset of $\mathcal{T}$ such that $X\in\mathcal{S}$ and for every $I\in\mathcal{S}$, $\mathcal{F}(I)\subseteq\mathcal{S}$. Next for every $I\in\mathcal{S}$ we define the set $$A_{I}=I~\backslash{\displaystyle\bigcup\limits_{J\in\mathcal{F}(I)}} J\label{e14}$$ and note that $\mu(A_{I})=\alpha\mu(I)$ and $I={\displaystyle\bigcup\limits_{_{\substack{J\in\mathcal{S}\\J\subseteq I}}}} A_{J}$ for every $I\in\mathcal{S}$. Also since $\mathcal{S}=\bigcup_{m\geq 0}\mathcal{S}_{(m)}$ where $\mathcal{S}_{(0)}=\{X\}$ and $\mathcal{S}_{(m+1)}=\bigcup_{I\in\mathcal{S}_{(m)}}\mathcal{F}(I)$, we can define rank$(I)=r(I)$ for $I\in\mathcal{S}$ to be the unique integer $m$ such that $I\in\mathcal{S}_{(m)}$ and remark that $\sum\limits_{\substack{\mathcal{S}\ni J\subseteq I\\r(J)=r(I)+m}}\mu(J)=(1-\alpha)^{m}\mu(I)$ for every $I\in\mathcal{S}$. Next for any $\lambda,\gamma>0$ we define the function$$\psi={\displaystyle\sum\limits_{I\in\mathcal{S}}} \lambda\gamma^{r(I)}\chi_{A_{I}}$$ and we have for any $I\in\mathcal{S}$ the following$$\frac{1}{\mu(I)}\int_{I}\psi d\mu=\frac{\lambda\alpha}{1-\gamma(1-\alpha )}\gamma^{r(I)}\text{.}$$ Hence taking $$\phi_{\alpha}={\displaystyle\sum\limits_{I\in\mathcal{S}}} \lambda_{1}\gamma_{1}^{r(I)}\chi_{A_{I}}\text{, \ }w_{\alpha}={\displaystyle\sum\limits_{I\in\mathcal{S}}} \lambda_{2}\gamma_{2}^{r(I)}\chi_{A_{I}}$$ we have for any $I\in\mathcal{S}$ $$\frac{w_{\alpha}(I)[w_{\alpha}^{-\frac{1}{p-1}}(I)]^{p-1}}{\mu(I)^{p}}=\frac{\alpha^{p}}{[1-\gamma_{2}(1-\alpha)][1-\gamma_{2}^{-\frac{1}{p-1}}(1-\alpha)]}$$ thus $w_{\alpha}$ is an $A_{p}$ weight but with respect to the *tree* $\mathcal{S}_{\alpha}$ on $(X,\mu)$ and with $[w_{\alpha}]_{p}$ equal to the right hand side of the above relation. Moreover$$M_{\mathcal{S}}\phi_{\alpha}\geq{\displaystyle\sum\limits_{I\in\mathcal{S}}} \frac{1}{\mu(I)}\int_{I}\phi_{a}d\mu\chi_{A_{I}}=\frac{\alpha}{1-\gamma (1-\alpha)}\phi_{\alpha}\text{.}$$ However the values of such functions on each $A_{I}$ where $r(I)=m$ is of the form $$\gamma_{m}=\frac{\lambda}{\alpha(1-\alpha)^{m}}\int_{(1-\alpha)^{m+1}}^{(1-\alpha)^{m}}u^{s}du\label{e16}$$ for some real numbers $\lambda,s>0$ and as it is proved in Lemma 3 of [@Mel5] these behave like functions of the form $\lambda t^{s}$ on $(0,1]$ as we approach the limit $\alpha\rightarrow0^{+}$. Hence by taking a sequence $\alpha_{m}\rightarrow0$ considering the trees $\mathcal{T}_{m}=\mathcal{S}_{\alpha_{m}}$ on $(X,\mu)$ and using the construction for the lower bound in the proof of Theorem 2, choosing the constants $k,b$ ($-1<b<p-1$) appropriately in (\[Ap4\]) according to the conditions $a=\frac{1}{p-1-b}$, $c=ka$, $\frac{k}{b+1}=z$, $\frac{1}{b+1}(\frac{p-1}{p-1-b})^{p-1}=h$ from the restrictions below which give $\frac{p-1-b}{p-1}=\omega_{p}(\frac{1}{h})$ we conclude the following. Given appropriate $F,f,h,z$ there exists a sequence of trees $\mathcal{T}_{m}$ on $(X,\mu)~$and two sequences $(\phi_{m})$ and $(w_{m})$ of positive measurable functions on $(X,\mu)$ such that $\int_{X}\phi_{m}d\mu\rightarrow f$,  $\int_{X}\phi_{m}^{p}w_{m}d\mu\rightarrow F$, each $w_{m}$ is an $A_{p}$ weight with respect to the tree $\mathcal{T}_{m}$ with $[w_{m}]_{p}\rightarrow h$ and $\int_{X}w_{m}d\mu\rightarrow z$ such that$$\lim_{m\rightarrow\infty}\int_{X}(M_{\mathcal{T}_{m}}\phi_{m})^{p}w_{m}d\mu\geq F\omega_{p}\left( \frac{zf^{p}}{hF}\right) ^{p}\omega_{p}(\frac {1}{h})^{-p}\text{.}$$ The above proposition implies a lower bound on the class of functions $\mathcal{B}_{p,w}^{\mathcal{T}}(F,f)$ when viewed over all trees $\mathcal{T}$ and corresponding $A_{p}$ weights $w$. [99]{} D. L. Burkholder, Martingales and Fourier analysis in Banach spaces, C.I.M.E. Lectures (Varenna (Como), Italy, 1985), *Lecture Notes in Mathematics* 1206 (1986), 61-108. D. L. 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--- abstract: 'We show that any element of the special linear group $\operatorname{SL}_2(\operatorname{R})$ is a product of two exponentials if the ring $\operatorname{R}$ is either the ring of holomorphic functions on an open Riemann surface or the disc algebra. This is sharp: one exponential factor is not enough since the exponential map corresponding to $\operatorname{SL}_2({\mathbb{C}})$ is not surjective. Our result extends to the linear group $\operatorname{GL}_2(\operatorname{R})$.' author: - Frank Kutzschebauch and Luca Studer title: Exponential factorizations of holomorphic maps --- introduction ============ For a Stein space $X$, a complex Lie group $G$ and its exponential map $\exp: \mathfrak{g} \to G$ we say that a holomorphic map $f:X \to G$ is a product of $k$ exponentials if there are holomorphic maps $f_1, \ldots, f_k:X \to \mathfrak{g}$ such that $$f=\exp(f_1)\cdots \exp(f_k).$$ It is easy to see that any map $f$ which is a product of exponentials (for some sufficiently large $k$) is null-homotopic. In the case where $G$ is the special linear group $\operatorname{SL}_n({\mathbb{C}})$ the converse follows from [@IK] as explained in [@DK]. However, it turns out to be a difficult problem to determine the minimal number $k$ of needed factors in dependence of the dimensions of $X$ and $\operatorname{SL}_n({\mathbb{C}})$. We solve this problem for $\dim X=1$ and $n=2$. \[Riemann\] Any holomorphic map from an open Riemann surface to the special linear group $\operatorname{SL}_2({\mathbb{C}})$ is a product of two exponentials. Theorem \[Riemann\] improves a result of Doubtsov and Kutzschebauch, who showed the same result with three instead of two factors in the conclusion, see Proposition 3 in [@DK]. Stated differently, Theorem \[Riemann\] says that every element of $\operatorname{SL}_2({{\ensuremath{\mathcal{O}}}}(X))$ can be written as a product of two exponentials, where ${{\ensuremath{\mathcal{O}}}}(X)$ denotes the ring of holomorphic functions on a given open Riemann surface $X$. Our second result is of similar flavor, but the ring ${{\ensuremath{\mathcal{O}}}}(X)$ is replaced by the disc algebra ${{\ensuremath{\mathcal{A}}}}$. By definition, the disc algebra ${{\ensuremath{\mathcal{A}}}}$ is the ${\mathbb{C}}$-Banach algebra of continuous functions on the closed disc $\{z \in {\mathbb{C}}: |z|\leq 1\}$ which are holomorphic on the interior, equipped with the supremum norm. \[disc\] For the disc algebra ${{\ensuremath{\mathcal{A}}}}$, any element of $\operatorname{SL}_2({{\ensuremath{\mathcal{A}}}})$ is a product of two exponentials. Recall that the exponential map $\exp: \mathfrak{sl}_2({\mathbb{C}}) \to \operatorname{SL}_2({\mathbb{C}})$ is not surjective. In this sense Theorem \[Riemann\] and \[disc\] are sharp. It is worth mentioning that $\operatorname{SL}_2({\mathbb{C}})$ is simply connected implying that holomorphic maps from open Riemann surfaces to $\operatorname{SL}_2({\mathbb{C}})$ and elements of $\operatorname{SL}_2({{\ensuremath{\mathcal{A}}}})$ are null-homotopic. This is the reason that the map in question being null-homotopic is a redundant assumption in Theorem \[Riemann\] and \[disc\]. As corollaries of Theorem \[Riemann\] and \[disc\] we get the analogous results if the special linear group is replaced by the linear group with the corresponding entries. \[Riemann2\] Any null-homotopic holomorphic map from an open Riemann surface to the linear group $\operatorname{GL}_2({\mathbb{C}})$ is a product of two exponentials. Let $X$ be an open Riemann surface and $\operatorname{M}_2({\mathbb{C}})$ the complex $2 \times 2$-matrices. If a given holomorphic map $A:X \to \operatorname{GL}_2({\mathbb{C}})$ is null-homotopic, then $\det A: X \to {\mathbb{C}}^\ast$ is null-homotopic as well. Therefore $\det A$ has a holomorphic logarithm $\log:X \to {\mathbb{C}}$, satisfying $e^{\log}=\det A$. In particular, if $D:X \to \operatorname{M}_2({\mathbb{C}})$ is the diagonal matrix with diagonal entries $\log/2$, $\exp(-D)A$ has values in $\operatorname{SL}_2({\mathbb{C}})$. By Theorem \[Riemann\] there are holomorphic $B, C: X \to \operatorname{M}_2({\mathbb{C}})$ such that $$A=e^De^{-D}A=e^De^Be^C=e^{D+B}e^C,$$ where we used in the last equality that $D$ commutes with all other matrices. This finishes the proof. Unlike in Theorem \[Riemann\], in Corollary \[Riemann2\] the assumption that $f$ is null-homotopic is not redundant. For instance, $$\begin{aligned} A(z)= \begin{pmatrix} z & 0 \\ 0 & z \end{pmatrix}, \ z \in {\mathbb{C}}^\ast\end{aligned}$$ is not null-homotopic since otherwise $\det A: {\mathbb{C}}^\ast \to {\mathbb{C}}^\ast, z \mapsto z^2$ would be null-homotopic as well. \[disc\] For the disc algebra ${{\ensuremath{\mathcal{A}}}}$, any element of $\operatorname{GL}_2({{\ensuremath{\mathcal{A}}}})$ is a product of two exponentials. This follows from Theorem \[disc\] in the same way as Corollary \[Riemann2\] follows from Theorem \[Riemann\]. Here, we need in addition that any unit in ${{\ensuremath{\mathcal{A}}}}$ has a logarithm, which follows from the fact that the disc (and thereby the domain of the elements of ${{\ensuremath{\mathcal{A}}}}$) is contractible. In particular, the map in question being null-homotopic is again a redundant assumption. Corollary \[disc\] improves a result of Mortini and Rupp, who showed the same with four instead of two factors in the conclusion, see Theorem 7.1 in [@MR]. Also Corollary \[Riemann2\] and \[disc\] are sharp in the sense that one exponential factor is not enough. An example is the matrix $$\begin{aligned} A(z)= \begin{pmatrix} 1 & 1\\ 0 & e^{4\pi i z} \end{pmatrix}, z \in \Delta.\end{aligned}$$ One can show that the second entry of any lift of $z \mapsto A(z)$, $|z|<1/2$ via the exponential map tends to infinity if $z \to 1/2$. For details see [@MR], Example 6.4. We would like to thank Sebastian Baader for helpful comments on a draft of this text. Proof of Theorem \[Riemann\] ============================ An important ingredient in the proof is an Oka principle due to Forstnerič, which follows essentially from Theorem 2.1 in [@Forstneric]. The version, which we use in this text is the below stated Theorem \[Oka principle 2\]. It is used to show Proposition \[vanishing trace\], which is the main ingredient in the proof of Theorem \[Riemann\]. Throughout this section $X$ denotes an open Riemann surface. \[vanishing trace\] Let $A: X \to \operatorname{SL}_2({\mathbb{C}})$ be holomorphic and assume that $A(x)$ has distinct eigenvalues for some $x \in X$. Then $A=BC$ for suitable holomorphic $B,C: X \to \operatorname{SL}_2({\mathbb{C}})$, both of which have vanishing trace. Note that the conclusion of Proposition \[vanishing trace\] is equivalent to finding a holomorphic $B:X \to \operatorname{SL}_2({\mathbb{C}})$ such that $B$ and $AB$ have vanishing trace, simply since taking the inverse of a $2\times 2$-matrix with trace zero has again trace zero. Expressed differently, Proposition \[vanishing trace\] is proved if we can show the existence of a global section of the bundle $$\begin{aligned} Z\coloneqq \{(x,B) \in X \times \operatorname{SL}_2({\mathbb{C}}): \operatorname{tr}(B)=\operatorname{tr}(A(x)B)=0\}\end{aligned}$$ over $X$. If $a,b,c,d$ denote the coefficients of $A$, and $u,w,v,-u$ denote the coefficients of $B$, we can express $Z$ more explicitly as $$\begin{aligned} \{(x,u,v,w) \in X \times {\mathbb{C}}^3: (a(x)-d(x))u+b(x)v+c(x)w=0, \ u^2+vw=-1\}.\end{aligned}$$ More concretely, Proposition \[vanishing trace\] is proved if we manage the prove the following reformulation. \[section\] Let $A: X \to \operatorname{SL}_2({\mathbb{C}})$ be holomorphic and assume that $A(x)$ has distinct eigenvalues for some $x \in X$. Then the restriction $h$ of the projection $X \times {\mathbb{C}}^3 \to X$ to $Z$ has a holomorphic section. For an open subset $U\subset X$, $Z|U$ denotes the restriction of the bundle $h: Z\to X$ to $h^{-1}(U)$. We start the proof of Proposition \[section\] with the following simple \[local\] For every $x \in X$ there is a neighborhood $U$ of $x$ and a holomorphic section $F:U \to Z|U$ of $Z|U$. After passing to a local chart we may assume that $X$ is the unit disc $\Delta \coloneqq \{z \in {\mathbb{C}}: |z|<1\}$ and $x=0$. Finding a local holomorphic section in a neighborhood of $0$ is equivalent to finding a neighborhood $0 \in U\subset \Delta$ and holomorphic maps $u,v,w:U \to {\mathbb{C}}$, which satisfy $$\begin{aligned} \label{1} (a-d)u+bv+cw=0, \ \ u^2+vw=-1.\end{aligned}$$ Local holomorphic solutions to () exist if and only if there are local holomorphic solutions to the less restrictive problem $$\begin{aligned} \label{2} (a-d)u+bv+cw=0, \ \ u^2+vw \in {{\ensuremath{\mathcal{O}}}}^\ast_0.\end{aligned}$$ The reason is that if $u,v,w$ are local solutions in a neighborhood of the origin to (), we can rescale these solutions with a local holomorphic square root of $u^2+vw$, or more precisely, by defining new solutions by $\tfrac{iu}{r}, \tfrac{iv}{r}, \tfrac{iw}{r}$ for some $r:U \to {\mathbb{C}}^\ast$ satisfying $r^2=u^2+vw$ defined on a sufficiently small neighborhood $U$ of the origin. To find solutions to () we distinguish three cases. Let $n(f) \in {\mathbb{Z}}_{\geq 0}$ denote the vanishing order of a holomorphic function $f:\Delta \to {\mathbb{C}}$ at the origin. The first case is $n(a-d)\geq n(b)$. Then $-\tfrac{a-d}{b}$ is holomorphic in a neighborhood of $0$ and $u=1$, $v=-\tfrac{a-d}{b}$ and $w=0$ is a solution to (). The second case $n(a-d)\geq n(c)$ we find similarly a solution $u=1$, $v=0$ and $w=-\tfrac{a-d}{c}$ to (). The remaining case is $n(a-d)<\min(n(b),n(c))$, which implies $n(a-d)<n(b+c)$ and hence $-\tfrac{b+c}{a-d}$ is holomorphic in a neighborhood of the origin and vanishes at the origin. Then $u=-\tfrac{b+c}{a-d}$, $v=1$, $w=1$ solves (). This finishes the proof. Let $D$ denote the discriminant of $A$, that is $D\coloneqq (a+d)^2-4$. By *isomorphic* fiber bundles we mean isomorphic as complex analytic fiber bundles. \[local trivialization\] Let $U\subset X \setminus (\{D=0\}\cup \{c=0\})$ be an open neighborhood where $D:U \to {\mathbb{C}}$ has a holomorphic square root $\sqrt D$, and set $f\coloneqq \tfrac{d-a + \sqrt{D}}{2c}.$ Then $Z|U$ is isomorphic to $U \times {\mathbb{C}}^\ast$, and an isomorphism is given by $$\phi:Z|U \to U \times {\mathbb{C}}^\ast, \ \phi(x,u,v,w)=(x, u+f(x)v).$$ First we do the necessary computations at the level of a single fiber. For this, we think of the coefficients $a,b,c,d$ of $A$ as elements of ${\mathbb{C}}$. We want to determine all $u,v,w \in {\mathbb{C}}$ such that $$\begin{aligned} (a-d)u+bv+cw=0, \ \ -u^2-vw=1.\end{aligned}$$ Since $c\not = 0$, we can solve for $w$ and get equivalently $$\begin{aligned} -1 &=u^2+vw \\ &=u^2+v\tfrac{(d-a)u-bv}{c} \\ &=u^2+\tfrac{d-a}{c}uv-\tfrac{b}{c}v^2 \\ &=\Big(u+\tfrac{d-a}{2c}v\Big)^2-\Big(\tfrac{(d-a)^2}{4c^2}+\tfrac{b}{c}\Big)v^2.\end{aligned}$$ Furthermore we have $$\begin{aligned} \frac{(d-a)^2}{4c^2}+\frac{b}{c}=\frac{(d+a)^2-4ad}{4c^2}+\frac{4bc}{4c^2}=\frac{(d+a)^2-4(ad-bc)}{4c^2}=\frac{D}{4c^2}.\end{aligned}$$ Fix a square root $\sqrt D$ of $D$ and note that $$\tilde u=u+\tfrac{d-a+\sqrt{D}}{2c}v, \ \ \tilde v=u+\tfrac{d-a-\sqrt{D}}{2c}v$$ defines a linear coordinate change of ${\mathbb{C}}^2$, which translates the above equation to $$\begin{aligned} -1 &=\Big(u+\tfrac{d-a}{2c}v\Big)^2-\tfrac{D}{4c^2}v^2 \\ &=\Big(u+\tfrac{d-a}{2c}v\Big)^2-\Big(\tfrac{\sqrt{D}}{2c}v\Big)^2 \\ &=\Big( u+\tfrac{d-a+\sqrt{D}}{2c}v\Big ) \Big(u+\tfrac{d-a-\sqrt{D}}{2c}v \Big) \\ &=\tilde u \tilde v.\end{aligned}$$ This shows that the fiber is given by $\{(\tilde u, \tilde v) \in {\mathbb{C}}^2: \tilde u \tilde v=-1\}={\mathbb{C}}^\ast$ and that $(u,v,w) \to u+\tfrac{d-a+\sqrt{D}}{2c}v$ is an isomorphism of the fiber onto ${\mathbb{C}}^\ast$. Moreover, our computations yield a trivialization of $Z|U$, which is defined similarly, or more precisely, as in the assumption of the Lemma. This is the case since our computations work out just the same way if we have a holomorphic dependence on $x \in U$. \[D\] Over $X\setminus \{D=0\}$, $h:Z \to X$ is a fiber bundle with fiber ${\mathbb{C}}^\ast$. At points $x \in X\setminus \{D=0\}$ with $c(x)\not =0$, choose a neighborhood $U\subset X$ of $x$ such that $c|U$ does not vanish, and such that $D$ has a square root on $U$. Then a trivialization of $Z|U$ is given by Lemma \[local trivialization\]. In the case $c(x)=0$, let us reduce the problem to the case $c(x)\not =0$ with the following observation. Our bundle is given by $$\begin{aligned} Z=\{(x,B) \in X \times \operatorname{SL}_2({\mathbb{C}}): \operatorname{tr}(B)=\operatorname{tr}(A(x)B)=0\}.\end{aligned}$$ Define for $P \in \operatorname{SL}_2({\mathbb{C}})$ a bundle $$\begin{aligned} Z_P &=\{(x,PBP^{-1}) \in X \times \operatorname{SL}_2({\mathbb{C}}): \operatorname{tr}(B)=\operatorname{tr}(A(x)B)=0\}.\end{aligned}$$ Clearly $Z$ and $Z_P$ are isomorphic over $X$. Since conjugation with a matrix does not change the trace, we obtain with the substitution $C=PBP^{-1}$ $$\begin{aligned} Z_P &=\{(x, C) \in X \times \operatorname{SL}_2({\mathbb{C}}): \operatorname{tr}(P^{-1}CP)=\operatorname{tr}(A(x)P^{-1}CP)=0\} \\ &=\{(x, C) \in X \times \operatorname{SL}_2({\mathbb{C}}): \operatorname{tr}(C)=\operatorname{tr}(PA(x)P^{-1}C)=0\}.\end{aligned}$$ Note that if the third entry $c$ of $A$ equals $0$ at $x$, then, since $D(x)\not =0$ and hence $A(x)\not = \pm id$, there is $P \in \operatorname{SL}_2({\mathbb{C}})$ such that the third entry of $PA(x)P^{-1}$ does not vanish. Using that $Z$ and $Z_P$ are isomorphic and that we can solve the problem for $Z_P$ close to $x$, the statement follows. To finish the proof of Propostion \[section\] we need the following special case of Theorem 6.14.6, p.310 in [@Francs; @book]. \[Oka principle 2\] Let $h:Z \to X$ be a holomorphic map of a reduced complex space $Z$ onto a reduced Stein space $X$. Let $X'\subset X$ be a complex analytic subvariety and let $Z'\coloneqq h^{-1}(X')$ and assume that the restriction $h:Z\setminus Z' \to X \setminus X'$ is an elliptic submersion. Moreover, let $f:X \to Z$ be a continuous section of $h$ which is holomorphic in a neighborhood of $X'$. Then $f$ is homotopic through continuous sections of $h$ which are holomorphic in a fixed small neighborhood of $X'$ to a holomorphic section of $h$. A consequence of this is the following \[Oka principle\] Let $h:Z \to X$ be a holomorphic map from a reduced complex space onto an open Riemann surface. Moreover, assume that there is a discrete set $X'\subset X$ such that for $Z'=h^{-1}(X')$, the restriction $h:Z \setminus Z' \to X \setminus X'$ is a fiber bundle with fiber ${\mathbb{C}}^\ast$ and assume that there is a local holomorphic section in a neighborhood of every point of $X'$. Then $h$ has a global holomorphic section $f:X \to Z$. First we show the existence of a continuous section which is holomorphic in a neighborhood $U$ of $X'$. By assumption there is a local holomorphic section $f:U \to Z$ of $h$ defined on a neighborhood $U$ of $X'$. By possibly shrinking $U$ we may assume that every connected component of $U$ contains exactly one point of $X'$ and is homeomorphic to a disc, and that $f$ extends continuously to $\overline{U}$. $X\setminus X'$ is an open Riemann surface and thus deformation retracts onto a $1$-dimensional CW-complex $K$, see e.g.[@Hamm]. After possibly modifying a fixed deformation retract $r$ of $X \setminus X'$ onto $K$ by a conjugation with a suitable homeomorphism of $X \setminus X'$ we can assume that $\partial U \subset K$. Since the fiber ${\mathbb{C}}^\ast$ of $Z$ is connected we can extend $f|\partial U$ to a section $\tilde f:K \to Z|K$. Since $K$ is a deformation retract of $X \setminus X'$ and $h:Z\setminus Z' \to X \setminus X'$ is a fiber bundle, the section $\tilde f$ extends to a continuous section $F:X \setminus X' \to Z \setminus Z'$, see e.g.Theorem 7.1, p.21 in [@Husemoller]. Since $f$ and $F|X \setminus U$ agree on $\partial U$, these two sections define a continuous section $X \to Z$ which agrees with the holomorphic section $f$ on the neighborhood $U$ of $X'$. The existence of a global holomorphic section follows now from the above Oka principle due to Forstnerič, see Theorem \[Oka principle 2\]. This finishes the proof. Let $h:Z\to X$ be the bundle over $X$ from Proposition \[section\]. With Lemma \[local\] we proved that there are local sections of $h$ at every point $x \in X$, in particular also at points of the discrete set $X'=\{D=0\}$. Moreover, with Lemma \[D\] we showed that $h$ is a locally trivial ${\mathbb{C}}^\ast$-bundle over $X\setminus \{D=0\}$. It follows now from Proposition \[Oka principle\] that there is a holomorphic section of $h$. This finishes the proof. \[log vanishing\] Let $X$ be an open Riemann surface and let $A:X \to \operatorname{SL}_2({\mathbb{C}})$ be holomorphic with vanishing trace. Then $A=e^B$ for some holomorphic $B: X \to \operatorname{M}_2({\mathbb{C}})$ with vanishing trace. The characteristic polynomial of $A$ equals $T^2+1$. In particular $\pm i$ are the eigenvalues (at every point $x \in X$). There are line bundles $E(i)$ and $E(-i)$ over $X$, whose non-vanishing sections correspond to holomorphic eigenvectors of $i$ and $-i$ respectively. Explicitly, we have $$\begin{aligned} E(i) & \coloneqq \{(x,z) \in X \times {\mathbb{C}}^2: A(x)z=iz\}, \\ E(-i) & \coloneqq \{(x,z) \in X \times {\mathbb{C}}^2: A(x)z=-iz\}.\end{aligned}$$ Since every line bundle over an open Riemann surface is trivial, we have $E(i) \cong X \times {\mathbb{C}}\cong E(-i)$ as complex analytic line bundles. This implies that there are two holomorphic eigenvectors $v: X \to E(i)$, $w:X \to E(-i)$ with $v(x)\not = 0 \not =w(x)$ for all $x \in X$. In particular $$P:X \to \operatorname{M}_2({\mathbb{C}}), \ P(x)\coloneqq (v(x) \ w(x))$$ takes values in $\operatorname{GL}_2({\mathbb{C}})$ since $v(x)$ and $w(x)$ are eigenvectors of $A(x)$ to the distinct eigenvalues $\pm i$. This implies that $A$ is holomorphically diagonalisable with $$\begin{aligned} A=PDP^{-1}, \ \ D\coloneqq \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}.\end{aligned}$$ For the diagonal matrix $\tilde D$ with entries $ \pm \tfrac{i\pi}{2}$ we have $e^{\tilde D}=D$. We get for $B \coloneqq P \tilde D P^{-1}$ the equality $$\begin{aligned} A=PDP^{-1}=Pe^{\tilde D}P^{-1}=e^{P\tilde D P^{-1}}=e^B, \end{aligned}$$ as desired. Note that $B$ has vanishing trace since $\tilde D$ has vanishing trace. This finishes the proof. Let $X$ be an open Riemann surface and let $A:X \to \operatorname{SL}_2({\mathbb{C}})$ be a holomorphic map. If the characteristic polynomial of $A$ equals $(T-1)^2$, then, since $(A-id)^2=\chi_A(A)=0$ by Cayley-Hamilton, we have $$\exp(A-id)=id +(A-id) =A.$$ Moreover, the trace of $A$ is equal to minus the second coefficient of the characteristic polynomial, which implies in our case that $\operatorname{tr}(A-id)=0$, as desired. This shows that $A$ can be written as a single exponential factor. If the characteristic polynomial is $(T+1)^2$, then the characteristic polynomial of $-A$ is $(T-1)^2$ and since $-id$ is equal to the exponential of the diagonal matrix with diagonal entries $\pi i$ and $-\pi i$, $A$ is a product of at most two exponentials with vanishing trace. Otherwise there is $x \in X$ such that $A(x)$ has distinct eigenvalues. In that case it follows from Proposition \[vanishing trace\] that $A=BC$ for holomorphic $B,C:X \to \operatorname{SL}_2({\mathbb{C}})$ with vanishing trace. In particular, the characteristic polynomials of $B$ and $C$ are both $(T-i)(T+i)$. Since $B$ and $C$ have a logarithm by Lemma \[log vanishing\], we are done. Proof of Theorem \[disc\] ========================= The proof depends essentially on three ingredients. The first ingredient is that the Bass stable rank of the disc algebra ${{\ensuremath{\mathcal{A}}}}$ equals $1$. This is needed to reduce the problem to matrices with an invertible first entry. The second and third ingredient are the simple facts that the elements of ${{\ensuremath{\mathcal{A}}}}$ are bounded, and that $\exp: {{\ensuremath{\mathcal{A}}}}\to {{\ensuremath{\mathcal{A}}}}$ is onto to units of ${{\ensuremath{\mathcal{A}}}}$. In the following $\overline \Delta\subset {\mathbb{C}}$ denotes the closed unit disc centered at the origin. We use the following notation. If $f:\overline \Delta \to {\mathbb{C}}$ is a function, then $|f|:\overline \Delta \to {\mathbb{R}}$ denotes the absolute value $z \mapsto |f(z)|$. In particular, the symbol $|f|$ should not be confused with the sup-norm on ${{\ensuremath{\mathcal{A}}}}$, which is not used explicitly in the proof. Moreover, for $f,g: \overline \Delta \to {\mathbb{R}}$ we write $f>g$ if $f(z)>g(z)$ for all $z \in \overline \Delta$. The proof depends on the following elementary lemma. \[square root\] Let $f \in {{\ensuremath{\mathcal{A}}}}$ be such that $|f|>2$. Then the polynomial $T^2-fT+1$ has roots $\lambda, \lambda^{-1} \in {{\ensuremath{\mathcal{A}}}}$ such that $|\lambda|>1$. First note that our assumption implies that the discriminant $f^2-4$ does not vanish. Therefore $f^2-4$ has a square root in ${{\ensuremath{\mathcal{A}}}}$, which implies that there are roots $\lambda, \lambda^{-1} \in {{\ensuremath{\mathcal{A}}}}$ of $T^2-fT+1$. We have to show that one of $|\lambda|$ and $|\lambda^{-1}|$ is strictly larger than $1$. Note that if $T^2-zT+1$, $z \in {\mathbb{C}}$ has a root $r \in {\mathbb{C}}$ with $|r|=1$, then we get $|z|=|r^2+1|/|r|=|r^2+1|\leq 2$. Expressed differently, if $|z|>2$, then $T^2-zT+1$ has no root on the unit circle. This implies that $\lambda$ and $\lambda^{-1}$ avoid the unit circle, and moreover – by continuity of $\lambda$ and $\lambda^{-1}$ – that exactly one of the two is strictly bigger than $1$ in absolute value. Let $$A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in \operatorname{SL}_2({{\ensuremath{\mathcal{A}}}}).$$ It is well-known that the Bass stable rank of ${{\ensuremath{\mathcal{A}}}}$ equals $1$, see [@JMW]. By definition of the Bass stable rank this means that for any pair $f,g \in {{\ensuremath{\mathcal{A}}}}$ with $f {{\ensuremath{\mathcal{A}}}}+ g {{\ensuremath{\mathcal{A}}}}={{\ensuremath{\mathcal{A}}}}$, there is $h \in {{\ensuremath{\mathcal{A}}}}$ such that $f +h g$ is a unit in ${{\ensuremath{\mathcal{A}}}}$. In particular, since $ad-bc=1$, there is $h \in {{\ensuremath{\mathcal{A}}}}$ such that $a+h c=1$. Consequently the first entry of $$\begin{aligned} \begin{pmatrix} 1 & h \\ 0 & 1 \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 1 & -h \\ 0 & 1 \end{pmatrix}= \begin{pmatrix} a+h c & \ast \\ \ast & \ast \end{pmatrix}\end{aligned}$$ is a unit. Since conjugation with matrices in $\operatorname{GL}_2({{\ensuremath{\mathcal{A}}}})$ does not change the number of needed exponential factors to represent a given matrix, this shows that it suffices to consider the case where the first entry $a$ of $A$ is a unit. For such $A$, the strategy is as follows: for $\delta>0$ set $$B \coloneqq \begin{pmatrix} \delta & 0 \\ 0 & 1/\delta \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} \delta a & \delta b \\ c/\delta & d/\delta \end{pmatrix} \in \operatorname{SL}_2({{\ensuremath{\mathcal{A}}}}).$$ If we find $\delta$ such that $B=B(\delta)$ has a logarithm, then – since $A$ is the product of the diagonal matrix with entries $1/\delta, 0, 0, \delta$ and $B$ – we know that $A$ is a product of two exponentials. Our claim is that $B$ has a logarithm for any sufficiently large $\delta>0$. To see this, let $\delta \geq 1$ be an upper bound of the (bounded) function $$\beta=\frac{3+|d|}{|a|}.$$ From the fact that $\delta \geq 1$ is an upper bound of $\beta$ it follows that $$|\operatorname{tr}(B)|= |\delta a +d/\delta| \geq \delta|a|-\frac{|d|}{\delta} \geq (3+|d|)-|d|>2.$$ By Lemma \[square root\] we know that the characteristic polynomial $\chi_B=T^2-\operatorname{tr}(B) T+1$ has roots $\lambda, \lambda^{-1} \in {{\ensuremath{\mathcal{A}}}}$ with $|\lambda|>1$. Since $\lambda$ is a unit in ${{\ensuremath{\mathcal{A}}}}$, the matrix $D$ with diagonal entries $\lambda$ and $\lambda^{-1}$ has a logarithm given by the diagonal matrix with diagonal entries $\log(\lambda) \in {{\ensuremath{\mathcal{A}}}}$ and $-\log(\lambda) \in {{\ensuremath{\mathcal{A}}}}$ for some fixed logarithm of $\lambda$. Moreover, since conjugation with an element in $\operatorname{GL}_2({{\ensuremath{\mathcal{A}}}})$ does not change the number of needed exponential factors, it suffices to find $P \in \operatorname{GL}_2({{\ensuremath{\mathcal{A}}}})$ with $$B=PDP^{-1}.$$ Our claim is that $$P= \begin{pmatrix} d/\delta-\lambda &-\delta b \\ -c/\delta & \delta a-\lambda^{-1} \end{pmatrix} \in \operatorname{M}_2({{\ensuremath{\mathcal{A}}}})$$ does the job. To show this it suffices to show that the columns $v$ resp.$w$ of $P=(v \ w)$ satisfy $(B-\lambda id)v=(B-\lambda^{-1} id)w=0$ and that $|\det B|\geq 1$. For the first part we get $$(B-\lambda id)v= \begin{pmatrix} \delta a -\lambda & \delta b \\ c/\delta & d/\delta -\lambda \end{pmatrix} \begin{pmatrix} d/\delta-\lambda \\ -c/\delta \end{pmatrix} = \begin{pmatrix} \chi_B(\lambda) \\ 0 \end{pmatrix} =0,$$ and similarly $$(B- \lambda^{-1} id)w= \begin{pmatrix} \delta a -\lambda^{-1} & \delta b \\ c/\delta & d/\delta -\lambda^{-1} \end{pmatrix} \begin{pmatrix} -\delta b \\ \delta a-\lambda^{-1} \end{pmatrix} = \begin{pmatrix} 0 \\ \chi_B(\lambda^{-1}) \end{pmatrix} =0 .$$ For the second part, we get with $ad-bc=1$ $$\det P=-\delta \lambda a - \delta^{-1} \lambda^{-1} d +2.$$ It follows from $|\lambda|>1$ that $$|\det P| \geq \delta |\lambda| |a|-\delta^{-1} |\lambda^{-1}| |d|-2 \geq \delta |a|-\delta^{-1}|d|-2.$$ Furthermore, the fact that $\delta \geq 1$ bounds $\beta=(3+|d|)/|a|$ from above yields $$\delta |a|-\delta^{-1}|d|-2 \geq (3+|d|)-|d|-2=1,$$ which shows that $|\det P|\geq 1$. This finishes the proof. 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--- abstract: 'Graph-specific computing with the support of dedicated accelerator has greatly boosted the graph processing in both efficiency and energy. Nevertheless, their data conflict management is still sequential in essential when some vertex needs a large number of conflicting updates at the same time, leading to prohibitive performance degradation. This is particularly true for processing natural graphs. In this paper, we have the insight that the atomic operations for the vertex updating of many graph algorithms (e.g., BFS, PageRank and WCC) are typically incremental and simplex. This hence allows us to parallelize the conflicting vertex updates in an accumulative manner. We architect a novel graph-specific accelerator that can simultaneously process atomic vertex updates for massive parallelism on the conflicting data access while ensuring the correctness. A parallel accumulator is designed to remove the serialization in atomic protection for conflicting vertex updates through merging their results in parallel. Our implementation on Xilinx Virtex UltraScale+ XCVU9P with a wide variety of typical graph algorithms shows that our accelerator achieves an average throughput by 2.36 GTEPS as well as up to 3.14x performance speedup in comparison with state-of-the-art ForeGraph (with single-chip version).' author: - bibliography: - 'sample-bibliography.bib' title: | An Efficient Graph Accelerator with Parallel\ Data Conflict Management --- Introduction ============ Graph processing plays an important role in many real-world applications, e.g., ranking the web sites [@shun2013ligra], analysing the social networks [@teixeira2015arabesque], and streaming applications [@liao]. Therefore, a large number of research efforts have been made to build the dedicated hardware that can execute graph applications with more efficiency than what the general-purpose processors and systems can provide [@nurvitadhi2014graphgen; @ham2016graphicionado; @ozdal2016energy; @dai2017foregraph]. Despite these efforts, the graph algorithms may still suffer from a considerable performance impact caused by the atomic protections. During the graph iteration, each vertex sends its value to all associated vertices. Therefore, it is common that many vertices may read/write the same vertex simultaneously, needing a significant number of atomic protections in existing graph accelerators for preserving the correctness. This performance overhead arising from the atomic operations can be as much as nearly half of total graph execution, as demonstrated in previous work [@nai2017graphpim; @wu2015g] and also witnessed in our motivating study in Section 2. Much effort has been put into reducing the atomic overhead. By offloading the atomic operations to specialized memory (e.g., hybrid memory cubes [@lee2015bssync; @nai2017graphpim]), data access overhead can be reduced. Speculative lock elision can expose the fine-grained parallelism due to inappropriate atomic protection [@Herlihy1993Transactional]. Recent studies also attempt to reduce the number of atomic operations by a series of sophisticated preprocessing, e.g., graph partition [@dai2017foregraph] and dynamic scheduling [@ozdal2016energy]. Unlike these previous work that concentrates on optimizing the individual atomic overhead, this work focuses on the totally-sequential performance impact between atomic operations, which is under-studied in graph processing. Interestingly, graph processing for many graph algorithms (e.g., BFS, PageRank and WCC) shows a significant, common feature for their atomic operations: 1) [*incremental*]{}–the atomic operations follow the commutative and associative law, 2) [*simplex*]{}–all atomic operations are similar. Instead of enforcing sequential execution of conflicting operations as traditional designs, this unique observation in graph processing enables to parallelize massive conflicting vertex updates in an accumulative manner in the sense of simultaneously processing multiple operations and merging the results in parallel. In this paper, we are addressing how we can design such an efficient accumulator for parallelizing the conflicting data accesses for vertex updating in graph processing. We propose a novel accelerator that can simultaneously process multiple atomic operations for parallelizing the vertex updates with a data conflict while ensuring the correctness. Considering that the real-world graphs generally follow the power-law distribution [@gonzalez2012powergraph], a specialized accumulator is designed to distinguish the processing of low-degree and high-degree vertices. Internally, it executes multiple low-degree vertices in parallel for efficient edge-level parallelism and limits the vertex parallelism for the high-degree vertices to avoid frequent synchronization. To keep the architecture balanced, our accelerator is built with a high-throughput on-chip memory to provide efficient vertex access for the accumulator. The memory evenly distributes the requests based on a rearranging mechanism and process them in an out-of-order manner to ensure an efficient throughput. The contributions of this work are summarized as follows: - We study a wide range of graph workloads and perform a detailed analysis on their atomic operations. We demonstrate that their distinct characteristics enable the parallel execution for conflicting vertex updates. - We propose a graph-specific accelerator which supports parallel execution of atomic operations. A parallel accumulator is designed to guarantee efficient process of vertices with different degrees. A high-throughput on-chip memory is also provided for the efficient use. - We compare our accelerator with the state-of-art ForeGraph. Experimental results with three graph algorithms on six real-world graphs show that our accelerator provides 2.36 GTEPS on average, outperforming ForeGraph by up to 3.14x speedup. The rest of this paper is organized as follows. In Section 2, we introduce the background of graph processing and provide our motivations and challenges in detail. Section 3 and Section 4 propose our parallel accumulator designs and optimizations in memory subsystem. The evaluation results are presented in Section 5. We survey related work in Section 6 and conclude the paper in Section 7. Background and Motivation ========================= This section first reviews the vertex updating mechanism of existing graph accelerators for the conflicting data accesses. We next discuss its potential deficiency for graph processing through a motivating study, finally presenting our approach. Modern Graph Accelerator and Its Data Conflict Management --------------------------------------------------------- Graph accelerator is a customized hardware that is specially designed for iterating the computation on graphs. In graph representation, each entity is traditionally defined as [*vertex*]{}, and its connection is defined as [*edge*]{}. The [*degree*]{} of a vertex denotes the number of connections it has. The degree distribution is the probability distribution of all degrees. In existing graph accelerators with shared memory architecture, all vertices in the graph are shared and also able to be accessed by multiple pipelines. As a result, there is a high coverage of data contentions for graph processing, particularly those vertices associated with a large number of edges. For ensuring the correctness of vertex updating, existing researches often seek to use atomic structures (e.g., content addressable memory [@ham2016graphicionado; @ozdal2016energy; @pagiamtzis2006content]), which tend to atomically protect the update of each vertex if a conflicting data access to this vertex has been detected at runtime. A typical procedure of data conflict management used in many graph accelerators [@ham2016graphicionado; @ozdal2016energy] is as follows. Multiple edges of the given vertices will be fetched and sent to the accelerator in each cycle. When receiving these edges, the accelerator will check the pipeline states at first. If an edge is connected with a vertex which is executing in the pipeline, its process would be stalled until the prior one finishes execution. In this way, the same vertex cannot appear in more than one of the pipeline stages for vertex execution at the same time, thus ensuring atomicity. Inefficiency in Graph Processing -------------------------------- Graph often exhibits the complex connections where any vertex may be shared among different vertices. This is particularly true and serious for nature graph that follows the power-law degree distribution, where most vertices have low degree while a few have extremely large degree [@gonzalez2012powergraph]. Thus, there may involve a high risk that a large number of low-degree vertices simultaneously access the same high-degree vertex, leading to serious data contention. Unfortunately, modern graph accelerators (e.g., ForeGraph [@dai2017foregraph] and Graphicionado [@ham2016graphicionado]) fall short in handling these highly-frequent data conflicts in graph processing due to its serial semantics with atomic protection for vertex updates. [**Atomic Protection Analysis**]{}Figure \[fig\_bfs\_code\] illustrates the pseudo-code of [*Breadth-First-Search*]{} (BFS). It starts from a root vertex $r$ and iteratively traverses the graph to calculate the shortest distance from the root vertex to other vertices. During the traversal, each vertex $v$ in the scheduling list will receive values $dis[u]$ from its neighboring vertices and update its own data based on these values (Line 7). In the end of the traversal, a new vector $Q^\prime$ is generated and used as the scheduling list of the next iteration. Because of the atomic protection, these received data from neighboring vertices has to be updated one-by-one in each cycle for preserving the correctness of final result. Figure \[fig\_bfs\_atomic\] shows the execution flow of BFS with atomic protection. Each scheduled vertex will access data from itself and one of its neighbors, and write back the updated data after finishing processing. The data of other neighbors is cached and will not be released to the pipeline before receiving the completion of prior process. In other word, the process inside each vertex is enforced to be sequential for reducing data contention at the cost of performance. {width="2.7in" height="1.6in"} \[memory\_syn\] [***Experimental Demonstration***]{}We further make a set of experiments to investigate how much performance impact may be incurred by atomic protection in graph processing. We use a cycle-accurate simulation to perform the vertex iteration with a parallel update for a maximal set of 16 edges[^1]. Figure \[memory\_syn\] depicts the comparative results. It is observed that the pure atomic protection leads to a significant performance degradation for all real-world graphs, with 45% extra memory overheads on average in contrast to 16-edge parallel vertex update. This is particularly true and serious for those graphs that have the greater average degree (e.g., [*Orkut*]{}). [**Remark**]{} There are also a number of potential solutions that can be used for reducing the performance impact arising from atomic operations. ForeGraph [@dai2017foregraph] proposes a shuffling mechanism to rearrange the edges with potential data conflicts.  [@ozdal2016energy] excessively schedules destination vertices and sends part of them to the processing unit based on a credit based mechanism. Similarly, the basic idea of the above mechanism is to avoid simultaneously scheduling edges with the same destination vertex. While they can reduce the pipeline stalls caused by atomic protection, they still have sequential process of different edges for the same destination vertex. Some work [@ahn2016scalable; @nai2017graphpim] uses novel [*processing-in-memory*]{} (PIM) technology [@gokhale1995processing] to offload the atomic operations to specialized memory region, which reduces the processing time of atomic operations. However, it needs to incorporate with specialized memory architecture and also increases the memory requests since all atomic operations needs to be sent to the memory. \[atomic\_type\] ----------------------------- ------------------------ [**Algorithm**]{} [**Operation Type**]{} Breadth-First Search CAS if less Weakly Connected Components CAS if less Shortest Path CAS if less PageRank Atomic add Triangle Counting Atomic add Degree Centrality Atomic add Collaborative Filtering Atomic add ----------------------------- ------------------------ : Atomic operation types for the vertex update in different graph algorithms Potential of Accumulator ------------------------ The key insight of this work is that atomic operations for many graph algorithms can be parallelized in an accumulative manner. Table \[atomic\_type\] illustrates the typical operations that need an atomic protection for seven popular graph algorithms. We can observe that these atomic operations as a whole have two aspects of significant properties. [***Observation 1***]{}: [*The atomic operations on different edges follow the commutative and associative law*]{}. The commutative law means that the execution sequence of the operations has no effect on the result. Associativity ensures the correctness of merging multiple operations. That is, any of the operations can be simultaneously merged without changing the final result. For example, [*PageRank*]{} follows the atomic-add operations. It updates every vertex by following $Rank(v) = \varepsilon + \sum_{u \in neighbor(v)} Rank(u) / |neighbor(u)|$, where $\varepsilon$ is a constant. Actually, no matter how we change the sequence of these atomic operations or merge successive atomic operations, the final result can be still consistent. [***Observation 2***]{}: [*The atomic operations for updating the value of conflicting vertex are simple and used repeatedly*]{}. Taking [*PageRank*]{} as the example, we find that all of its atomic operations use the same atomic-add to sum up their values to the final result. This similarity allows to use a unique structure to merge all atomic operations. ![Architecture of Graph Accelerator. $P_i$ denotes the $i$th pipeline stage[]{data-label="architecture_overview"}](fig/architecture_overview){width="2.7in" height="1.6in"} These two observations consequently enable us to leverage existing well-developed accumulator to parallelize the vertex update conflicts. Accumulator is a hardware component that merges the inputs into a set of results with specific function. Nevertheless, designing such accumulator for large-scale graph processing remains tremendously challenging. First, the real-world graph topology is often sparse with a low averaged degree. Although traditional accumulator designs [@knowles2001family; @ladner1980parallel; @blelloch1989scans] can provide desirable throughput, they often establish a fixed mapping relationship between the inputs and the results. The reality is that the degree of vertices is dynamically changing during the iteration. The accumulator may get incorrect results when simultaneously processing multiple vertices. Therefore, the traditional accumulator can only accumulate the atomic operations of the single low-degree vertex at the same time, leading to extremely low parallelism for graph processing. There remains a significant gap in applying the accumulation ideology into graph processing without losing a wealth of edge-level parallelism. Second, natural graphs often follow a power-law distribution. When processing the low-degree vertex, the accumulator is expected to simultaneously process multiple vertices. However, for the high-degree vertices with a large number of edges that can be easily more than millions (e.g., [twitter]{}), an accumulator with limited width is extremely difficult to handle so many edges simultaneously. If multiple vertices are simultaneously processed in this case, the accumulator will be invoked several times at the cost of increased synchronization overheads. Moreover, it may lead to massive random edge accesses since the edges of these vertices are more likely to be non-sequential. Therefore, there still lacks an effective technique that can improve the synchronization overheads and random accesses for an efficient accumulation. Third, it is also extremely difficult to predict the non-sequential neighboring vertices of each vertex in real-world graphs. A large number of random accesses have be incurred before invoking the accumulator. Although the accumulator can largely reduce the atomic overheads and provide desirable execution performance, the vertex access remains to be a potential bottleneck and significantly limits the throughput. Architectural Overview ---------------------- Figure \[architecture\_overview\] shows an overview of our accelerator, which is designed in pipeline with six stages in total. These stages basically serve as two major objectives as follows: [**How to Design an Efficient Accumulator**]{} (Section 3): As explained in the challenge discussions, the accumulator generally suffers from the sparse topology and power-law degree distribution in real-world graphs. To achieve desirable performance, the accumulator is expected to efficiently process both of the low-degree and high-degree vertices. When processing the low-degree vertex, the accumulator is expected to simultaneously process multiple vertices for efficient parallelism. Since the vertex degrees are mutable during the process, the accumulator should establish a dynamic relationship between the input vertices and the final results to ensure the correctness. When processing the high-degree vertex, the number of vertices scheduled should be decreased to avoid random access. Therefore, the accumulator should be dynamically aware of the changes in degree and distinguish the process of different vertices. Furthermore, there is a significant synchronization overhead between the multiple accumulations of the same high-degree vertex, which requires an efficient synchronization mechanism. [**How to Use Accumulator Efficiently**]{} (Section 4): While the accumulator could provide high execution efficiency, the on-chip memory is likely to be a potential performance bottleneck. To keep with the throughput of accumulator, the on-chip memory is required to be partitioned into independent parts to process multiple accesses. Furthermore, considering the randomness in vertex access, the address values of vertices may follow an unbalanced distribution. Consequently, multiple requests will be sent to the same memory part in each cycle, leading to significant throughput degradation. To ensure a high throughput, a specialized mechanism is required to dynamically balance the memory requests for on-chip memory. Parallel Accumulator Design =========================== This section discusses the design guideline for a parallel accumulation as well as its core components for the efficiency. Design Philosophy ----------------- Since accumulator is bounded with fixed width, it generally needs to consider two situations where skewed graph vertices with different degrees that can be greater or less than accumulator width, involving different parallel designs. ### **Accumulation Design for Low-Degree Vertex** As is known, most of vertices for a natural graph have a very few degree which can be often no more than the fixed number of ports for a typical accumulator. It is clear of a necessity to simultaneously process the update values of multiple low-degree vertices at a time for high parallelism. [**Problem Definition:**]{} Assuming $N$ update values, belonging to $M$ vertices, need to be processed at once. This problem can be described by $p_j = \sum_{1 \le i \le N} a_i \cdot b_{ij}, 1 \le j \le M$, where $p_j$ denotes the accumulated result of vertex $j$. $a_i$ denotes the update value $i$, and $b_j^i$ denotes whether $a_i$ belongs to vertex $j$. The objective is to get $p$ with minimal latency. Considering the locality of graph traversal, this problem can be further simplified. During traversal, edges of the same destination vertex are sequentially accessed in common graph representations, e.g., CSR/CSC [@shun2013ligra]. It ensures that update values of the same destination vertex are sequentially received by the accumulator. Therefore, assuming that $C_j = [c_j^1, c_j^2]$ denotes the interval of vertex $j$’s update values in all $a_i$, the function of accumulator could be simplified by $p_j = f(c_j^2)$, where $$\begin{aligned} \label{compressed_dp} f(i) = \left \{ \begin{aligned} f(i - 1) + a_i, & \quad i \notin \{c_1^1, c_2^1, \ldots, c_M^1\} \\ a_i, & \quad i \in \{c_1^1, c_2^1, \ldots, c_M^1\} \end{aligned} \right.\end{aligned}$$ [**Solution Discussion:**]{} A naive method for solving this problem is to use a Multi-N-Way [@ma2017garaph] accumulator, which reserves a N-Way accumulator with the binary tree architecture for each vertex. However, its hardware overhead is unacceptable for graph applications. First, its fanouts are too large to implement, which can be up to 8192 when processing a cacheline-width data for 16 vertices. Second, its resource utilization is extremely low since only $N$ among $N \times M$ received values are useful for the real accumulation. In Equation (\[compressed\_dp\]), we find that $f(i) = f(i - 1) + a_i$ is a typical prefix-sum problem, which has been extensively studied in previous work [@sklansky1960conditional; @kogge1973parallel; @ladner1980parallel; @brent1982regular; @knowles2001family]. Beyond the prefix-sum problem, a significant problem is that we still need to consider solving the otherwise case. This needs to 1): dynamically recognize the breakpoints that [*break*]{} the sequential computation and cancel the related operations, and (2) select the results in appropriate ports since not all outputs are required. These are what we have additionally contributed to cope with. ### **Accumulation Design for High-degree Vertex** There are also many high-degree vertices that over-fit the width of an accumulator. Invoking the accumulator multiple times can be considered a useful approach by dividing these edges into multiple parts and processing one of them at the same time, but this costs more overhead. First, iteratively reading the temporary vertex data and writing it back after merging with the accumulated result can lead to an extra synchronization. Second, the graph edges are sequentially stored with common data structure (e.g., [*CSR/CSC*]{} or [*adjacency list*]{}), which means that these edges are distributed to many continuous cachelines. When multiple vertices are simultaneously processed with a high-degree vertex, their edges may be located in non-adjacent cachelines, leading to performance degradation. We present a potential design with an efficient accumulation for solving these problems. For the first problem, the update values of the same destination vertex come in sequence. It ensures that the results of multiple accumulations for the same high-degree vertex are also continuously generated. Therefore, the write back of the vertex data can be delayed before the accumulator sending a different vertex. For the second problem, the inefficiency mainly comes from fixed granularity for vertex scheduling. Without considering the differences in the vertex degree, it schedules fixed number of vertices and simultaneously accesses their edges in each cycle. Instead of accessing the edges based on the scheduled vertices, the viable method is to sequentially access all edges and dynamically schedule the vertices based on the accessed edges. Parallel Accumulator Architecture ---------------------------------- {width="3.2in" height="1.8in"} \[src\_acc\] Figure \[src\_acc\] shows the overview of a parallel accumulator, consisting of four parts. The [*source vertex accumulator*]{} simultaneously accumulates update values of different destination vertices. The [*multiplexer*]{} is responsible for dynamically selecting accumulated data from appropriate ports of the source vertex accumulator. The [*destination vertex accumulator*]{} receives the selected data and fully accumulates each destination vertex. The [*degree-aware accumulation*]{} dynamically decides the number of vertices to be scheduled. [**Source Vertex Accumulator:** ]{} The research of prefix-sum has been extensively studied since 1960s [@sklansky1960conditional; @kogge1973parallel; @brent1982regular; @ladner1980parallel]. In this work, we choose Ladner-Fischer Adder [@ladner1980parallel] as the basis of our accumulator among a large number of previous efficient accumulators for three reasons as follow. First, our main objective is to get the accumulated results in minimal latency, which filters the networks with depth larger than log($N$). Second, among all networks with minimal latency, it has relatively fewer adders, which means that we could add fewer extra resources for breakpoint recognition and result selection. Finally, although its fanouts are relatively larger than others, it does not increase the length of critical path since its delay and route time is much smaller comparing to that of on-chip memory access. Ladner-Fischer Adder opens a great opportunity for our graph-specific source vertex accumulator. In Ladner-Fischer Adder’s original design, it establishes a fixed mapping relationship between the inputs and outputs, which leads to incorrect results when multiple vertices with mutable degrees are processed. As a result, we complement a breakpoint recognizing mechanism. We add a new vector $V = (v_1, v_2, \ldots, v_N)$ where $v_i$ denotes the destination vertex that $a_i$ belongs to. With the vector $V$, the recognition conditions could be easily implemented by comparing the destination vertices of two inputs: $$\begin{aligned} \label{new_dp} f(i) = \left \{ \begin{aligned} f(i - 1) + a_i, & \quad v_i = v_{i - 1} \\ a_i, & \quad v_i \neq v_{i - 1} \end{aligned} \right.\end{aligned}$$ We attach each update value with the ID of its destination vertex in our source vertex accumulator. To further reduce resource usage, we compress the destination vertex ID by only using its last log$m$ bits, where $m$ denotes the width of the accumulator. Based on Formula (\[new\_dp\]), the adder nodes (refer to the gray nodes) are modified to compare the IDs of two inputs at first. If two IDs are the same, the behaviors of the adder nodes are the same as the original design which directly accumulates the input values. Otherwise, they will recognize the second destination vertex as breakpoint and send its update value to the output. [**Multiplexer:** ]{} Once the results are accumulated, the next is to dynamically select the accumulated results for each destination vertex from the output ports of source vertex accumulator. We use a $N \times M$ multiplexer to implement such a logic. Instead of directly comparing the destination vertex IDs, the multiplexer selects the data based on edge offsets to simplify the conditional logic. When the edges in pipeline stage P2 are accessed, each scheduled vertex is attached with its right edge offset, indicating the last edge connected to it. Based on this information, the multiplexer is thus able to naturally select the data for each scheduled destination vertex in the ports related to its last edge. For example, if the updated values $a_1, a_2, a_3$ belong to the vertex, the multiplexer would select the accumulated data from the third port of the source vertex accumulator. {width="3.3in" height="1.2in"} \[fig\_edge\_parallel\] [**Destination Vertex Accumulator:** ]{} In light of the sequential arrival of accumulated values, this opens an opportunity to avoid synchronization on the temporary vertex data by delaying the write back of the destination vertex data until the accumulated value of a different vertex is received. We design a destination vertex accumulator to merge different accumulated results of the same vertex. The accumulator holds the destination vertex ID and the accumulated value in private registers. In each cycle, if the IDs in the input and register are found to be the same, the accumulator would accumulate the vertex data in the input and register. Otherwise, the vertex data in the register will be written back and replaced by the input data. Furthermore, since the source vertex accumulator may simultaneously process multiple destination vertices, we replicate the destination vertex accumulators and use a crossbar switch to connect them with multiplexer. The crossbar switch routes the vertex data based on the destination vertex. That is, the last log($m$) bits in its ID are used for $m$ replications. [**Degree Aware Accumulation:** ]{} Figure \[fig\_edge\_parallel\] shows the specific design of degree aware accumulation. The basic idea is to sequentially access all edges and dynamically schedule vertices based on the runtime information of their edge offsets (e.g., edge ID table in CSR/CSC [@ham2016graphicionado] which denotes the location for the edges of each vertex). To make sure that multiple vertices could be accessed in each cycle, we replicate vertex units in stage P1 and P2. Furthermore, a special matching mechanism is implemented in the vertex units of stage P2 to dynamically decide the vertices to be scheduled. More specifically, we use a specialized generator to automatically generate memory address to sequentially access all edges. In each cycle, every vertex unit stores received edge offsets, and compares generated memory address with the top data in its FIFO. If the memory address is within the range of two edge offsets, the top vertex would be scheduled and sent to the next stage. Moreover, if the memory address is equal to the right edge offset, which means all edges of the vertex have been read, the top vertex in the FIFO would be removed. In this way, the number of scheduled vertex is ensured to be the same with that of vertex contained in requested cacheline. Furthermore, the edge units could be shared among all vertex units to improve resource utilization. Optimizations For Efficient Use =============================== In this section, we present several optimizations that are the key for using the proposed parallel accumulator efficiently. {width="2.6in" height="1.5in"} \[fig\_memory\_partition\_overhead\] Source Vertex Access Parallelization ------------------------------------ While the above accumulator can provide reasonable execution efficiency, the memory access is likely to be a potential performance bottleneck. In practice, the neighbors of every vertex are discontinuous, leading to significant randomness in vertex access. Consequently, the vertex data is typically stored in on-chip memory (e.g., BRAM in FPGA) [@dai2017foregraph; @nurvitadhi2014graphgen; @ozdal2016energy] to improve memory performance. Despite that it could efficiently reduce the latency of vertex access, the throughput of on-chip memory is hard to keep with that of accumulator. For example, assuming that the accumulator runs at 250MHz with a DDR4-2400 memory. In each cycle, the accelerator would receive 16 32-bits edges and generate memory requests based on their source vertices, which means the on-chip memory need to simultaneously process 16 random read requests. Nevertheless, the standard RAM module could only process one read and write request in each cycle. Considering the limitation of capacity and frequency for on-chip memory in typical FPGA chips, memory partitioning [@cong2011automatic; @wang2013memory] is the most practical method to implement such multi-ported memory. Typical memory partitioning mechanisms divide the memory into $n$ independent parts and shuffle the requests to achieve a maximal throughput of $n$. Nevertheless, due to the randomness in vertex access, we find a significant number of requests are shuffled to the same memory partition in each cycle, which means that the memory needs more than one cycle to process these requests. As shown in Figure \[fig\_memory\_partition\_overhead\], the unbalanced shuffling increases up to 70% cycles, even if we partition the memory into 128 parts. [**Optimizations:** ]{} Through analysing the graph data, we find that such inefficiency is caused by the unbalanced edge values: 1) the edge values are not evenly distributed when accessing in the cacheline-width granularity, 2) the edge values themselves are unbalanced when processing in the single-vertex granularity. Algorithm \[alo\_rearrange\] represents the pseudocode of our mechanism for solving the first problem. The basic idea is to rearrange the edges of each vertex to ensure that the address values are relatively balanced in cacheline-width granularity before processing the graph. Assuming that the memory is partitioned to 16 dependent parts, we would also maintain 16 queues for each vertex to store the edges based on the connected vertex’s ID. During rearranging, we would iteratively select edges from each queue in sequence for every vertex. The overhead of rearrangement is about O(|E|), which is the same as that of compressing algorithms commonly used in graph processing (e.g., CSR/CSC). With the mechanism, the address values could be evenly rearranged, thus improving the memory performance. {width="3in" height="1.6in"} \[fig\_source\_access\] For the second problem, we find that even though address values of single vertex are unbalanced, those of the whole graph are relatively balanced. Therefore, we try to change processing granularity to deal with such imbalance. More specifically, we allow the on-chip memory to process the requests in an unblocking (out-of-order) manner. Through unblocked process, the idle memory ports could be utilized by the latter requests, thus improving memory efficiency. Figure \[fig\_source\_access\] shows the work flow of our mechanism. In each cycle, stage P3 receives $N$ edges from memory, and shuffles them to different request FIFOs based on their values. The FIFOs cache these edges and send the requests generated by the top ones to the on-chip memory. To avoid the unblocked requests breaking sequentiality of edge access and further leading to incorrect results, a reorder stage is involved after accessing the source vertex data. The reorder stage caches the accessed vertex data, reorders them to match the sequence of original requests, and sends reordered data to stage P4. To implement such reordering logic, each memory request would be attached with a token based on the last log$(m)$ of original edge memory address, where $m$ denoted the size of buffer in reorder stage. All accessed data with the same token would be stored in the same location in reorder stage. Once the top data finishes reordering, i.e., all data of the first request has been received, it would be sent to the next stage. Source-Based Graph Partition ---------------------------- While storing vertex data in on-chip memory could avoid costly random access in main memory, it might require a large number of resources that exceed the capacity of the chip. Assuming the 4-byte width of vertex data and 8 M vertices, the on-chip memory is desired to be larger than 32 MB, which is unpractical for most of FPGAs. To enable process of large-scale graphs without losing the benefit of on-chip memory usage, we partition the graph into several parts and process a single part at a time. To ensure that all vertex data needed to be processed in each graph parts could be held in on-chip memory, we use a source-based partition mechanism [@gonzalez2012powergraph]. The partition mechanism works as follows. Firstly, the vertices of the input graph are divided into $K$ parts based on their vertex IDs. The value of $K$ depends on the number of vertex and the capacity of on-chip memory. For each part, the out-edges of each vertex are also included. After the input graph is partitioned, our accelerator sequentially processes each graph part in each iteration. Since every edge would be partitioned to the graph part which includes its destination vertex, no edges need to be processed twice. The graph partition does incur some extra memory overhead, since the same destination vertex data might be read and written more than once. More specific impacts would be discussed in Section 5.4. Evaluation ========== This section evaluates the effectiveness and efficiency of our graph accelerator on a wide variety of graph algorithms with real-world graph datasets. Experimental Settings --------------------- [**Evaluation Tools:** ]{} We implement our accelerator on Xilinx Virtex Ultrascale+ XCVU9P-FLGA2104 FPGA with -2L speed grade. The target FPGA chip provides 1.18 M LUTs, 2.36 M registers, and 9.49 MB on-chip BRAM resources. We verify the correctness and get the clock rate as well as resource utilization using Xilinx Vivado 2017.1. All these results have passed post-place-and-route simulations. Our target off-chip memory is Micron 4GB DDR4 SDRAM (MT40A256M16GE-083E). We use DRAMSim2 [@rosenfeld2011dramsim2] to simulate the cycle-accurate behavior of the off-chip access. The memory has a running frequency of 1.2 GHz and a peak bandwidth of 19.2 GB/s. \[Graph\_datasets\] =0.1cm --------------- --------------------- ------------------ --------------------- [**Names**]{} [**\# Vertices**]{} [**\# Edges**]{} [**Description**]{} Slashdot 0.08 M 0.95 M Link Graph DBLP 0.32 M 1.05 M Collaboration Graph Youtube 1.13 M 2.99 M Social Network Wiki 2.39 M 5.02 M Website Graph LiveJournal 4.85 M 69.0 M Follower Graph Orkut 3.07 M 117 M Social Network --------------- --------------------- ------------------ --------------------- : Graph datasets [**Graph Algorithms:** ]{} We implement three well-known graph algorithms on our accelerator, covering both CAS-if and atomic-add operation types in Table \[atomic\_type\]. - [*Breadth First Search (BFS)*]{} is a basic traversal algorithm utilized by many graph algorithms. It iteratively traverses the input graph and calculates the distance of shortest path from root to every vertex. - [*PageRank (PR)*]{} is an important graph algorithm used to rank web pages according to their importance. It updates every vertex based on the formula $Rank(v) = \varepsilon + \newline \sum_{u \in in-neighbor(v)} Rank(u) / |out-neighbor(u)|$ in each iteration, where $\varepsilon$ is a constant. - [*Weakly Connected Components (WCC)*]{} is an algorithm that checks the connectivity between two vertices in a graph. During the traverse, every vertex would receive the labels from all neighbors and update itself with the minimal one. [**Graph Datasets:**]{} The graph datasets for the experiments are summarized in Table \[Graph\_datasets\]. All these graphs are real graph data sets collected from SNAP [@snapnets] and TAMU [@DvisSparse]. In our implementation, each undirected edge is treated as two directed edges between source vertex and destination vertex by being processed twice. Therefore, the number of edges for undirected graphs ([*DBLP*]{}, [*Youtube*]{}, and [*Orkut*]{}) is considered double in our evaluation. ![Our accelerator normalized to the ForeGraph performance. YT denotes graph [*Youtube*]{}, Wk denotes graph [*Wiki*]{}, and LJ denotes graph [*LiveJournal*]{}. AVG presents the average speedup of all tested graphs[]{data-label="fig_performance_compare_foregraph"}](fig/performance_compare_foregraph){width="2.8in" height="1.5in"} Overall Performance ------------------- [**Resource utilization:** ]{} Table \[resource\_utilization\] shows the resource utilization and clock rate of the FPGA design with 8 vertex pipelines and 16 edge pipelines, which maximizes throughput given the peak DRAM bandwidth. First of all, because of the shared edge pipeline design described in Section 3.2, the number of resources required is reduced. Therefore, the logic resource (LUT and register) consumption of our accelerator is relatively low. Secondly, we implement the on-chip memory with BRAM resources to maintain vertex data. Similar to prior work [@dai2017foregraph], we use 1 byte integer to represent the depth value in BFS, single-precision floating point (4 bytes) in PR, and 4 bytes integer in WCC. In this way, the maximal memory requirement is 1 $\times$ 4.85 $=$ 4.85 MB for 1 byte data and 4 $\times$ 4.85 $=$ 19.4 MB for 4 bytes data. Therefore, we hold all vertex data when running BFS and about 1.7 M vertex data for other algorithms, which consumes 57.9% and 69.9% of available BRAM resources, respectively. The UltraRAM resources are not used in our implementation. \[resource\_utilization\] ----------------------- ------------- ------------ ------------- [****]{} [**BFS**]{} [**PR**]{} [**WCC**]{} LUT 7.39% 10.1% 8.26% registers 2.53% 4.47% 3.02% BRAM 57.9% 69.9% 69.9% Maximal clock rate 256 MHz 211 MHz 251 MHz Simulation clock rate 250 MHz 200 MHz 250 MHz ----------------------- ------------- ------------ ------------- : Resource utilization and clock rate [**Throughput:**]{} Figure \[fig\_performance\_compare\_foregraph\] shows the normalized performance comparing to ForeGraph, which is one of the fastest graph processing accelerator implemented on FPGA, with respect to throughput. By throughput, we refer to the number of [*traversed edges per second*]{} (TEPS) [@Graph500], which is a performance metric frequently used in graph processing. As described above, ForeGraph is a representative accelerator that sequentially processes different edges of the same destination vertex to ensure atomicity. Since ForeGraph has not been open-sourced, we execute the same graph algorithms (BFS, PR, and WCC) and datasets ([*youtube*]{}, [*wiki-talk*]{} and [*LiveJournal*]{}) used by its evaluation on our accelerator, and compare the results with the performance reported in its work (just as previous work has also done [@dai2017foregraph; @zhou2016high]). When running PR and WCC on [*Wiki*]{}, the BRAM resources available in the FPGA chip used in ForeGraph is large enough to (up to 16.6 MB) hold all vertex data on-chip, which is unreliable for that of our FPGA chip (9.49 MB). Therefore, we compress the vertex data to 2 bytes when running PR and WCC on [*Wiki*]{} for fair comparison. As shown in Figure \[fig\_performance\_compare\_foregraph\], our accelerator achieves 1.36x $\sim$ 3.14x speedup compared to the ForeGraph. As analysed in Section 2.2, the speedup mainly comes from the reduced synchronization overheads by simultaneously processing atomic operations. Moreover, our accelerator could achieve better load-balance using degree-aware accumulation by dynamically deciding the number of vertices scheduled. For the results of different algorithms, we find that the speedup of PR is smaller. This is because of the lower clock rate caused by complex floating units. Since the number of edge pipelines is fixed in our implementation, the clock rate directly influences the overall performance. Moreover, the floating point units significantly increase the length of pipelines, thus would need more cycles when recovering from pipeline stalls. Therefore, the algorithms that use integer values could achieve slightly higher performance. Sensitivity Study ----------------- To get a more comprehensive performance result, we execute all graphs described in Table \[Graph\_datasets\] on our accelerator. The structures of these graphs significantly differ from each other (e.g., number of vertices and edges, average degree), thus providing an in-depth overview on the performance. As shown in Figure \[fig\_throughput\], our accelerator achieves 1.4 GTEPS $\sim$ 3.5 GTEPS over all graph algorithms and datasets. Among all graph datasets, [*Wiki*]{}’s throughput is particularly low when executing on our accelerator. This is because [*Wiki*]{} is extremely sparse and makes the accelerator exhibits unbalance between the vertex and edge pipelines. With low average, the edges accessed from [*Wiki*]{} in each cycle prefer to belong to multiple vertices (more than 8). Therefore, the vertex pipelines might need more than one cycle to process these edges, leading to lower performance. As shown in Figure \[fig\_averge\_throughput\], the performance is almost linearly increased when the average degree is less than 16. This is because that the percentage of low-degree vertex ($\le 2$) decreases. Moreover, the performance improves slightly when increasing the average degree from 16 to 76. This is because that the memory bandwidth becomes the potential bottleneck in these cases, since it could only send a cacheline-width edges in each cycle. In summary, the performance improves as the average degree increases before reaching the limitation of maximal memory bandwidth. Lastly, we find obvious performance degradation for PR and WCC when average degree is about 14 ([*LiveJournal*]{}). Moreover, the performance of PR and WCC is significantly lower than that of BFS when average degree is larger than 14 ([*LiveJournal*]{} and [*Orkut*]{}). This is because that the vertices data is too large to be all held in on-chip memory in these cases. Therefore, the graph partition mechanism is used when executing PR and WCC on these graphs, which involves in more vertex access. More detailed analysis of degree distribution and graph partition is presented in Section 5.4. Benefit Breakdown ----------------- We next break down the respective benefits of our different graph accelerator designs as follow: [**Benefits from Parallel Accumulation:** ]{} Figure \[performance\_accumulator\] presents the normalized performance results. The baseline represents the basic design without any optimizations described in Section 3 and 4. It sequentially processes each edge, and accumulates its values to the final result in each cycle. CFG 1 represents source vertex accumulation. CFG 2 further uses destination vertex accumulation based on CFG1. {width="2.6in" height="1.5in"} \[performance\_accumulator\] It is shown that CFG1 achieves 1.9x$\sim$ 5.2x speedup compared to the baseline. Note that [*Wiki*]{} is lowest performance among all graph workloads. This is because that the number of vertex pipelines to set to one, leading to the fact that only one vertex can be scheduled in each cycle for CFG 1. Therefore, the number of edges sent to the accumulator in each cycle is directly depended on the average degree. In a word, the graphs with higher degree could experience higher speedup when using source vertex accumulator. For CFG 2, destination vertex accumulator achieves about 1.3x speedup in most of graphs, except for [*Slashdot*]{} (2.0x speedup). This is because that [*Slashdot*]{} has self-loops, which means that some edges connect a vertex to itself. When processing these self-loops, the memory requests of source and destination vertex would be assigned to the same on-chip memory partition, leading to increased memory cycles. With the source vertex accumulator, the request of destination vertex could be avoid, thus improving the overall performance. [**Benefits from Degree-aware Accumulation:** ]{}Secondly, we explore the impact of degree aware accumulation on above accumulators. Figure \[fig\_performance\_vertex\_parallel\] presents the results which assume that on-chip memory could process any 16 memory requests in each cycle. For the performance, we analyse the speedup brought by different number of vertex pipelines, which denotes the maximal parallelism of the accumulation[^2]. We make the observation that the performance improves sub-linearly as the number of vertex pipelines increases. This is because of the power-law degree distribution of graphs. Assuming that the number of vertex pipelines is $N$, our degree aware mechanism could cover the vertices with degree $\ge 16 / N$ with 16 edge pipelines. As depicted in Figure \[degree\_distribution\], the percentage of the covered edges for most graphs increases sub-linearly because that high-degree vertices have most of the edges. While for [*Wiki*]{}, the skewness of its degree distribution is low, thus leading to an almost linear increment. [**Benefits from Vertex Access Parallelization:** ]{} Figure \[fig\_performance\_memory\] explores the impact of different optimization for parallel accumulations, without ignoring the influence of the on-chip memory’s throughput. The left most bar in Figure \[fig\_performance\_memory\] represents the baseline case where only parallel accumulation is applied. CFG 3 represents that degree aware accumulation is involved in with 8 vertex pipelines based on CFG2. CFG 4 shows the effects of rearranging mechanism and CFG 5 shows the effects of reordering discussed in Section 4.1. The first observation is that the speedup of degree aware accumulation is decreased to about 1.3x when considering the influence of on-chip memory’s throughput. Without any optimizations, there would be a significant amount of increased memory requests caused by the unbalanced edge values, thus decreasing the impact of degree aware accumulation. Another observation is that our rearranging mechanism could achieve 1.5x speedup and reordering mechanism could achieve another 1.5x $\sim$ 2.8x speedup. With these mechanisms, the increased memory requests could be reduced to $\le 10\%$, which significantly improves the memory efficiency. {width="2.6in" height="1.5in"} \[fig\_performance\_memory\] [**Benefits from Graph Partition:** ]{} Figure \[fig\_performance\_partition\] explores the impact of graph partition described in Section 4.2. The leftmost bar represents the case where the on-chip memory size is enough to hold all vertex data, denoted as partition number = 1. The other bars represent cases where on-chip memory size is only enough to hold $1 / N$ of the total vertex data where $N$ represents the number of partitions. In general, partitioning the graphs into 4 parts would result in around 40% performance degradation. Among all workloads, the [*Wiki*]{} experiences the largest performance degradation which reaches about 61%. This is because that we would traverse all vertex in each sub-iteration when processing each graph partition. As the average degree decreases, the increased vertex access overheads would account for a significant percentage of total overheads. Therefore, the performance of graphs with lower average degree would be more sensitive to the partition number. Related Work ============ A wealth of recent studies [@guo2014well; @beamer2015locality; @guo2015empirical] indicate that even with extensive optimizations, graph processing still subjects to the underlying limitation of general-purpose processors. A vast body of research efforts have been therefore put into making the graph-specific architectural innovations to improve the execution efficiency. Graphicionado [@ham2016graphicionado] proposes a pipelined graph accelerator which efficiently utilizes large on-chip scratchpad memory. GraphGen and Graphops [@nurvitadhi2014graphgen; @oguntebi2016graphops] propose FPGA-based frameworks which automatically compile graph algorithms to specialized graph processors. Compared with these prior researches with strict atomic protection, we argue that the heavy reliance on atomic operations leads to significant performance degradation and propose a novel accelerator to reduce atomic overhead. There are also a large number of attempts that aim at reducing the number or the execution time of atomic operations for graph processing. ForeGraph [@dai2017foregraph] partitions the input graph in a grid-manner [@zhu2015gridgraph] to avoid simultaneously scheduling edges with the same vertex.  [@ozdal2016energy] proposes a specialized synchronizing mechanism to avoid scheduling conflicting edges. Shijie et al [@zhou2016high] use a combing network to avoid the same vertex being simultaneously scheduled through filtering the unnecessary edges before processing. In general, their basic idea is to avoid scheduling the edges with conflict vertices through preprocessing. Speculative Lock Elision [@rajwar2001speculative] speculatively remove the lock operations and enable highly concurrent execution. As a comparison, we focus on the performance impact between multiple atomic operations, instead of the performance of atomic operation itself. We find that these atomic operations could be parallelized according to distinct characteristics in vertex updates of graph processing. We thus propose an efficient graph-specific accumulator to exploit the potential benefits of this insight. Many other efforts also have been put into improving the execution time of atomic operations. Tesseract [@ahn2016scalable] offloads all graph operations to memory-based accelerator to ensure atomicity without requiring software synchronization primitives. There are also some researches [@ahn2015pim; @nai2017graphpim] enables offloading operations at instruction-level. They statically or dynamically detect the atomic instructions during processing and directly map them into PIM region  with minor extension to the host processors. Compared to these PIM-enabled graph architecture, our accelerator can achieve efficient management on shared data conflicts without introducing special memory components. Moreover, our parallel data conflict management can be also integrated into PIM-enabled graph accelerators and help to reduce the memory requests. {width="2.6in" height="1.5in"} \[fig\_performance\_partition\] Conclusion ========== In this paper, we present a pipelined graph processing accelerator to enable massive parallelism of vertex updates. Our accelerator provides a parallel accumulator to simultaneously schedule and process multiple destination vertices without losing edge-level parallelism. Moreover, the accumulator is designed to be degree-aware and can adaptively adjust the vertex parallelism to different kinds of graphs. We also present vertex access parallelization and source-based graph partition for better supporting the efficient use of graph accelerator. Our evaluation on a variety of graph algorithms shows that our accelerator can achieve the throughput by 2.36 GTEPS on average, and up to 3.14x speedup compared to the stat-of-the-art FPGA-based graph accelerator ForeGraph with its single-chip version. [^1]: The simulation is conducted with a pipelined architecture that is similar to ForeGraph [@dai2017foregraph]. While data width of edges is usually 32-bits in BFS, we set 16-edge parallelism according to the memory access granularity (512-bits). Edge shuffling optimization [@dai2017foregraph] is not covered in our simulation. [^2]: When the number of vertex pipelines is set to $N$, the mechanism dynamically schedules $1 \sim N$ vertices based on the degree.

--- abstract: '**In this paper we use quantum magnetohydrodynamic (MHD) as well as magnetohydrostatic (MHS) models for a zero-temperature Fermi-Dirac plasma to show the fundamental role of Landau orbital ferromagnetism (LOFER) on the magnetohydrostatic stability of compact stars. It is revealed that the generalized flux-conserved equation of state of form $B=\beta \rho^{2s/3}$ only with conditions $0\leq s\leq 1$ and $0\leq \beta< \sqrt{2\pi}$ can leads to a stable compact stellar configuration. The distinct critical value $\beta_{cr}=\sqrt{2\pi}$ is shown to affect the magnetohydrostatic stability of the LOFER ($s=1$) state and the magnetic field strength limit on the compact stellar configuration. Furthermore, the value of the parameter $\beta$ is remarked to fundamentally alter the Chandrasekhar mass-radius relation and the known mass-limit on white dwarfs when the star is in LOFER state. Current findings can help to understand the role of flux-frozen ferromagnetism and its fundamental role on hydrostatic stability of relativistically degenerate super-dense plasmas such as white dwarfs.**' author: - 'M. Akbari-Moghanjoughi' title: 'Orbital Ferromagnetism and the Chandrasekhar Mass-Limit' --- Introduction ============ **Since the pioneering discovery by Chandrasekhar in 1939 [@chandra1] concerning the mass-limit on compact stars and the hydrostatic stability mechanisms in such stars due to the relativistic degeneracy of electrons, there has been a growing interest towards the study of hydrodynamic properties of degenerate ionized matter, the so-called zero-temperature quantum plasmas [@bohm; @pines; @levine; @Markowich; @shukla]. It is well known that the matter under compression exerts enormous pressure called the degeneracy pressure due to the Pauli exclusion principle when the interparticle distances are lowered to become comparable to the de Broglie thermal-wavelength $\lambda_D = h/(2\pi m_e k_B T)^{1/2}$ [@landau]. Quantum peculiarities such as quantum tunneling, quantized Hall effect, magnetic quantization etc. ubiquitously appear as the degeneracy limit is reached. Such peculiar features prove to be of fundamental significance and applications in ordinary metallic and semiconductor materials. Many recent investigations based on the quantum hydrodynamics (QHD) and quantum magnetohydrodynamics (QMHD) models [@gardner; @Marklund1; @Marklund2; @Brodin1; @Brodin2; @manfredi; @haas1; @haas2; @akbari2; @akbari3] indicate that the incorporation of the quantum electron-tunneling and degeneracy pressure can lead to quite different nonlinear dynamic effects in plasmas. It has also been remarked that the relativistic degeneracy caused by large-scale gravitational forces in stars which causes the gravitational collapse in stellar objects [@chandra2; @chandra3] may also lead to distinctive nonlinear hydrodynamic features [@akbari4; @akbari5; @akbari7] due to the change in the thermodynamical quantities in the Fermi-Dirac statistics [@kothary].** **Among the greatest challenges today is the problems associated with the origin of strong magnetic fields present in many compact astrophysical entities such as white dwarfs, pulsars, neutron stars, etc. and its formidable role on the stellar chain of evolution. It is also believed that the magnetic field has a fundamental role in the formation and the dynamical processes in astrophysical environments [@weeler]. There has been extensive past studies on the thermodynamical behavior of degenerated electron gas under arbitrarily high magnetic field [@can1; @can2; @can3; @can4; @can5; @can6; @can7; @can8]. Such investigations have revealed that the high magnetic field can lead to the anomalous quantization and spiky features in the electronic density of states (DoS) affecting all the thermodynamical properties of the Fermi-Dirac gas. It was suggested that under such a quantizing field the magnetic transverse collapse of the gas is possible [@chai; @akbari1] where the Fermi-Dirac gas may become one-dimensional. A review of the current findings on the properties of matter in strong magnetic field has been reviewed in some recent literature [@dong; @harding]. Recent studies based on the QMHD including the magnetization effect confirm the significant role of the electron spin-orbit magnetization effects on the nonlinear properties of degenerated quantum plasmas [@marklund3; @marklund4; @brodin0; @brodin3; @misra1; @misra2; @martin; @mushtaq; @zaman; @vitaly; @akbari9]. Particularly, a more recent study remarks distinctive paramagnetic nonlinear features of a Fermi-Dirac plasma due to the relativistic electron degeneracy [@akbari10].** **Another outstanding feature of a degenerated plasma under a strong magnetic field is that a ferromagnetic solution called the Landau orbital ferromagnetism (LOFER) is possible [@lee; @con1; @con2; @con3; @burk1; @burk2] which may account for the large magnetic fields (as high as $10^8G$) estimated for some astrophysical compact objects. In the present investigation we use both magnetohydrodynamic and magnetohydrostatic (MHS) models to explore the role of Landau orbital ferromagnetism on compact stellar characteristics such as hydrodynamic quantum collapse, hydrostatic stability and the mass-radius relation comparing the results with that of previous ones. The presentation of the paper is as follows. The QMHD model including Bohm potential and the spin-orbit magnetization effects is introduced in Sec. \[equations\] and the possible regimes for a quantum collapse is explored in Sec. \[calculation\]. The hydrostatic stability of a LOFER state is investigated and the Chandrasekhar mass-limit is calculated based on the generalized LOFER equation of state in Sec. \[discussion\]. Finally, a summary is given in Sec. \[conclusion\].** Magnetohydrodynamic Model and Degeneracy Collapse {#equations} ================================================= In this section we show that for a flux-conserved degenerate plasma the total pressure can vanish in the transverse direction making the MHD wave unstable due to the effect of plasma magnetization. In this case a transverse collapse may occur without the need for presence of the gravitational force. Let us consider the magnetohydrodynamic (MHD) equations for a completely degenerate dense **singly-ionized and quasineutral ($n_i\simeq n_e = n$) helium plasma with the center of mass density $\rho=m_i n_i + m_e n_e \simeq m_i n = 2m_p n$ ($m_p$ is the proton mass)**. We have for the continuity equation $$\label{cont} \frac{{\partial \rho }}{{\partial t}} + \nabla \cdot(\rho {\bf{u}}) = 0,$$ where, ${\bf{u}} = ({m_i}n_i{{\bf{u}}_{\bf{i}}} + {m_e}n_i{{\bf{u}}_{\bf{e}}})/\rho$ is the center of mass speed of plasma. Therefore, the generalized momentum equation including the quantum degeneracy pressure, dipole force, magnetization and electron nonlocality effects can be written in cgs units in the following form [@brodin0] $$\label{mom} \rho \frac{{d{\bf{u}}}}{{dt}} = ({\bf{B}}\cdot\nabla) \left( {{\bf{B}} - {\bf{M}}} \right) - \nabla \left( {\frac{{{B^2}}}{2} - {\bf{M}}\cdot{\bf{B}}} \right) - \nabla {P_d} + \frac{{\rho {\hbar ^2}}}{{2{m_e}{m_i}}}\nabla \frac{{\Delta \sqrt \rho }}{{\sqrt \rho }},$$ **in which $P_d$ is the electron degeneracy pressure and we have neglected the Bohm force on ions and the pressure due to them. The magnetization, $\bf{M}$ and the induced field, $\bf{B}$, are related through; $\bf{H}=\bf{B}-4\pi \bf{M(B)}$, with $\bf{H}$ being the magnetic field due to physical currents. The metastable Landau orbital ferromagnetism (LOFER) for the flux-conserved degenerate plasma model is given by $\bf{H}=0$ which leads to $\bf{B}=4\pi \bf{M(B)}$ [@sho]. On the other hand, it has been shown that the LOFER condition for a magnetized degenerate electron-gas leads to the field/density equation of state (EoS) of form; $B(r) = \alpha(\rho_6/\mu_e)^{2/3}$ [@con1], where, $\rho_6=\rho/10^6$ and $\mu_e$ is the number of nucleon per electron (in this calculation we use $\mu_e=2$ for helium). The parameter $\alpha$ is the normalizing factor for magnetic field and will be found to be of the order $10^8$ for the white-dwarf mass-density ranges ($\rho_6\simeq 1$). However, this parameter is known to relate to some other parameters such as the plasma temperature, electron exchange interactions etc. [@burk1]. There has been many reports of compact star with strong internal or external magnetic fields [@crut; @kemp; @put; @jor]. On the other hand, many reports confirm the role of flux-conservation on star formation [@bra] with the similar density dependence of magnetic field. In 1964 a theory based on flux conservation was suggested simultaneously by Ginzburg [@ginz] and Woltjer [@wolt] to explain the presence of intense magnetic fields in some young compact stars born in supernova explosions [@mestel]. As it will be apparent later in discussion the parameters $s$ and $\alpha$ in EoS of form $B = \alpha(\rho_6/\mu_e)^{2s/3}$ are central to the stability criteria of compact stellar configurations. There are some theoretical discrepancies on the values of these parameters. For instance, some calculations show that [@burk2] the value of $s$ in LOFER EoS, should be $4/3$ rather than $2/3$ calculated for this parameter previously [@con1]. Also, the calculations of the parameter $\alpha$ leads to different values due to the oscillatory nature of the spin-orbit magnetization elements. However, in this calculation we introduce a more general field/density EoS as $B = \alpha(\rho_6/\mu_e)^{2s/3}$ to show that only the restricted values of $0<s\leq1$ is consistent with the magnetohydrostatic (MHS) stability of compact stars. Also, we will find an upper limit on the value of $\alpha$ (or the strength of the magnetic field) for the known LOFER state ($s=1$) of stellar configuration.** In a strongly magnetized Fermi-Dirac electron gas the equation of state (EoS) is quantized and we may write for the electron number-density in terms of Hurwitz zeta functions [@claud] $$\label{h} \begin{array}{l} n_e(x,\gamma ) = {n_c}{(2\gamma )^{3/2}}{H_{ - 1/2}}\left( {\frac{{{x^2}}}{{2\gamma }}} \right), \\ {P_{e\parallel}}(x,\gamma ) = \frac{{{n_c}{m_e}{c^2}}}{2}{(2\gamma )^{5/2}}\int_{0}^{\frac{{{x^2}}}{{2\gamma }}} {\frac{{{H_{ - 1/2}}(q)}}{{\sqrt {1 + 2\gamma q} }}} dq, \\ {H_z}(q) = h(z,\{ q\} ) - h(z, q + 1 ) - \frac{1}{2}{q^{ - z}}, \\ h(z,q) = \sum\limits_{n = 0}^\infty {{{(n + q)}^{ - z}}} . \\ \end{array}$$ where $h(z,\{q\})$ is the Hurwitz zeta-function of order $z$ with the fractional part of $q$ as argument and $P_{e\parallel}$ denote the degeneracy pressure parallel to the magnetic field. Note also that, $n_c=m_e^3 c^3/2\pi^2 \hbar^3$, $\gamma=B_0/B_c$ with $B_c=m_e^2c^3/e\hbar\simeq4.41\times 10^{13}G$ being the fractional critical-field parameter, $\varepsilon _{Fe}=\sqrt{1+x^2}=E_{Fe}/m_e c^2$ is the normalized Fermi-energy and $x=p_{Fe}/m_e c$ is the normalized Fermi-momentum the so-called relativity parameter. It is evident that, the electron degeneracy pressure is field dependent on both magnitude and direction so that $P_{e\perp}\neq P_{e\parallel}$. It has been shown that in such plasmas two distinct quantum and classical degeneracy regimes based on the parameter $x^2/2\gamma$ can be defined [@akbari11]. In our case the magnetic fields of interest are of the order $10^8G$ leading to the classical regime $x^2/2\gamma\gg 1$ for the white dwarf star density ranges. Hence, one may assuredly use the Chandrasekhar classical EoS for the degeneracy pressure as [@chandra1] $$\label{p} {P_d(x)} = \frac{{\pi m_e^4{c^5}}}{{3{h^3}}}\left\{ {x\left( {2{x^2} - 3} \right)\sqrt {1 + {x^2}} + 3\ln \left[ {x + \sqrt {1 + {x^2}} } \right]} \right\},$$ **where, the well-known relativity parameter is $x=p_{Fe}/m_e c=(h/m_e c)(3n/8\pi)^{1/3}=(n/n_0)^{1/3}$ ($n_0=8\pi m_e^3 c^3/3h^3\simeq 5.9\times 10^{29}/cm^3$) [@kothary], with $p_{Fe}$ being the relativistic Fermi momentum. We may also write the relativity parameter in terms of the plasma mass-density $\rho=2m_p n$ and $\rho_0=2m_p n_0=2m_p/3\pi^2\mathchar'26\mkern-10mu\lambda_c^{3}\simeq 2\times 10^6 gr/cm^3$ as $x=(n/n_0)^{1/3}=(\rho/\rho_0)^{1/3}$ with $\mathchar'26\mkern-10mu\lambda_c=\hbar/m_e c\simeq 3.863\times10^{-11}cm$ being the scaled electron Compton-wavelength. Note that in the forthcoming algebra we will use the normalized mass-density $\bar\rho=\rho/\rho_0$ dropping the bar notation for simplicity, hence, $x=\rho^{1/3}$.** Therefore, the normalized momentum equation, Eq. (\[mom\]) may be cast in terms of the effective potentials [@akbari12] as $$\label{momq} \rho \frac{{d{\bf{u}}}}{{dt}} = - \nabla \left( {{\Psi _m} + {\Psi _d}} \right) + \frac{{\rho {\hbar ^2}}}{{2{m_e}{m_i}}}\nabla \frac{{\Delta \sqrt \rho }}{{\sqrt \rho }},$$ where $\Psi_m$ and $\Psi_d$ are the corresponding flux-conserved magnetic and electron degeneracy effective potentials (the normalization is given elsewhere [@akbari12]). Therefore, the inward magnetic-force, ${{\bf{F}}_m} = \nabla \Psi_m = - sC_s^2{\beta ^2}\nabla {x^{(4s - 3)}}/2\pi (4s - 3)$ (we have made use of $\alpha=\beta C_s$ with $C_s=c\sqrt{m_e/2m_p}$ being the quantum sound speed) and the outward degeneracy force, ${{\bf{F}}_d} = \nabla \Psi_d = {C_s^{2}}\nabla \sqrt {1 + x^2}$ oppose each other. It has been shown that on some conditions the total force is canceled giving rise to the quantum collapse for LOFER case [@akbari12]. However, for the general EoS considered here there exists a surface for which a quantum collapse can occur which is shown in Fig. 1 for the parameter range of $3/4\le s\leq1$. It is clearly observed that for a quantum degeneracy collapse to occur in the LOFER state ($s=1$) the value of $\beta$ should be above a critical value $\beta_{cr}=\sqrt{2\pi}$ [@akbari12]. It is also remarked that quantum degeneracy collapse is not possible for $s<3/4$. In the proceeding section we use the MHS stability of a magnetized degenerate plasma model to find some limits on the value of parameters $s$ and $\beta$ which is also a limit on the magnetic field of flux-conserved compact star. Hydrostatic Stability in Flux-Frozen Stellar Plasmas {#calculation} ==================================================== Using a classical treatment, in this section, we consider the MHS equilibrium for homogenous spherical magnetized plasma. We also ignore the quantum tunneling effect on electrons due to large size of system compared to the interparticle distances. To begin with, let us consider a spherical shell-element of mass-density $\rho$ which is normalized to $\rho_0$ defined in previous section. The equilibrium for the shell-element is defined as $\bf{F}_{in}=\bf{F}_{out}$, where, the inward force consist of the gravity and the magnetic force and the outwards force is the electron degeneracy force. In a simpler argument we may write $$\label{eq1} \frac{{d{P_{tot}}}}{{dr}} = - \frac{{G\rho(r) M(r)}}{{{r^2}}},\hspace{3mm}{P_{tot}} = {P_d} + {P_m},$$ where, $G$ is the gravitational constant and $M(r)$ is the mass of plasma within the radius $r$, defined as $$\label{eq2} \frac{{dM(r)}}{{dr}} = 4\pi r^2\rho(r),$$ where, again, $\rho$ is the normalized local density. Thus, the above definitions lead to the MHS stability condition of the form $$\label{eq} \frac{1}{{4\pi {r^2}G}}\frac{d}{{dr}}\left( {\frac{{{r^2}}}{\rho(r) }\frac{{d{P_{tot}(\rho)}}}{{d\rho}}\frac{{d\rho(x)}}{{dx }}\frac{{dx(r) }}{{dr}}} \right) + \rho(r) = 0,$$ with the parameter $x=\rho^{1/3}$ being the Chandrasekhar relativity parameter. From the definitions for the degeneracy and magnetic pressures one obtains $$\label{pr} \frac{{d{P_d}(\rho)}}{{d\rho }} = \frac{{C_s^2}}{6}\frac{{{x^2}}}{{\sqrt {1 + {x^2}} }},\hspace{3mm}\frac{{d{P_m}(\rho)}}{{d\rho }} = - \frac{s{C_s^2{\beta ^2}}}{{12\pi }}{x^{{{2s - 1}}}},$$ where, we have used a general form of $B=\alpha\rho^{2s/3}$ ($s$ is real) for the magnetic equation of state previously defined. Hence, we have $$\label{prt} \frac{{d{P_{tot}(\rho)}}}{{d\rho }} = \frac{{C_s^2}}{6}\left[ {\frac{{{x^2}}}{{\sqrt {1 + {x^2}} }} - \frac{{s{\beta ^2}}}{{2\pi }}{x^{{{2s - 1}}}}} \right].$$ Thus, in terms of the relativity parameter, we rewrite Eq. (\[eq\]) as $$\label{eqx} \frac{{3\pi \mathchar'26\mkern-10mu\lambda _c^2{n_h^2}}}{{16{r^2}}}\frac{d}{{dr}}\left[ {{r^2}\left( {\frac{x(r)}{{\sqrt {1 + {x(r)^2}} }} - \frac{{s{\beta ^2}}}{{2\pi }}{x(r)^{{{2(s - 1)}}}}} \right)\frac{{dx(r)}}{{dr}}} \right] + {x(r)^3} = 0,$$ where, $n_h=\sqrt{\hbar c/G}/2 m_p\simeq 1.3\times 10^{19}$ is the dimensionless hierarchy-number defined based on three fundamental constants, namely, the scaled Plank-constant, $\hbar$, the speed of light in vacuum, $c$ and the gravitational constant, $G$. It is clearly evident that, for the value of $s=1$ the magnetic field pressure gradient term in Eq. (\[eqx\]) becomes independent of the relativity parameter. Therefore, each stable model is obtained by integrating Eq. (\[eqx\]) from the center ($r=0$) outwards until $x(r)$ vanishes for some $r=r_s$ which would be the surface (of the star). This is done by employing initial conditions $x(r=0)=x_c$ and $dx/dr(r=0)=0$ so that for every given value of the relativity parameter at the center ($x_c$) we find a distinct solution. However, it is convenient to put the Eq. (\[eqx\]) in a more friendly shape by introducing dimensionless parameters $y=x/x_c$ and $\eta=r/r_c$ with $r_c=\sqrt{3\pi}n_h\mathchar'26\mkern-10mu\lambda_c/4x_c$ so that at the center ($\eta=0$) we will have $y=1$. Therefore, the Eq. (\[eqx\]) in dimensionless form reads as $$\label{eqy} \frac{1}{{{\eta ^2}}}\frac{d}{{d\eta }}\left[ {{\eta^2}\left( {\frac{{{x_c}y(\eta)}}{{\sqrt {1 + x_c^2{y(\eta)^2}} }} - \frac{{s{\beta ^2}}}{{2\pi }}x_c^{2(s - 1)}{y(\eta)^{2(s - 1)}}} \right)\frac{{dy(\eta)}}{{d\eta }}} \right] + {y(\eta)^3} = 0,$$ It is noticed that at the very high central density ($x_c\rightarrow \infty$) Eq. (\[eqy\]) reduces to the famous Lane-Emden equation of index 3, which for $\beta=0$ (unmagnetized case) leads to the well-known Chandrasekhar mass-limit of $M_{Ch}\simeq1.43M_S$ ($M_S$ being the mass of the sun). A standard integration algorithm such as the Runge-Kutta fourth-order (as I have carried out the calculations with mathematica software) may be used to find, for instance, the value of $\eta=\eta_s$ at which $y$ vanishes (the surface of star) from which the radius and the mass of the stable configuration (star) can be calculated via the following relations $$\label{rm} {r_s} = r_c{\eta _s},\hspace{3mm}M = 4\pi \int_0^{{r_s}} {\rho(r) {r^2}dr} = \frac{{\sqrt {3\pi } }}{4}{m_p}n_h^3\int_0^{{\eta _s}} {{y(\eta)^3}{\eta ^2}d\eta}.$$ It is interesting, however, to note the unique scaling, i.e. $n_h\mathchar'26\mkern-10mu\lambda_c\simeq 0.787 R_E$ ($R_E$ being the earth’s radius) and $2m_p n_h^3\simeq 1.849 M_S$ ($M_S$ being the sun’s mass). These relations for $\beta=0$ (unmagnetized) case, following the Chandrasekhar’s pioneering work, has been reviewed in Ref. [@gar]. The main goal of next section is to explore the effect of Landau orbital ferromagnetism on magnetohydrostatic stability and consequences on the Chandrasekhar mass-limit in completely degenerate magnetized plasmas. The Chandrasekhar Limit for Flux-Frozen Compact Star {#discussion} ==================================================== Equation (\[eqx\]) can be solved for given values of $\beta$, $s$ and $x_c$. The solution $y(\eta)$ for the simplest case of $\beta=0$ (or $s=0$) which corresponds to the unmagnetized (uniformly magnetized) plasma case is shown in Fig. 2 for different values of central fractional density $x_c$. The values of $\eta_0$ where $y(\eta_0)=0$ indicate the surface of plasma from which the radius and mass of the configuration can be calculated. It is observed that as the value of $x_c$ is increased $\eta_0$ also increases and in the limit it approaches a liming value $\eta_{Ch}$ called the Chandrasekhar value. Figure 3 shows the famous Chandrasekhar mass-radius plot (in terms of sun’s mass and earth’s radius) and the corresponding Chandrasekhar mass-limit $M_{Ch}\simeq1.43M_S$ for unmagnetized case. Following is the evaluation of the problem for different values of parameters $s$ and $\beta$. Case $s>1$ ---------- For the case $s>1$, it is observed (e.g. see Fig. 4 for $s=2$) that, as one increases the value of $x_c$ to $x_c=1$, the value of $\eta_0$ increases and approaches to a liming value. However, it is remarked that, there are no stable solutions for $x_c>1$ in this case. In other words, in this model, i.e. $s>1$, no dense-centered stable configuration such as observed for a star is allowed. Therefore, this case can not correspond to a physical solution. Case $s<0$ ---------- On the other hand, for negative values of $s$, when $x_c$ is finite no $\eta_0$-value can be obtained, hence, the plasma must be infinite. However, for $x_c=+\infty$ the same Chandrasekhar-limit is obtained, regardless of the value of parameter $\beta$, as it is evident from Eq. (\[eqx\]). For instance, Fig. 5 shows the solution $y(\eta)$ and the variation with respect to different values of $x_c$ for the case $s=-1$, indicating that, the case $s<0$ is unphysical. Case $0<s<1$ ------------ For the range of $0<s<1$ the plasma is stable for large values of $x_c$. However, as $x_c\rightharpoonup 0$ it is observed from Eq. (\[eqx\]) that the integrand diverges and the plasma becomes unstable. Therefore, in this model there may be a radius-limit to the plasma evaluation of which is beyond the scope of this paper. Figure 6 for $s=1/2$ shows the similar features as Fig. 2 as the central density of plasma is increased. Also, evaluation of Eq. (\[eqx\]) reveals that the same Chandrasekhar mass-limit is obtained and that in this model the mass-limit is invariant under the change in $\beta$ parameter (e.g. see Fig. 7). Another important feature in this model is the existence of critical value for $\beta$. Evaluation of the solutions to Eq. (\[eqx\]) for this case shows that stable configurations exist only below a critical value $\beta_{cr}=2\pi$. The LOFER Case $s=1$ -------------------- In this model which corresponds to the equation of state of form $B=\beta\rho^{2/3}$ and is known as LOFER state, unlike the previous case, the plasma is stable for the whole range of the parameter $x_c$. However, the stability of the plasma is removed for values of $\beta>\beta_{cr}$. It is also remarked from Fig. 8 that in this model the chandrasekhar-limit decreases as the value of $\beta$ is increased. From the previous discussion it may be concluded that the only stable and physical model corresponds to the cases $s=1$ and $\beta<2\pi$. The magnetogravity collapse condition ($\beta<2\pi$) is to be compared with the one obtained for the quantum collapse mentioned above ($\beta>2\pi$). Therefore, in a dense ferromagnetic plasma depending on the value of $\beta$ only one of possible collapses is possible of which the magnetogravity collapse can occur in hydrostatically stable configurations. It is noted that, although the LOFER-state is a metastable configuration and requires a sufficient condition to operate, however, once it occurs the quantum collapse will inevitable. The variation of Chandrasekhar limit with the LOFER parameter $\beta$, shown in Fig. 9 confirms that the mass-limit decreases with the increase in the value of $\beta$ until it vanishes at critical value $\beta=\sqrt{2\pi}$. Also, Fig. 10 compares the Chandrasekhr mass-radius curves for the tentative values of $\beta^2/2\pi=0,0.1,0.5$ and $s=1$. It is remarked that, the flux frozen assumption used in this investigation can lead to fundamental effects of the Chandrasekhar mass-radius relation for the case of $s=1$ and the mass-limit on magnetic white dwarfs can be much lower than $1.43217$. This stellar model is also consistent with supernovae type I theory of Hoyle and Fowler [@hoyle] which requires stellar mass between 1.01 and 1.40 solar mass. It is obvious that for a complete treatment of stellar stability the effects such as Thomas-Fermi, Coulomb, electron exchange and ion correlation have to be considered [@salpeter]. Also, the electron-capture (inverse beta-decay) has been shown to alter the known EoS and limits on the compact stars [@hamada] investigations of which is out of the scope of the current study. Conclusion and Summary {#conclusion} ====================== Using the quantum magnetohydrodynamics and magnetohydrostatic models we showed that in a relativistically degenerate magnetized stellar objects such as white dwarfs the magneto-gravitational collapse is possible. The collapse was shown to be consistent with Landau orbital ferromagnetism with magnetic equation of state $B=\beta \rho^{2s/3}$ on some conditions. It was also revealed that, a critical $\beta_{cr}=\sqrt{2\pi}$ exists above which the plasma is hydrostatically unstable which comparing with the previous results indicates that in the LOFER plasma state quantum collapse and magnetogravity collapse can not take place at the same time. Furthermore, in a stable LOFER plasma it was shown that the Chandrasekhar mass-limit may critically depend on the value of the $\beta$ and $s$-parameters. These findings reveal the inevitable role of magnetism on the evolution of dense stellar objects. 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Figure-4 Variation of the value of $\eta_0$ with respect to change in the central density parameter $x_c$ for magnetized case ($B=\beta\rho^{2s/3}$) with ($s=2$) showing magnetohydrostatic instability of this model for $x_c>1$. The thickness of the curves is used as a measure for the value of the varied parameter. Figure-5 Variation of the value of $\eta_0$ with respect to change in the central density parameter $x_c$ for magnetized case ($B=\beta\rho^{2s/3}$) with ($s=-1$) showing unphysical results for this model. The thickness of the curves is used as a measure for the value of the varied parameter. Figure-6 Variation of the value of $\eta_0$ with respect to change in the central density parameter $x_c$ for magnetized case ($B=\beta\rho^{2s/3}$) with ($s=1/2$) showing the magnetohydrostatic stability for the range of central plasma density parameter, $x_c\neq 0$. The thickness of the curves is used as a measure for the value of the varied parameter. Figure-7 Variation of the value of $\eta_0$ for fixed large central density parameter $x_c$ and varied magnetic parameter, $\beta^2<2\pi$, for magnetized case ($B=\beta\rho^{2s/3}$) with ($s=1/2$). The plot indicates that in this model the Chandrasekhar mass-limit does not depends on the value of the magnetic parameter, $\beta$. The thickness of the curves is used as a measure for the value of the varied parameter. Figure-8 Variation of the value of $\eta_0$ for fixed large central density parameter $x_c$ and varied magnetic parameter, $\beta^2<2\pi$, for magnetized case ($B=\beta\rho^{2s/3}$) with ($s=1$) showing the magnetohydrostatic stability for the whole range of central plasma density parameter, $x_c$. The plot also shows that in this model the Chandrasekhar mass-limit depends strongly on the value of the magnetic parameter, $\beta$. The thickness of the curves is used as a measure for the value of the varied parameter. Figure-9 Variation of the value of the Chandrasekhar mass-limit with the value of the ferromagnetic parameter, $\beta$ for the case of $s=1$. The horizontal line indicates ordinary mass-limit $M/M_S\simeq1.43217$. Figure-10 The Chandrasekhar mass-radius curves for ordinary LOFER ($s=1$) with $\beta/\sqrt{2\pi}=0$ (rectangles), $\beta/\sqrt{2\pi}=0.1$ (empty circles), and $\beta/\sqrt{2\pi}=0.5$ (filled circles) for the case $s=1$.

--- abstract: 'We analyze a directed variation of the book embedding problem when the page partition is prespecified and the nodes on the spine must be in topological order (upward book embedding). Given a directed acyclic graph and a partition of its edges into $k$ pages, can we linearly order the vertices such that the drawing is upward (a topological sort) and each page avoids crossings? We prove that the problem is NP-complete for $k\ge 3$, and for $k\ge 4$ even in the special case when each page is a matching. By contrast, the problem can be solved in linear time for $k=2$ pages when pages are restricted to matchings. The problem comes from Jack Edmonds (1997), motivated as a generalization of the map folding problem from computational origami.' author: - 'Hugo A. Akitaya' - 'Erik D. Demaine' - Adam Hesterberg - 'Quanquan C. Liu' bibliography: - 'ref.bib' title: Upward Partitioned Book Embeddings --- Introduction {#sec:intro} ============ Definitions {#sec:definitions} =========== UPBE is NP-Complete {#sec:main} =================== Acknowledgements {#acknowledgements .unnumbered} ================ We thank Jack Edmonds for valuable discussions in August 1997 where he described how <span style="font-variant:small-caps;">Upward Matching-Partitioned $k$-Page Book Embedding</span> generalizes the map folding problem. We also thank Therese Biedl for valuable discussions in 2007 about the complexity this problem. This research was conducted during the 31st Bellairs Winter Workshop on Computational Geometry which took place in Holetown, Barbados on March 18–25, 2016. We thank the other participants of the workshop for helpful discussion and for providing a fun and stimulating environment. We also thank our anonymous referees for helpful suggestions in improving the clarity of our paper. Supported in part by the NSF award CCF-1422311 and Science without Borders. Quanquan Liu is supported in part by NSF GRFP under Grant No. (1122374). Appendix {#appendix .unnumbered} ========